Poster Abstracts

Chaos in Taylor-Dean flow
Ait Aider Aomar
University of Tizi Ouzou

Long memory and large deviations for weak chaos.
Roberto Artuso
Center for Nonlinear and Complex Systems, Universita; dell'Insubria

We consider dynamical systems lacking full hyperbolicity, and point out known problems about analytic and numerical analysis of time correlations decay, focusing in particular on intermittent and area-preserving mappings. We report on recent studies where it was shown that quantitative informations can be obtained either by considering time statistics (like Poincare' recurrences), or by large deviations estimates of the distribution of finite time Lyapunov exponents. References: R Artuso, C Manchein, Phys.Rev. E 80, 036210 (2009); submitted (2012) M Sala, C Manchein, R Artuso, Int.J.Bifurcations Chaos, to appear (2012)

Performance of digital chaos-based communication systems in a noisy environment
Greta Augat Abib
Federal University of the ABC Region

Chaotic signals are deterministic, aperiodic and sensitive to initial conditions [1]. They are suited for use in applications that require security due to its difficulty of prediction and because they can be mistaken for noise in the channel [2].

In the last decades many works describing communication schemes based on chaotic signals have been published [2]. However, the performance of these systems under non-ideal conditions is much less studied.

In this paper, we evaluate the performance of a binary communication system based on a discrete-time version of the Wu and Chua's synchronization method [3,4] when the communication channel introduces additive white gaussian noise.

Simulations are performed using different maps and two pairs of coding and decoding functions were considered, one using sum and the other using product. The results are measured for each map and encoding function in terms of bit error rate.

For comparison, we also show the performance of a communication system with the conventional Phase Shift Keying (PSK) [5]. When using product as coding function, the chaos-based communication system performance is closer to the optimal performance of the PSK.

[1] K. T. Alligood, T. D. Sauer, J. A. Yorke. Chaos - An Introduction to Dynamical Systems, Springer, New York, 1996.

[2] W. M. Tam, F. C. M. Lau, and C. K. Tse, Digital Communications with Chaos: Multiple Access Techniques and Performance. Elsevier Science, New York, NY, USA , 2006.

[3] C. W. Wu and L. O. Chua. A simple way to synchronize chaotic systems with applications to secure communication systems. International Journal of Bifurcation and Chaos, vol. 3, no. 6, pp. 1619-1627, 1993.

[4] M. Eisencraft, R. D. Fanganiello, and L. A. Baccala, Synchronization of discrete-time chaotic systems in bandlimited channels. Mathematical Problems in Engineering, vol. 2009, 2009.

[5] S. S. Haykin, Communication systems, 4 ed. New York: Wiley, 2000.

Enhancing Synchrony in Chaotic Oscillators by Dynamic Relaying
Ranjib Banerjee
Gargi Memorial Institute of Technology

We report a phenomenon of enhancing of synchronization in arrays of coupled chaotic oscillators when mediated by an oscillator with a mismatch in parameter. In a chain of mutually coupled oscillators, the coupling threshold for synchronization between the outermost identical oscillators decreases when a type of impurity or a disorder as a parameter mismatch is introduced in the inner oscillator(s). We define the lowering of critical coupling of complete synchronization as an enhancing effect.

Chaotic trajectories in coupled nonlinear dynamical systems are known to synchronize when the strength of the coupling exceeds a critical value. Such complete synchronization (CS) occurs only when the oscillators are identical. In case of mismatched oscillators, lag synchronization (LS) may arise at a lower value of the critical coupling when the amplitude of oscillations in the coupled systems remains correlated but shifted in time. An interesting consequence of this lag synchrony is that it can promote complete synchrony in a chain of diffusively coupled chaotic oscillators when there are isolated impurities, namely, mismatched oscillators. The simplest case arises when three oscillators are mutually coupled in an open-ended chain in which the outer two oscillators are identical to each other but mismatched with the intermediate (central) oscillator. The central oscillator with either positive or negative mismatch induces a common time lag or lead with the outer oscillators, leading to a LS scenario at a lower critical coupling. As a result, the outer oscillators synchronize at a critical coupling lower than the critical coupling for three identical oscillators. Note that the outer oscillators are not directly connected but basically interacting through the central oscillator via a dynamic relaying. The effect is found true for both unidirectional and bidirectional coupling. An immediate consequence is that a central oscillator can drive many identical oscillators in a star-like configuration into a state of enhanced synchrony. All the driven oscillators synchronize at a lower critical coupling without being connected directly and affecting the original dynamics of the driving oscillator. The enhancing effect is also true for a ring of oscillators when the identical oscillators in symmetric positions to the mismatched oscillator synchronize at a lower critical coupling but, at the same time, do not synchronize with its nearest neighbors.

We used examples of Lorenz systems, Rössler system, Mackey-Glass system to confirm the effect and thereby establish that it is true for both instantaneous coupling and delay coupling. However, the enhancing effect is only seen when the coupled dynamics is chaotic. For periodic coupled dynamics, a diminishing effect is rather observed when the critical coupling is increased. We provide experimental support to this effect using electronic analog of Rössler oscillators.

Defibrillation mechanisms on a one-dimensional ring of cardiac tissue
Jean Bragard
University of Navarra

"Defbrillation is a medical treatment used to terminate ventricular fibrillation or pulseless ventricular tachycardia. An electrical device via a pair of electrodes delivers controlled amount of electrical energy to the heart in order to reestablish the normal heart rhythm. First generation of defibrillators applied monophasic shock, in which electrodes did not change polarity during the application of the shock. Later it was found that changing the polarity of the electrodes during the shock leads to better result with less energy applied. Optimal monophasic and biphasic shock release approximately 200 J and 150 J, respectively. It is desirable to use as less energetic shock as possible in order to reduce the damage done to the tissue by the strong electric current. However, to this day, there is no full understanding why biphasic shocks are better than monophasic shocks. To assess this question, we have used a bidomain model for cardiac tissue with modified Beeler-Reuter model1 for transmembrane currents. Modifications account for anode break phenomena and electroporation effect known to happen during defibrillation. We have studied three different types of protocols for shock application (i.e. monophasic; symmetric biphasic; and asymmetric biphasic shock) in a one-dimensional ring of cardiac tissue. The size of the ring was chosen to exhibit a discordant-alternans dynamics. Results of the numerical simulations reveal that monophasic shocks defibrillate with higher rate of success than the two biphasic shock protocols at lower energies. On the contrary for higher shock energies, the biphasic shock are significantly more efficient than monophasic shocks. This latter result confirms the medical common wisdom about defibrillators. Moreover, in this study, we were able to identify and classify the different defibrillation mechanisms that happen in this system. One identifies four different types: direct block, delayed block, annihilation and direct activation. Which defibrillation mechanism prevails depends on the energy level, the current dynamic state of the system and the shock protocol. This study has permitted to uncover and confirm the experimental fact stating that biphasic shocks are more efficient (at high energy) than monophasic shock to defibrillate cardiac tissue.

An experimental exploration of the dynamics of kneading
Allison Brown
University of Colorado Boulder

Kneading is often used as a metaphor in discussions of chaos. These discussions include various claims about how kneading demonstrates different aspects of nonlinear dynamics and chaos, especially state-space mixing and sensitive dependence on initial conditions. While the kneading metaphor is useful, the associated claims do not appear to have been experimentally verified. Indeed, many of those claims---regarding the culinary aspects of kneading and their effects on the dynamics of bread dough---are actually incorrect. To test these claims, we designed two experiments with homemade dough. The choice of dough was important; bread and other pastry doughs have complex properties (e.g., the formation of gluten chains, which change the bulk properties of the material) that complicate the kneading dynamics. Since most discussions that involve this metaphor are more concerned with the generic kneading process, we chose to avoid these additional dynamics, and the associated issues, by using homemade play dough. Using a plexiglass compression device and colored beads to track the deformation of this dough, we found that the dynamics of the kneading process definitely demonstrate sensitive dependence on initial conditions, as well as other properties that are characteristic of chaos.

Sensing and information storage capabilities of optoelectronic oscillators
Kristine Callan
Duke University

Optoelectronic oscillators (OEOs) are versatile experimental devices that have been used for exploring both fundamental aspects of nonlinear physics [1,2] and applications in chaos-based communications [3]. In this poster, we propose two novel applications for an OEO: (1) A node in a sensing network and (2) a dynamical information storage unit. In both cases, we investigate the best operating conditions for an OEO to perform these tasks and find that statistical complexity measures [4] can be used as design-oriented metrics. The sensing properties of an OEO can be attributed to changes in complexity of the dynamics of a single node that occur when the strength of interaction with other nodes in the network changes. Precisely how the complexity of the dynamics changes depends on both the coupling strength and the sensing node's parameter values. For dynamical information storage, we exploit the presence of multi-stable periodic solutions that arise due to the relatively large feedback delay time. We hypothesize that the complexity of the chaotic regime provides hints as to the best parameter values to choose for an optimal information storage capacity.

[1] Callan, K.E. et al. Phys. Rev. Lett. 104, 113901 (2010).

[2] Larger, L. and Dudley, J. Nature 465, 41-42 (2010).

[3] Argyris, A et al. Nature 438, 343-346 (2005).

[4] Bandt, C. and Pompe, B. Phys. Rev. Lett. 88, 174102 (2002).

Multi-scale modeling of cancer cell chemotaxis: Role of receptor dynamics and gradient formation
S. Laura Chang
University of Michigan

Cancer cell migration in gradients of chemotactic factors is thought to be a key function for invasion and metastasis. The chemokine receptor, CXCR4, is overexpressed in many cancers, including breast cancer. Cells expressing CXCR4 can chemotax in response to its ligand, CXCL12, which is upregulated in the tumor stromal environment as well as common metastatic sites, such as the bone. The availability of CXCL12 within the tumor microenvironment can be altered due to scavenging by its second receptor, CXCR7. In order to untangle the role of CXCR7 within breast cancer and how it may affect the microenvironment to promote cancer cell migration, we have constructed a multi-scale hybrid agent-based model that simulates cancer cell migration within a microfluidic chemotaxis assay. CXCR4 and CXCR7-expressing cells, along with CXCL12-secreting cells move and interact on a lattice. On the single-cell level, ordinary differential equations have been implemented to describe the internalization, recycling, and degradation of CXCL12 and its receptors for both CXCR4 and CXCR7-expressing cells. Therefore, the cells can respond to, as well as update, their environment. Latin Hypercube Sampling coupled with Partial Rank Correlation Coefficients has been used to determine the critical parameters that can be targeted to inhibit cancer cell migration. The results from this model will be used to further guide experiments within the microfluidic chemotaxis assay and to interpret experimental data.

A superparamagnetic agent map of science
Markus Christen
University of Notre Dame & University of Zurich

Visualization of high-dimensional data by means of a low-dimensional embedding is an important technique in data analysis. We present a novel approach of this problem based on an agent-based simulation using superparamagnetic clustering. Our method can deal with nonlinear structures since it is essential local. Moreover, the nonparametric characteristics and the robustness of the superparamagnetic self-organization approach allow distinguishing between different clusters and background noise. In our contribution, we first briefly introduce the concept of superparamagnetic clustering and the algorithm of superparamagnetic agent maps. We exemplify latter by two standard problems of data visualization: the noisy ring problem and the PCA-mixer problem.

As main application, we present a superparamagnetic agent map of data emerging from a large survey on the (dis-)similarity of scientific disciplines represented as ISI subject categories, which classify journals included in the Science Citation Index. In this research, we use a bottom-up approach based on a standard procedure of classification psychology that has been translated in a binary comparison task suitable for a web interface. In this data collection procedure, scientists from all disciplines are able to provide their assessments of the similarity of subject category triplets, leading to a distance matrix that integrates a conditional structure (i.e. concept X is attributed to concept Y under the condition that Y has been presented with concept Z). Furthermore, we outline the mathematical issues related to such a distance function and we show how to deal with the problem of combinatorial explosion. In this way, we map the topology of science using a new paradigm, not relating on citations or keyword-based comparisons of publications, but using the expert opinions of humans. The survey, which is based on a server infrastructure able to handle several million entries, is currently set up and will be finished in a few weeks.

Multi-scale Modeling Reveals the Influence of Opposing Dynamics of TNF and IL-10 in MTb Infection
Nicholas Cilfone
University of Michigan

Mycobacterium tuberculosis (Mtb) infection leads to the formation of a granuloma, a spherical collection of immune cells that immunologically constrain and physically contain bacteria. These structures are mainly comprised of lymphocytic and monocytic cells surrounding a core of infected macrophages. Tumor necrosis factor-α (TNF) is an inflammatory cytokine critical to the formation and maintenance of granulomas. Interleukin-10 (IL10), an anti-inflammatory cytokine, mediates processes including the down regulation of TNF. We have developed a hybrid multi-scale agent based model of Mtb infection that simulates Mtb infection over tissue, cellular, and molecular scales and allows us to explore how the dynamics of TNF and IL10 affect infection outcome. We use the model to identify the importance of simultaneous TNF and IL10 production and infection induced IL10 production. Our results indicate that the opposing actions of TNF and IL10 are central to the ability of the granuloma to contain the infection. Also, the simulation can be used to explore pharmacologic manipulation of TNF and/or IL10 levels to improve infection outcome.

Design, Implementation and Applications of a Nonlinear Microwave Oscillator
Hien Dao
University of Maryland- College Park

We describe here the design and implementation a chaotic frequency-modulated microwave source with time-delayed feedback that produces dynamical behaviors ranging from periodic to high-dimensional chaos, depending on the feedback gain and filter bandwidth. We characterize the dynamical behavior under different conditions, and use the ambiguity function to assess its potential for use in radar-based range-finding and velocity sensing applications.

A priori low-dimensional models for porous medium convectoin
Navid Dianati
University of Michigan

We develop a class of low-dimensional models for porous medium convection using adapted a priori bases inspired by energy-stability and upper-bound theories. A natural and systematic truncation scheme emerges from the method. Models of various orders are examined, with a focus on the time-averaged convective heat transport or the Nusselt number.

Discovery of dozy chaos and discovery of quanta: analogy being in science and perhaps in human progress
Vladimir Egorov
Russian Academy of Sciences, Photochemistry Center

Dozy chaos was introduced into science by the author at the beginning of the 21st century as a novel physical substance to describe extended multiphonon transitions. The necessity for introducing this substance stems from the presence of inherent singularity in the probability of extended transitions as a result of transit beyond the adiabatic approximation in the quantum mechanics of electron-nuclear motion. Dozy chaos is absent in the initial and final states, and arises in the transient state alone. In the initial and final states, where the adiabatic approximation works well, the electrons do not exchange their motion and energy with the nuclei. In the transient state, where the adiabatic approximation does not work at all, the electrons share their motion and energy with the nuclear subsystem. The dynamics of the interchange is so intense that leads to chaos in the motion of both electrons and nuclei. This chaos is called dozy chaos. Dozy chaos is a mix of chaotic motions of the electronic charge, nuclear reorganization, and the electromagnetic field via which the electrons and nuclei interact in the transient state. By dozy chaos the light electrons succeed in controlling the motion of extremely heavy nuclei in the transient state, making it chaotic. The introduction of a novel physical substance into science in order to eliminate a singularity is not new to physicists. It is common knowledge that the quantum hypothesis was introduced into science at the beginning of the 20th century in order to eliminate a singularity in the distribution function of black light at high frequencies. The novel quantum theory originated in the hypothesis of dozy chaos gives an insight into a whole series of the basic experimental results in chemistry, which defied understanding in the context of the standard quantum mechanics of electron-nuclear motion for many decades. The most impressive results are in theoretical optical band shapes fitted by the author to the basic experimental data on polymethine dye monomers, dimers, H-, H*-, and J-aggregates and also theoretical band shapes fitted to the well-known data on the monomer-dimer and monomer-J-aggregate concentration equilibriums. The author believes that dozy chaos is of primary importance to the dynamic self-organization of any living organism and is concentrated in its brain. A hypothesis of the physical origin of cancer is put forward. A cancerous growth is associated with an abnormally large concentration of dozy chaos in it. In this respect, the cancerous growth is similar to the brain. But they differ essentially in structure. The cancerous growth, lowly organized in structure, uses a huge amount of dozy chaos for destructive purposes. It is the high organization of the brain in structure allows him to use a huge amount of dozy chaos for constructive purposes. The appearance of living nature and the diversity in its structural and dynamic self-organization, which results from its evolution, are ultimately a consequence of the dynamic self-organization of quantum transitions through dozy chaos. In the process of evolution of living organisms, it is dozy chaos that sets a primary tone, according to which living organisms gain the structure that allows them to perform any necessary dynamic functions. All the above suggests that trends in the future development and applications of the dozy-chaos theory of quantum transitions may be such immense as were in the case of the standard quantum mechanics not long ago.

Labyrinthic standard non-twist map and its variations
Ricardo Egydio de Carvalho
UNESP-State University of São Paulo

We introduce the labyrinthic standard non-twist map, which is a modification of the non-twist standard map, with an additional perturbation. This perturbation has a period which is multiple of the original perturbation period. This map present several sets of invariant meandering curves due to the reconnection process of isochronous resonances. Inside each meander set we find an invariant curve called shearless curve, which generates a wide stickiness effect when it is broken. This effect interferes significantly on the dynamical transport properties of the system. As non-twist maps as transport barriers play important roles for theoretical plasma fusion.

Direct evidence of macroscopic chaos in epileptic seizure Electroencephalography
Jianbo Gao
Wright State Univ

Electroencephalography (EEG) signals provide a wealth of information about brain dynamics, especially related to cognitive processes and pathologies of the brain such as epileptic seizures. Detecting chaos in EEG has been a classic issue. Through surrogate data testing, which is an indirect method, the current consensus is that normal EEG signals may not be distinguished from linear stochastic processes, while epileptic seizure EEGs contain considerable nonlinear aspects. Here, using scale-dependent Lyapunov exponent and an adaptive denoising algorithm, we show direct evidence of chaos in EEGs with epileptic seizures. Moreover, we show that the approaches developed here can be conveniently and effectively used to detect epileptic seizures.

A systems biology approach to understanding the efficiency of the lymph node in producing primed T cells
Chang Gong
University of Michigan

Dendritic cells (DCs) ingest foreign material (antigen) present during infection, process antigen for display on their cell surface, and migrate to T cell zones of lymph nodes (LNs) where millions of circulating T cells are present. A small fraction of these T cells (cognates) have receptors that bind to the displayed antigen on the DCs, initiating a cascade of events leading to priming. These primed T cells then return to the site of infection to lead the body's defense. Recent studies employing two-photon microscopy (2PM) have significantly advanced our knowledge of T cell motility and the behavior of cognate T cells in the presence of antigen-bearing DCs within LNs, but many unanswered questions remain. For example, it is difficult to relate the short length- and time-scale measurements of 2PM to efficiency of LNs in producing primed T cells. We therefore developed a 3 dimensional (3D) agent-based model representing the T cell zone of LNs, allowing for rapid in silico simulation of T cell zone function. We used the model to explore the effect of T cell zone morphology on LN efficiency and used uncertainty and sensitivity analysis [1] to predict which mechanisms contribute significantly to the production of primed T cells. Monkey LNs were obtained, cross-sectioned, and fluorescently labeled to reveal the size and shape of T cell zones, density of T cells, and distributions of high-endothelial venules (HEVs) which serve as T cell entry and cortical and medulary sinuses which are connected to efferent lymphatics (ELs) and serve as T cell exit ports, respectively. These data were used to build an agent-based model in C++ on a 3D toroidal lattice populated with T cells and DCs that interact according to prescribed stochastic rules. System parameters (e.g., cell lifespans) were taken from the literature or otherwise calibrated with known behaviors. System outputs (e.g., number of primed T cell) were recorded and analyzed using uncertainty and sensivity analysis. Preliminary data shows T cells in model simulations exhibit motility in close accord with experimental data, moving at an average speed of about 12 μm/min with a motility coefficient of 45 μm^2/min, and an average transit time of about 17 hrs. Introduction of antigen-bearing DCs induces an in silico immune response, leading first to the production of CD4+ T cells followed by the production of CD8+ T cells. Analysis of simulations also allows us to identify additional parameters that influence LN output. Overall, our systems biology approach provides a platform not only to understand but also to guide manipulation of LN function in the context of disease.

Stochastic Analysis of Rogue Waves
Ali Hadjihosseini
ForWind- Oldenburg University

The waves appear from nowhere and disappear without trace. They may occur on the surface of a relatively calm sea and not reach very high amplitudes, but still be fatal for ships due to their unexpectedness an abnormal features. This work presents an analysis of rogue wave data, with a stochastic approach based on the theory of Markov processes. In many cases Markov processes can be described completely by a Fokker-Planck or Langevin equation with parameters derived directly from experimental data. The method was applied to rogue wave measurement data. Their Markov properties were shown and first estimations for the parameters of the Fokker-Planck equations were performed.

Electronic Repressilator with Quorum Sensing Feedback
Edward Hellen
University of North Carolina at Greensboro

Quorum sensing (QS), a typical process of communication in a bacterial colony, has been used as a mechanism for coupling synthetic genetic oscillators. Prior to studying the dynamics of multiple oscillators interacting via QS it is important to understand the dynamics of a single synthetic oscillator under the influence of QS feedback. We investigate the dynamics of an electronic ring oscillator, an analog of the synthetic genetic network repressilator (e-Rep) with repressive QS feedback. The repressilator is a 3 gene inhibitory loop which displays oscillations in protein concentration. The electronic circuit allows complete control of the synthetic network's model parameters by varying the circuit parameters. We find: multi-stability of limit cycles (LC) with steady-state (SS); multi-stability of two different SS's (one with all 3 repressilator protein concentrations low, the other with one high and two low); LCs with period-doubling and period-halving; and infinite period bifurcation (IPB) transitions for both increasing and decreasing strength of QS feedback. Noise amplification near the IPB is demonstrated. The background of IPB is the appearance of a stable SS which may coexist with a stable LC. The size of the hysteretic loop between LC and SS is determined by the lifetime ratio of QS protein to repressilator protein, and by the level of transcription-stimulating activity of the QS protein. At the extremes of QS, zero feedback gives the LC of the single e-Rep, and large feedback is characterized by the SS with one protein high and the other two low. In addition to hysteresis, increasing the QS protein-dependent stimulation of transcription may lead to the complex dynamic behavior which is characterized by the appearance of period doubling cascades.

Modeling and Experimental Verification of Cyclic Variability and Thermal Runaway in Advanced Internal Combustion Engines
Anna G. Stefanopoulou
Erik Hellstrom, Co-Author
Mechanical Engineering, University of Michigan

Dynamical instabilities in the autoignition combustion are investigated. A novel low order dynamical model is derived for predicting the experimentally observed combustion phasing at the edge of stable autoignition in homogeneous charge compression ignition (HCCI) engines. These engines rely on large amounts of residual gases and are intrinsically very efficient and clean, but their operating range (load and speed) is very limited. At fixed speed and load the edges of stable combustion are defined, from one side, by late and highly varying combustion phasing, and from the other side, early combustion phasing that can result in thermal runaway.

The proposed model utilizes two discrete states that capture the coupling between engine cycles due the thermal energy in the recycled residual gases and the recycled chemical energy in the unburned fuel due to incomplete combustion. A stability analysis of this model with respect to the amount of recycled residual gases shows that for low amounts of residual gas, a cascade of period-doubling bifurcations occurs, which eventually leads to chaotic behavior, whereas a runaway behavior occurs for high amounts of residual gas.

Numerical simulations of the second-order system, using physically reasonable parameters and random variations in the residual gas amount, show remarkable qualitative agreement with the experimental observations. The model and analysis provides a foundation for a fuel controller design that mitigates the combustion variability and can improve the commercial feasibility of these advanced engines.

How to obtain extreme multistability in coupled system?
Chittaranjan Hens
CSIR-Indian Institute of Chemical Biology

We present a method for designing an appropriate coupling scheme for two dynamical systems in order to realize extreme multistability. We achieve the coexistence of infinitely many attractors for a given set of parameters by using the concept of partial synchronization based on Lyapunov function stability. We show that the method is very general and allows a great flexibility in choosing the coupling. Finally we demonstrate its applicability in different models, such as the R ̈ssler system and a chemical oscillator. The extreme multistability is robust to parameter mismatch.

Experimental network synchronization of Chua's circuits using plastic optical fiber: application to communications
Everardo Inzunza-Gonzalez
Baja California Autonomous University

In nature and the technology are many systems composed by dynamical units highly interconnected, where their collective behaviors are completely different to the individual behaviors of each dynamical unit, such systems generate very complicate dynamics, the so-called complex systems, see e.g. [1]. These systems have been observed in different areas of the science, for instance in physics, biology, computer science, economy, chemistry, engineering, social sciences, etc. Among the important issues on collective behavior of connected oscillators is synchronization [2-7].

On the other hand, the synchronization problem between two chaotic oscillators had been widely studied, see e.g. [8-13] and references therein. More recently, synchronization of complex dynamical networks (with many coupled oscillators) has received a great deal of interest from the scientific community. Particularly interesting is the case where the connected oscillators have chaotic dynamics. This synchronization property in complex dynamical networks is supposed to have interesting applications in different fields, see e.g. [3, 4, 14-20].

On synchronization in complex networks with Chua's circuits like nodes, we can mention the following works in [21] are obtained sufficient conditions to synchronize complex networks with different coupling topologies, the results are illustrated by means of computer simulations. The possibility of coloring graphs by means of synchronized coupled Chua's circuits is studied in [23]. Sufficient conditions to synchronize complex networks of coupled linearly and diffusively nodes are reported in [16,22]. In [23] are shown numerical results of the synchronization of four coupled Chua's circuits in different coupling schemes.

In this work, we concentrate on experimental synchronization in complex networks of linearly coupled identical dynamical systems, using techniques reported in [16,23] where synchronization in complex networks is shown analytically. In particular, in [23] we have given some preliminary results for synchronization in a network globally coupled of three Chua's circuits. Nevertheless, there are few works on experimental network synchronization, see e.g. [7].

In this work, an experimental study on practical realization of synchronization in globally coupled networks with Chua's circuits like nodes is presented, in particular, Plastic Optical Fiber (POF) is used in the network like communications channels among chaotic nodes to achieve synchronization, the signal through a fiber optical coupler with corresponding electrical/optical and optical/electrical stages. Synchronization of coupled multiple Chua's circuits is achieved by appealing to results from complex systems theory. In particular, we design and implement electro-optically complex dynamical networks composed by three coupled Chua's circuits.

In this work, network synchronization of coupled Chua's circuits in global configuration is experimentally studied. In particular, Plastic Optical Fiber (POF) is used in the network like communication channels among chaotic circuits to achieve network synchronization. The master signal is sent to multiple slaves through a fiber optical coupler with corresponding electrical/optical and optical/electrical stages. An application to encrypted chaotic communication to transmit analogical signal and image messages to multiple receivers is also given.

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[2] Posadas-Castillo C, Cruz-Hernández C, López-Mancilla D. Hybrid Intelligent Systems Analysis and Design. Studies in Fuzziness and Soft Computing, vol. 208. Springer; 2007.

[3] Posadas-Castillo C., López-Gutierrez R. M and Cruz-Hernández C. (2008) 'Synchronization of Chaotic Solid-State Nd:YAG lasers: Application to Secure Communication', Communications in Nonlinear Science and Numerical Simulations 13(2008), 1655-1667, Elsevier Editorial, ISSN: 1007-5704.

[4] Zhang H-F, Wu R-X, Fu X-C. Chaos, Solitons & Fractals 2006;28(2):472-9.

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[5] Lu J, Ho DWC. Chaos, Solitons & Fractals in press, doi:10.10.16/j.chaos.2006.10.030.

[6] Wang J, Zhou T. Chaos, Solitons & Fractals 2007;33(1):163-70.

[7] Posadas-Castillo C, Cruz-Hernández C, López-Gutiérrez RM. Experimental realization of synchronization in complex networks with Chua's circuits like nodes. Chaos, Solitons & Fractals, 2009; 40(4):1963-1975.

[8] Pecora LM, Carroll TL. Phys Rev Lett 1990;64:821-4.

[9] Nijmeijer H, Mareels IMY. IEEE Trans Circ Syst I 1997;44(10):882-90.

[10] Cruz-Hernández C, Nijmeijer H. Int J Bifurct Chaos 2000;10(4):763-75.

[11] Sira-Ram´ırez H, Cruz-Hernández C. Int J Bifurct Chaos 2001;11(5):1381-95. And in: Proceedings of American Control Conference, Chicago, USA, 2000; p. 769-73.

[12] López-Mancilla D, Cruz-Hernández C. Nonlinear Dyn Syst Theory 2005;5(2):141-56.

[13] López-Mancilla D, Cruz-Hernández C. Chaos, Solitons & Fractals 2008;37(4):1172-1186.

[14] Posadas-Castillo C., López-Gutiérrez RM, Cruz-Hernández C. Synchronization of chaotic solid-state Nd:YAG lasers: Application to secure communication. Commun Nonlinear Sci Numer Simul 2008; 13(8):1655-67

[15] XF Wang, G. Chen, Synchronization in small-world dynamical networks, Int. J. Bifurc. Chaos 12(1) (2002) 187-192.

[16] XF Wang, Complex networks: Topology, dynamics and synchronization, Int. J. Bifurc. Chaos 12(5) (2002) 885-916.

[17] López-Gutiérrez R. M., Posadas-Castillo C., López-Mancilla D., Cruz-Hernández C. (2009). Communicating via robust synchronization of chaotic lasers. Chaos, Solitons & Fractals, 2009; 42(1):277-285.

[18] Posadas-Castillo C, Cruz-HernándezC, Lóopez-Gutiérrez RM. (2008). Synchronization of 3D CNNs in irregulars arrays, Procs. of the 16th Mediterranean Conference on Control and Automation, June 25-27, Ajaccio, France, 321-325.

[19] Posadas-Castillo C, Cruz-Hernández C, López-Gutiérrez RM. Synchronization of chaotic neural networks with delay in irregular networks. Applied Mathematics and Computation. 2008;205(1):487-496.

[20] Serrano-Guerrero H, Cruz-Hernández C, López-Gutiérrez RM, Posadas-Castillo C, Inzunza-González E. Chaotic synchronization in star coupled networks of three-dimensional cellular neural networks and its application in communications. International Journal of Nonlinear Sciences andNumerical Simulation 2010;11(8):571-580.

[21] Wu CW, Chua LO. Synchronization in an array of linearly coupled dynamical systems. IEEE Trans Circ Syst I 1995;42(8):430-47.

[22] Wang XF. Int J Bifurct Chaos 2002;12(5):885-916.

[23] Posadas-Castillo C, Cruz-Hernández C, López-Gutiérrez RM. In: Proceedings of 4th IASTED, International Conference on Circuits, Signals, and Systems, San Francisco, California, USA, 2006:236-241.

Learning chaotic dynamics by Runge-Kutta Generalized Orthogonal Forward Regression
Stanislaw Jankowski
Warsaw University of Technology

This study is devoted to modeling of nonlinear chaotic dynamics by learning systems, as e.g. recurrent neural networks, recurrent kernel machines etc. In general, it is difficult to obtained the satisfactory long-term prediction by these solutions. We present a learning system that is designed to approximate the unknown functions of the right-hand side of the Runge-Kutta numerical algorithm for solving ordinary differential equations. Hence, this is a model of the nonlinear system derivatives that is able to generate directly the system states step by step starting from the initial condition. In our approach we apply the generalized orthogonal forward regression (GOFR) scheme as model. The training set contains the observed system trajectory in a given time interval. The idea of the GOFR consists of the following steps:

a) construct a library of the basis functions, e. g. RBF;

b) select the function of the library that is the most correlated with respect to the goal trajectory;

c) tune the parameters of the selected function by using gradient optimization;

d) orthogonalize the trajectory and the library by using the Gram-Schmidt algorithm.

This method was tested on 3 data sets: a) chaotic Chua's circuit, b) local volcanic activity (Etna), c) identification of the specific unmanned aircraft vehicle.

The obtained model of dynamical system is set up of the minimal number of components and is superior in long-term trajectory prediction as compared to recurrent neural networks andurrent support vector machines.

Transitions to chaos in a diode laser with phase-conjugate feedback from a semiconductor photorefractive crystal
Andreas Karsaklian Dal Bosco
Supélec, OPTEL Research Group, Laboratoire Matériaux Optiques, Photonique et Systèmes

Optical chaos is easily observed in a laser diode with optical feedback, that is when part of the emitted light is sent back into the own laser cavity after being partially reflected by a mirror. We will focus on the case of a phase-conjugate feedback (PCF) which is performed by four-wave mixing in a SPS (Sn2P2S6) photorefractive crystal using a self-pumped ring-cavity. Qualitatively different dynamics are triggered when increasing the phase-conjugate mirror reflectivity or the injection current level [1]. We will detail the route to chaos undertaken by the laser using spectral and temporal analysis. Those experimental results will be confronted to the theoretical bifurcation scenario computed from the Lang-Kobayashi laser equations model [2].

This study fundamentally aims at demonstrating the relationship between the external laser cavity round-trip time - also called cavity delay - and the laser dynamical behaviour when the feedback strength is increased. The external cavity delay has a great influence on the laser dynamics and when the feedback strength is changed. We can experimentally notice that whenever a nonlinear dynamics is reached, the external cavity length is somehow printed in the laser time trace and spectrum, thus compelling the laser to oscillate at specific frequencies. For instance, the laser spectrum shows periodical spikes at frequencies inversely proportional to the external cavity length. For some feedback values one might see quasiperiodic dynamics combining both relaxation oscillations and time-delay time-scales. For higher feedback values, the laser can exhibit a peculiar behaviour called low-frequency fluctuation regimes (LFF) [3] in which regular power dropouts are clearly noticeable despite a chaotic background signal. Statistical analysis of those LFF regimes will be presented, unveiling a new form of deterministic resonance of chaotic fluctuations induced by varying the amount of optical feedback.

Besides the dynamical study, our work is interesting in that it is the first one undertaken using a SPS crystal to perform a phase conjugation mirror. Its high gain and low absorbance make it suitable for lasers operating in the near-infrared domain [4]. The novelty of this material is its response time which is close to 1 ms, hence is three orders of magnitude faster than what has been reported in previous dynamical studies of chaos in lasers with PCF using BaTiO3 photorefractive crystals [1]. The influence of the crystal phase-conjugation build-up time on the laser dynamics will be discussed as well.

The authors thank A. A. Grabar for providing them with the SPS crystal and the Conseil régional de la Lorraine for its financial support.

[1] J. S. Lawrence, D. M. Kane. Phys. Rev. A 63, 033805 (2001)

[2] M. Virte, A. Karsaklian Dal Bosco, D. Wolfersberger, M. Sciamanna. Phys. Rev. A 84, 043836 (2011)

[3] Ch. Risch, C. Voumard. J. Appl. Phys. 48, 2083 (1977)

[4] M. Jazbinšek, D. Haertle, G. Montemezzani, P. Gunter. J.Opt.Soc. Am. B. Vol.22, No.11 p. 2459 (2005)

General anesthesia disrupts hub-like nodes and efficient information integration in human brain networks
Heonsoo Lee
University of Michigan

Background: General anesthesia induces significant changes of functional connectivity in the brain, accompanied by unconsciousness. Although graph theory is an important method for study brain networks, there has been relatively little graph-theoretical analysis of the anesthetized brain.

Method: Electroencephalogram (EEG) was recorded twice from 20 normal subjects with one-week interval. The functional connections among 21 EEG channels were defined with phase lag index (PLI), which is a measure of phase coherence, in three states: wakefulness, general anesthesia and recovery. An intravenous bolus of 2.0 mg/kg (Propofol) was used to induce anesthesia. The change of typical network properties (characteristic path length, modularity and hub-structure) was investigated in five EEG frequency bands with weighted networks across states of consciousness.

Results: The functional connectivity measured by PLI was significantly decreased in most frequency bands after anesthesia, but spindles increased. The characteristic path length and the modularity significantly increased. The anesthetic mainly affected the nodes with higher connection strengths (hub like structures). Nodes with higher connection strength lost strength, whereas nodes with lower connection strength gained strength, transforming a hub structure into a random structure. The most significant change took place at right parietal cortex (P6) in alpha, spindle and beta bands. The changed network properties reversed at return of consciousness (ROC) in specific frequency bands (alpha and spindle).

Conclusion: Propofol reduces the efficiency of information transmission, strongly modularizing the functional connectivity. Furthermore, it selectively disrupts the hub-like nodes in the brain network, reducing the efficient information integration in the conscious state. These results emphasize the important role of hub structures in the cortex for conscious state, and imply the functional mechanism of anesthetic-induced unconsciousness.

Ketamine anesthesia selectively reduces the fractal dimension of periphery nodes in brain networks constructed from the electroencephalogram (EEG)
UnCheol Lee
Dept. of Anesthesiology, University of Michigan Medical School

Background: Fractality is an essential property to explain global functional and structural mechanisms of the brain. In a previous study, we introduced a method to estimate the fractal dimension from minimal spanning tree (MST) constructed from multivariate time series. Here we applied this method to anesthesia EEG data, and quantified the fractal dimensions for the core and periphery nodes in MST separately.

Method: Eight channel EEG data were recorded from six surgical patients undergoing ketamine anesthesia. The EEG data contains two states, waking and anesthetized. First, the eight-channel EEG time series was converted to a weighted network: an EEG topography (the distribution of EEG values at eight channels) at a time point becomes a node, and the Euclidean distance between two EEG topographies at two different time points becomes a weighted edge. Second, based on these definitions of node and edge, we constructed a MST, and measured the fractal dimension. Since the MST consists of EEG topographies and its similarities, the fractal dimension of MST reflects the complexity of temporal organization of EEG topographies for a state. The fractal dimension was calculated with EEGs recorded on the frontal and parietal regions, and also for the core and periphery nodes in the MST, respectively. The core nodes were defined as the nodes whose distance is smaller than the average distance over all nodes in MST.

Results: In the regional analysis, the fractal dimension drops considerably after anesthesia, in particular, in the parietal region. Regarding the core and periphery nodes, the fractal dimension of the periphery nodes significantly decreased, whereas the core nodes were preserved after anesthesia.

Conclusion: This core-periphery node analysis may separate the persistent and adaptive brain activities during general anesthesia. The reduction of fractal dimension of the peripheral nodes may imply that the activity of EEG topographies processing external stimuli was selectively inhibited by anesthetics, particularly in the parietal region.

Influence of time delays and of stochastic fluctuations on the dynamics of a self-repressing gene
Marc Lefranc
Université Lille 1, Lille

The simplest genetic regulatory network consists of a single gene which is repressed by its own protein. As such, it provides an invaluable testbed to understand the influence of various effects on the dynamics of gene circuits. In particular, assuming a finite gene response time allows one to investigate the consequences of a non-trivial transcriptional dynamics. It was shown previously that this response time is equivalent to a time delay, and favors the appearance of oscillations in this system [1].

Here we consider the interaction of this response time with explicit or implicit time delays associated with reversible or irreversible molecule transport between cytoplasm and nucleus. We obtain an analytical expression of the oscillation threshold, which allows us to study how the two types of delay combine to enhance or repress oscillations. To understand the role of fluctuations, we have also constructed a modified version of the model of Ref. [1], where both the average values and variances of gene activity and molecule concentrations are dynamical variables. This allows us to study how nonlinearities and stochastic fluctuations interact, and to show that the latter can induce oscillatory behavior in this system without requiring nonlinear degradation.

[1] Morant et al., Phys. Rev. Lett. 102, 068104 (2009).

Stochastic competitive population dynamics: A study on evolutionarily stable dispersal rate in heterogeneous spaces
Presenting Author: Yen Ting Lin
Co-author: Hyejin Kim, Charles R. Doering

We propose two individual-based patch models to study competitive population dynamics of two species with identical birth and death rates, but distinct dispersal rates and compete for limited resourses in heterogeneous environments. The objective is to explore whether the evolutionarily stable dispersal rate exists, and if so, how it depends functionally on various parameters of the systems. Combining conventional asymptotic analysis with a novel asymptotic analysis proposed by Lin et al[1], we obtained closed forms of asymptotic solutions of both systems, as well as the insight of the detail dynamical mechanisms.

The essential parameter of the system is identified to be the carrying capacity of each patch times the environmental variance. The conclusions are: (1) Given fixed dispersal rates of both species, the slower dispersers will always have evolutionarily advantage over a long period of time if the parameter is greater than a critical value that depends upon the ratio of the birth/death rates and the dispersal rates. In other words, slower dispersers have evolutionary advantages in more heterogeneous environments, as well as in a system with larger characteristic population size, and vice versa. (2) The evolutionarily stable dispersal rate exists only when the parameter is greater than a uniquely defined critical value which depends solely on the ratio of the birth/death rates . (3) We obtained asymptotically closed form of the evolutionarily stable dispersal rate as the response of various parameters of the systems. Most importantly, we understand how the evolutionarily stable dispersal rates depend on environmental variance.

Demographic fluctuations, which are often neglected in deterministic models, are identified to be the fundamental mechanisms for such regime shifts. Our analytical results are supported by large-scale exact numerical simulations. The limit behaviors reported by previous studies [2-4] confirm our general analysis, which provides a comprehensive understanding of such type of individual-based competitive population dynamics.

1. Y.T. Lin, H. Kim, C.R. Doering, Features of Fast Living: On the Weak Selection for Longevity in Degenerate Birth-Death Processes, Journal of Statistical Physics, Doi: 10.1007/s10955-012-0479-9
2. J. Dockery, V. Hutson, K. Mischaikow, and M. Pernarowski, The evolution of slow dispersal rates: a reaction diffusion model, Journal of Mathematical BIology 37, 61-83 (1998).
3. D.A. Kessler and L.M. Sander, Fluctuations and dispersal rates in population dynamics, Physical Review E 80, 041907 (2009).
4. J.N. Waddell, L.M. Sander, and C.R. Doering, Demographic stochasticity versus spatial variation in the competition between fast and slow dispersers, Theoretical Population Biology 77, 279-286 (2010).

The Nonlinear Dynamics of Running: Symmetry, Stability, and the Effects of Amputation
Nicole Look
University of Colorado

In traditional gait biomechanics studies, kinematic data - e.g., the positions of the limbs and joints of a human runner - are normalized and averaged across multiple strides. This kind of analysis effectively ignores the dynamical details of locomotion, which are key elements in stability. Nonlinear time-series analysis of gait data can bring out these details quite effectively. Most existing studies of the nonlinear dynamics of human gait have focused on walking, however, and none have involved people with amputations - a population to whom stability is uniquely important. Using nonlinear time-series analysis on motion-capture data from 17 subjects running on a treadmill over a wide range of speeds (3-9 m/s), we studied runners with (6) and without (11) unilateral transtibial amputations. We used standard delay-coordinate embedding techniques to reconstruct the dynamics from scalar time-series traces of the positions of various anatomical markers (e.g., the height of the sacrum or the fore-and-aft position of the right knee), then calculated the maximal Lyapunov exponent - an established proxy for stability - of each of the resulting trajectories. We found that lower-limb mechanics were less stable (viz., higher lambda) for the affected leg of runners with amputations than their unaffected leg - or than either leg of the runners with two intact biological legs. The lambdas increased with running speed, but the inter-leg and inter-subject relationships remained the same. Surprisingly, the results showed that the center of mass of runners with intact biological legs is slightly less stable than for runners with unilateral transtibial amputations. This suggests that asymmetries may lead to instability in the legs that are compensated for by increased control of the core.

Quantification of dynamic stability has significant potential in assessment, diagnosis, and treatment: e.g., identifying people who may be at a greater risk of falling, understanding the role of asymmetries in stability, and evaluating efficacy of new prosthetic devices or rehabilitation programs (which attempt to increase stability in people with amputations by addressing asymmetries between the affected and unaffected leg).

Classical, quantum and scattering properties of irrational triangular billiards
Flavio M. de Aguiar
Universidade Federal de Pernambuco

Classical billiards in polygons are not chaotic, yet there are numerical evidences they might be ergodic. Here we use numerical and microwave techniques to investigate a number of particle and wave features in triangles whose sides are consecutive integers (N, N+1, N+2). As recently demonstrated [1], all angles in a triangle belonging to this one-parameter family are irrational with pi if 3 < N < ∞. The lower limit is the 3-4-5 right triangle and the upper limit is the equilateral triangle. For the classical dynamics, we calculate the relative measure r(t) of the occupied cells in a discretized surface of section as a function of the discrete time t, as well as position and speed correlation functions, for varying N. Results are compared with the universal random model (RM) [2]. For small N, the calculated r(t) is very close to the analytic result of the RM, and the correlation functions decay with an exponent close to -1. A slow departure from the thus characterized ergodicity is observed for N > 1000. For the quantum dynamics, we numerically solve the Schroedinger equation with a boundary method. Short range (nearest neighbor spacing distribution) and long range (Dyson-Metha) statistics are calculated for the first 145 000 energy eigenvalues beyond the first 5 000 ones in the unfolded spectra. In contrast with the classical result, a relatively fast departure from Random Matrix Theory is observed for 50 < N < 100. Scars were searched and not found in high-lying wavefunctions for N < 50, in qualitative agreement with the so called quantum unique ergodicity. Ghosts of classical periodic orbits could be found for quasidoublets which appear in the spectra for N > 50. In addition, we are currently making vectorial scattering experiments in flat microwave resonators in the 1-20 GHz range, at room temperature. We are particularly interested in measuring the one-channel S matrix (Poisson kernel) [3] for representative values of N and the first results will be presented at the conference.

[1] F.M. de Aguiar, Phys. Rev. E 77, 036201 (2008).

[2] M. Robnik, J. Dobnikar, A. Rapisarda, T. Prosen, and M. Petkovšek J. Phys. A: Math. Gen. 30, L803 (1997); G. Casati and T. Prosen, Phys. Rev. Lett. 83, 4729 (1999).

[3] U. Kuhl, M. Martínez-Mares, R.A. Méndez-Sánchez, and H.-J. Stoeckmann, Phys. Rev. Lett. 94, 144101 (2005); J.-H. Yeh, J.A. Hart, E. Bradshaw, T.M. Antonsen, E. Ott, and S.M. Anlage, Phys. Rev. E 81, 025201(R) (2010).

* Work supported by CNPq, CAPES and FACEPE (Brazilian Agencies)

Wavelet Based Method for Phase Detection of Chaos
Elbert Macau
National Institute for Space Research – INPE
Co-authors: Maria Teodora Ferreira; Rosangela Follmann Bageston; Margarete Oliveira Domingues; Epaminondas Rosa

We present here a new technique for phase detection in chaotic systems based on the dual-tree complex wavelet transformation. This transformation is a recent enhancement to the discrete wavelet transform that implies in nearly shift invariant and directionally selectivity in two and higher dimensions. In the context of chaotic dynamics, our methodology requires as the input a scalar measured signal and allow us to detect phase for phase coherent and also phase noncoherent chaotic oscillators. This approach is robust to noise level and computational efficient. Based on it, phase synchronization between chaotic signals can be detected. The applicability of our method is shown in simulated and in experimental situations.

Chaotic Hydromagnetic Convection
Wieslaw M. Macek
Space Research Centre, Polish Academy of Sciences

"We consider convection in a horizontally magnetized viscous fluid layer in the gravitational field heated from below with a vertical temperature gradient. Following Rayleigh-Benard scenario and using a general magnetohydrodynamic approach we obtain a simple set of four ordinary differential equations. In addition to the usual three-dimensional Lorenz model a new variable describes the profile of the induced magnetic field. We show that nonperiodic oscillations are influenced by anisotropic magnetic forces resulting not only in an additional viscosity but also substantially modifying nonlinear forcing of the system. On the other hand, this can stabilize convective motion of the flow. However, for certain values of the model parameters we have identified a deterministic intermittent behavior of the system resulting from bifurcation. In this way, we have identified here a basic mechanism of intermittent release of energy bursts, which is frequently observed in space and laboratory plasmas. Hence we propose this generalized Lorenz model as a useful tool for analysis of intermittent behavior of various environments, including convection in planets and stars, and possibly magneto-confined plasmas in tokamaks, nanodevices and microchannels in nanotechnolgy. Therefore, we hope that our simple but still a more general nonlinear model could shed light on the nature of hydromagnetic convection.

W. M. Macek and M. Strumik, Model for Hydromagnetic Convection in a Magnetized Fluid, Phys. Rev. E 82, 1, 027301 (2010), doi: 10.1103/Phys.RevE.82.027301.

Experimental Confirmation of Discontinuous Bifurcation in Digital Sliding-Mode Controlled Step-Down Power Converters
Somnath Maity
National Institute of Technology, Rourkela

Power electronics converters are known to exhibit undesirable subharmonic and chaotic behavior beyond certain parameter range. Theory of border-collision bifurcation has been developed for piecewise smooth continuous (PWS) maps and has been successfully applied to explain nonsmooth bifurcation phenomena in such systems. However, there exists some dynamical systems, e.g., discretized sliding mode (DSM) controlled power converters, where the steady state dynamical behaviors cannot be understood using 2-D PWS continuous maps. Occurrence and existence of typical complex behaviors observed in these systems are still unknown. This paper addresses the bifurcation phenomena of 2-dimensional SM controlled step-down converter when the measurement of the state feedback is sampled. The concept of discontinuous border-collision bifurcation theory is introduced to investigate the inherent dynamical properties of DSM controlled systems. Experimental results are presented to verify the theory of nonsmooth border-collision bifurcation for PWS discontinuous maps.

Visualization of Structural and Functional Discovery in Dynamic Graphs
Shawn Mankad
Department of Statistics, University of Michigan

Visual analytics, a class of methods that combine analytical and visualization techniques, contain many important tools to extract information from large, complex data. We present methodology for visual analytics on graphs evolving over time. The algorithms, which employ penalized matrix decompositions, uncover embedded structures and latent node-level functions. When coupled with classical graph layout and concept visualization techniques, the methodology yields filtered graphs and decompositions that are interpretable, informative and facilitate analytic reasoning. The methodology is illustrated on several simulated and real-world data sets.

Immunotherapy and Vaccination Strategies for latently Tuberculosis infection: a hybrid multi-organ modeling approach.
Simeone Marino
University of Michigan

Tuberculosis is a world wide health problem with 2 billion people infected with Mycobacterium tuberculosis (Mtb, the bacteria causing TB). The hallmark of infection is the emergence of organized structures of immune cells forming primarily in the lung in response to infection. Granulomas physically contain and immunologically restrain bacteria that cannot be cleared. We have developed several models that spatially characterize the dynamics of the host-mycobacterial interaction, and identified mechanisms that control granuloma formation and development. In particular, we published several agent-based models (ABMs) of granuloma formation in TB that include many subtypes of T cell populations, macrophages as well as key cytokine and chemokine effector molecules. These ABM studies emphasize the important role of T-cell related mechanisms in infection progression, such as magnitude and timing of T cell recruitment, and macrophage activation. In these models, the priming and recruitment of T cells from the lung draining lymph node (LN) was captured phenomenologically. In addition to these ABM studies, we have also developed several multi-organ models using ODEs to examine trafficking of cells between, for example, the lung and LN. While we can predict temporal dynamic behaviors, those models are not coupled to the spatial aspects of granuloma. To this end, we have developed a multi-organ model that is hybrid: an ABM for the lung compartment and a non-linear system of ODE representing the lymph node compartment. This hybrid multi-organ approach to study TB granuloma formation in the lung and immune priming in the LN allows us to dissect protective mechanisms that cannot be achieved using the single compartment or multi-compartment ODE system. The main finding of this work is that trafficking of important cells known as antigen presenting cells from the lung to the lymph node is a key control mechanism for protective immunity: the entire spectrum of infection outcomes can be regulated by key immune cell migration rates. Our hybrid multi-organ implementation suggests that effector CD4+ T cells can rescue the system from a chronic persistent infection and lead to clearance once a granuloma is fully formed. This could be effective as an immunotherapy strategy for latently infected individuals.

Distinghising determinism from stochasticity: ordinal analysis of the structure of the spiking activity of semiconductor lasers with optical feedback
Cristina Masoller
Universitat Politecnica de Catalunya

When a semiconductor laser is submitted to optical feedback it exhibits dynamical instabilities. These instabilities, or Low Frequency Fluctuations (LFFs), consist in sudden, irregular power dropouts followed by a gradual recovery. The mechanisms underlying the LFFs are both deterministic and stochastic.

We analyze experimental time traces of the dropouts activity of the interdropout time intervals into series of words. We study the statistics, the ordering and the transitions of these words to distinguish between determinism and stochasticity. We also separate the time traces into trains of dropouts separated by fixed point stays to analyze the underlying dynamics of the system.

The method of "channels and jokers" in the research of bioelectrical activity of the brain
Oleg Mayorov
Kharkiv Medical Academy of Postgraduate Education

Rather recently has appeared a nonlinear dynamics approach to the development of new methods in EEG analysis as well as an increasing interest in the study of bio-electric activity of the brain through nonlinear analysis. The special value of this approach allows not only steady and exponential estimations of brain functions, but also provides for new diagnostic tools in the assessment of various pathological processes in the cerebral cortex.

Unfortunately, at present, this approach is mostly of theoretical interest. The main reason of which being the fact that EEG rhythms represent a superposition of signals coming from different structures of the brain, representing different sometimes unrelated functions. The ‘randomness' of such an activity cannot be reliably analyzed because of the inevitable background noise and limitations due to violations of conditions of stationarity of the time interval accessible in the analysis.

We offer a new approach to the study of EEG rhythms by nonlinear analysis by using the concept of ‘channels and jokers ‘, based on the EEG as a structured segment-stable process.

As known, at present, the EEG-signal consists of practically stationary segments ‘connected' by fast transients. As it is unavoidable to tackle models of higher dimensions, the presence of these fast transients reduces the constraints on the application of nonlinear dynamics in EEG analysis.

The phase space of such dynamic systems is inhomogeneous. The state of the system of stationary segments can be rather exactly described by few variables, that is, components constituting a projection of small dimension. Such areas are called ‘channels' and areas in which the construction of a projection of small dimensions is not possible are called ‘jokers'. The operator of the brain dynamic system in area of a channel represents a dynamic system which includes also random components. The determined part reproduces the observable regular dynamics adequate to a corresponding range of scales, and the irregularity brought about by both fast small-scale interior factors, and uncontrollable exterior actions on a system, is described as a stochastic process.

Using this approach from the perspective of deterministic chaos we are planning to construct an adequate mathematical EEG model for subsequent analysis. This will increase the precision and reliability of the estimation of parameters of chaos in the corresponding dynamical brain systems.

Practical Applications of the Hausdorff-Besicovitch Fractal Dimension to Real World Data
Mike McCormack
Fractical Solutions Inc.

The Hausdorff-Besicovitch fractal dimension offers a powerful technique to investigate the fractal structure of real world data. This presentation/paper presents examples of the analysis of time series data from petroleum geological well log information and stock market indices. The technique is based on the recognition that when the Hausdorff-Besicovitch fractal dimension is applied to real world data that the calculated fractal dimensions vary along the time axis of the data as well as with the relative scale of investigation. A grid of Hausdorff-Besicovitch dimension is calculated with time and scale as x,y components. The value of the Hausdorff-Besicovitch fractal dimension at each grid node is then used as a z value and is contoured to give a 3 dimensional representation of the data. In essence, the technique takes 2 dimensional data and creates a 3 dimensional representation using the Hausdorff-Besicovitch fractal dimension as a window on the connections between the various data elements. The results can be surprising, showing regions with high fractal dimensions that appear deceptively smooth to the eye. There are also “cusp” type data structures that although subtle, can result in very significant changes in the fractal dimensions over narrow time spans. These “cusp” structures tend to be associated with changes in geological horizons from petroleum well log data. The presentation will also include a summary of the calculation routines employed.

Chaos and the Quantum: How Nonlinear Dynamics Produces Classical Correlations Analogous to Quantum Entanglement
Wm. C. McHarris
Dept. of Chemistry and Physics/Astronomy, Michigan State University

Replicate Periodic Windows in the Parameter Space of Driven Oscillators.
Everton Medeiros
Instituto de Física, Universidade de São Paulo

We obtain the two-dimensional parameter spaces of two nonlinear dissipative oscillator, namely, Duffing and Josephson junction. For these well-known oscillators, the parameters for which these systems behave periodically are aggregated in shrimp-shaped periodic windows immersed in chaotic regions. By applying a weak harmonic perturbation to the original system we replicate shrimp-shaped periodic windows giving rise to new parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have the same periodicity and pattern of stable and unstable periodic orbits already existent for the unperturbed oscillator. Moreover, these unstable periodic orbits are embedded in chaotic attractors in phase space regions where the new stable orbits are identified. Thus, the observed periodic window replication is an effective oscillator control process, once chaotic orbits are replaced by regular ones.

Secure communication of digital messages using synchronization of a response system with a chaotic drive system based on linear and time independent error dynamics
Ashok Mittal
University of Allahabad

Several communication schemes have been suggested that make use of synchronization of a response system with a chaotic drive system for communication of messages. Typically the synchronization errors, which we may denote by e, are governed by a system of nonlinear equations. For synchronization to take place, a signal from the drive is coupled to the response system in such a way that this error system has a stable fixed point at e = 0. This stability is demonstrated by techniques like the Lyapunov method or by obtaining the Lyapunov exponents of the variational equation. If the synchronization error system is nonlinear, the coefficients of the variational equation depend on the variables of the chaotic drive system. As a consequence, the synchronization errors fluctuate chaotically on time scales of the chaotic drive oscillation period. Synchronization time has to be larger than the oscillation time.

We present variants of some of the existing schemes so that the difference between the response and a chaotic driving system is governed by a system of linear and time independent equations. The synchronization time can be made smaller than oscillation time leading to faster synchronization, which can be used to transmit digital messages by modulating a parameter. The value of the forcing parameter is not known to the response system, but a parameter adaptation mechanism allows this parameter to be determined rapidly. The rate of parameter adaptation is chosen to be proportional to the synchronization error. This ensures that the augmented system of synchronization errors in variables and parameters is linear and time independent. There are several advantages of this scheme of communication. Not only do the response variables and parameters converge rapidly to the drive values, they also converge smoothly, instead of fluctuating chaotically before convergence. Consequently the parameters of the drive, which can be used for modulation, can be more closely spaced than would otherwise have been possible. This makes it possible for a larger number of symbols to be transmitted. Thus messages can be communicated with greater speed. As synchronization takes place on a time scale smaller than the period of the driving chaotic oscillations, the modulating parameter can be switched in a time less than the time interval between two successive maxima of a drive variable. Therefore an intruder cannot reconstruct the chaotic system using a return map technique. Thus the scheme presented allows for faster and more secure communication of digital messages. The only drawback of the scheme is that it requires all the variables of the drive to be transmitted to the response system. However, the advantages of speed, smooth convergence of the response parameter and enhanced security outweigh this drawback.

Determining Memory of a Dynamical System from Observed Time Series
Chetan Nichkawde
Department of Physics and Astronomy, Macquarie University

Deterministic dynamics theoretically have infinite memory as the present state can be traced back exactly to its initial condition. Chaotic system are however characterised by sensitive dependence to the initial condition. This is coupled with the finite resolution of the measurement apparatus and various sources of noise. This means that causal connection with the past becomes increasingly diminished and practically chaotic systems especially those with broadband spectra have finite memory. An information-theoretic approach to determining memory of a system from observed time series is proposed.

Simulated data from Mackey-Glass equation with added noise and experimental data from semiconductor laser is used to demonstrate the approach.

Characterization of the chaotic and fractal dynamics of a predator-prey model
Chantawan Noisri
Purdue University

A classic issue in ecology is the study of the evolution of population of species, including prey and predator populations. Recently, a simple yet fascinating 2-D model has been proposed for the study of the chasing of a prey by a predator (Zhdankin and Sprott, Phys. Rev. E, V82, 056209 (2011)). It has been shown that the model can exhibit quasi-periodic and chaotic motions. We study the general correlation between the predator and the prey's trajectories, in particular, their fractal properties. We also explore the emerging new dynamical behaviors when the 2-D model is generalized to 3-D. Such a generalization is important for modeling predator-prey dynamics in 3-D, including eagles catching rabbits, birds catching insects, and so on. We also explore the new behavior of the model by synthesizing certain stochasticity with the shortest/quickest path principle.

Detecting connectivity pattern of brain activity in different consciousness level using wavelet transform and regression analysis
Yeo Jung Park
University of Michigan

An important challenge in neuroscience is uncovering the relationship between changes in consciousness level induced by anesthesia and brain activity. This has been studied by measuring dynamic changes in electroencephalography (EEG) from several regions of the brain. We propose a novel methodology that combines multiresolution wavelet transformation and a regression based measure that represents the strength of association between two EEG recordings. EEG data are obtained from 18 patients before and after anesthesia. We compare the pairwise relationships among 6 different brain channels (4 frontal and 2 parietal) in two states -- awake and anesthesia. Correlations between raw EEG series indicate that coupling patterns in the brain activity are systematically altered in the anesthesia state. Building on this observation, a multiresolution decomposition of the raw EEG allows us to determine the temporal scales at which coupling is either maintained or lost under anesthesia. We also discuss extensions to our methodology that assesses a potential unobserved effect that could be responsible for the previous experimental findings.

Bifurcation and chaos in driven simple pendulum under the effect of nonlinear damping
Vinod Patidar
Sir Padampat Singhania University

Nonlinearity is abundant in nature. It is having an increasingly important impact on a variety of applied subjects ranging from the study of turbulence and the behaviour of weather, through the investigation of electrical and mechanical oscillations in engineering systems, to the analysis of various biological, ecological and economic phenomena. The nonlinearity in the dynamical systems may exist in various forms e.g., in a mechanical system: the nonlinearity may be due to the presence of nonlinear elastic/spring elements, nonlinear damping, systems with fluids, nonlinear boundary conditions etc., in an electromagnetic system: the nonlinear resistive, inductive, capacitive elements, hysteresis properties of ferromagnetic materials, nonlinear active elements like vacuum diode, transistor etc. may be responsible for nonlinearities in the system. The notion of such nonlinear systems is studied under the banner of nonlinear dynamics and its sub-filed chaos theory, which particularly discusses the bounded, aperiodic behaviour of certain nonlinear systems that are highly sensitive to initial conditions (deterministic chaos). The driven simple pendulum is one of the most common examples of nonlinear systems (under the large amplitude oscillations) exhibiting chaotic motion and it is also isomorphic to many nonlinear systems such as Josephson junctions and the phase-locked-loop configuration of a voltage-controlled oscillator (VCO) etc.

In this communication, we discuss the dynamical behaviour of driven simple pendulum under the effect of nonlinear damping. We particularly consider the nonlinear damping term proportional to the power of velocity and focus our attention on how the damping exponent affects the global dynamical behaviour of the forced pendulum. We obtain analytically the threshold condition for the occurrence of homoclinic bifurcation using Melnikov technique and compare the results with the computational results. We also identify the regions of the 2D parameter space (consists of external forcing amplitude and damping coefficient) corresponding to the various types of asymptotic dynamics under linear (viscous or friction like) and nonlinear (drag like) damping. We also analyse how basin of attraction patterns corresponding to various attractors change with the introduction of nonlinear damping as well as damping strength.

Signatures of Non-Monotonicity in the Quantum-Classical Transition
Arjendu Pattanayak
Carleton College

The transition between a system behaving completely classically and behaving completely quantum-mechanically depends on various factors including the size of the system, the temperature and environmental effects. The behaviors can be very different in the two limits; for example, it is typically assumed that chaos only exists in the classical limit and quantum effects only serve to reduce the degree of chaos in the problem. In this talk we present results from a theoretical analysis of a double-well system in a regime of currently experimentally accessible nano-mechanical systems. Various measures are shown to behave non-monotonically across the quantum-classical transition. In particular, Lyapunov exponents are shown to behave non-monotonically, and in some situations signal the presence of chaos in the intermediate regime when chaos does NOT exist at either the quantum or classical limit.

A new phase space method for radar/acoustic target identification.
Frederic Rachford
Naval Research Laboratory

We have developed a method for target identification employing chaos based waveforms and a nearest neighbor separation metric. Rf Simulation and acoustic range data were acquired by scattering RF or acoustic chaotic waveforms from four different but similar objects. In simulation the objects were a 130mm artillery shell and various metal cylinders and ogival objects of similar dimensions. In the acoustic case the objects were four different plastic water bottles. In either case nearest neighbor strands were located in the 2D embedded data of the designated reference objects. The nearest neighbor strand coordinates where then used to select strands in data from other targets. The various targets act as unique filters transforming the scattered waveforms and affecting the relative location of the strands and their separations. In general we find that data taken from the same object as employed for the reference set will have a smaller average strand separation than that acquired from one of the other objects. We propose to use the relative average distance between strands to discriminate between targets.

Evolving Heterogeneous Social Fabrics For The Solution Of Real Valued Optimization Problems Using Cultural Algorithms
Robert Reynolds
Wayne State University and University of Michigan

In this paper we investigate the performance of Cultural Algorithms over the complete range of Langtons' optimization problem complexities, from fixed to chaotic. In order to apply the Cultural Algorithm over all complexity classes we generalize on its co-evolutionary nature to keep the variation in the population across all complexities. We produced a new version of the Cultural Algorithms Toolkit, CAT 3.0, which supported a variety of co-evolutionary features at both the Knowledge and Population levels. We then applied the system to the solution of a 150 randomly generated problems that ranged from simple to chaotic complexity classes. As a result we were able to produce the following conclusions: No homogeneous Social Fabric tested was dominant over all categories of complexity. As the complexity of problems increased, so did the complexity of the Social Fabric or network that was needed to deal with it efficiently. In other words, there was experimental evidence that social structure can be related to the frequency and complexity class of the problems that presented to a cultural system. Next, we allowed the system to evolve heterogeneous Social Fabrics for the solution of these problems classes. The results are compared with those that used the homogeneous SocialFabrics.

Topology of toroidal chaos produced by a driven active double pendulum
Martin Rosalie
CORIA, Universite de Rouen

A topological analysis of some toroidal chaotic behaviors produced by an active double pendulum are performed. The dynamical system is a mechanical double pendulum driven by a DC motor to supply energy to the system to compensate for the friction. This motor which is controlled by a speed feedback applies a torque to the first arm of the double pendulum : the torque depends on the angle made by the first arm with the vertical and the associated angular speed. This system is therefore a dissipative system which leads to sustained oscillations. Some of them are sensitive to initial conditions and developed in the neighborhood of a torus : this is thus toroidal chaos. Such a type of chaotic attractor has already been observed in various dynamical systems but its topological characterization remains an open problem. In the case of our double pendulum, the phase portrait is reconstructed from angle θ_1 and its successive derivatives. From this attractor the inversion symmetry — which complicates the analysis — has been removed by using a coordinate transformation to provide an 'image attractor' as discussed in [1,2]. We identified a period-doubling cascade by varying one of the parameter. We choose to investigate the dynamics slightly after the accumulation point of this cascade, where there is a banded chaos [3,4]. This was motivated by the need to avoid the situation where the trajectory visits the whole toroidal surface, thus forbidding to draw up a template. This is the main drawback when a toroidal chaos is investigated using a topological approach. This contribution is therefore a first step in the topological characterization of toroidal chaos which, in its general perspective, remains an open problem. Doing so, we constructed a template of the image attractor and then obtained the template for the symmetrical attractor by a systematic procedure we developed. Such a procedure allows us to draw up the template of any equivariant systems — bounded by a genus-1 torus — starting from its image. We thus obtained the template of banded toroidal chaos produced by a mechanical driven double pendulum whose experimental data were also investigated.

1. C. Letellier & R. Gilmore, Covering dynamical systems : Two-fold covers, Physical Review E, 63, 16206, 2001.

2. C. Letellier, R. Gilmore & T. Jones, Peeling bifurcations of toroidal chaotic attractor, Physical Review E, 76, 066204 (2007)

3. C. Letellier, V. Messager & R. Gilmore, From quasiperiodicity to toroidal chaos : analogy between the Curry-Yorke map and the van der Pol system, Physical Review E, 77, 046203 (2008)

4. M. Rosalie & C. Letellier, Vers une topologie du chaos toroïdal, Comptes-Rendus des Rencontres du Non Linéaire, 14, 151-156 (2011)

Multi-lag Synchronization in Coupled Chaotic Oscillators
Prodyot Kumar Roy
Presidency University

When two oscillators are coupled through conventional coupling schemes, during lag synchronization, all the pairs of state variables develop same amount of lag (or delay time) which may depend on mismatch and coupling strength. In contrast to such phenomenon, we report in this paper, a multi-lag synchronization between two chaotic mismatched oscillators coupled through a nonlinear coupling scheme. By multi-lag synchronization we mean that the coupled system develops a state of synchronization where the various pairs of state variables are simultaneously synchronized but with different lag time (or delay) in each pair of them. The amount of these lags can be predetermined by the user. We discuss the design procedure of the nonlinear coupling and apply the method to establish multi-lag synchronization in Hindmarsh-Rose neuron model and Lorenz systems. We realize the multi-lag synchronization in an electronic circuit of a coupled Sprott system and present experimental results. In addition, we show how this scheme can be used such that the amount of lag varies at different length of time. Such type of time varying multi-lag synchronization is important in respect of neuron dynamics.

Experimental deviations from the Strogatz-Lorenz model for the Malkus waterwheel
George Rutherford
Illinois State University

The Malkus waterwheel is well-known as a simple mechanical system that can exhibit chaotic behavior. Our experimental version of this wheel consists of a polycarbonate base with 36 cylindrical cells placed around the edge of the tilted wheel. A trough at the top of the cells directs all the incoming water into the cells that pass under the input stream, and water leaks from each cell through a long outlet of small diameter. A thin aluminum ring at the periphery of the wheel passes through variable gap magnets, allowing for adjustable eddy-current braking that is proportional to the wheel’s angular velocity. We acquire angular time series data with a rotary encoder, and we then smooth the data and calculate angular velocity and acceleration using Fourier transforms. The magnetic brake strength is used as the primary variable to construct a bifurcation plot. Strogatz derived a set of coupled first-order differential equations with simplifying assumptions, and these equations can be solved numerically to yield the wheel’s motion for a given set of parameter values. While simulations based on the Strogatz model give a reasonable facsimile of the wheel’s behavior, experiments have shown some significant disagreement. For example, the model predicts a sharp Hopf bifurcation at low brake strength as the wheel transitions from rotation to oscillatory motion. The experiments also show this behavior but indicate a noticeable shift in the position of the bifurcation with changing initial angular velocity of the wheel, an effect that is not shown by the model simulations. We believe this shift is related to the model’s assumption that the leak rate from each cell is strictly proportional to the mass of water in the cell, while experiments show that the relationship is linear but with a significant offset. The data also show that the wheel’s behavior at higher brake strengths does not contain the large periodic windows predicted by the model. We will show the results of the application of Gottwald’s 0-1 test for chaos to the data, which indicate that all data at brake strengths higher than the period doubling cascade are chaotic.

Resonancy width distribution for open chaotic quantum systems
Gavriil Shchedrin
Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

Random Matrix Theory is a conventional approach for exploring a wide variety of complex quantum systems exhibiting signatures of quantum chaos. In case of neutron low-energy elastic scattering off heavy nuclei, the root of quantum chaos lies in a complex structure of the wave functions of the compound system neutron $+$ nucleus.

Under the standard assumption that the decay of an unstable state serves as an analyzer that singles out a specific component of the intrinsic chaotic wave function, the decay amplitudes reflect the Gaussian distribution of corresponding components. Then the widths, being proportional to the squares of the amplitudes, should obey the natural chi-square $chi^{2}_{1}$, or the Porter-Thomas distribution.

Recent measurements of resonance widths for low-energy neutron scattering off heavy nuclei show significant deviations from the Porter-Thomas distribution. The unstable nucleus is an open quantum system, where the intrinsic dynamics has to be supplemented by the coupling of chaotic internal states through the continuum. This is governed by the effective non-Hermitian Hamiltonian which changes statistics of levels and widths described now as poles of the scattering matrix in the complex plane.

We propose [arXiv:1112.4919] a new analytical resonance width distribution based on the random matrix theory for an open quantum system with a single open channel. For practically important cases, the new distribution is represented by ${cal P}(Gamma)simeqchi^{2}_{1}sqrt{sinh{kappa}/{kappa}}$ with $kappa={piGamma}/{2D}$ and $D$ the mean energy level spacing.

In the limit of weak continuum coupling $kappall1$, our result naturally recovers the Porter-Thomas distribution. The new distribution suppresses small widths and increases the probabilities of larger widths. Along with this, the real resonance energies do not repel each other at small spacings. In the limit of strong continuum coupling, the resonances overlap which leads to the collectivization of the widths analogous to the super-radiance in quantum optics.

Human capital accumulation in a Growth Model: when chaos emerge.
London Silvia
Instituto de Investigaciones Económicas y Sociales del Sur

We analyze the importance of the accumulation and government funding of human capital through the formal educational system for economic growth. We base our analysis on two previous models (London: 2005, London: 2006) in which, starting with the framework in (Lucas: 1988) we add a parameter of distortion on the formal educational sector. Since the model is built upon a logistic equation and it is not analytically solvable, we run simulations for different scenarios, using data from Argentine´s economy. We conclude that the stability of the growth process depends critically on a fine tuned combination of the main variables, which involve government funding as well as technical and institutional parameters. Classification JEL: C63, I28 Growth - Education - Chaotic Systems

On the reconstruction of a signal from its unthresholded recurrence plot subject to disturbances
Aloys Sipers
Centre of Expertise in Life Sciences, Zuyd University

"We examine the information content of unthresholded recurrence plots (URPs) subject to disturbances. This is important for making valid inferences from URPs. In our recent work [1], we provided joint conditions on the embedding parameters (embedding dimension and the time-delay) which guarantee that a zero mean signal can be uniquely recovered from its URP up to a sign factor. For signals with a known frequency content, this permits the computation of values for the embedding parameters which lead to maximally informative URPs. When these uniqueness conditions are not satisfied, then it is possible for different morphologies in a signal to give rise to identical patterns in the URP or RP. The information content of binary (thresholded) RPs has been studied in [2] and [3].

We also refine the graph theoretic procedure of [1] which was developed to characterize the uniqueness conditions. Here, the reconstruction methods of [1] are employed to investigate the reconstructibility of a signal from a disturbed URP in particular for values of the time delay near a value which not satisfies the uniqueness conditions. The influence of disturbances on URPs or RPs is illustrated by several examples.


[1] A. Sipers, P. Borm, and R. Peeters, Phys. Lett. A 375, 2309 (2011).

[2] M. Thiel, M.C. Romano, J. Kurths, Phys. Lett. A 330, 343 (2004).

[3] G. Robinson, M. Thiel, Chaos 19, 023104 (2009).

Network recruitment to coherent oscillations in a hippocampal computer model
William Stacey
University of Michigan

Coherent neural oscillations > 30 Hz are present in normal functions, such as gamma binding, as well as network pathology such as epilepsy. Similarly, High Frequency Oscillations (HFOs) > 60 Hz have been identified in both normal and epileptic tissue. However, the mechanisms by which these phenomena form and the distinctions between normal and abnormal HFOs are unknown.

This work uses a physiological computational model to demonstrate that high levels of random synaptic activity can generate HFOs in a locally-coupled network, as well as help spread them to neighboring tissue. Measuring network recruitment required the development of a novel statistical method, which quantified the recruitment generated via preexisting physiological connections such as interneurons, gap junctions, and recurrent axons.

'Epileptic' HFOs, produced by active pyramidal cells with high levels of gap junctions or recurrent axons, recruited neighboring cells provided those cells were exposed to elevated levels of random synaptic activity. 'Normal' HFOs, produced by coherent inhibitory oscillations and sparse pyramidal cell firing, suppressed surrounding tissue and were unable to induce anodal break firing.

These findings suggest that synaptic noise and physiological coupling are important for generating and propagating HFOs, and point to a key potential difference between the mechanisms and behavior of normal and epileptic HFOs.

Analysis of regular and chaotic dynamics of the buck converter
Zbigniew Szymanski
Warsaw University of Technology

The theoretical analysis of the buck converter is presented. The initial parts of current pulses depend on the receiver voltage and the terminal parts depend on the periodic pulses of the control clock. For one-dimensional LR buck converter the influence of the supplier source voltage and the receiver resistance on the intervals of stable periodic states of the clock period or its multiplicity as well as the intervals of chaotic states were determined. Also, the properties of chaotic dynamics of this systems are defined: attractors, their basins of attraction, the Lyapunov exponent.

The analysis of two-dimensional LCR buck converter focused on the effect enhanced the qualitative modifications of the system functionality caused by the capacity parallel to the receiver resistance.

The conclusions are twofold: the conditions of occurring chaos is of theoretical interest while the suggestions of avoiding chaos in the buck converter are technically sounding.

Discontinuous spirals in the parameter plane of a semiconductor laser model
C. Abraham Torrico Chávez
Universidade Federal do Rio Grande do Sul

The existence of spirals domains of stable periodic orbits (SPO's) in the parameter plane of nonlinear dynamical systems was anticipated theoretically [ C. Bonatto and J.A.C. Gallas Phys. Rev. Lett. 101, 054101 (2008)] and then confirmed experimentally in a circuit family [ R. Stoop, et al. Phys. Rev. Lett. 105, 074102 (2010)].

In this talk we report a new kind of discontinuous spiral of SPO's in a laser model, compound of the intercalation of rounded and cuspidal structures (the two normal forms of cubic dynamics), and we discuss its features and relations with another two kinds of spirals also present in the system. Its appearance in a concrete physical system, as is the semiconductor laser, incentivates its experimental verification and future applications; as well as motivates additional theoretical study about their origin and the possibility of the emergence of the so-called Mandelbrot-like sets [ A. Endler and J.A.C. Gallas Comptes Rendus Mathematique 342, 681-684 (2006)] in a concrete vector field as is the semiconductor laser rate equations.

Turbulence driven particle transport analyses in a plasma toroidal device
Dennis Toufen
University of São Paulo

In this work, we analyze the plasma turbulence driven particle transport in Texas Helimak [1] and investigate how alterations on the radial electric field, through an external voltage bias, modify the turbulence and the transport [2]. We employ spectral analysis to identify the main changes on the power spectra due to the external alterations on the radial electric field profile. Moreover, the use of biasing also allows observing the dependence of turbulence spectrum and transport on the flow velocity shear. Thus, we calculate the transport radial profile and its dependence on the electric bias and observe that the external bias value changes spectral plasma characteristics, inducing a dominant frequency for negative bias values and a broad band frequency spectrum for positive bias values. Because of these differences, the shots with negative and positive bias values are analyzed separately. For negative biased shots, the transport reaches a maximum where the waves propagate with phase velocities approximately equal to the plasma flow velocity. We use an almost integrable Hamiltoninan model to interpret this maximum as evidence that the transport is strongly affected by a wave particle resonant interaction. On the other hand, for positive bias values, the plasma has a reversed shear flow and we observe that the transport is almost zero in the shearless radial region indicating an evidence of a transport barrier in this region.

[1] K. W. Gentle and H. He, Plasma Sci. Technol. 10, 284 (2008).

[2] D. L. Toufen, et al, Phys Plasmas 19, 012307 (2012).

Detecting variations in dynamical systems by cross-scale-dependent Lyapunov exponent
Wen-wen Tung
Department of Earth and Atmospheric Sciences, Purdue University

Detecting variations in dynamical systems is an important issue in the study of chaos and complexity. Scenarios where this topic is crucial include 1) detecting the general correlation between two dynamical systems, 2) detecting whether and when a change has occurred in a chaotic system, and 3) rapid signal matching in engineering, whose purpose is to evaluate whether a measured complicated signal is from some chaotic system in the database.

To solve this challenging problem, we propose a new measure, the cross-scale-dependent Lyapunov exponent (cross-SDLE), which quantifies the cross-divergence between the signals x and y, through monitoring the co-evolution of the x and y dynamics. When this cross-divergence is the same as the auto-divergence of x and y, which can be obtained by monitoring, separately, the self-evolution of x and y dynamics, then the two signals x and y can be concluded to come from the same dynamical system. Otherwise, they come from different systems. The scheme is extremely resilient to noise. It effectiveness is illustrated by studying a few model dynamical systems.

Nontwist Phenomena in Area-Preserving Maps
Celso Vieira Abud
University of São Paulo

Many discretized Hamiltonian systems can be basically described by two-dimensional area-preserving maps. Some area-preserving maps present topological differences that can be provided by the so-called twist condition; i.e., a condition that asserts the nondegeneracy of the frequencies, classifying maps as twist and nontwist. The compositions of twist and nontwist phase spaces have some similarity, but also special differences. Phenomena as reconnection scenarios, periodic-orbit collisions, and shearless tori have been recognized as features of nontwist maps. Of particular interest, shearless tori are robust tori with maximum or minimum frequency, assuring the violation of the twist condition. However, recent studies have shown that nontwist phenomena may appear, secondarily, on both kinds of area-preserving maps.

In this work, we present a numerical investigation of the onset of twistless tori around elliptic fixed points of standard nontwist and twist maps. Owing to the topological similarity to the global shearless structure, we refer to twistless tori as secondary shearless (s-shearless) tori. As we are interested in secondary islands, our numerical procedure consists of identifying the emergence of the s-shearless torus in the standard map phase space by the onset of a maximum or minimum in the internal rotation number profile.

Our results have shown that s-shearless tori can emerge from secondary resonances in a neighborhood of the elliptic fixed point for both twist and nontwist standard maps. In the present work we show, for these maps, similar cases in the neighborhood of a tripling bifurcation of a fixed point that creates a s-shearless torus. Furthermore, in the interval where the s-shearless torus exists, the internal rotation number may reach a rational value where there should be nontwist bifurcations as reconnection process.

Theoretically, a twistless torus implies the local violation of important theorems that assume the nondegeneracy of the frequency; e.g., the KAM and Poincaré-Birkoff theorems. Apart from the theoretical implications, we have discussed a possible application of the emergency of a s-shearless tori within island of stability. There is numerical evidence that the breakdown of a tripling separatrix, at the edge of the chaotic sea, generates stickiness stronger than that generally observed. As tripling bifurcation comes always from secondary s-shearless torus, this increases the interest on theses twistless tori, especially because several dynamical systems have reported stickiness phenomena.

Control of the coupled logistic map
Charles Villet
University of Johannesburg

The dynamics of the two-dimensional coupled logistic map is described in parameter space. It is shown how the OGY method can be used to control unstable orbits of periods greater than one. A novel targeting procedure using bisection in two dimensions is then introduced and implemented numerically.

Adaptive Channel Equalization using Synchronized Chaotic Systems
Christian Wallinger
Institute of Electrical Measurement and Measurement Signal Processing, Graz University of Technology

In the last two decades, the synchronization of chaotic systems has received a great deal of attention in the areas of signal processing and communication engineering. Recently, the synchronization of chaotic systems is used to design Time-of-Flight (ToF) measurement systems [1] being a primary measurand in many metrological applications such as distance measurement, localization, and tracking. In this context the beneficial properties of signals generated by chaotic systems are their unpredictability and their noise-like appearance. From the metrological point of view such applications are required to deliver estimates with low measurement uncertainties in the presence of small signal-to-noise ratios, different kind of disturbers and bandwidth (BW) limitations. The communication channel impacts on the signal due to effects such as BW limitations, time dependent fading, and dispersion. Consequently, the ability of the receiver to synchronize with the transmitter will be impaired by such a channel [2], [3], [4].

In this work we address the problem of channel equalization in order to compensate for those distortions using the self-synchronization property of chaotic systems. Specifically, we present a channel equalization method based on the work of Sorrentino and Ott [5]. We extend this work for estimating and compensating the communication channel. A comparison of the proposed method with a classical channel equalization method based on an adaptive least-mean-square algorithm as used by Cuomo et al. [6] is given. In our experimental setup we compare the synchronization performance of different chaotic systems which are connected using a range of channel models. We further present an experimental estimation of an ultrasonic channel.

[1] C.F. Wallinger and M. Brandner, ‘Time-of-Flight Estimation using Synchronized Chaotic Systems', Advances in Nonlinear Dynamics & Synchronization with Selected Applications in Theoretical Electrical Engineering, Springer-Verlag, Germany, 2012

[2] T.L. Carroll, ‘Synchronizing chaotic systems using filtered signals', Phys. Rev. E, 50(4), 2580-2587 (1994).

[3] A.S. Dimitriev, A.I. Panas, S.O. Starkov and L.V. Kuzmin, ‘Experiments on RF Band Communications using Chaos', Int. J. Bifurcation and Chaos, 7(11), 2511- 2527 (1997).

[4] N.F. Rulkov and L.S. Tsimring, ‘Synchronization Methods for Communication with Chaos over Band-Limited Channels', Int. J. Cir. Theor. Appl., 27, 555-567 (1999).

[5] F. Sorrentino and E. Ott, ‘Adaptive Synchronization of Dynamics on Evolving Complex Networks', Phys. Rev. Lett., 100, 114101 (2008)

[6] K.M. Cuomo, A.V. Oppenheim and R.J. Barron, ‘Channel Equalization for Self-Synchronizing Chaotic Systems', Conf. Proc. 1996 IEEE Int. Conf. Acoustics, Speech and Signal Processing (ICASSP-96), 3, 1605-1608 (1996).

Ensemble Kalman Filter Observability of Neuronal Networks
Andrew Whalen
Penn State University

Ensemble Kalman filters (EnKF) have many useful applications in nonlinear dynamics from weather prediction to neuronal systems. A well-designed Kalman filter can optimally estimate states and track uncertain parameters of a chaotic system with sparse and unreliable measurements of the system states. A fundamental question that arises when utilizing the EnKF to estimate the future states of a system is how to choose a model and measurement function that faithfully captures the system dynamics and can predict future states. The underlying form of a Kalman filter is that of an observer, which is a model of a system or process with measurements from the natural system being modeled. The key concept to employ a "well designed" observer, is observability, which asks the question: is there sufficient information contained in the measurement to adequately reconstruct the full system dynamics? From the theories of differential embeddings and nonlinear reconstruction we have a nonlinear measure of observability comprised of the measurement function and its higher Lie derivatives: the so called differential embedding map. The differential embedding map of an observer provides the information contained in a given measurement function and model, which can be quantified by a measure: the observability index. Computed from the Jacobian of the differential embedding map, the observability index is a matrix condition number which quantifies the perturbation sensitivity (closeness to singularity) of the mapping created by the measurement function used to observe the system. Singularities in the map cause observability to decrease and information about the system is lost. We quantify observability in small (3 node) neuronal networks as a function of 1) the connection topology and symmetry, and 2) the presence or absence of inhibition. We find that typical observability indices for 3 neuron motifs range over several orders of magnitude, depending upon topology. For 4 neuron systems, similar ranges of observability were calculated, and adding an inhibitory layer, typically not observable in experiments, the network observability decreases. Our findings demonstrate that such networks are observable, and lead to their use in reconstructing network dynamics from limited observation data. How well such strategies can be used to reconstruct the underlying, and immeasurable, network topology in experimental settings is a subject for future experimental work.

Group Synchrony in an Experimental System of Delay-coupled Optoelectronic Oscillators
Caitlin R. S. Williams
University of Maryland

The study of group synchronization of delay-coupled dynamical systems is of interest in the context of physical and biological systems. The delay-coupled nodes or oscillators are placed into groups based on different parameters or governing equations. In this case, it has been shown theoretically that nodes in the same group may identically and isochronally synchronize with the other nodes in the group, even if there is no direct intra-group coupling. We report experimental observations of group synchrony in a network of four nonlinear optoelectronic feedback loops that are divided into two groups of two nodes, distinguished by different parameters for each group. Each node is delay coupled to the nodes not in its same group, but is not coupled to the other node in its own group. We find that each node will identically synchronize with the other node in its group, but will have distinctly different dynamics than the other nodes, to which it is directly coupled. We compare the experimental results and numerical simulations with theoretical predictions for the existence and stability of group synchrony based on a master stability function framework.

Return phase-lag mapping approach uncover multi-rhythmicity in 3-cell CPGs with mixed synapses
Jeremy Wojcik
Georgia State University

We describe a novel computational approach to reduce detailed models of central pattern generation to equationlesss return mapping for the phase lags between the constituting bursting interneurons.

Such mappings are then studied geometrically as the model parameters, including coupling properties of inhibitory and excitatory synapses, or external inputs are varied. Bifurcations of the fixed points and invariant circles of the mappings corresponding to various types of rhythmic activity are examined. These changes uncover possible biophysical mechanisms for control and modulation of motor-pattern generation.

Our analysis does not require knowledge of the equations that model the system, and so provides a powerful new approach to studying detailed models, applicable to a variety of biological phenomena beyond motor control.

Motifs of three coupled cells are a common network configuration including models of biological central pattern generators. We demonstrate our technique on a motif of three reciprocally coupled, inhibitory and excitatory, cells that is able to produce multiple patterns of bursting rhythms. In particular, we examine the qualitative geometric structure of two-dimensional maps for phase lag between the cells. This reveals the organizing centers of emergent polyrhythmic patterns and their bifurcations, as the asymmetry of the synaptic coupling is varied. The presence of multistability and the types of attractors in the network are shown to be determined by the duty cycle of bursting, as well as coupling interactions.

Coincidences and thermodynamic invariances: new phenomena in complex chemical kinetics
Gregory Yablonsky
Parks College, Department of Chemistry, Saint Louis University

New phenomena, i.e. coincidences and thermodynamic time invariances, have been discovered theoretically and verified experimentally.

(1) Coincidences and intersections. These properties of transient concentration curves have been discovered. The classical consecutive mechanism is used as an example. We identify six different special points, and analyze and classify the 6 possible (out of 612 combinations) patterns of concentration peak and intersection times and values that distinguish the parameter subdomains and sometimes can eliminate the mechanism. This developed theory is tested on examples (multi-step radioactive decay, isomerization reaction). The mathematical analysis relies on a combination of elementary and symbolic techniques, special functions and numerical approximations.

(2) Thermodynamic invariances in dual kinetic experiments.

A new class of kinetic experiments, so called dual kinetic experiments, is proposed.

In this experiments performed in non-steady-state batch- or steady-state plug flow reactors from reciprocal initial conditions, time (space) invariances for all reversible single reactions of the first and second order have been found explicitly. In all analyzed cases, these invariances are quotient-like functions of concentrations. These functions are equal to the equilibrium constant of the reaction during the whole course of the experiment, and not only at the end, i.e., under equilibrium conditions.

The obtained invariances can be used as simple fingerprints for distinguishing the types of reactions.

For multistep reactions: (a) a similar invariance was obtained for the two- step catalytic reaction (single route complex reaction) under pseudo-steady-state assumption; (b) for the two-step non-steady-state reaction, an approximation for the thermodynamic invariance was found to be valid in two domains, (I) at the very beginning of the reaction (II) at the end of the reaction, near equilibrium conditions. We hypothesize that such two-domain validity of the thermodynamic invariance is a general feature of dual kinetic experiments performed in complex chemical systems.

The developed theory was justified experimentally using catalytic water-gas shift reaction as an example. The Temporal-Analysis-of-Product (TAP)-pulse response experiment was performed.

(1) G.B.Marin,G.Yablonsky 'Kinetics of Complex Reactions. Decoding Complexity', J.Wiley-VCH (2011) 428p.

(2) G. S. Yablonsky, D. Constales, G. Marin, 'Coincidences in Chemical Kinetics: Surprising News about Simple Reactions', Chem. Eng. Sci. 65(2010)2325-2332

(3) G. Yablonsky, D. Constales, G. Marin, 'Equilibrium relationships for non-equilibrium chemical dependences', Chem. Eng.Sci.;66, 1,111-114(2011) arXiv: 1007.4642v1[math-phys]

(4) G. Yablonsky, A. N. Gorban, D. Constales, V. Galvita and G.B. Marin, 'ReciprocalRelations Between Kinetic Curves', EuroPhysics Letters, EPL, 93 (2011) 20004-20007 [cond-mat.stat-mech]

(5) D. Constales, G. Yablonsky, V. Galvita, and G.B.Marin, 'Thermodynamic time-invariances: theory of TAP pulse-response experiments', Chem. Eng. Sci., 66 (2011) 4683-4689

(6) D. Constales, G.S. Yablonsky, G.B. Marin, 'Thermodynamic time invariances for dual kinetic experiments: nonlinear single reactions and more', 2012 ('Chemical Engineering Science',accepted for publication)

Rential attachment network models without "rich-get-richer" effects
James Bagrow
Northwestern University

Growing network models with preferential attachment, where new nodes are injected into the network and form links with existing nodes proportional to their degree, have been well studied for some time. Extensions have been introduced where nodes attach proportional to arbitrary fitness functions. However, in these models attaching to a node increases the ability of a node to gain more links in the future. We study network growth where nodes attach proportional to the clustering coefficients, or local densities of triangles, of existing nodes. Attaching to a node typically lowers its clustering coefficient, in contrast with preferential attachment or rich-get-richer models. This simple modification naturally leads to a variety of rich phenomena, including non-poissonian bursty dynamics, cluster and community formation, aging, and renewal. This shows that complex network structure can be modeled without artificially imposing multiple dynamical mechanisms. Potential applications include modeling social networks and understanding the appearance and disappearance of fads and fashions.

Nonlinear dynamics of endocrine feedback in hypothalamico-pituitary-ovarian (HPO) axis and its prospective role in understanding etiology of ovarian cancer
Chinmay Basu
Netaji Subhash Chandra Bose Cancer Research Institute

Deficiency of omics , generally believed to be founding stone of Systems biology, is that it gives an idea of actual happening in a snapshot, a momentary fashion which is often two dimensional or at best three dimensional. But it ignores time which is fourth dimension. Thus it ought to be less realistic than one system which could incorporate time. Systems biology as an application of dynamical systems theory in vivo is one which deals with time series data of changes in variables or vectors which are varying biochemical marker levels in living body. They generally show nonlinearity and feedback control. But they are less discussed and tested in systems science. Nonlinear dynamics of feedback controls both nervous and endocrine systems which are information system of body negating entropy. Though neural communication is more rapid, endocrine communication is also very important. Unlike neural function, in endocrine function determination of hormone fluctuations is difficult and expensive, requiring drawing blood at frequent intervals and performing expensive assays on the blood samples. But there are some circumstances where nonlinear endocrine dynamical feedback system may be scientifically important, both medically and clinically. Postmenopausal ovarian cancer is one such place where there is aberrant accumulation of follicle stimulating hormone receptor (FSHR) in ovarian surface epithelial (OSE) cells and concomitant significant lowering of FSH level without any reductionist apparent cause. Endocrine reaction in hypothalamic-pituitary-ovarian (HPO) axis that will follow Michaelis-Menten-Hill (MMH) kinetics can be looked at as signal that moves in the phase space of our body and can be plotted in graph that can be digitized and analyzed using nonlinear science. Time series data of hormone fluctuation including life events should influence such lowering of FSH and high expression of FSHR promoting G-protein linked signal transduction and malignant transformation. Solving MMH equation, use of Lyapunov exponent, correlation dimension will be relevant here for time dependent quantitative calculation of constituent chemicals in the equation. Chemo-prophylaxis of ovarian cancer by oral contraceptive, proposed major protective effect of this cancer due to late birth indicate that time series of life event in hormonal fluctuation of HPO axis may be better understood and described by the use of systems biology of nonlinear dynamical system of endocrine feedback. Thus more appropriate diagnosis and prevention of this cancer may be possible by easier hormonal manipulation.

Soliton Solution as Inverse Problem for Coefficient Identification
Rossitza Marinova
Concordia University College of Alberta

In this work, we propose a numerical method for finding soliton solutions of nonlinear equations. Stationary localized waves are considered in the frame moving to the right. The original ill-posed bifurcation problem is transferred into an inverse problem for coefficient identification from over-posed boundary data, in which the trivial solution is excluded. The inverse problem solution method used in this work is the Method of Variational Imbedding, a generalization of the Least-Squares Method. The idea consists of replacing the incorrect problem with a well-posed problem for the minimization of a quadratic functional from the original equations. The Euler-Lagrange equations for minimization comprise higher-order equations for the solution of the original equation and for the unknown coefficient. In other words, the original incorrect problem can be embedded in a well-posed higher-order boundary value problem. A difference scheme for solving the embedded problem is developed. The idea is illustrated by several examples.

Keywords: Inverse Problem, Soliton, Bifurcation, Coefficient Identification, Variational Imbedding.

A non-ordinary route to chaos and complexity
Ued Maluf
Researcher at the Universidade Federal Fluminense

I will present unusual routes into the domains of chaos and complexity in the set of natural numbers

Noise reduction by dynamical coupling
Maria Eugenia Mera
Universidad Complutense de Madrid

In the study of real world dynamical phenomena it is common to record the values of a feature of the phenomenon as time varies. Typically, the measuring device used to record the time series has limited accuracy, and consequently the recorded signal differs from the clean signal by a measurement error.

We say that several noisy univariate time series are dynamically coupled if the corresponding unknown clean time series are the time ordered values either of some of the state variables or of real functions of the state variables of the same smooth dynamical system.

We show that substantial reduction of measurement noise can be achieved by crossing a given time series with other time series with which it is dynamically coupled. The noise reduction is made using a noise reduction algorithm (see Ref. 1) designed for multivariate time series, able to exploit efficiently (see Ref. 2) the different levels of uncertainty in each of the univariate noisy time series dynamically coupled. The method is especially advantageous when the uncertainty of the univariate time series under study is greater than that of the time series dynamically coupled to it, and for very short time series. For instance, in a laboratory time series generated by the logistic map corrupted by a measurement noise with a standard deviation of 50% of the standard deviation of the clean signal, we can obtain (see Ref. 3) noise reduction levels of up to 80%. The good behavior of the dynamical coupling procedure for short time series makes it possible to extend the scope of application of noise reduction techniques to the social sciences, where many of the methods used in nonlinear time series analysis have not yet been applied because of the short lengths of the available time series.

[1] M. E. Mera and M. Morán, Reduction of noise of large amplitude through adaptive neighborhoods. Phys. Rev. E 80, 016207 (2009)

[2] W. A. Fuller, Measurement Error Models, Wiley Series in Probability and Statistics, (John Wiley & Sons, New York, 2006).

[3] M. E. Mera and M. Morán Noise reduction by recycling dynamically coupled time series. Chaos 21, 043110 (2011)

A new theory of the anharmonicvity of nonlinear period behavior
Patrick Hanusse
CRPP CNRS / University of Bordeaux

We address the description of nonlinear periodic phenomena as they appear in a very large class of systems and situations. They are ubiquitous, and yet, surprisingly enough, no specific and relevant theory of the morphology of the signals they produce is available.

We should expect that the morphology of the signals produced by the system will reflect the physical interactions and internal dynamics or structure of the system. Unfortunately, we have so far no simple concepts and measurements, in small number, to describe the large variety of shapes they exhibit.

Typically, a system entering into self-oscillation produces initially a simple harmonic behavior, then more anharmonic signals, if not very so, and possibly more complex behaviors. We consider only such ``simple'' oscillations, often call ``relaxations oscillations'', with one maximum and one minimum per period, by their morphology may be extremely anharmonic.

In essence we aim at answering the following question: is it possible to define a small number of universal morphological parameters which capture the essential aspects of the behavior and characterize the state of the system, its evolution, the distance to harmonicity, the relationship between the various morphologies and the corresponding physical situations and the pathways between them. In other words, can we design a generic theory of anharmonicity ? The answer is yes !

We don't forget that Fourier series provide a well defined mathematical description, but the morphology described by a large number of Fourier modes does not bear much physical significance and requires a quantity of information that does not seem to match the apparent low complexity of the behavior, even when extremely anharmonic.

By describing the phase dynamics near generic harmonic situations and by considering the various ways symmetry unfoldings can be introduced, we build a new general description of the phase dynamics and obtain the general solution for the phase evolution, hence for the morphology of the signal. The description turns out to be extremely compact and well suited even in very anharmonic situations, such as near global bifurcations.

New mathematical tools have been designed, the importance of which goes beyond the present domain. A new factorization theorem of periodic functions as been established, and new nonlinear trigonometric functions have been defined, opening a new field of ``nonlinear trigonometry''

This general solution leads very naturally to the definition of new measurable quantities. A remarkably small number of them suffices for a very good qualitative and quantitative description. They provide a signature of the dynamical situation of the system, allowing easy comparison between them.

Being able to describe accurately the behavior with a few notions and measurements is of great importance for modeling, analyzing, understanding, comparing and coding such signals and has therefore a great impact on many applications in various domains. We will discuss some of them.

Not only do we have a new theory and new highly significant measurements but we also have a new language to speak about nonlinear periodic phenomena.