# Presentations

## Subject-driven birdsong synthesizer based on a model of the bird's vocal organ

Ezequiel M. Arneodo

Physics Department, Universidad de Buenos Aires

The complex vocalizations composing birdsong emerge from the interaction between the nervous system and a nonlinear peripheral device, its vocal organ.

It is an area of intensive work to elucidate how much of the complexity of it is due to the huge dimensionality of the dynamics of the nervous system, and how much due to the mechanical response of the periphery.

This last point is of particular interest for applied physics, as biomechanic systems used by living things are overwhelmingly non-linear, and capable of presenting complex behavior even as a response to simple instructions.

We present a bioprosthetic device based on a model for the zebra finch vocal organ, which is capable of producing the subtleties present in the actual song of the birds.

This model is simple to the extent that its solution can be computed by digital signal processors (DSP), when driven by actual physiological motor instructions fed by a freely behaving subject.

This allows for the design of a new strategy for biomimetic prosthetics: real-time integration of simple models, controlled by few physiological signals.

## Assessing atrial fibrillation as a chaotic dynamical state of coupled oscillators

Guillaume Attuel

IHU liryc / sigma Espci

Clinical Background:

Fibrillation is an electrical pathology of the heart responsible for sudden death, when striking the ventricles. The more common atrial fibrillation (AF), which though in itself not lethal highly increases the risk of stroke.

The possibility that AF stems from a quasi periodic route to chaos has been first pointed out by Garfinkel (1999). However, it is generally believed that fibrillation is a consequence of the substrate abnormalities alone, such as fibrosis, and therefore is due to fractionated propagation, thereby inflicting a kind of wave turbulence, the so called leading circle theory of functional reentry (Allessie 1973). This vision has been challenged more recently with the discovery of foci, especially near the pulmonary veins, triggering AF in certain occasions, after the electrical isolation of which normal sinus rhythm is spontaneously recovered in patients suffering paroxysmal AF, i.e. episodes of AF lesser than 24h (Haissaguerre 1998). Since then, an intense debate holds on the identification of the zones to target in the case of sustained AF, i.e. episodes lasting weeks. Targeting coherent behaviors, such as rotors, or on the contrary zones of fractionation, has not come yet to a consensus. The precise means to quantify these hypothesized areas are not clear either.

Motivations:

We are addressing the clinical and physiological concern of trigger and perpetuation of fibrillation as being a manifestation of one single phenomenon.

Methods:

Our aim is to assess a fibrillatory state as a state past a dynamical phase transition of a system of many coupled oscillators. Collective motion in these systems is known to show synchronized or desynchronized phases (Kuramoto 1981), as well as intermediate more complex phases (Daido 1992).

We would like to present a non exhaustive overview of this class of systems, in particular to insist on the importance of the mean field interaction among the oscillators, and its influence on the collective behavior, such as the possibility to ignite a trapping instability that we will describe shortly. In this respect, we will also cast a bridge with the aspects of spatio-temporal intermittency, seen in Rayleigh-Bénard turbulence.

The open question, of great relevance with regards to the Coumel triangle, deals with the classification of the possible dynamical states arising in systems of coupled oscillators with quenched disorder, and external modulation.

We will constantly put forward during the presentation as many key experimental facts as known to us which support this approach.

## Cardiac tissue heterogeneity mediates electrical stimulation effects

Philip Bittihn

Max Planck Institute for Dynamics and Self-Organization

Electric shocks are used to terminate life-threatening cardiac arrhythmias. However, the biophysical mechanisms that cause extracellularly applied electric fields to induce excitation in cardiac tissue still pose a scientific challenge. One common aspect of existing studies on this subject is that heterogeneity is a necessary condition for the proposed mechanisms to produce any effect on the membrane potential of cardiac cells. This heterogeneity includes the non-uniformity of the intracellular space (introduced by gap junctions), changes in fiber direction, varying electrical conductance and the inhomogeneity of the electric field itself. In this study, we focus on the heterogeneity that is caused by the complex geometry of the heart, namely its internal and external boundaries which present obstacles to electrical conduction. In numerical simulations and experiments, we determine the field strength necessary to induce excitation from different kinds of boundaries and examine the influence of pacing parameters on the stimulation result. Moreover, we present a theory that predicts the selection of specific anatomical features as wave emitting sites. The observed mechanisms are shown to exhibit the strongest effects at near-threshold field strengths, which may have important implications for the development and optimization of low-energy pacing strategies.

## Transport Analysis of Spatiotemporal Oceanographic Dynamical from Remote Sensing

Erik Bollt

Clarkson University

A broad range of scientific fields, such as climatology, oceanography, and fluid dynamics produce large data sets in the form of digital images or continuous-time, spatiotemporal video data from remotely sensed hyperspectral satellite data. There have been terrific advancements in variational methods for image processing, and likewise in dynamical systems, there have been tremendous advancements in analyzing transport in complex spatiotemporal dynamical systems. Nonetheless, there has been little specialization of the methods of image processing to develop techniques specifically suited to the complex dynamical systems typical of fluid systems, and the tools of dynamical systems have not been brought to bear on data inferred directly from movies. The Frobenius-Perron operator for a dynamical system known allows transport modeling and phase decomposition into almost invariant sets and relatively coherent sets. A particular application which interests us is remotely sensed ecological systems such as biological products including algae blooms, from which we will discuss modeling, transport analysis, and filtering.

## Analysis &Prediction of Computer Performance Dynamics

Elizabeth Bradley

University of Colorado

Modern computers are deterministic nonlinear dynamical systems---despite designers' opposing views, it is both interesting and useful to treat them as such. While being careful not to disturb the dynamics, we used a custom measurement infrastructure to observe the computer performance, gathered time-series data from a variety of simple programs running on several microprocessors, and then used delay-coordinate embedding to study the associated dynamics. Results from multiple corroborating methods indicated low-dimensional dynamics of simple loops running on these processors---including the first experimental evidence of chaos in real computer hardware. Both hardware and software play a role in these dynamics: the memory usage of the same three-line loop was chaotic on an Intel Core2 extregistered ~machine and periodic on a Pentium~4 extregistered~machine, for instance, and swapping the loop indices caused similar bifurcations.

This leads naturally to a model of computer dynamics as an iterated map with two components: one dictated by the hardware and one dictated by the software. Under the influence of this map, the computer's state moves on a trajectory through its high-dimensional state space as the clock cycles progress and the program executes. The second component of the map is time varying: as the execution moves through different parts of the code, the dynamics undergo a series of bifurcations. The dimensions of the resulting dynamics--attractors with dimension $approx 1$ in state spaces with tens of axes---are somewhat surprising in a system as complex as a microprocessor, which contains billions of transistors. Computers are organized by design into a small number of tightly coupled subsystems, however, which effectively restricts the dynamics to a lower-dimensional manifold.

All of this is not only interesting from a dynamics standpoint, but also important for the purposes of computer design. If one can effectively model a computer as a deterministic dynamical system, then there exists a deterministic forecasting rule for its dynamics. Even a short-term prediction of the important variables in a computer could help a controller tailor the system resources to the changing needs of a computing application: e.g., saving power by turning off unneeded processing units when the predicted loads are low. Building forecast models in embedding space is completely impractical for 'on the fly' prediction in nonstationary systems like running computers, however, because the process of verifying values for the two free parameters (the delay and the dimension) cannot be automated. We conjecture, however, that a full formal embedding---which is necessary for detailed dynamical analysis---(this not always necessary for the purposes of prediction). We have verified this conjecture by constructing accurate prediction models in various two-dimensional projections of the full embedded dynamics. These models allowed us to predict the processor and memory loads of running programs many millions of clock cycles into the future---results that could be leveraged to greatly improve computer performance. From a mathematical standpoint, this success also brings into question the need for full topological conjugacy in forecasting schema.

## Reduction of Chaotic Particle Transport in Nontwist Plasmas

Ibere Caldas

University of Sao Paulo

## Detecting differences in noisy chaotic signals

Thomas Carroll

Naval Research Lab

If the output of an experiment is a chaotic signal, it may be useful to detect small changes in the signal, but there are a limited number of ways to compare signals from chaotic systems, and most known methods are not robust in the presence of noise. One may calculate dimension or Lyapunov exponents from the signal, or construct a synchronizing model, but all of these are only useful in low noise situations. I introduce a method for detecting small variations in a chaotic attractor based on directly calculating the difference between vector fields in phase space. The differences are found by comparing close strands in phase space, rather than close neighbors. The use of strands makes the method more robust to noise and more sensitive to small attractor differences. I demonstrate this method with simulated signals and data from a Rossler circuit experiment

## Acoustic Ranging Using Solvable Chaos

Ned Corron

U.S. Army RDECOM

Acoustic experiments are presented that demonstrate an improved approach to ranging and detection by exploiting the unusual properties of an analytically solvable chaotic oscillator. It has long been recognized that the wide bandwidth and aperiodic properties of chaos naturally suggest benefits for high-resolution, unambiguous ranging in radar, sonar, and ladar systems. An obvious, conventional approach might be to simply substitute chaos for the noise source in, for example, a random-signal radar. In such a system, a segment of the transmitted waveform is sampled and stored, using a resolution defined by the signal bandwidth and the Nyquist sampling criterion. The stored signal is then used in a correlation receiver to detect a return signal and determine time of flight. The cross-correlations are usually done digitally, using a digital signal processor (DSP) and fast-Fourier transforms (FFT). In this approach, the distinguishing properties of a chaotic waveform are not used: chaos is simply a wide-bandwidth, random source. In contrast, an alternative approach to ranging has recently been developed that truly exploits the properties of a chaotic waveform to alleviate the most expensive parts of random-signal radar - i.e., sampling, digital memory, and digital signal processor - while still maintaining the performance of a correlation receiver. This new approach uses chaotic waveforms generated by a solvable nonlinear oscillator comprising both an ordinary differential equation and a discrete switching state. This hybrid oscillator admits an exact analytic solution as the linear convolution of a symbolic dynamics and a basis function, which enables coherent reception using a simple analog matched filter and only a few stored symbols. For the acoustic ranging experiments, an amplified speaker emits an audio-frequency waveform generated by an electronic realization of the oscillator, which sounds like noise. A complementary receiver circuit incorporates a matched filter for the basis function, which is mathematically equivalent to a correlation receiver. At repeated intervals, a sequence of twelve symbols detected in the symbolic dynamics of the emitted waveform is captured, thereby defining a transmitted signal for ranging. The captured symbol sequence is provided to the receiver, where it defines a matched filter for the transmitted signal. Practically, the symbols define weights applied to elements of a microphone array, the outputs of which are summed and passively filtered. The output of the matched filter is a continuous signal that is proportional to the cross-correlation of the transmitted and received signal. In operation, a consistent peak in the output of the matched filter indicates a detected target, from which ranging is derived by the time of flight. The entire experimental system is realized using simple analog and digital electronic circuit components. Importantly, the receiver does not require waveform sampling or digital signal processing for detection. Real-time measurements using only an oscilloscope provide visible evidence of detection and ranging with the system. This successful demonstration enables the development of new, low-cost sonar and radar technologies using chaotic waveforms.

## Percolation and misfire in spark-ignited engines

Charles Finney

Oak Ridge National Laboratory

The nonlinear dynamics of cycle-to-cycle variability in spark-ignited combustion engines have been studied experimentally for over two decades. Numerous investigators have confirmed that, under conditions of high dilution, noisy cycle-to-cycle bifurcations in combustion intensity can develop as critical levels of dilution (e.g., due to high excess air or high exhaust gas recirculation) are approached. The deterministic features of these cycle-to-cycle variations have been successfully replicated with relatively simple nonlinear maps, but the basic nature and causes of the main combustion nonlinearity itself and the source of the noisy (stochastic) component of the dynamics are still poorly understood. We propose a 3D percolation model that appears to explain both the nature of the combustion nonlinearity and the dominant cause of the stochastic component. We demonstrate that the model captures key features of a widely used empirical correlation for engine combustion and also replicates key features of experimentally measured intra-cycle combustion trajectories and probability distributions. We expect this model to have practical value in understanding how engine design and operation can be modified to reduce the negative impact of dilute-limit combustion instabilities on fuel efficiency and emissions.

## Phase synchronization of coupled chaotic noncoherent oscillators

Rosangela Follmann

Escola Politécnica da Universidade de São Paulo

In this work we present a method for measuring the phase of chaotic systems. This method has as input a scalar time series and operates by estimating a fundamental frequency for short segments, or windows, along the whole extension of the signal. It accomplishes that by minimizing the square error of fitting a sinusoidal function to the series segment.

This approach does not require following the trajectory on the attractor, works well over a wide range of adjustable parameters, is very easy to be implemented, and is particularly useful for experimental settings with single signal outputs since there is no need of attractor reconstruction. It is applicable to both coherent and noncoherent chaotic oscillators. We show the applicability of the method by using two coupled chaotic Ressler oscillators in coherent and noncoherent cases in order to identify their phase synchronous regimes. We also apply the method to experimental time series obtained from two coupled Chua circuits. The approach shows flexibility and in principle is applicable to any time series suitable for sinusoidal fittings. The use of least square parameter estimation makes the method intrinsically resilient to the presence of noise.

## Nonlinear Time-Reversal in a Wave Chaotic System

Matthew Frazier

University of Maryland

Time reversal mirrors are particularly simple in wave chaotic systems and form the basis for a new class of sensors [1-3]. These sensors apply the quantum mechanical concepts of Loschmidt Echo and fidelity to classical waves. These sensors make explicit use of time-reversal invariance and spatial reciprocity in a wave chaotic system to remotely measure the presence of small perturbations to the system. The operation of the time-reversal mirror itself benefits from the wave chaotic scattering in the system. We extend our time-reversal mirror to include an element with a nonlinear dynamical response. The initially injected pulse interacts with the nonlinear element, generating new frequency components originating at the element. By selectively filtering for and applying the time-reversal mirror to the new frequency components, we focus a pulse only onto the element. Furthermore, we demonstrate two possible applications: (a) transmission of arbitrary patterns of pulses to the element, creating a communication channel with the nonlinear element, to the exclusion of 'eavesdroppers' at other locations in the system; and (b) reconstruction of an amplified pulse, focusing sufficient energy to disable the element. Further possible applications include wireless power transfer and pin-point hyperthermia treatment of tumors.

Work funded by the Intelligence Community Postdoctoral Research Fellowship Program, ONR MURI grant N000140710734,and by the AFOSR under grant FA95501010106

[1] B. T. Taddese, et al., Appl. Phys. Lett. 95, 114103 (2009).

[2] B. T. Taddese, et al., J. Appl. Phys. 108, 1 (2010).

[3] S. M. Anlage, et al., Acta Physica Polonica A 112, 569 (2007).

## Chimera States in a Spatial Light Modulator Feedback System

Aaron Hagerstrom

University of Maryland

We construct a dynamical imaging system which includes a liquid crystal spatial light modulator (SLM) and camera, connected in a feedback loop. This system is configured as a cellular automaton which models an excitable medium. The cells in this automaton correspond to pixels on the SLM screen, and are updated according to the Greenberg-Hastings rules[1]. The cells can be excited, refractory, or quiescent. If a cell is excited, a corresponding area of the camera's detector will be illuminated. Cells will be excited if they are in a quiescent state and the intensity detected by the corresponding camera pixel is above a threshold. After firing, a cell enters a refractory period, and must wait for some number of iterations before it can fire again. Due to optical spillover, which is a characteristic of the imaging system, there is coupling between adjacent cells. This system supports spiral waves, target waves, and incoherent and synchronized firing patterns.

We also observe chimera states in this system when it is suitably configured and prepared. These states are characterized by two groups of cells with very different levels of synchrony [2]. To achieve these states, we compute a spatial average of the intensity recorded on each side (top and bottom) of the camera's screen. The threshold for a cell to become is then adjusted based on this average intensity. Coupled in this way, pixels in each group have a tendency to synchronize within a group. The degree of synchronization can be dramatically different between the two groups, in spite of symmetric coupling and homogeneous initial conditions. Experimental results and numerical simulations will be presented.

[1] J.M. Greenberg and S.P. Hastings, Bull. Am. Math. Soc. 6, 84 (1978); G. Bub, A. Shrier, and L. Glass, Phys. Rev. Lett. 94, 028105 (2005) [2] D.M. Abrams, R. Mirollo, S.H. Strogatz, and D.A. Wiley. Physical Review Letters 101, 084103 (2008)## Crenelated slow-fast oscillations in a dual delay nonlinear photonic dynamics

Maxime Jacquot

FEMTO-ST institute

## Minimal configuration models for experiment-based central pattern generator of melibe

Sajiya Jalil

Georgia State University

Central pattern generation in vivo determines complex behaviors seen in animals. A relatively small group of neurons in the central nervous system may play critical roles in controlling specific behaviors in the animal, so that blocking constituent neurons results in that the animal loses the corresponding behavior. Following this idea, many central pattern generators (CPGs), that are groups of neurons forming such small networks via synaptic connections, can be identified; examples are the leech heart rhythm, melibe swimming, and lobster pyloric networks. Analysis of various CPG systems with the aid of mathematical modeling, computational, and statistical tools has provided useful insights into synergetic mechanisms of neuronal networks that underlie the central pattern generation. In our study, we explore a plausible network that represent swim CPG in the marine invertebrate - melibe. We employ mathematical models developed using Hodgkin-Huxley formalism with parameter estimation from leech heart interneurons. The model of leech interneurons has been studied extensively and shown, both mathematically and experimentally, to have the ability to transition into a number of distinct patterns including square wave bursting, spiking, and chaos. In addition, multistability with two or more coexisting stable patterns has been reported for this model. Due to its versatility, and physiological derivation, the leech heart interneuron model is a generic candidate for modeling melibe swim CPG interneurons, for which physiologically accurate model is yet to be identified. Based on experimental recordings, electrical activity patterns and the hypothesis concerning the number of constituent neurons and the nature of connections, a number of CPG models are proposed. Phases and phase relations between bursting interneurons are imperative for representing repetitive nature of activity patterns of the CPGs. We design the models inspired by the specific phase relations seen in the experimental voltage traces. We include four neurons, connected via fast non-delayed inhibitory synapses modeled by fast threshold modulating function (FTM), which are grouped into two pairs of half center oscillator (HCO) configurations. In the HCO, neurons reciprocally inhibit each other, leading to activity patterns that alternate; anti-phase bursts emerge, consistent with experimental recordings. We introduce inhibitory and excitatory connections between the pairs of HCOs, and find phase-locked states that are idiosyncratic of the experimental system. Our research indicates that weak excitatory connections play no significant role in the generation of this particular phase-locked pattern. We identify control parameters for the pattern in questions, which corresponds to a single attractor for the phase-lag return mapping on a 3D torus, based on sixty cycle long simulations of eight thousand initial conditions. Our goal is to explain the mechanism that causes the particular phase-locked states and explore parametric regime for sensitivity and emergence of additional patterns in the system. In the future, we plan to enhance the CPG models by including extra interneurons of other types, introducing heterogeneity in network connections and by increasing physiological fine details that are currently neglected. Mechanistic understanding of CPGs is important for engineering equipments that are dynamically controlled by circuits, such as in robotics and prosthetics.

## Information Theory for Tralfamadorians: The Anatomy of a Bit

Ryan James

UC Davis

How is information organized within an observation? By placing the present observation within the context of its past and future, we provide three different answers to this question in the form of information-theoretic decompositions. The later two of these decompositions utilize information in the future, and so we interpret them as being how Vonnegut's Tralfamadorians might perform time series analysis. We associate each component of these decompositions with the extensive part of multivariate information measures when applied to increasing contiguous blocks of observations. Finally, this decomposition framework isolates information shared by the past and future yet not observed in the present. This component motivates the notion of a process's effective (internal) states and indicates why one must build models.

## The Onset of Chaos in Vortex Sheet Flow

Robert Krasny

University of Michigan

Vortex sheets are used in fluid dynamics to model thin shear layers in slightly viscous flow. Examples include the trailing wake of an aircraft, and the spiral vortex ring formed by ejecting fluid from a circular tube. Here we present regularized point-vortex simulations of vortex sheet roll-up in planar and axisymmetric geometry, leading to the formation of a vortex pair and a vortex ring, respectively. At late times the sheet develops irregular small-scale features and these are attributed to the onset of chaotic dynamics typical of Hamiltonian systems. These findings are supported by a numerically computed Poincare section indicating the presence of resonances and heteroclinic tangles in the flow dynamics. Interestingly, chaos develops naturally in these simulations, without external forcing, due to self-initiated and self-sustained oscillations in the vortex core, arising from the combined effect of rotation and strain. A key issue is to determine whether such dynamical features occur in real fluids and can be seen in experiments.

## Afterbounce instability and period doubled emission in single-bubble sonoluminescence

Mogens Levinsen

Niels Bohr Institute, UCHP

We report the first direct and long time stable observation for a single sonoluminescing bubble of the afterbounce instability that is believed to be one of the ways for a sonoluminescing bubble to lose stability and eventually break up. Furthermore we show that the instability is directly linked to the curious phenomenon of flash by flash spatially asymmetric period doubling of the sonoluminescent emission as the afterbounce instability is always period doubled whenever the emission is. This lends credit to a hot core picture coupled with refraction in the surface of the bubble. A complete theoretical understanding of this peculiar coupling between the two phenomena is still missing.

## Explosive phase synchronization of chaotic oscillators

Inmaculada Leyva

Center for Biomedical Technology, Universidad Politecnica de Madrid.

The understanding of the spontaneous emergence of collective behavior in ensembles of networked dynamical units constitutes a fascinating challenge in science. Despite the fact that critical phenomena in networks have been intensively studied, the physics literature almost exclusively reports continuous phase transitions.

However, very recent studies on critical phenomena in complex networks have proved the emergence of dynamical abrupt transitions in the macroscopic state of the system. In particular, it has been recently shown that discontinuous synchronization transitions can take place in networks of periodic oscillators, with the only condition of a positive correlation between the heterogeneity of the connections and the natural frequencies of the oscillators.

In this work, broadening the extent of validity of such a concept, we give the first numerical and experimental evidence of an explosive phase synchronization in networks of chaotic units. In particular, we use both extensive simulations of networks made up of chaotic Ressler units, and an experiment with equivalent electronic circuits in a star configuration, to demonstrate the existence of a first order transition towards synchronization of their phases. As corresponds to its first-order character, the transition is characterized by a hysteresis in the order parameter.

Such a discovery is the first evidence of this new paradigm of synchronization in chaotic systems. We show that, as it happens in periodic systems, the correlation between oscillation frequencies and structural heterogeneity is a necessary condition for such an explosive phenomenon. However, we also establish that, in chaotic systems, additional features like the specific dynamical state at which the units operate are determinant. We numerically and experimentally verify the robustness of the phenomenon, which does not need to a fine-tune of the coupling strength or suppressing the noise to get the transition, but to vary the system parameters in a wider region.We then expect that our work will open the path to the use of explosive synchronization phenomena in many relevant applications.

## Minimal agent model for economic critical behavior: application to financial stability

Pedro Lind

Center for Theoretical and Computational Physics, University of Lisbon

We address the controversy in the study of financial observables, sometimes taken as brownian-like processes and other as critical systems with fluctuations of arbitrary magnitude. We translate basic economical principles into physical properties and interactions and construct a financial network composed of economical agents which establish trade connections among them. Our model reproduces the evolution of macroscopic quantities (indices) and correctly retrieves the common exponent value characterizing several indices in financial markets. Furthermore, we show how these drops and crises emerge as a natural result of local economical principles ruling trades between economical agents. In particular, we present evidence that heavy-tails of the return distributions are bounded by constraints associated with the topology of trade connections among the agents. Finally, we apply our model for discussing recent directives from Basel III, namely those concerning the raise of minimum capital levels of banks on an individual basis, with the aim of lowering the probability for a large crash to occur. We show that the probability of large crisis does not necessary decrease and may even increase in certain situations.

## Burning Invariant Manifolds in Advection Reaction Diffusion

John Mahoney

University of California, Merced

Invariant manifolds are well known organizing structures in chaotic advection. We consider reaction diffusion dynamics within a fluid that is simultaneously undergoing such advection. We then construct the invariant manifolds of this extended system and find that they play a critical role as boundaries for front propagation. Unlike their passive counterparts, these manifolds are one-sided boundaries and may develop self-intersections and cusps. These manifolds also allow for an elegant description of mode-locking and front patterns. These findings are corroborated by detailed experiments on MHD forced fluid containing BZ reagents.

## Information dynamics in the Kinouchi-Copelli model

Leonardo Maia

USP

The hypothesis that the notion of criticality may be relevant to the understanding of some biological systems is steadily getting experimental support. Indeed we live exciting times when models poised near the boundary between different regions in their parameter spaces are "naturally" built from data collected in experiments performed in systems as diverse as swarms of animals and cultures of cortical tissues [1]. It has been argued that systems set in such "edge of chaos" can exhibit optimal performance of some measures essential to their biological function.

In this context, the Kinouchi-Copelli (KC) model [2] has attract some interest. It is a network of excitable elements (each unit is a Greenberger-Hastings automaton) and was conceived as a model for sensory processes. Remarkably it can be tuned to a critical point where the dynamic range - a key quantity in psychophysics and in general information processing systems that indicate the sensitivity of a system to arbitrary stimuli - achieves a maximum. This optimization was also seen in experiments with cortical slices selectively exposed to neuromodulators [3,4]. Besides that, the KC model generates bursts of activity characterized as power-laws in the critical state, with exponents also compatible with experiments [3,4,5]. These empirical findings gave rise to statements of optimization of "computational capabilities" of the KC model in the critical state. Notwithstanding the experimental support for the dynamic range, those claims still remain wishful thinking, since there are many more aspects of information processing in the model that remain unknown.

In this work we report our preliminary theoretical studies of information processing in the Kinouchi-Copelli model based on methodologies recently developed, mainly the local information dynamics framework of Lizier and coworkers [6].

[1] T. Mora and W. Bialek, "Are Biological Systems Poised at Criticality?", J. Stat. Phys. 144, 268 (2011)

[2] O. Kinouchi and M. Copelli, "Optimal Dynamical Range of Excitable Networks at Criticality", Nat. Phys. 2,348 (2006)

[3] W. L. Shew, H. Yang, T. Petermann, R. Roy and D. Plenz, "Neuronal Avalanches Imply Maximum Dynamic Range in Cortical Networks at Criticality", J. Neurosc. 29, 15595 (2009)

[4] W. L. Shew, H. Yang, S. Yu, R. Roy and D. Plenz, "Information Capacity and Transmission Are Maximized in Balanced Cortical Networks with Neuronal Avalanches", J. Neurosc. 31, 55 (2011)

[5] J. M. Beggs and D. Plenz, "Neuronal Avalanches in Neocortical Circuits", J. Neurosc. 23, 11167 (2003)

[6] J. T. Lizier, S. Pritam and M. Prokopenko, "Information Dynamics in Small-World Boolean Networks", Art. Life 17, 293 (2011)

## Climate network analysis based on statistics of ordinal patterns and symbolic dynamics

Cristina Masoller

Universitat Politecnica de Catalunya

We analyze climatological data from a complex networks perspective, using techniques of nonlinear time series symbolic analysis. Specifically, we employ ordinal patterns and binary representations to analyze monthly averaged surface air temperature (SAT) anomalies. By computing the mutual information of the time series in regular grid points covering the Earth's surface and then performing global thresholding, we construct climate networks that uncover short-term memory processes, as well as long ones (5-6 yr). Our results suggest that the time variability of the SAT anomalies is determined by patterns of oscillatory behavior that repeat from time to time with a periodicity related to intraseasonal variations and to El Nino on seasonal to interannual time scales. The present work is located at the triple intersection of three highly active interdisciplinary research fields in nonlinear science: symbolic methods for nonlinear time series analysis, network theory, and nonlinear processes in the earth climate. While a lot of effort is being done in order to improve our understanding of natural complex systems, with many different methods for mapping time series to network representations being investigated and employed in complex systems such as the human brain, our work is the first one aimed at characterizing the global climate network in terms of the statistics of ordinal patterns and binary patterns constructed at various time scales. The analysis reveals that the structure of the network changes drastically depending on the pattern time scale. M. Barreiro, A. C. Martã? and C. Masoller, Inferring long memory processes in the climate network via ordinal pattern analysis, Chaos 21, 013101 (2011).

## The Immunomics of Lymphocyte repertoires

Ramit Mehr

Bar-Ilan University, Israel

The immune system is a highly complex system, capable of cognitive tasks such as pathogen recognition, decision, learning and memory. This is a distributed system, in which the tasks are performed by an interacting network of specialized cells - primarily lymphocytes. Immune learning and memory are embedded in the dynamical states of the complete lymphocyte repertoire, and cannot be understood by studying the behavior of single cell types. This complexity, further increased by the non-linear behavior of each component, can only be elucidated by using theoretical tools to complement experimental and clinical studies. I shall review several examples of the use of such tools in immunological research.

## GeoChaos: Engineered Chaotic Advection in Porous Media Enhances Reactive and Thermal Transport Rates

Guy Metcalfe

CSIRO Australia

Many activities in the terrestrial subsurface need to recover or emplace scalar quantities (dissolved phase concentration or heat) from/in volumes of saturated porous media. Scalars can be targeted by pump-and-treat technologies, where target fluids are abstracted from the porous medium, or by amendment technologies, were specific chemicals or substrates are injected into the porous medium. Application examples include solution mining, recovery of contaminant plumes, or harvesting geothermal heat energy. Transient switching of the pressure at different wells can, perhaps surprisingly, intimately control subsurface flow, generating a range of 'programmed'' flows. While some programs produce chaotic mixing and rapid transport, others create encapsulating flows (islands) confining pollutants for in situ treatment. In a simplified model of an aquifer subject to balanced injection and extraction pumping, chaotic flow topologies have been predicted theoretically and verified experimentally. Mixing enhancement due to chaotic advection and kinematic confinement of aquifer volumes are key features of the chaotic dynamics. We show theory and data from two types of experiments. First we use a Hele-Shaw cell to visualize and quantitatively confirm the simplest examples of the theory of transport in open chaotic flows of constant permeability. Second we use a 1 meter tank filled with glass beads of different sizes and emplace salt 'deposits'. The porous media tank permits the application of many different well switching protocols. Data indicates a doubling of salt extraction rate from particular stirring protocols over the rate obtained with a steady flow. Other flow protocols clearly demonstrate confinement of the salt concentration, which can be released and reconfined as we turn particular stirring protocols on and off. Understanding these phenomena may form the basis for new efficient technologies for many applications in porous media scalar transport.

## Ionization of kicked Rydberg atoms via a turnstile mechanism

Kevin Mitchell

University of California, Merced

We present a theoretical and experimental study of the chaotic ionization of quasi-one-dimensional potassium Rydberg wavepackets via a phase space turnstile mechanism. Turnstiles form a general transport mechanism for numerous chaotic systems, and this study is the first to explicitly illuminate their relevance to atomic ionization. We create time-dependent Rydberg wavepackets, subject them to alternating applied electric fields (kicks), and measure the ionization fraction. We show that the ionization of the electron depends not only on the initial electron energy, but also on the phase space position of the electron with respect to the turnstile--that part of the electron packet inside the turnstile ionizes quickly, after one period of the applied field, while that part outside the turnstile ionizes after multiple kicking periods. The dependence of the ionization on the kicking period can also be understood in terms of the turnstile geometry. Finally, this geometric understanding suggests a related mechanism to de-excite Rydberg states.

## Patterns of Neuronal Synchronization

Epaminondas Rosa

Illinois State University

Neuronal synchronization in the brain can be desirable, as is the case for the different synchronous frequencies during sleeping stages, and can also be undesirable, as is the case for Parkinson's disease, for example. In this presentation we show simulation results displaying various synchronous patterns directly related to the firing regimes of the neurons and dependent on how they are coupled. Our model for the neuron is based on a modified version of the Hodgkin-Huxley equations, and is capable of generating a wide range of firing patterns. We argue that neuronal synchronization may be induced, or hindered, with the action of excitatory/inhibitory drugs capable of regulating neuronal firing rates. We also show synchronous behaviors observed in coupled experimental electronic neurons, consistent with the numerical simulations results.

## Coherent fluid structures and biological invasions

Shane Ross

Virginia Tech

The language of coherent structures provides a new means for discussion of transport and mixing of atmospheric pathogens, paving the way for new modeling and management strategies for the spread of infectious diseases affecting plants, domestic animals, and humans. Atmospheric dynamical structures have an influence on aeroecology, namely the population structure of airborne microbe species. We report on a recent integration of experimental biology and applied mathematics uncovering how Lagrangian coherent structures and their phase space complements, strongly connected (almost-invariant) regions, provide a framework for understanding how airborne microbe populations are dispersed and mixed. Applications to identification of frontiers between qualitatively different kinds of behavior in other areas will also be discussed.

## Experimental observations and numerical studies of the microbunching instability in storage rings

Eléonore Roussel

Laboratoire de Physique des Lasers, Atomes et Molécules

Electron bunches circulating at a relativistic speed in a storage ring are used to produce intense radiation from far infrared to X-rays. However, at high charge, electron bunches typically undergo a spatio-temporal instability, which leads to the formation of patterns (at a millimeter scale) and irregular evolutions in space and time. Here we present detailed observations and numerical investigations of this electron bunch dynamics in the French Synchrotron SOLEIL storage ring. Experimental signatures are intense terahertz bursts which are detected using a bolometer. The modeling used is based on a one-dimensional Fokker-Planck-Vlasov equation, and a major ingredient of the pattern formation is the (nonlocal) interaction between the electrons, via electromagnetic interactions. Detailed and combined analysis of experimental and numerical data reveals several characteristic instability thresholds, associated to the appearance of specific regimes. We also present interpretations for the occurrence of these irregular regimes in terms of damping and diffusion processes, as well as a set of open questions.

## Co-adaptation and the Emergence of Structured Environments

Robert Savit

University of Michigan

We wish to address the question of which of the common features seen in complex adaptive systems (e.g. the formation of hierarchical structures, the generation of various kinds of inter-agent interactions, and the formation and exploitation of niches) are a result of the meta-dynamics of co-adaptation of co-evolution and which are contingent on details of the system at hand.

As a first step in this project, we consider an abstract model which is a priori as random and structureless as we can make it, but has the meta-dynamics of co-adaptation. Specifically, we consider a system in which N heterogeneous adaptive agents act on and alter their environment. The environment can be in any one of E states, and an agent, when he acts on the environment changes its state. An agent's actions at a given point in the game is determined by one of a set of S strategies possessed by that agent. Different agents possess different sets of strategies. When an agent acts, he receives a payoff (positive, negative or zero) associated with his action and dictated by his current strategy. Thus, each strategy is a look-up table that indicates what action the agent must perform when he sees a given environmental state (i.e. what the new state must be after his action), and what his payoff will be for taking that action. Actions and payoffs of each strategy are random, although the payoffs associated with a given strategy must sum to zero. Agents can adapt, in that they can change which of their strategies they use, choosing those that return the highest payoffs given the history of that agent's actions. Over time, agents see environmental states which are determined by the actions of all other agents, who are also (co-) adapting, i.e. "jockeying for position".

We find that this model, although it is ab initio, as random as we can make it, has a very rich non-random emergent structure. In particular, as a function of N and E, there are regions in which the system shows the formation of ordered environments, which can be interpreted as the emergence of niches and, simultaneously, the generation of agents to exploit those niches. There are also regions in which the agents accrue net positive payoffs, even though each strategy is reward neutral. There is also a very important phase transition in the system which, we argue, is of the replica-symmetry breaking type seen in physical models of disordered systems. Although our system is highly simplified and abstract, we argue that the underlying structure we observe may be expected to have echos in real complex adaptive systems, since the underlying meta-dynamics of co-adaptation are so ubiquitous. We also hope to stimulate a discussion of how these insights can be tested by experiments on, or observations of, real systems.

## The Observability of Biological Networks

Steven Schiff

Pennsylvania State University

In the past decade, we have seen the concurrent development of sophisticated control theoretic techniques suitable for nonlinear and chaotic systems, as well as computational models of neuronal systems that have improving fidelity to the behavior of neuronal ensembles in health and disease. Using nonlinear ensemble Kalman filters, we have in recent years demonstrated that we can fuse computational neuroscience models with data from single cells, small network motifs, and larger scale neuronal dynamics including ring and spiral waves in experiments. Simultaneously, the ability to quantify both analytically and numerically the formal observability of nonlinear dynamical systems has been developed using several approaches: the rank and determinate of the Jacobian of the delay embedding map, the equivalent measures of the differential embedding map when equations are known, and an indirect approach using a distortion matrix. Such metrics of observability define how much of the experimentally inaccessible variables of a complex system can be reconstructed from measurements of only a subset of the state variables, and whether different system trajectories are discriminable from measurement observations. We here examine small neuronal network motifs using these metrics of observability. Using 3 and 4 neuron networks, with a variety of network topologies, we demonstrate that the observability of neuronal networks can be computed as for other physical systems. The presence of symmetries is directly related to the degree of observability. In addition, the inclusion of inhibitory with excitatory neuronal layers diminishes the observability of such networks. Since such small network motifs are highly overrepresented in other biological systems, from gene regulatory networks to ecological food webs, such observability investigation may have broad impact on biological sciences.

## On the balance between segregation and integration in complex modular networks

Irene Sendiña-Nadal

Unviersidad Rey Juan Carlos

Many physical and biological systems (such as electronic devices, communications networks, and the human brain) face similar constraints as they interact with complex environments, and organize their structure and function along similar principles of resource allocation [1]. On the one hand, the need for fast and reliable responses to changes in the environment naturally favors the emergence of segregated modules of specialized computation (e.g. sensory systems in the brain). On the other hand, interactions among modules become essential when an information processing whose complexity exceeds the capacity of the single modules is required. For instance, perceptual systems in the brain need to bind information from different brain areas to produce a single coherent percept [2]. Therefore, segregation into specialized modules and integration into global coherent activity present an inherent trade-off, and an appropriate balance between these two tendencies has been shown to be necessary for efficient functioning, particularly in neural systems [3]. In fact, an exceedingly segregated or integrated functioning of the brain has been associated with various pathological conditions, e.g. autism or schizophrenia [4-6], and epilepsy [7] respectively.

In this work, we introduce a measure that quantifies the balance between segregation and integration in a modular network which is easily computable from the network topology [8]. Segregation corresponds to the degree of network modularity, while integration is expressed in terms of the algebraic connectivity of an associated hyper-graph. The rigorous treatment of a simplified case, where clusters are of equal size, allows to extract an optimal balance condition and the associated connectivity organization. We show that such a configuration is indeed the one producing a perfect balance between cluster and complete synchronization of a network of coupled phase oscillators, which occurs precisely at the transition thresholds of the two collective dynamics. This result unveils information on the relationships between modular organization and optimal performance of networked systems [9], and opens the path for a new perspective in the evaluation of the dynamical behavior of different biological, technological and social interacting systems.

[1] S. Laughlin and T. Sejnowski, Science 301, 1870 (2003).

[2] F. Varela, J. Lachaux, E. Rodriguez, and J. Martinerie, Nat. Rev. Neurosci. 2, 229 (2001).

[3] G. Tononi, O. Sporns, and G. M. Edelman, Proc. Natl. Acad. Sci. 91, 5033 (1994).

[4] P. Fletcher, J. McKenna, K. Friston, C. Frith, and R. Dolan, Neuroimage 9, 337 (1999).

[5] G. Tononi, A. R. McIntosh, D. P. Russell, and G. M. Edelman, NeuroImage 7, 133 (1998).

[6] M. Just, V. Cherkassky, K. T.A., and N. Minshew, Brain 127, 1811 (2004).

[7] C. Stam, Clinical neurophysiology 116, 2266 (2005).

[8] A.-A. Rad, I. Sendina-Nadal, D. Papo, M. Zanin, J.M. Buldu, F. del Pozo, S. Boccaletti, under review in Phys. Rev. Lett.

[9] S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D.-U. Hwang, Phys. Rep. 424, 175 (2006).

## Experimental Bouncer Ball Model

Mário Silva

São Paulo State University

We studied experimentally the classical dynamics of a steel ball moving vertically under the action of the gravitational field and having inelastic collisions with a wall, also made of steel, forced to move periodically along the vertical axis by a shaker. In order to correct unwanted excitement in any other direction or even small vertical misalignment of normal to a base surface, the movement of the ball was limited in vertical direction by a guide tube of glass glued to the base, in which vacuum was made, allowing drag forces to be disregarded. The impacts were detected from the response of a microphone attached to the base and the displace of motion was measured directly by a micrometer screw which eclipses a light beam established between a pair emitter and a sensor attached to the base or indirectly from a accelerometer also attached to the base and also used in control system of excitement. Considering this controlling scheme and the small ratio of the ball mass and the oscillating masses, especially the mass of the vacuum sealing system also used for fix the tube to the shaker, we assumed that the movement of the base is not affected by the impacts. To obtain a impact map (velocity of the particle and phase of the movement), the velocity of the particle immediately after the shock was obtained indirectly from the elapsed time between impacts and the phases of the movement base on the instant of the shocks were obtained from the time intervals between a reference phase and the impacts. These time intervals were acquired by "software" and "hardware" developed in reconfigurable logic and incorporates a finite state machine capable of discriminating the impacts of the ball with the metal base of the impact with the guide tube of glass, based on the spectral components of microphone response to these different impacts. It was possible to observe a strange attractor in the impact map as previously observed by Tufillaro in 1986. Indeed they showed plots of the dynamics obtained in a memory oscilloscope that recorded the intensity of the response of a piezoelectric sensor to the impacts. Our experiment can indeed now acquire date with greater accuracy afforded by the digital acquisition of temporal intervals. The availability of numerical series acquired also allows statistics of the growth mean speed after impact as traditionally is done in computer simulations. The amplitude of base acceleration possible in these experimental set was from 2.5 g to 5 g and the restitution coefficient was measured in 0.69 and it can change only by choice of different combination of materials for ball and base.

Keywords: Chaos. Bouncer Ball. Experimentation.

## Time-resolved estimation of direct directed influences

Linda Sommerlade

Freiburg Institute for Advanced Studies

The inference of interaction structures from multidimensional time series is a major challenge.

Knowledge about the interactions between processes promises deeper insights into mechanisms underlying network phenomena, e.g. in the neurosciences where the level of connectivity in neural networks is of particular interest.

To gather information about the connectivity in a network from measured data, several parametric as well as non-parametric approaches have been proposed and widely examined.

We discuss two shortcomings, which are often faced in applications, i.e.~nonstationarity of the processes generating the time series and contamination with observational noise.

To overcome both, we present a new approach by combining renormalized partial directed coherence with state space modeling. This method allows studying the evolution of the network connectivity in time that might contain information about ongoing tasks in the brain or possible dynamic dysfunctions.

Statistical issues are addressed.

The performance is illustrated by means of model systems and in an application to neurological data.

## Master stability function approach to unveil the complex dynamics of an experimental ring of coupled optoelectronic oscillators

Francesco Sorrentino

University of New Mexico

We experimentally study the complex dynamics of a unidirectionally coupled ring of four identical optoelectronic oscillators. The coupling between these systems is time-delayed in the experiment and can be varied over a wide range. We observe that as the coupling delay is varied, the system may show different synchronization states, including complete isochronal synchrony, cluster synchrony, and a splay-phase state. We are interested in understanding how these different states may emerge as the delay is varied.

We analyze the stability problem through a master stability function approach, which we show can be effectively applied to all the different states observed in the experiment, including cluster synchrony and splay-phases. Our analysis points out the existence of multistability in the system. Our theoretical approach can be easily generalized to rings of arbitrary length and possibly bidirectional coupling.

## A Lagrangian interpretation of advective-diffusive scalar transport

Michel Speetjens

Eindhoven University of Technology

Transport of scalar quantities as e.g. heat and chemical species in fluid systems by the interplay of advection and diffusion is key to many processes in industry and Nature. Examples include mixing and thermal processing of viscous fluids, compact processing equipment, emerging lab-on-a-chip applications, pollutant transport in environmental flows as well as nutrient transfer in physiological flows. An important aspect that remains elusive is the role of (chaotic) advection in the scalar transfer. Studies to date investigate this issue primarily in terms of the spatio-temporal evolution of the Eulerian scalar field by e.g. eigenmode analyses. An alternative exists in considering scalar transfer in terms of the 'total scalar flux,' i.e. the net scalar flow due to advection and diffusion combined, and 'scalar transport paths' as its associated Lagrangian trajectories. In this Lagrangian picture, scalar redistribution becomes the 'motion' of scalar quantities along transport paths in a similar way as fluid flow is the motion of fluid parcels along fluid paths. This analogy admits application and generalisation of well-known Lagrangian concepts from mixing studies to advective-diffusive scalar transport. Key to these concepts is analysis and representation of transport by way of the topology and geometry of transport paths.

Scalar transport paths are, similar to fluid trajectories, organized into coherent structures due to the underlying conservation laws. This yields a 'scalar flow topology' that, similar to its fluid counterpart, the flow topology, geometrically determines the transport of a scalar quantity throughout the flow domain. The existence of such a scalar flow topology enables visualization of scalar transport in a way essentially similar to flow and transport visualization by e.g. 'scalar streamline patterns' or 'scalar Poincaré sections.' Moreover, it facilitates interpretation of scalar transport from a Lagrangian perspective and thus affords new fundamental insights into its workings.

The Lagrangian representation and interpretation of scalar transport is demonstrated by way of heat transfer in 2D (un)steady channel flows. The 'thermal topologies' are visualized by the proposed ansatz and, together with the flow topology and the temperature field, provide a complete the image of the heat transfer. Two promising results are that the Lagrangian representation (i) offers a rigorous definition of convective heat transfer and (ii) enables direct visualization and investigation of heat-transfer enhancement by the fluid motion. This may greatly benefit further exploration of the still ill-understood role of chaotic advection in heat transfer.

## State of Play for Analysing the Nonlinear Dynamics of Laser Systems from Output Power Time Series

Joshua Toomey

Macquarie University

It is an exciting time in the field of analysing output power time series from nonlinear laser systems. Advances in real time oscilloscopes mean that the time resolution of the output power is sufficient to study all but the fastest semiconductor lasers. Advances in computer control of experimental laser systems mean high density data sets, to support the generation of high resolution maps of the dynamics, are increasingly available. Advances in applying chaos data analysis tools to experimental output power time series are starting to generate almost complete dynamic maps for systems based on correlation dimension, and/or related dimension analyses; and permutation entropy. Additionally, analysing other experimental measurands such as the peak to peak amplitude, average period/frequency for pseudo-pulsed data, fluctuations in in the 'period', and searching for regions where the dynamics show transients all add to a more complete evaluation of an experimental system, and support robust comparison with theoretical models for the systems. We will present a summary of this set of nonlinear time series analysis tools that we have used successfully to characterise chaotic experimental laser systems.

The output power of a laser is known to display a wide variety of dynamics, ranging from constant stable emission, regular periodic oscillations and unstable chaotic fluctuations. We have characterised several different complex laser systems: an optically injected Nd:YVO4 solid state laser [1], an optically injected vertical cavity surface emitting laser (VCSEL) [2] and an edge-emitting semiconductor laser with optical feedback [3]. The control variables used to drive the change in dynamics are the injection strength and frequency detuning in the injection systems (solid state laser and VCSEL), and diode injection current and optical feedback level in the feedback system (semiconductor laser). Laser systems are an ideal nonlinear system on which to test these chaos analysis techniques, owing to the level of adjustment and control an experimenter has over the system parameters and data collection procedures.

The maps produced from this set of experimental data represent one of the most detailed representations of the dynamic diversity of each of these laser systems to date. Combining the information provided in each of the maps gives a comprehensive overall picture of the dynamics within the parameter space investigated.

The tools and methods presented here can be applied to any nonlinear system from which time series can be recorded and, as such, will be of much interest to those studying complex dynamics in a wide range of research fields. The new insights they give into nonlinear laser systems will be summarised.

1. J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, "Automated correlation dimension analysis of optically injected solid state lasers," Opt. Express 17, 7592-7608 (2009).

2. J. P. Toomey, C. Nichkawde, D. M. Kane, K. Schires, I. D. Henning, A. Hurtado, and M. J. Adams, "Stability of the nonlinear dynamics of an optically injected VCSEL," Under peer review (2012).

3. J. P. Toomey, D. M. Kane, M. W. Lee, and K. A. Shore, "Nonlinear dynamics of semiconductor lasers with feedback and modulation," Opt. Express 18, 16955-16972 (2010).

## On the frequency content of chaotic time series

Lawrie Virgin

Duke University

The characterization of chaos as a random-like response from a deterministic dynamical system with an extreme sensitivity to initial conditions is well-established, and has provided a stimulus to research in nonlinear dynamical systems in general. In a formal sense, the computation of the Lyapunov Exponent (LE) spectrum establishes a quantitative measure, with at least one positive LE (and generally bounded motion) indicating a local exponential divergence of adjacent trajectories. Other measures are associated with certain geometric features of a chaotic attractor, e.g., the fractal dimension, and broad- band frequency content. However, although the extraction of LE's can be accomplished with (necessarily noisy) experimental data, this is still a relatively data-intensive and sensitive endeavor. We present here an alternative, pragmatic approach to identifying chaos as a function of system parameters, based on extending the concept of the spectrogram.

Given a time series, we extract the frequency content using a standard Fourier transform: for a periodic signal we expect a spectrum dominated by a single peak (harmonic) or a finite set of peaks for a more complicated but still periodic, or quasi-periodic, signal. For a chaotic response the spectral content is spread over a relatively broad range of frequencies. A spectrogram or waterfall plot then displays this information as a function of a system parameter, providing a useful alternative to the standard bifurcation digram. The new heuristic criterion consists of choosing a threshold value (some fraction of the maximum peak height of the frequency spectra). Then the number of peaks (local maximums) above this threshold are counted. If the number of peaks counted is above a second threshold the system is considered chaotic, otherwise it is labeled non-chaotic. In the case of chaotic signals the noise produces many peaks on the same order of magnitude or close to the maximum peak. Therefore in practice it is relatively easy to choose the two threshold values.

This work will describe this approach applied to systems of increasing complexity, ranging from direct numerical simulations of the pendulum equation under a single control parameter, and then two control parameters (in which this approach is especially useful), to experimental data generated from mechanical systems (related to Duffing's equation). The accuracy and utility of the approach, including the effect of noise, is tested relative to the behavior of the LE's.

## Water retention on random surfaces

Robert Ziff

University of Michigan

The retention of water rained down on a random surface with open boundaries is a form of invasion percolation which self-organizes to a critical state closely related to the regular percolation critical point. Focusing on systems with discrete rather than continuum heights, a direct mapping to regular percolation is found, and several properties of the retention problem can be stated in terms of percolation processes on two-level systems of various low-level fractions. Explaining a surprising non-monotonic retention with the number of levels reveals many complex and subtle features of this model. This analysis has applications to understanding properties of watersheds and their drainage regions.

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