Learning about dynamical systems from an ensemble approach

Antonio Politi

Short Abstract
Nonlinear dynamical systems are often studied by following single trajectories (or pairs of them) in phase space to quantify, e.g., Fourier power-spectra, fractal dimensions and Lyapunov exponents. However, especially in the perspective of unveiling and characterizing emergent collective properties, it is more instructive to follow the simultaneous evolution of (large) ensembles of trajectories. This approach implicitly corresponds to studying suitable evolution operators (Frobenius-Perron, Liouville) in some functional space and it is particularly suited to analyse and characterize the emergence of collective properties.

As a result, in the context of standard low-dimensional chaos, it is possible to provide a complementary description to the usual "microscopic" one, based on the analysis of single trajectories. In particular, it is possible to determine additional dynamical invariants that are related to the convergence properties of the corresponding functional operator, without the need to integrate it.

This approach is potentially useful both in the perspective of characterizing the emergence of collective behaviour (when the different trajectories are coupled with one-another) and while studying the the response of a chaotic system to a given class of external modulations.

I will both discuss ``simple" chaotic systems (such as the Roessler attractor or the Hindmarsh-Rose neuron) and a set of experimental data that correspond to some chaotic laser dynamics.