A heteroclinic cycle is a structure in a dynamical system composed of a sequence of invariant sets---such as equilibria, periodic orbits, or even chaotic sets---and orbits which connect them in a cyclic manner. Near an attracting heteroclinic cycle, trajectories visit each invariant set in turn and, as time evolves, spend increasingly longer periods of time near each set, before making a rapid switch to the next one. A heteroclinic network is a connected union of heteroclinic cycles.

We describe some results in dynamics, symmetry and geometric analysis which have applications to the theoretical study of machine learning.
Most of the first half of the talk will emphasise the mathematics and be introductory; at appropriate points, brief
indications will be made concerning the
motivating question from statistical learning.
 

We study the minimal distance between two orbit segments of length $n$, in
a random dynamical system with sufficiently good mixing properties. For the annealed version of this problem, the asymptotic behavior is given by a dimension-like quantity associated to the invariant measure, called its correlation dimension. We study the analogous quenched question, and show that the asymptotic behavior is more involved: two correlation dimensions show up, giving rise to a non-smooth behavior of the associated asymptotic exponent.

We present a skew system made of two unimodal maps of the interval coupled in a master-slave configuration. This system arises in the physics literature as an explanatory model of the so-called topological synchronization, i.e. the mechanism believed to be at the basis of emerging collective phenomena in coupled chaotic systems.

This talk is divided in two parts.

Firstly, we analyze the effect of the seasonality in the SIR model, realized through the addition of a sinusoidal map in the transmission rate. Our major findings are the following:

Non-autonomous or random dynamical systems provide useful and flexible models to investigate systems whose evolution depends on external factors, such as noise and seasonal forcing. In the last fifteen years, transfer operators have been combined with multiplicative ergodic theory to shed light on ergodic-theoretic properties of random dynamical systems, through the so-called Lyapunov--Oseledets spectrum. While the scope of this framework is broad, in practice it is challenging to identify and even approximate this spectrum.

The expanding Lorenz map, which is obtained by taking a Poincaré section of the geometric Lorenz attractor and quotienting along stable leaves, has been studied in the literature.

We study the discrete centralizers of smooth flows and their time-t maps. Centralizers describe the symmetries of the respective dynamics. Given a $C^1$ -flow $(\varphi_t)_t$ on a compact manifold preserving an ergodic probability measure $\mu$, we prove that there exists a Baire generic subset of values of $t \in \mathbb{R}$ such that the centralizer of the time-$t$ map coincides with the discrete centralizer of the flow on $supp(\mu)$. The result relies on the ergodicity of generic time-t maps, established by Pugh and Shub in the seventies.

We consider nonuniformly hyperbolic dynamical systems with exponential decay of correlations, and show that exponential decay holds for pairs of observables v in L^p, w in L^q for which the corresponding local Holder constants also lie in L^p and L^q, with 1/p+1/q<1. Along the way, we give a particularly simple proof of quasicompactness for the one-sided transfer operator.

Title: About the stability of heteroclinic cycles: a new approach

Speaker: Alexandre Rodrigues (ISEG, CMUP) 

Abstract: