The study of hyperbolic structures (uniform and nonuniform ones)
is a central subject in Dynamical Systems.
Nowadays, there are many notions of weak hyperbolicity, and
here I am interested in the setting of flows with singularities (e.g.,
Lorenz systems).
In this talk I am going to talk about some notions of (nonuniform)

We show that quenched limiting hitting distributions are
compound Poisson distributed for certain random dynamical systems with
targets.
Targets are random and assumed to have well-defined return statistics
of certain type, which turn out to characterize the said compound
Poissonian limit.
Moreover, quenched and annealed polynomial decay of correlations are
assumed, whereas annealed Kac-time normalization is adopted.
Examples discussed are one-dimensional random piecewise expanding systems.

We consider a renewable resource distributed in a periodic environment, which is taken as n-dimensional torus. The dynamics of the resource is described by Kolmogorov-Piskunov-Petrovsky-Fisher equation [1], [7], [8] in divergent form

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.