What is the expected number of iterates of a point needed for a plot of these iterates to approximate the attractor of the dynamical system up to a given scale delta (i.e., the orbit will have visited a delta-neighbourhood of every point in the attractor)?  This question has analogues in random walks on graphs and Markov chains and can be seen as a recurrence problem.  I'll present joint work with Natalia Jurga (St Andrews) where we estimate the expectation for this problem as a function of delta for some classes of interval maps using ideas from Hitting Time Statistics and transfer operato

Dynamical systems generated by scalar reaction-diffusion equations enjoy special properties that lead to a very simple structure for the semiflow. Among these properties, the monotone behavior of the number of zeros of the solutions plays an essential role. This discrete Lyapunov functional contains important information on the spectral behavior of the linearization and leads to a Morse-Smale description of the dynamical system.

The variational principle states that the topological entropy of a compact dynamical system is a supremum of measure-theoretic entropies of invariant measures supported on this system. Therefore, one may ask whether we can get a similar formula for the topological entropy of a dynamical system restricted to the level sets, which are usually not compact.

Apresentaremos o estado da arte sobre o estudo da conjectura da estabilidade para fluxos geodésicos, que ainda é um problema aberto devido à ausência do closing lemma no âmbito dos fluxos geodésicos. 

This presentation is divided into three parts where we analyze three different epidemiological models.

In the first part we analyze a periodically-forced system SIR model. We prove that the condition R0 < 1 is not enough to guarantee the elimination of the disease. Using the Theory of rank-one attractors, we prove persistent strange attractors for an open subset in the space of parameters where R0 < 1.

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.