The billiard map of a polygonal billiard with the standard refection law is conservative and non-chaotic. Completely different dynamics arises when the reflection law is contracting, i.e. when the reflection angle measured from the normal is a contraction of the incidence angle. In this case, the billiard map is dissipative: the Liouville measure is no longer preserved. In this talk I will discuss recent results obtained in collaboration with Gianluigi Del Magno, João Lopes Dias and Pedro Duarte concerning the existence of SRB measures and their ergodic properties.
In the context of heteroclinic networks the term ’switching’ refers to a particular form of complex dynamics near the network: Trajectories follow any possible sequence of connections that can be prescribed given the network architecture. We consider simple heteroclinic networks in Rn and give sufficient conditions for the absence of a weak form of switching (i.e. along a connection that is common to two cycles), generalizing a similar result in the work of M. Aguiar (Physica D 240, 1474-1488, 2011).
The Polymatrix replicators form a simple class of o.d.e.’s on prisms defined by
simplexes, which describe the evolution of strategical behaviours within a population
stratified in n>=1 social groups. This class of replicator dynamics contains well known
classes of evolutionary game dynamics, such as the symmetric and asymmetric (or bimatrix)
replicator equations, and some replicator equations for n-person games. In the
1980’s Raymond Redheffer et al. developed a theory on the class of stably dissipative
We consider stochastic processes arising from dynamical systems simply by evaluating
an observable function along the orbits of the system and study marked point
processes associated to extremal observations of such time series. In particular, we consider
marked point processes measuring the sum of the excesses over the threshold (AOT)
or measuring the sum of the largest excesses in a cluster of exceedances (POT). We provide
conditions to prove the convergence of such marked point processes to a compound
In dimension three, the Shilnikov cycle to a saddle-focus is one of the most famous and rich examples in the dynamical systems
theory, in which a simple conﬁguration generates a very complicated behaviour around the neighbourhood of the cycle.
Holomorphic dynamical systems are frequently non-uniformly hyperbolic systems.
I will discuss in the context of ergodic theory, some recent progress on the statistical
behavior of holomorphic dynamics in one and several complex variables.
We consider stochastic processes arising from dynamical systems by evaluating an observable
function along the orbits of the system. We review some results about the existence of Extreme Value
Laws for such processes with special emphasis on the clustering effect associated to observables achieving
a global maximum at a periodic point. Then we consider recent developments where the observables have
multiple maximal points which are correlated or bound by belonging to the same orbit of a certain chosen point.
Discutiremos alguns exemplos no bordo do conjunto dos sistemas uniformemente hiperbólicos. Destacaremos algumas propriedades geométricas e ergódicas desses sistemas, relacionando-as com o tipo de degeneração da hiperbolicidade.
Maria Joana Torres (CMAT e DMA, Universidade do Minho)
Friday, 15 May, 2015 - 13:30
L.A. Bunimovich and B.Z. Webb developed a theory for isospectral graph reduc- tion. This procedure maintains the spectrum of the network’s adjacency matrix up to a set of eigenvalues known beforehand from its graph structure. We make a simple observa- tion regarding the relation between eigenvectors of the original graph and its reduction, that sheds new light on this theory. As an application we propose an updating algorithm for the maximal eigenvector of the Markov matrix associated to a large sparse dynamical network.