Time crystals are crystalline structures in the time domain. In contrast to the usual crystals which manifest in spatial dimensions, the formation of a time crystal or even a proposal of such, can be tricky if not controversial.

The reason is closely related to the time translation symmetry. Time translation symmetry not only underlies the invariance of laws of physics, but also in the standard dynamical framework, is directly related to the con- servation of energy. It was thought until very recently that time translation symmetry could not be spontaneously broken.

We present a method of combining coupled cell systems to get dynamics supporting robust simple heteroclinic networks given by the product of robust simple heteroclinic networks (cycles). We consider coupled cell networks, with no assumption on symmetry, and combine them via the join operation. Assuming that the dynamics of the component networks supports robust simple heteroclinic cycles or networks, we show that the join dynamics realizes a more complex heteroclinic network given by the product of those cycles or networks.

We propose an approach to construct Bernoulli trials {Xi}i∈N combining dependence and independence periods, and we call it Bernoulli sequence with random dependence (BSRD). The structure of dependence in the past 

Patients infected with the human immunodeficiency virus (HIV) are more vulnerable to develop various types of cancer, in particular, Hodgkin’s lymphoma, Kaposi’s sarcoma and vulvar cancer. Furthermore, cancers progression tends to be more aggressive in HIV-positive individuals than in HIV-negative ones. In this work, we develop mathematical models to describe the dynamics of cancer growth and HIV infection, when chemotherapy and treatment for HIV, namely, highly active antiretroviral therapy (HAART) are included.

See attached file.

We discuss some recent and ongoing works on the dynamics of flows with various expansive measures. In particular, we present a measurable version of the Smale’s spectral decomposition theorem for flows. More precisely, we prove that if a flow φ on a compact metric space X is invariantly measure expanding on its chain recurrent setCR(φ) and has the invariantly measure shadowing property on CR(φ) then φ has the spectral decomposition, i.e. the nonwandering set Ω(φ) is decomposed by a disjoint union of finitely many invariant and closed sets on which φ is topologically transitive.

We discuss some recent and ongoing works on the dynamics of flows with various expansive measures. In particular, we present a measurable version of the Smale’s spectral decomposition theorem for flows. More precisely, we prove that if a flow φ on a compact metric space X is invariantly measure expanding on its chain recurrent setCR(φ) and has the invariantly measure shadowing property on CR(φ) then φ has the spectral decomposition, i.e. the nonwandering set Ω(φ) is decomposed by a disjoint union of finitely many invariant and closed sets on which φ is topologically transitive.

We consider stochastic processes arising from dynamical systems by evaluating an observable function along the orbits of the system. The Extremal Index, which is responsible for the appearance of clusters of exceedances, usually coincides with the reciprocal of the mean of the limiting cluster size distribution. In this talk, we show how to build dynamically generated stochastic processes with an Extremal Index for which that relation does not hold.

Discutiremos a dinâmica de grupos de difeomorfismos (analíticos) do círculo que são (localmente) não discretos. Uma técnica já bem estabelecida de aproximação desses grupos por campos de vectores locais permite uma descrição bastante detalhada das dinâmicas (extremamente caóticas) correspondentes.

Given a heteroclinic network, there is an associated graph such that the vertices of the graph correspond to the equilibria of the network and an edge corresponds to a connection between equilibria. Classification of ac-networks is carried out by describing all possible types of asso- ciated graphs.