Seminars

Abstract: The purpose of this talk is to describe some recent work (joint with Pedro Duarte and Joana Torres) where the technique of isospectral reduction, introduced by L. Bunimovich and B. Webb, is applied in the context of certain operators related to stochastic processes in order to show the existence of stationary measures and also get some approximations of the measure by means of the iteration of a dynamical procedure. We also want to discuss some possible new applications of the method.

Abstract: In this seminar, we present a mechanism for the emergence of strange attractors (observable chaos) in a two-parametric periodically-perturbed family of differential equations on the plane. The two parameters are independent and act on different ways in the invariant manifolds of consecutive saddles in the cycle. The first parameter makes the two-dimensional invariant manifolds of consecutive saddles in the cycle to pull apart; the second forces transverse intersection. These relative positions may be determined using the Melnikov method. Generalising [1], we prove the existence of many complicated dynamical objects in the two-parametric family, ranging from rank-one attractors to Hénon-type strange attractors. We also draw a plausible bifurcation diagram associated to the problem under consideration and we show that the \\\"occurrence of heteroclinic tangencles\\\" is a prevalent phenomenon. [1] I. Labouriau, A. Rodrigues, Periodic forcing of a heteroclinic network, Journal of Dynamics and Differential Equations 2021 (to appear), DOI 10.1007/s10884-021-10054-w