Francisco Balibrea

RECENT DEVELOPMENTS IN DYNAMICAL SYSTEMS: THREE PERSPECTIVES

This paper aims to an present account of some problems considered in the past years in Dynamical Systems, new research directions and also provide some open problems.

Minimal Sets on Graphs and Dendrites

The topological structure of minimal sets of continuous maps on graphs, dendrites and dendroids is studied. A full characterization of minimal sets on graphs and a partial characterization of minimal sets on dendrites are given. An example of a minimal set containing an interval on a dendroid is given.

TAMING CHAOS IN A DRIVEN JOSEPHSON JUNCTION

We present analytical and numerical results concerning the inhibition of chaos in a single driven Josephson junction by means of an additional weak resonant perturbation. From Melnikov analysis, we theoretically find parameter-space regions, associated with the chaos-suppressing perturbation, where chaotic states can be suppressed. In particular, we test analytical expressions for the intervals of initial phase difference between the two excitations for which chaotic dynamics can be eliminated. All the theoretical predictions are in overall good agreement with numerical results obtained by simulation.

Weak mixing and chaos in nonautonomous discrete systems

Abstract The paper is devoted to a study of chaotic properties of nonautonomous discrete systems (NDS) defined by a sequence f ∞ = { f i } i = 0 ∞ of continuous maps acting on a compact metric space. We consider such properties as chaos in the sense of Li and Yorke, topological weak mixing and topological entropy, all defined in a way suitable for NDS. We compare these concepts with the case of a single map (discrete dynamical system, DS for short) and relate them to recent results in the topic. While previous research of various authors were focusing on analogues to the DS case, we show that in general the dynamics of NDSs is much richer and quite different than what is expected from the DS case. We also provide a few new tools that can be used for the successful investigation of their qualitative behavior.

Topological entropy of Devaney chaotic maps

Abstract The infimum respectively minimum of the topological entropies in different spaces are studied for maps which are transitive or chaotic in the sense of Devaney (i.e., transitive with dense periodic points). After a short survey of results explicitly or implicitly known in the literature for zero and one-dimensional spaces the paper deals with chaotic maps in some higher-dimensional spaces. The key role is played by the result saying that a chaotic map f in a compact metric space X without isolated points can always be extended to a triangular (skew product) map F in X×[0,1] in such a way that F is also chaotic and has the same topological entropy as f. Moreover, the sets X×{0} and X×{1} are F-invariant which enables to use the factorization and obtain in such a way dynamical systems in the cone and in the suspension over X or in the space X× S 1 . This has several consequences. Among others, the best lower bounds for the topological entropy of chaotic maps on disks, tori and spheres of any dimensions are proved to be zero.

Iteration Theory: Dynamical Systems and Functional Equations

The aim of this paper is to give an account of some problems considered in the past years in the setting of Discrete Dynamical Systems and Iterative Functional Equations, some new research directions and also state some open problems.

Global Periodicity Of

We show that where C is a positive constant and , and where k is an odd natural number, are the unique -cycles having the form where f j are continuous selfmaps of the interval , .

ROLE OF PARAMETRIC RESONANCE IN THE INHIBITION OF CHAOTIC ESCAPE FROM A POTENTIAL WELL

Abstract This paper shows how a periodic parametric modulation can inhibit chaotic escape of a driven oscillator from the cubic potential well that typically models a metastable system close to a fold. Melnikov analysis shows that, depending on its amplitude, period, and initial phase, a periodic parametric modulation of the linear potential term suppresses chaotic escape when certain resonance conditions are met. In particular, it is shown that chaotic escape suppression is impossible under a period-1 parametric perturbation. The effect of nonlinear damping on the inhibition scenario is also studied.

Local bifurcations of continuous dynamical systems under higher order conditions

Abstract In this paper we provide higher order conditions which imply the appearance of non-standard local bifurcations in uniparametric families of one-dimensional continuous-time dynamical systems. By the Center Manifold Theory, they also describe generalizations of local bifurcations of uniparametric families of systems on R n .

On ω-limit sets of antitriangular maps☆

Abstract We give a topological characterization of ω-limit sets of continuous antitriangular maps, that is, maps F :[0,1] 2 →[0,1] 2 with the form F(x,y)=(f2(y),f1(x)), (x,y)∈I2. We also point out some differences between ω-limit set of antitriangular and one-dimensional maps.