Dissonant points and the region of influence of non-saddle sets

Abstract

The aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non-saddle set. We see, in particular, that some topological conditions are sufficient to guarantee that these sets are attractors or repellers. We study in detail the existence of dissonant points of the flow, which play a key role in the description of the region of influence of a non-saddle set. These points are responsible for much of the dynamical and topological complexity of the system. We also study non-saddle sets from the point of view of the Conley index theory and consider, among other things, the case of flows on manifolds with trivial first cohomology group. For flows on these manifolds, dynamical robustness is equivalent to topological robustness. We carry out a particular study of 2-dimensional flows and give a topological condition which detects the existence of dissonant points for flows on surfaces. We also prove that isolated invariant continua of planar flows with global region of influence are necessarily attractors or repellers. MSC: 37C75, 37B30.Non-saddle set, region of influence, dissonant point, Conley index, homoclinic orbit, robustness.

Čech cohomology, homoclinic trajectories and robustness of non-saddle sets

“…This is motivated by some results from [23,24] about certain unstable attractors. In addition, motivated by the results in [6] and [3] about the continuation properties of isolated non-saddle sets we find necessary and sufficient conditions for the property of being non-saddle to be robust for families of smooth flows defined on smooth manifolds without further assumptions about the dimension or cohomology of the phase space.…”

“…This is especially rich in presence of the so-called dissonant points, a phenomenon that does not appear in the basin of attraction of an attractor neither stable nor unstable. The region of influence of an isolated non-saddle set has been deeply studied in [5,6].…”

“…More specifically, small perturbations of a flow preserve attractors and some of their basic topological properties [25] while small perturbations may transform isolated non-saddle sets into saddle ones with distinct topological structures [13]. However, as established in [3,6] there are some situations in which the robustness of some topological properties is equivalent to the robustness of non-saddleness.…”

“…This set is an open subset of the phase space and its topological and dynamical structures have been extensively studied in [6]. Although these structures share many features with those of the basin of attraction of an isolated attractor without external explosions, the global structure of I(K) may be much richer.…”

“…We denote by A * (K), R * (K) and H(K) the sets of purely attracted, purely repelled and homoclinic points respectively. The three sets are invariant subsets of M and they satisfy that A * (K) ∪ K and R * (K) ∪ K are closed in I(K) and H(K) \ K is open (see [6,Proposition 12]). The situation one would expect at this point would be that I(K) \ K was decomposed as a finite union of components comprised entirely of purely attracted points, components comprised entirely of purely repelled points and components comprised entirely of homoclinic points.…”