Theory and examples of attractors with basins which are of positive measure, but contain no open sets, are developed; such basins are called riddled. A theorem is established which states that riddled basins are detected by normal Lyapunov exponents. Several examples, both mathematically rigorous and numerical, motivated by applications in the literature, are presented.
Abstract.We announce the discovery of a diffeomorphism of a three-dimensional manifold with boundary which has two disjoint attractors. Each attractor attracts a set of positive 3-dimensional Lebesgue measure whose points of Lebesgue density are dense in the whole manifold. This situation is stable under small perturbations.
IntroductionThrough the numerical experiments and theoretical works of many researchers we have gained some understanding of a dynamical system which is "typical" in the "standard" setting; that is, the dynamical system / : M -► M is a smooth map and M is a smooth «-dimensional manifold. Special circumstances change the meaning of "typical" and add possibilities to the list of what we are likely to see, say, by requiring / to be in some particular subset of smooth maps on M. For example, if / is symplectic, then we see the KAM tori and other phenomena of Hamiltonian dynamics; or if / has symmetries, we are likely to see "symmetry-breaking" bifurcations. The setting we examine here is where / maps some fixed submanifold K c M of dimension k < m to itself ( / is invariant on K ). That is, we discuss some new phenomena which are "typical" in the sense that they occur for some large set of perturbations of / which continue to map K to itself. This is the natural setting for modeling physical systems with constraints or reflectional symmetries.We find some unexpected new limiting behaviors in the new setting. More precisely, the co-limit set of a point x g M is the set eo(x) = f)k (J¡>kfl{x) > the forward limit set of x under iteration by /. We say that a compact set A is an attractor if {x e M : eo(x) = A} has positive «-dimensional Lebesgue measure and the basin of attraction of A is 3$ (A) = {x e M :
Recently it has been shown that there are chaotic attractors whose basins are such that every point in the attractor's basin has pieces of another attractor's basin arbitrarily nearby (the basin is "riddled" with holes). Here we report quantitative theoretical results for such basins and compare with numerical experiments on a simple physical model. PACS numbers: 05.45.+b
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