Riddled Basins

Abstract

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.

The control of chaos: theory and applications

“…The description of riddled basins was introduced in Ref. [90] where it was shown that for certain class of dynamical systems with an invariant subspace: (i) if there is a chaotic attractor in the invariant subspace; (ii) if there is another attractor in the phase space; and (iii) if the Lyapunov exponent transverse to the subspace is negative, then the basin of the chaotic attractor in the invariant subspace can be riddled with holes belonging to the basin of the other attractor. That is, for every initial condition that asymptotes to the chaotic attractor in the invariant subspace, there are initial conditions arbitrarily nearby that asymptote to the other attractor.…”

“…Rigorous results on the dynamics of riddled basins for discrete maps were presented in Refs. [90,91]. The dynamics of riddled basins was subsequently investigated in [92] using a more realistic physical model.…”

“…The dynamics of riddled basins was subsequently investigated in [92] using a more realistic physical model. A more extreme type of basin structure referred to as`intermingled basinsa in which the basins of more than one chaotic attractors are riddled, was also studied using both discrete maps [90] and a more realistic physical system [93]. Riddled basins have been veri"ed in experiments conducted using coupled electrical oscillators [94,95].…”

Estimating invariant measures and Lyapunov exponents

“…Finally, in Section 4 we apply this procedure to a specific example: the quadratic map with a particular parameter value studied previously by Ruelle [15]. In particular, we estimate a time average for this map which can be shown to be a Lyapunov exponent for a two-dimensional map studied in [2,1]. In the latter paper it is vital to know rigorously that the Lyapunov exponent is negative.…”

“…One exponent will simply be the Lyapunov exponent of the one-dimensional map, and the other will reflect the rate of contraction or expansion transverse to A for the two-dimensional map. This planar map, originally studied in [12], has been of great interest recently as a fundamental example of the phenomenon of "riddled" and "intermingled" basins of attraction [2,1]. In order to verify mathematically the properties of this map which were discovered with a computer, it is necessary to verify that the "transverse" Lyapunov exponent for A is negative, which in particular implies that A attracts a set of positive Lebesgue dimensional measure.…”