Viability kernels and capture basins for analyzing the dynamic behavior: Lorenz attractors, Julia sets, and Hutchinson’s maps

Abstract

Some issues on chaotic solutions, to Lorenz systems, for instance, are related to the concepts of viability kernels of subsets under continuous time systems, or in the case of Julia or Cantor sets, for instance, under discrete time systems. It happens that viability kernels of subsets, capture basins of targets and the combination of the twos provide tools for the analysis of the local behavior around equilibria (local stable and unstable manifolds), the asymptotic behavior and the fluctuation of evolutions between two areas of a domain, etc. Since algorithms and softwares do exist for computing the viability kernels and the capture basins, as well as evolutions viable in the viability kernel until they converge to a target in finite time, we are able to localize the attractor, to compute local stable and unstable manifolds, heteroclinical points, etc.