On α-Limit Sets in Lorenz Maps

Abstract

The aim of this paper is to show that α-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on [0,1]. On the basis of provided examples, we also present how the performed study on the structure of α-limit sets is closely connected with the calculation of the topological entropy.

Spike patterns and chaos in a map-based neuron model

“…We have now obtained a picture of periodic and chaotic properties for Lorenz maps. At the end of this theoretical part, let us mention that in their recent works Cholewa and Oprocha (2021a;2021b) developed the theory of limit sets and renormalization for Lorenz maps.…”

Information Geometry, Complexity Measures and Data Analysis

“…Cholewa and Oprocha, in their article, focus on α-limit sets in Lorenz maps [ 6 ]. Lorenz maps are interval maps which are realized as Poincaré sections in the Lorenz attractor.…”