On α-Limit Sets in Lorenz Maps

Łukasz Cholewa,Piotr Oprocha

doi:10.3390/e23091153

Abstract

The aim of this paper is to show that -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 . On the basis of provided examples, we also present how the performed study on the structure of -limit sets is closely connected with the calculation of the topological entropy.

Highlights

Lorenz maps are interval maps which appear in a natural way as Poincaré sections in the Lorenz attractor
Families of Lorenz maps are usually derived from the so-called geometric Lorenz model, where, by definition, the Poincaré section leads to a map f : [0, 1] → [0, 1] satisfying the following three conditions: 1
As Theorem 1 shows, this is not always the case, and backward invariance needs additional checking. This comes with a surprise, since as we mentioned earlier, Lorenz maps are derived from the Lorenz model whose discretization is invertible; all α-limits sets are completely invariant

Summary

Introduction

Lorenz maps are interval maps which appear in a natural way as Poincaré sections in the Lorenz attractor. This approach is probably the most popular one It appears in the work of Coven and Nitecki [17], who showed that for a continuous interval map, a point x is nonwandering if and only if x ∈ α( x ), or, in a more recent paper, Cui and Ding [18] studied α-limit sets of unimodal interval maps. As Theorem 1 shows, this is not always the case, and backward invariance needs additional checking This comes with a surprise, since as we mentioned earlier, Lorenz maps are derived from the Lorenz model whose discretization is invertible (and smooth); all α-limits sets are completely invariant. We use these examples as a testing ground to apply a few techniques to calculate the entropy of interval maps in concrete cases

Symbolic Dynamics

Construction of Expanding Lorenz Map f : Proof of Theorem 1

Decomposition of Nonwandering Set of f and Entropy

Unimodal Example

Continuous Piecewise Affine Maps

When α-Limit Sets Are Invariant