One-dimensional Dynamics for a Discontinuous Singular Map and the Routes to Chaos

TL;DR

This paper explores a discontinuous two-parameter map, revealing unique routes to chaos different from the logistic map, including non-period-doubling pathways and complex aperiodic behaviors with infinite accumulation points.

Contribution

It provides a detailed numerical analysis of a novel discontinuous map, highlighting new routes to chaos and complex dynamics not seen in continuous maps.

Findings

Routes to chaos without period-doubling

Existence of aperiodic maps with infinite accumulation points

Complex cobweb structures with x = infinity

Abstract

We visit a previously proposed discontinuous, two-parameter generalization of the continuous, one-parameter logistic map and present exhaustive numerical studies of the behavior for different values of the two parameters and initial points. In particular, routes to chaos exist that do not exhibit period-doubling whereas period-doubling is the sole route to chaos in the logistic map. Aperiodic maps are found that lead to cobwebs with x = infinity as accumulation points, where every neighborhood contains infinitely many points generated by the map.