Dynamics of a map with power-law tail

TL;DR

This paper investigates a one-dimensional map with a power-law tail, revealing complex bifurcation structures, chaos transitions, and phenomena like interior crises, through numerical and analytical methods.

Contribution

It provides a detailed analysis of bifurcations and chaos in a map with a power-law tail, highlighting the role of the power-law component in these dynamics.

Findings

Abrupt transition from order to chaos via infinite limit cycles

Presence of interior crises and crisis-induced intermittency

Power-law tail induces unique bifurcation structures

Abstract

We analyze a one-dimensional piecewise continuous discrete model proposed originally in studies on population ecology. The map is composed of a linear part and a power-law decreasing piece, and has three parameters. The system presents both regular and chaotic behavior. We study numerically and, in part, analytically different bifurcation structures. Particularly interesting is the description of the abrupt transition order-to-chaos mediated by an attractor made of an infinite number of limit cycles with only a finite number of different periods. It is shown that the power-law piece in the map is at the origin of this type of bifurcation. The system exhibits interior crises and crisis-induced intermittency.