Shrinking shrimp-shaped domains and multistability in the dissipative asymmetric kicked rotor map

TL;DR

This paper explores the structure and behavior of shrimp-shaped domains and multistability in the dissipative asymmetric kicked rotor map, revealing scaling laws and the effects of dissipation strength on system dynamics.

Contribution

It demonstrates the repeating nature of shrimp-shaped domains under strong dissipation and analyzes how their size and spacing scale with system parameters, also uncovering multistability at weaker dissipation.

Findings

Shrimp-shaped domains repeat with increasing nonlinearity under strong dissipation.

The length of periodic domains follows a power law with a universal exponent.

Multistability emerges within periodic domains as dissipation weakens.

Abstract

An interesting feature in dissipative nonlinear systems is the emergence of characteristic domains in parameter space that exhibit periodic temporal evolution, known as shrimp-shaped domains. We investigate the parameter space of the dissipative asymmetric kicked rotor map and show that, in the regime of strong dissipation, the shrimp-shaped domains repeat themselves as the nonlinearity parameter increases while maintaining the same period. We analyze the dependence of the length of each periodic domain with the nonlinearity parameter, revealing that it follows a power law with the same exponent regardless of the dissipation parameter. Additionally, we find that the distance between adjacent shrimp-shaped domains is scaling invariant with respect to the dissipation parameter. Furthermore, we show that for weaker dissipation, a multistable scenario emerges within the periodic domains. We find that as the dissipation gets weaker, the ratio of multistable parameters for each periodic domain increases, and the area of the periodic basin decreases as the nonlinearity parameter increases.