Complexity of Coexistence Regions in the GRHT Map

TL;DR

This paper studies the geometric complexity of coexistence regions of stable periodic orbits in the GRHT map, revealing polygonal convex regions and how their properties change with parameters.

Contribution

It introduces a parameterized non-invertible GRHT map, develops an algorithm to analyze overlapping regions, and explores how these regions' geometry varies with parameters.

Findings

Coexistence regions are convex polygons in parameter space.

Number of vertices increases with the number of coexisting orbits.

Area of the overlapping region decreases as the number of coexisting orbits increases.

Abstract

GRHT map refers to a planar map which showcases the coexistence of infinitely many stable periodic orbits via the phenomenon of Globally Resonant Homoclinic Tangencies. This paper investigates the geometric properties of coexistence regions in the case of codimension-three scenario. We introduce parameters into the non-invertible GRHT map to understand the unfolding behavior of the GRHT. Near the infinite coexistence regions, there exists series of codimension-one saddle-node and period-doubling bifurcations. The most common overlapping region in the parameter space reveals the parameters with which there can be coexisting periodic orbits. Various slices of parameter space are considered to understand the coexistence regions in two-dimensional parameter space. We show that the parameter region of coexistence are polygons and are convex sets. We develop an algorithm that detects the number of vertices of the most common overlapping regions via optimization techniques. We study the variation of the number of vertices and area of the most common overlapping region with the variation in parameters of the map. It illustrates that when the number of coexisting stable periodic orbits increase, the area of the most common overlapping region decreases. Moreover the number of vertices of the most common overlapping region increases as the number of coexisting stable periodic orbits increases. We also explore the variation of the area and the number of vertices of the most common overlapping region with the simultaneous variation of two parameters of the GRHT map. It reveals that the variation in the number of vertices of the most common overlapping region is not trivial and varies in a highly nonlinear fashion illustrating the geometric complexity of the coexisting regions of stable periodic orbits in the parameter space.