Complexity of Coexistence Regions in the GRHT Map

GRHT map refers to a planar map which showcases the coexistence of infinitely many stable periodic orbits via the phenomena of Globally Resonant Homoclinic Tangencies. This paper investigates the geometric properties of coexistence regions in the case of codimension-3 scenario. We introduce parameters into the noninvertible GRHT map to understand the unfolding behavior of the GRHT. Near the infinite coexistence regions, there exist a series of codimension-1 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 regions 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 increases, 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.