Unstable periodic orbits and hyperchaos in 2D quadratic memristor map

Abstract

This study explores the characteristics of the simplest invariant sets within a symmetric two-dimensional quadratic memristor map, exhibiting pinched hysteresis. Our analysis reveals critically stable fixed points within the map. Through computation of critical curves, we demonstrate the map’s non-invertibility and categorize it as type $Z_1-Z_3$. We observe that successive iterates of the critical curve delineate both disjoint and merged hyperchaotic attractors. Utilizing the usual period-doubling route, we establish the manifestation of hyperchaos within the map. Furthermore, employing a multi-dimensional Newton–Raphson method, we extend saddle periodic orbits to the realm of hyperchaos. Notably, our investigation, supported by eigenvalue computations, elucidates that most of the periodic orbits absorbed into the hyperchaotic attractor transform into repellers while the remaining periodic orbits are saddles representing unstable dimension variability (UDV).