Multistability and bifurcation phenomena in a four-partitions piecewise linear map

This paper investigates the global dynamics and bifurcation structures of a two-dimensional piecewise linear map defined by four partitions, involving absolute value terms for both state variables. An analysis of fixed point stability demonstrates that the emergence of period-2 dynamics is initiated by a Flip bifurcation. A detailed analysis of these cycles reveals a unique branch manifesting in two topologically distinct configurations, connected via a Border Collision Bifurcation. Furthermore, results show that higher-period cycles (period 3 and 4) emerge in stable-saddle pairs through Fold Border Collision Bifurcations, leading to regions of multistability organized by the stable manifolds of saddle orbits. Finally, the study examines the degenerate parameter regime where the system reduces to a one-dimensional discontinuous map. In this case, the existence of a forward-invariant absorbing interval is proved, and parameter regions exhibiting chaotic dynamics are identified numerically.