Behavioral analysis of two-dimensional difference equations in the third quadrant

Piecewise linear systems of difference equations have gained significant attention for their ability to model complex behavior in population dynamics, economics, and electronics fields. Despite their simple structure, these systems can exhibit diverse behaviors, including convergence to equilibrium points, periodic solutions, and chaotic outcomes under specific conditions. This paper investigates the long-term behavior of a specific piecewise linear system of difference equations. The primary goal is understanding how initial conditions and parameter values influence the system's behavior. The research focuses on identifying and analyzing equilibrium points, periodic solutions (cycles), and the conditions under which these behaviors occur. Building on previous work by Grove et al., we study a family of two-dimensional difference equations containing absolute value terms. The analysis focuses on initial conditions in the third quadrant, divided into three distinct regions: A, B, and C. The behaviors within each region are explored to characterize the system's outcomes. Region A: The system converges to an equilibrium point. The number of iterations required for convergence varies depending on the sub-region. Region B: The system converges to an equilibrium point in exactly two iterations. Region C: The system exhibits more complex behaviors, with potential outcomes including convergence to a 4-cycle or an equilibrium point. Behavior in this region suggests that initial conditions may lead to one of two prime period-4 cycles. Regions A and B consistently lead to equilibrium points, while Region C displays more varied outcomes, including periodic cycles. These findings emphasize the complexity of piecewise linear systems and stress the need for further research to fully understand the behaviors in Region C.