4-CYCLES AND AN EQUILIBRIUM POINT IN A PIECEWISE LINEAR MAP WITH INITIAL CONDITIONS ON THE POSITIVE Y-AXIS

Abstract

This article examines the behavior of a piecewise linear map, particularly when initial conditions are set along the positive y-axis. Our focus is on the emergence of 4-cycles and an equilibrium point within the map. We identify specific regions on the y-axis where the solutions tend to move toward these 4-cycles and the equilibrium point. By partitioning the positive y-axis into smaller segments, we analyze the solution behavior through a combination of direct calculation and induction methods. This approach allows us to demonstrate that the solutions consistently reach a prime period of 4 and an equilibrium point. Notably, this finding holds true regardless of the application of stability theorems, indicating a robust pattern in the solution dynamics. This study looks closely at how these solutions change over time, giving a clear picture of their paths. By carefully dividing the positive y-axis and studying each part, we find out why the solutions move towards the 4-cycles and the equilibrium point. This detailed look shows that the map's behavior is predictable and follows a certain pattern no matter where we start on the positive y-axis. Our research helps us understand piecewise linear systems better, and these insights could be useful for other similar systems. Our findings prove that periodic behavior and equilibrium are common in these maps, making it easier to predict how they will act over a long time.