Stable Attractors on a Certain Two-dimensional Piecewise Linear Map

Abstract

In this article we study the behaviors of a piecewise linear map with initial condition in the second quadrant. There is a unique equilibrium point and two 4-cycles of the map. We found regions of initial condition that solutions become equilibrium point or 4-cycles. We divided the second quadrant into sub-regions and identify behaviors of solutions in each sub-region by direct calculations, and formulated inductive statements to explain the behaviors of the map without using stability theorems.