7th ISF

MATH014 - The System’s Behavior of a Certain Type of Piecewise Linear Difference Equations

A first-order system of the piecewise difference equations of the form x_(n+1)=|x_n |-ay_n-b and y_(n+1)=x_n-c|y_n |+d for n≥0 and a,b,c,d∈R, with given initial values x_0 and y_0, has been studied by several researchers. Such a system is a generalization of the Lozi map and Devaney’s Gingerbreadman map. Both are studied as examples of two-dimensional chaotic systems, exhibiting complex behavior that is of interest in studying chaos theory and dynamical systems.The objective of our project is to investigate the behavior of the solution of this system of piecewise linear difference equations where a=c=d=1 and b∈(5,6), which can be written as x_(n+1)=|x_n |-y_n-b and y_(n+1)=x_n-|y_n |+1. Our analysis reveals that the behavior of the solution can be classified into two types: equilibrium point and prime period 5. To understand the dynamics of the system, we need to consider the range of values for b. Our findings show that when b∈[16/3,6) and (x_0,y_0) is in the first quadrant, the solution has a certain behavior according to the range of x_0-y_0.

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