Analysis of Stability and Chaos of Discrete Time System with Local Bifurcation

This article deals with a two-dimensional time-distributed system. Three non-linear terms are included in the system's seven basic terms. The fundamental properties of the system are analyzed by means of fixed points, we obtained two fixed points which are real and two are complex. stability analysis by using (Characteristic Equation Roots Test, Jury Test and Lyapunov Function Test) the turbulence and chaotic behavior of the system indicated by phase portrait, hoph bifurcation, dispativity, and Lyapunov dimensions. Also, simulation was done by fixed point iteration method to get the optimal solution with (x,y)=(-0.0568, 0.0012). The maximal Lyapunov exponent and Lyapunov dimension are (0.0012878294) and (1.46271321), respectively, indicating the system is chaotic. Finally, using an adaptive control technique, a stable system path was established with an optimum controller.