Stability, Chaos Diagnose and Adaptive Control of Two Dimensional Discrete - Time Dynamical System

Abstract

In this paper 2D discrete time dynamical system is presented. The fixed points were found. The stability of fixed points is measured by characteristic roots, jury criteria, Lyapunov function. All show that the system is unstable, and analyzing the dynamic behavior of the system finds bifurcation diagrams at the bifurcation parameter. Newton’s Raphson numerical method was used the roots of the system with the minimum error. Then, chaoticity is measured by the phase space; maximum Lyapunov exponent is obtain as (Lmax=2.394569); Lyapunov dimension is obtain as (DL=3.366413); binary test (0 - 1) is obtain as (k = 0.982). All show that the system is chaotic. Finally, the adaptive control was performed. Moreover, theoretical and graphical results of the system after control show the system is stable and Lyapunov exponent is obtained as: L1=-0.390000, L2=-0.500000, so the system is regular.