Quantifying Stability and Chaoticity of One-dimensional and Two-dimensional Discrete Dynamical Systems using Stability Analysis and Lyapunov Exponents

Discrete dynamical system is a system that evolve dynamically with discrete time. In this paper, we consider two discrete systems which exhibit chaotic behaviour. We show that the chaoticity of a system is depend on the values of parameter in the system. The objective of this paper is to investigate both stability and chaoticity of the systems using stability analysis and Lyapunov exponent, respectively. The results show that there is only one fixed point for the one-dimensional system, while for the two-dimensional system, there exist four possible fixed points. We have proved the stability conditions for each fixed point obtained. The Lyapunov results show that the one-dimensional system is stable when r<π/2 and chaotic when r>π/2. Whereas for two-dimensional systems considered, the system is chaotic for the whole range of parameter α∈(0,1]. This study shows that it is significant to consider various values of parameter to study the stability of a dynamical systems in particular to control the chaos in a system.