A Study of Discrete-time Dynamical Systems with One Stable Equilibrium Using Standard and Nonstandard Difference Schemes

Chaotic systems with a single stable equilibrium can exhibit rich and unexpected dynamics that are important for both theoretical perceiveing and practical applications of chaos. In this study, we investigate a three-dimensional chaotic system of this type and compare the dynamics of its original continuous-time form with two discrete-time versions obtained using the standard forward Euler scheme and a nonstandard finite-difference scheme based on Mickens’ method. Local stability is inspected using the Jury criterion, where global dynamics are looked into through numerical simulations, including one-parameter and two-parameter bifurcation diagrams and attractor visualizations. The results show that although both discretization schemes reproduce the qualitative features of the continuous system, they differ in their bifurcation structures and stability regions: the Euler scheme tends to delay the onset of chaotic dynamics, whereas the Mickens scheme produces earlier bifurcations and maintains better boundedness and robustness for larger step sizes than the other methods. These findings highlight the significant influence of discretization choices on the accurate numerical representation of the chaotic systems.