Poincare maps for detecting chaos in fractional-order systems with hidden attractors for its Kaplan-Yorke dimension optimization

The optimization of fractional-order (FO) chaotic systems is challenging when simulating a considerable number of cases for long times, where the primary problem is verifying if the given parameter values will generate chaotic behavior. In this manner, we introduce a methodology for detecting chaotic behavior in FO systems through the analysis of Poincare maps. The optimization process is performed applying differential evolution (DE) and accelerated particle swarm optimization (APSO) algorithms for maximizing the Kaplan-Yorke dimension (D-KY) of two case studies: a 3D and a 4D FO chaotic systems with hidden attractors. These FO chaotic systems are solved applying the Grunwald-Letnikov method, and the Numba just-in-time (jit) compiler is used to improve the optimization process's time execution in Python programming language. The optimization results show that the proposed method efficiently optimizes FO chaotic systems with hidden attractors while saving execution time.

Automated summary

This study introduces a method for detecting chaotic behavior in fractional-order chaotic systems through the analysis of Poincare maps. The optimization process is performed using differential evolution and accelerated particle swarm optimization algorithms to maximize the Kaplan-Yorke dimension of chaotic systems with hidden attractors. The results show that the proposed method efficiently optimizes fractional-order chaotic systems while saving execution time.