Incommensurate fractional-order analysis of a chaotic system based on interaction between dark matter and dark energy with engineering applications

Chaotic systems, characterized by their sensitivity to initial conditions and the presence of deterministic unpredictability, have garnered significant attention in various fields of science and engineering. Researchers have demonstrated a growing interest in recent years in identifying various physical systems as chaotic system, including but not limited to fluid dynamics, electrical circuits, and biological oscillators. In this study, a cosmological chaotic system that portrays the interaction between dark matter and dark energy is examined using incommensurate fractional-order analysis. The effects of incommensurate fractional orders are evaluated by means of phase portraits, bifurcation diagrams and Lyapunov exponents spectra. It is dedicated that wider chaotic regions are observed once the values of these incommensurate fractional orders are changed. Meanwhile, a hidden attractor is found based on the stability analysis and basin of attraction in the incommensurate fractional-order dark matter and dark energy (IFODMDE) chaotic system. These findings highlight the richer dynamic characteristics of the IFODMDE chaotic system. Furthermore, the identified hidden attractor is effectively employed in this study for various engineering applications, including chaos control, random number generator (RNG) design, and image encryption. The analytical results offer a more accurate description of real-world physical phenomena, thanks to the inclusion of incommensurate fractional orders. The application results prove the good randomness of the IFODMDE chaotic system.