On the chaotic nature of the Rabinovich system through Caputo and Atangana–Baleanu–Caputo fractional derivatives

Abstract

AbstractThe Rabinovich system can describe different physical interactions, including waves in plasmas, a convective fluid flow inside a rotating ellipsoid, and Kolmogorov’s flow interactions. This study considers the Rabinovich system through Caputo and Atangana–Baleanu fractional derivatives to detect its chaotic nature. First, the existence and uniqueness of the solutions of the fractional-order systems are proved using the combination of the Picard–Lindelöf theorem and the Banach contraction principle. Then, a numerical approximation of the fractional systems is developed. The fractional Rabinovich system is found to exhibit a chaotic behavior verified via Lyapunov exponents. However, the fractional-order models do not enter into chaotic behavior at the same fractional-derivative order. Bifurcation diagrams referring to variation of the fractional-order derivatives are provided. Chaotic attractors for both cases of the fractional-derivative representation of the system are depicted. The two fractional-order models of the system show sensitivity to initial conditions. A master–response synchronization was developed in the context of the Atangana–Baleanu fractional derivative. The master and the response systems showed a strong correlation, proving the system’s applicability in solving real problems, including secure communications.