Chaos and hidden chaos in a 4D dynamical system using the fractal-fractional operators

Abstract

Although fractional calculus is about three centuries old, it has become the key to understanding many complex real-world phenomena. During the past few decades, many fractional derivatives have appeared. Among these, the fractal-fractional derivatives have shown acceptance in describing some real-world problems. In this paper, the Caputo, Atangana-Baleanu, and Caputo-Fabrizio fractal-fractional operators were applied to generate complex dynamics in a 4D dynamical system. Some conditions for the exact solutions' existence and uniqueness were demonstrated when the fractal-fractional operators are implemented into the mentioned 4D dynamical system. Some Ulam-Hyers stability results were demonstrated in the indicated fractal-fractional systems. Computation processes were carried out to demonstrate some graphical results that showed the existence of several complex dynamics in the considered system as the fractal-fractional operators are implemented. Furthermore, the computations of the system's Lyapunov exponents and the bifurcation diagrams were used to illustrate the wide range of chaotic dynamics that exist in the considered fractal-fractional 4D system. Existence of hidden chaotic attractors were also found. This interesting dynamical phenomenon was validated by the bifurcation diagrams and basin set of attraction.

Keywords

• fractal-fractional operators
• a 4d dynamical system
• chaos
• lyapunov exponents
• hidden chaotic attractor