Fractional Routh-Hurwitz conditions and nonlinear dynamics in some 3D and 4D dynamical systems modeled by Caputo-Fabrizio operators

Abstract

This work presents stability conditions in some 2D, 3D and 4D dynamical systems modeled by Caputo-Fabrizio operators. Some theorems about fractional Routh-Hurwitz criteria are presented and proven in two-, three-, four- and n-dimensional dynamical systems governed by the Caputo-Fabrizio derivatives. The stability analyzes are presented for all equilibrium states of four examples representing two 3D and two 4D chaotic systems governed by the Caputo-Fabrizio derivatives. The conditions of Hopf bifurcations in these systems are also discussed in the considered chaotic systems. In addition, the chaotic dynamics in these systems are illustrated via numerical simulations that show existences of periodic orbits and several types of chaotic dynamics, such as one-scroll chaos, double scroll-chaos, self-excited chaos, and an alien face chaotic attractor. The Lyapunov exponents and bifurcation diagrams are successfully utilized to measure these chaotic and hyperchaotic states.