Analytical solutions for a class of variable-order fractional Liu system under time-dependent variable coefficients

Abstract

In this article, we present a nonlinear model of the Liu system that includes fractional derivatives of variable-order. Due to the nonlocality of the dynamical system, we introduce the fractional derivative with power laws, exponential decay laws, and generalized Mittag-Leffler functions as kernels. We provide a detailed analysis of the existence and uniqueness of the proposed model, as well as the stability of these equations. Due to the existence of time-varying fractional derivatives, the proposed variable-order fractional system exhibits more complex characteristics and more degrees of freedom than an integer or conventional constant fractional-order chaotic Liu oscillator. Different chaotic behaviors can be obtained by using different smooth functions defined within the interval (0,1] as variable orders for fractional derivatives in the simulations. Furthermore, simulations demonstrate that fractional chaotic systems with variable orders can be synchronized.

Keywords

• Fractional derivatives
• Nonlinear equations
• Simulation
• Numerical results
• Iterative method
• Time varying control system