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

• Published in: Results in physics Vol. 56; p. 107311
• Main Authors: Ahmed, Khalid I.A., Adam, Haroon D.S., Almutairi, Najat, Saber, Sayed
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.01.2024; Elsevier
• Subjects: Fractional derivatives; Iterative method; Lyapunov functions; Nonlinear equations; Numerical results; Simulation; Time varying control system; Nonlinear equations; Lyapunov functions; Simulation; Iterative method; Time varying control system; Fractional derivatives; Numerical results
• ISSN: 2211-3797; 2211-3797
• DOI: 10.1016/j.rinp.2023.107311

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. •The Atangana–Baleanu–Caputo–Fabrizio fractional derivative with variable order is applied to synchronized chaotic systems.•A state-observer is proposed to simultaneously control fractional chaotic systems.•The influence of variable-order on synchronization is demonstrated numerically.