The Multi-Switching Sliding Mode Combination Synchronization of Fractional Order Non-Identical Chaotic System with Stochastic Disturbances and Unknown Parameters

Abstract

This paper deals with the issue of the multi-switching sliding mode combination synchronization (MSSMCS) of fractional order (FO) chaotic systems with different structures and unknown parameters under double stochastic disturbances (SD) utilizing the multi-switching synchronization method. The stochastic disturbances are considered as nonlinear uncertainties and external disturbances. Our theoretical part considers that the drive-response systems have the same or different dimensions. Firstly, a FO sliding surface is established in terms of the fractional calculus. Secondly, depending on the FO Lyapunov stability theory and the sliding mode control technique, the multi-switching adaptive controllers (MSAC) and some suitable multi-switching adaptive updating laws (MSAUL) are designed. They can ensure that the state variables of the drive systems are synchronized with the different state variables of the response systems. Simultaneously, the unknown parameters are assessed, and the upper bound values of stochastic disturbances are examined. Selecting the suitable scale matrices, the multi-switching projection synchronization, multi-switching complete synchronization, and multi-switching anti-synchronization will become special cases of MSSMCS. Finally, examples are displayed to certify the usefulness and validity of the scheme via MATLAB.

Time-Delay Fractional Variable Order Adaptive Synchronization and Anti-Synchronization between Chen and Lorenz Chaotic Systems Using Fractional Order PID Control

“…The Fractional-Order MatLab (Simulink) method, is used for solving the dynamics (3) and ( 4) with nonlinear control laws ( 14), (17), and (22) for time-delay synchronization, and nonlinear control laws (35), (40), and (45) for time-delay anti-synchronization. For chaotic behavior of the fractional-order Chen system (3) and Lorenz system (4), their controllers with the fractional-order PID control law, together, show the synchronization and anti-synchronization behavior, and the modeling errors tend to zero, as can be seen in Figures 1 and 2 with 𝛼 = 0.9, and in the Figures 3 and 4 the variable-order fractional with 𝛼 = 0.9, 𝛼 = 0.8, and 𝛼 = 0.5.…”

“…The time-delay synchronization and anti-synchronization between the Chen and Lorenz systems are obtained by means of the control laws ( 12), (17), and (22), and the analysis of the convergence of the approximation errors of these systems is guaranteed via the Lyapunov stability analysis for systems of fractional order and the fractional order PIDtype control law. The fractional-order derivative in this paper is using α = 0.9 and α = 0.9, α = 0.8, α = 0.5, and the time delay 𝜏 = 5 seg, and the initial conditions for simulation are 𝑥(0) = [−10, 0, 37] and 𝑦(0) = [10, 0, 10] for synchronization and anti-synchronization and the graphs of the Lorenz values, which are the laws of adaptation.…”

A fractional-order quantum neural network: dynamics, finite-time synchronization

“…Adaptive synchronization of fractional system with uncertainties was adopted in [22]. The multi-switching sliding mode synchronization of fractional systems was studied in [23]. The finite -time generalized synchronization of fractional order systems and its application was employed in [24].…”

A Numerical Method for Simulating Viscoelastic Plates Based on Fractional Order Model

“…Therefore, fractional order has a wide range of applications, especially in control systems. Using the multi-switch synchronization method, Pan et al [11] considered the sliding-mode combinatorial synchronization of fractional-order chaotic systems under double random disturbances. Zhang et al [12] introduced fractional order into sliding mode control of the system.…”