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

In this research work, time-delay adaptive synchronization and adaptive anti-synchronization of chaotic fractional order systems are analyzed via the Caputo fractional derivative, and the prob-lem of synchronization and anti-synchronization of chaotic systems of variable fractional order is solved by using the fractional order PID control law, the adaptive laws of variable-order frac-tional calculus, and a control law deduced from Lyapunov’s theory extended to systems of time-delay variable-order fractional calculus. In this research work, two important problems are solved in the control area: The first problem is described in which deals with syn-chro-nization of chaotic systems of adaptive fractional order with time delay, this problem is solved by using the fractional order PID control law and adaptative laws. The second problem is de-scribed in which deals with anti-synchronization of chaotic systems of adaptive frac-tional order with time delay, and this problem is solved by using the fractional order PID con-trol law and adaptative laws.