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.

Automated summary

In this research, time-delay adaptive synchronization and adaptive anti-synchronization of chaotic fractional order systems are analyzed using the Caputo fractional derivative. The problem of synchronization and anti-synchronization of chaotic systems with variable fractional order is solved using the fractional order PID control law, adaptive laws of variable-order fractional calculus, and a control law derived from Lyapunov's theory extended to systems of time-delay variable-order fractional calculus. This research solves two important problems in the control area: synchronization of chaotic systems with adaptive fractional order and time delay using the fractional order PID control law and adaptive laws, and anti-synchronization of chaotic systems with adaptive fractional order and time delay using the fractional order PID control law and adaptive laws.