synchronization between two different fractional chaotic systems using particle swarm optimization PID-controller

This work discusses the synchronization of two non-equivalent fractional chaotic systems. The optimized Proportional-Integral-Derivative (PID) controller is designed. The Particle Swarm Optimization (PSO) algorithm is employed to determine the appropriate controller gains: the proportional gain Kp, the derivative gain Kd, and the integral gain Ki. The paper proves the complex chaotic behavior of the two chosen systems (fractional Chen system and fractional Lorenz system) by exposing the Lyapunov exponent, the chaotic attractor, and the bifurcation diagram. Finally, synchronization is successfully achieved and demonstrated through numerical simulations.