In this paper, the Adomian decomposition method (ADM) semi-analytical solution algorithm is applied to solve a fractional-order entanglement symmetrical chaotic system. The dynamics of the system are analyzed by the Lyapunov exponent spectrum, bifurcation diagrams, poincaré diagrams, and chaos diagrams. The results show that the systems have rich dynamics. Meanwhile, sliding mode synchronizations of fractional-order chaotic systems are investigated theoretically and numerically. The results show the effectiveness of the proposed method and potential application value of fractional-order systems.

The self-adaptive terminal sliding mode synchronization of fractional-order nonlinear chaotic systems is investigated under uncertainty and external disturbance. A novel non-singular terminal sliding surface is proposed and proved to be stable. Based on Lyapunov stability theory, a sliding mode control law is proposed to ensure the occurrence of sliding-mode motion. In addition, two methods of the controller and the self-adaptive rules are used to establish the sliding mode function, and two sufficient conditions for achieving self-adaptive terminal sliding-mode synchronization of fractional-order uncertain nonlinear systems are identified. The results show that designing appropriate control law and sliding-mode surface can achieve self-adaptive terminal sliding mode synchronization of the fractional high-order systems with uncertainty. The effectiveness and applicability of the sliding mode control technique are validated through numerical simulation.

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Particle swarm optimization (PSO) is an effective robust and simple method to solve many problems proposed in science and engineering. How does the particle motion and how the particles in a swarm find the optimal solutions are an open problem. This paper investigates the particle trajectories for the standard PSO based on difference equations theories. Equilibrium point and asymptotically stable are used to study the particle trajectories. The theoretical analysis shows that the particle will just stop at the stable point with the proper PSO parameters.

Based on stability theory and Lyapunov method. a schemes is provide for synchronization of a hyperchaotic system. The scheme synchronizing the hyperchaotic system with linear controller, this scheme is more simple and important significance on using chaos synchronization for applications. numerical simulations have verified the effectiveness of the method.