## Anti-synchronization between different chaotic complex systems.(English)Zbl 1225.37117

In this work, basic dynamic properties of the complex Wang system $\begin{cases}\dot{z}_1=a_1(z_1-z_2),\\ \dot{z}_2=-c_1z_2+z_1z_3,\\ \dot{z}_3=-b_1z_3+\frac12 d_1(\bar{z}_1z_2+z_1\bar{z}_2),\end{cases} \tag{1}$ (where $$z_1=u_{11}+ju_{21}$$, $$z_2=u_{31}+ju_{41}$$ are complex state variables; $$j=\sqrt{-1}$$; $$z_3=u_{51}$$ is a real state variable; $$\bar{z}$$ represents the complex conjugate of $$z$$) are studied. Further, the anti-synchronization between the system (1) and a complex Lorenz system, and the system (1) and a complex Lü system is investigated via active control. The control design process is that, based on the Lyapunov function constructed, the designed controller meets the Lyapunov stability conditions. In the application of nonlinear control, the nonlinear error system in written as a linear system; then anti-synchronization is achieved using the stability conditions of linear control theory. The anti synchronization of complex systems is achieved via both methods, nonlinear control for personal purposes and simpler for computations. Numerical simulations verify that these methods are effective in achieving anti-synchronization of the system (1).

### MSC:

 37N35 Dynamical systems in control 34D06 Synchronization of solutions to ordinary differential equations 93C10 Nonlinear systems in control theory 37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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