Control Of Random Phase Disturbance For Some Chaotic Systems

In nonlinear dynamical systems, Chaos is widespread. We want to control the chaos to inhibit or eliminate it in many cases, so various methods of chaos control and application are catching more and more attention. At present, the methods of chaos control are divided into two categories: feedback control and non-feedback control. In recent years, as a non-feedback control method, random phase control has got wide attention, and because the method is easy to implement in practice, to study the effect on the chaos control is more valuable.In this paper, the chaotic phenomena in a class of coupling of the Double-Well Duffing system, automotive suspension systems and Duffing-Ueda system are studied, and by adding random phase noise into the system equations, the method achieves controlling of three chaotic systems. In fact, we use the largest Lyapunov exponent as the basis for identification of chaotic, through numerical simulation, making the original system Lyapunov exponent map: If the Lyapunov exponent is positive, it indicates that the system is chaotic. At the same time, combined with Poincarésurface of section, phase diagram and time history we confirm the chaotic state. On the other hand, we have get the map of Lyapunov exponents with the system after random phase perturbation, finding within the some certain parameters, Lyapunov exponents change from positive to negative, so it means the system in the random phase perturbation is controlled. And, drawn Poincarésurface of section in different noise intensities, and further confirmed that the system can be controlled under the random phase. Computing the top Lyapunov exponent is based on Khasminskii's spherical coordinates transform for linear stochastic systems. In this paper, the three systems in physics and engineering are widely used. Therefore, the results not only have profound theoretical significance, but also have important practical value.