Stochastic chaos in stochastic Duffing systems and its control by delayed feedback

The problem of stochastic chaos and its control by delayed feedback in a Duffing system with bounded random parameters (a stochastic Duffing system in short) under harmonic excitations is considered in detail. At first, the stochastic Duffing system is transformed into its equivalent deterministic nonlinear system by the Gegenbauer polynomial approximation. Thus, the problem of chaotic response and its control in stochastic Duffing system can be reduced to that in an equivalent deterministic system. So the available effective mathematical methods and control strategies can be applied to the latter. Then, the main feature of stochastic chaos is fully explored, where the top Lyapunov exponent of the equivalent system obtained by Wolf's algorithm is used to identify the dynamic behavior of stochastic Duffing system. Finally, the control strategy of delayed feedback is applied to suppress or to induce chaotic response in the system. The results of numerical simulation show that by proper choice of feedback intensity and time delay, either suppressing or inducing stochastic chaos can be achieved. Hence, the strategy of delayed feedback control is also effective to stochastic chaos.