Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation |
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| Authors: | CS Feng SL Chen |
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| Institution: | 1. Saint-Petersburg State University, Universitetskaya nab., 7–9, St. Petersburg, 199034, Russia;2. Institute of Problems of Mechanical Engineering RAS, V.O., Bolshoj pr., 61, St. Petersburg, 199178, Russia;3. Department of Mathematics and Statistics, P.O. Box 68, University of Helsinki, 00014 Helsinki, Finland;1. Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt;2. Department of Mathematics, Faculty of Science, Taif University, Taif, Saudi Arabia |
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| Abstract: | The asymptotic Lyapunov stability with probability one of Duffing–Mathieu system with time-delayed feedback control under white-noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. Meanwhile, the stability conditions for the system with different time delays are also obtained. The theoretical results are well verified through digital simulation. |
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