Analysis of stochastic bifurcation and chaos in stochastic Duffing--van der Pol system via Chebyshev polynomial approximation |
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Authors: | Ma Shao-Juan Xu Wei Li Wei and Fang Tong |
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Institution: | Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China; Department of Information & Computation Sciences, the Second Northwest University for Nationalities, Yinchuan 750021, China |
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Abstract: | The Chebyshev polynomial approximation is applied to investigate the stochastic
period-doubling bifurcation and chaos problems of a stochastic Duffing--van
der Pol system with bounded random parameter of exponential probability
density function subjected to a harmonic excitation. Firstly the stochastic
system is reduced into its equivalent deterministic one, and then the
responses of stochastic system can be obtained by numerical methods.
Nonlinear dynamical behaviour related to stochastic period-doubling
bifurcation and chaos in the stochastic system is explored. Numerical
simulations show that similar to its counterpart in deterministic nonlinear
system of stochastic period-doubling bifurcation and chaos may occur in the
stochastic Duffing--van der Pol system even for weak intensity of random
parameter. Simply increasing the intensity of the random parameter may
result in the period-doubling bifurcation which is absent from the
deterministic system. |
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Keywords: | stochastic Duffing--van der Pol system Chebyshev polynomial approximation stochastic period-doubling bifurcation stochastic chaos |
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