Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer--van der Pol system |
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| Authors: | Zhang Ying Xu Wei Fang Tong and Xu Xu-Lin |
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| Affiliation: | Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China; Vibration Research Centre, Northwestern Polytechnical University, Xi'an 710072, China; Department of Automation, Nankai University, Tianjin 300071, China |
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| Abstract: | In this paper, the Chebyshev polynomial approximation is applied to
the problem of stochastic period-doubling bifurcation of a stochastic
Bonhoeffer--van der Pol (BVP for short) system with a bounded random
parameter. In the analysis, the stochastic BVP system is transformed
by the Chebyshev polynomial approximation into an equivalent
deterministic system, whose response can be readily obtained by
conventional numerical methods. In this way we have explored plenty
of stochastic period-doubling bifurcation phenomena of the stochastic
BVP system. The numerical simulations show that the behaviour of the
stochastic period-doubling bifurcation in the stochastic BVP system
is by and large similar to that in the deterministic mean-parameter
BVP system, but there are still some featured differences between
them. For example, in the stochastic dynamic system the
period-doubling bifurcation point diffuses into a critical interval
and the location of the critical interval shifts with the variation
of intensity of the random parameter. The obtained results show that
Chebyshev polynomial approximation is an effective approach to
dynamical problems in some typical nonlinear systems with a bounded
random parameter of an arch-like probability density function. |
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| Keywords: | Chebyshev polynomial approximation stochastic Bonhoeffer--van der Pol
system hspace*{1 9cm} stochastic period-doubling bifurcation bounded random parameter |
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