scholarly journals Period-doubling bifurcation analysis of stochastic van der Pol system via Chebyshev polynomial approximation

2005 ◽  
Vol 54 (8) ◽  
pp. 3508
Author(s):  
Ma Shao-Juan ◽  
Xu Wei ◽  
Li Wei ◽  
Jin Yan-Fei
Keyword(s):  
Chebyshev Polynomial ◽  
Polynomial Approximation ◽  
Bifurcation Analysis ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Van Der Pol ◽  
Chebyshev Polynomial Approximation

scholarly journals Period-Doubling Bifurcation of Stochastic Fractional-Order Duffing System via Chebyshev Polynomial Approximation

2017 ◽  
Vol 2017 ◽  
pp. 1-12
Author(s):  
Youming Lei ◽  
Yanyan Wang
Keyword(s):  
Fractional Order ◽  
Chebyshev Polynomial ◽  
Polynomial Approximation ◽  
Period Doubling ◽  
Random Parameter ◽  
Fractional Order System ◽  
Period Doubling Bifurcation ◽  
Order System ◽  
Integer Order ◽  
Chebyshev Polynomial Approximation

Fractional-order calculus is more competent than integer-order one when modeling systems with properties of nonlocality and memory effect. And many real world problems related to uncertainties can be modeled with stochastic fractional-order systems with random parameters. Therefore, it is necessary to analyze the dynamical behaviors in those systems concerning both memory and uncertainties. The period-doubling bifurcation of stochastic fractional-order Duffing (SFOD for short) system with a bounded random parameter subject to harmonic excitation is studied in this paper. Firstly, Chebyshev polynomial approximation in conjunction with the predictor-corrector approach is used to numerically solve the SFOD system that can be reduced to the equivalent deterministic system. Then, the global and local analysis of period-doubling bifurcation are presented, respectively. It is shown that both the fractional-order and the intensity of the random parameter can be taken as bifurcation parameters, which are peculiar to the stochastic fractional-order system, comparing with the stochastic integer-order system or the deterministic fractional-order system. Moreover, the Chebyshev polynomial approximation is proved to be an effective approach for studying the period-doubling bifurcation of the SFOD system.

Stochastic period-doubling bifurcation analysis of stochastic Bonhoeffer–van der Pol system

2007 ◽  
Vol 16 (7) ◽  
pp. 1923-1933 ◽  
Author(s):  
Zhang Ying ◽  
Xu Wei ◽  
Fang Tong ◽  
Xu Xu-Lin
Keyword(s):  
Bifurcation Analysis ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Van Der Pol

scholarly journals Numerical Method for a Markov-Modulated Risk Model with Two-Sided Jumps

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Hua Dong ◽  
Xianghua Zhao
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Chebyshev Polynomial ◽  
Polynomial Approximation ◽  
Risk Model ◽  
Arbitrary Distribution ◽  
System Of Differential Equations ◽  
Numerical Result ◽  
Chebyshev Polynomial Approximation ◽  
Markov Modulated

This paper considers a perturbed Markov-modulated risk model with two-sided jumps, where both the upward and downward jumps follow arbitrary distribution. We first derive a system of differential equations for the Gerber-Shiu function. Furthermore, a numerical result is given based on Chebyshev polynomial approximation. Finally, an example is provided to illustrate the method.

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