Applications of Singularity Theory of Explosion Theory

Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

Advances in Difference Equations ◽

10.1186/s13662-019-2356-1 ◽

2019 ◽

Vol 2019 (1) ◽

Author(s):

Yajie Li ◽

Zhiqiang Wu ◽

Guoqi Zhang ◽

Feng Wang ◽

Yuancen Wang

Keyword(s):

Time Delay ◽

White Noise ◽

Gaussian White Noise ◽

Singularity Theory ◽

Van Der Pol Oscillator ◽

Order System ◽

Damping Force ◽

Van Der Pol ◽

Restoring Force ◽

White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

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Bifurcations of a Nonlinear Two-Degree-of-Freedom System Under Narrow-Band Stochastic Excitation

Journal of Vibration and Acoustics ◽

10.1115/1.2930228 ◽

1992 ◽

Vol 114 (1) ◽

pp. 24-31

Author(s):

R. Lin ◽

K. Huseyin ◽

C. W. S. To

Keyword(s):

Relaxation Time ◽

Correlation Time ◽

Averaging Method ◽

Narrow Band ◽

Singularity Theory ◽

Random Parameter ◽

Degree Of Freedom ◽

Stochastic Excitation ◽

Static Approach ◽

Two Degree Of Freedom

In this paper, bifurcations of a nonlinear two-degree-of-freedom system subjected to a narrow-band stochastic excitation are investigated. Under the assumption that the correlation time greatly exceeds the relaxation time, a quasi-static approach combined with averaging method is adopted to obtain the bifurcation equations, and the singularity theory is applied to analyze the bifurcations. It is demonstrated that bifurcation patterns jump from one to another due to the influence of a random parameter. The probabilities of the jumping bifurcation patterns are given.

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The Ermakov equation: A commentary

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm0802146l ◽

2008 ◽

Vol 2 (2) ◽

pp. 146-157 ◽

Cited By ~ 63

Author(s):

P.G.L. Leach ◽

S.K. Andriopoulos

Keyword(s):

Nonlinear Systems ◽

Differential Equations ◽

Ordinary Differential Equations ◽

English Translation ◽

Singularity Theory ◽

Discrete Math ◽

Previous Article ◽

Short History ◽

History Of ◽

Modern Context

We present a short history of the Ermakov equation with an emphasis on its discovery by thewest and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the east. We present the modern context of the Ermakov equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete math., 2 (2008), 123-145) for an english translation of Ermakov's original paper.

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