Noise Induced Jumping Dynamics Between Synchronized Modes

2015 ◽  
Vol 25 (12) ◽  
pp. 1530034 ◽  
Author(s):  
Shannon D. Algar ◽  
Thomas Stemler ◽  
Bernard De Saedeleer
Keyword(s):  
Dynamical System ◽  
Lyapunov Exponents ◽  
Van Der Pol Oscillator ◽  
External Forcing ◽  
Common Phenomenon ◽  
Periodic Forcing ◽  
Van Der Pol ◽  
Forced Oscillator

Synchronization is a common phenomenon whereby a dynamical system follows the pacemaker provided by an external forcing. Often, such systems have multiple synchronization modes, which are equivalent solutions. We investigate the specific case of two to one synchronization produced by the periodic forcing of a van der Pol oscillator where two possible modes, shifted by one period of the modulation, exist. By studying the flow and the local Lyapunov exponents along the orbit we give an explanation of the noise induced jumps observed in a stochastic forced oscillator. While this investigation gives results that are specific to this system, the framework presented is more general and can be applied to any system showing similar jumping dynamics.

TOWARD AN UNDERSTANDING OF STOCHASTIC HOPF BIFURCATION

1996 ◽  
Vol 06 (11) ◽  
pp. 1947-1975 ◽  
Author(s):  
LUDWIG ARNOLD ◽  
N. SRI NAMACHCHIVAYA ◽  
KLAUS R. SCHENK-HOPPÉ
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Lyapunov Exponents ◽  
Variational Equation ◽  
Van Der Pol Oscillator ◽  
Stochastic Bifurcation ◽  
Van Der Pol ◽  
Truncated Normal ◽  
Stochastic Hopf Bifurcation ◽  
Bifurcation Behavior

In this paper, asymptotic and numerical methods are used to study the phenomenon of stochastic Hopf bifurcation. The analysis is carried out through the study of a noisy Duffing-van der Pol oscillator which exhibits a Hopf bifurcation in the absence of noise as one of the parameters is varied. In the first part of this paper, we present an introduction to the theory of random dynamical systems (in particular, their generation, their invariant measures, the multiplicative ergodic theorem, and Lyapunov exponents). We then present the two concepts of stochastic bifurcation theory: Phenomenological (based on the Fokker-Planck equation), and dynamical (based on Lyapunov exponents). The method of stochastic averaging of the nonlinear system yields a set of equations which, together with its variational equation, can be explicitly solved and hence its bifurcation behavior completely analyzed. We augment this analysis by asymptotic expansions of the Lyapunov exponents of the variational equation at zero. Finally, the stochastic normal form of the noisy Duffing-van der Pol oscillator is derived, and its bifurcation behavior is analyzed numerically. The result is that the (truncated) normal form retains the essential bifurcation characteristics of the full equation.

The Lyapunov exponents of the Van der Pol oscillator

2005 ◽  
Vol 28 (10) ◽  
pp. 1131-1139 ◽  
Author(s):  
Johan Grasman ◽  
Ferdinand Verhulst ◽  
Shagi-Di Shih
Keyword(s):  
Lyapunov Exponents ◽  
Van Der Pol Oscillator ◽  
Van Der Pol

EFFECT OF GAUSSIAN WHITE NOISE ON THE DYNAMICAL BEHAVIORS OF AN EXTENDED DUFFING–VAN DER POL OSCILLATOR

2006 ◽  
Vol 16 (09) ◽  
pp. 2587-2600 ◽  
Author(s):  
XIAOLI YANG ◽  
WEI XU ◽  
ZHONGKUI SUN
Keyword(s):  
White Noise ◽  
Lyapunov Exponents ◽  
Random Noise ◽  
Gaussian White Noise ◽  
Harmonic Excitation ◽  
Basins Of Attraction ◽  
Van Der Pol Oscillator ◽  
Necessary Condition ◽  
Van Der Pol ◽  
Dynamical Behaviors

The influence induced by random noise on dynamical behaviors is a classical yet challenging subject. This paper discusses the influence of Gaussian white noise on the dynamics of a self-excited triple well extended Duffing–Van der Pol oscillator already subjected to harmonic excitation. Firstly, the condition for the rise of hom/heteroclinic chaos is derived by random Melnikov's technique under its corresponding mean-square criterion and the result indicates that the threshold amplitude of harmonic excitation is lowered by the appearance of Gaussian white noise. Moreover, the threshold is decreased as the noise intensity increases. Since the Melnikov's criterion is only a necessary condition for the occurrence of chaotic motion, this prediction is tested against numerical simulations of the basins of attraction and the Lyapunov exponents. By vanishing the largest Lyapunov exponents, another criterion for the onset of chaos is obtained which is accorded with the theoretical one. Finally, how the noise effects the structure of periodic or chaotic attractor is investigated by simulating Poincare maps of the original system and rich transition states displayed by the considered extended Duffing–Van der Pol oscillator are observed.

NUMERICAL SOLUTION FOR DUFFING-VAN DER POL OSCILLATOR VIA BLOCK METHOD

2020 ◽  
Vol 10 (1) ◽  
pp. 1857-8365
Author(s):  
A. F. Nurullah ◽  
M. Hassan ◽  
T. J. Wong ◽  
L. F. Koo
Keyword(s):  
Numerical Solution ◽  
Van Der Pol Oscillator ◽  
Block Method ◽  
Van Der Pol

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

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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