scholarly journals Suppressing Chaos of Duffing-Holmes System Using Random Phase

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Li Longsuo
Keyword(s):  
Lyapunov Exponent ◽  
Time Evolution ◽  
Stochastic Systems ◽  
Random Noise ◽  
Dynamical Behavior ◽  
Random Phase ◽  
Bifurcation And Chaos ◽  
Analysis Phase ◽  
Linear Stochastic Systems ◽  
Map Analysis

The effect of random phase for Duffing-Holmes equation is investigated. We show that as the intensity of random noise properly increases the chaotic dynamical behavior will be suppressed by the criterion of top Lyapunov exponent, which is computed based on the Khasminskii's formulation and the extension of Wedig's algorithm for linear stochastic systems. Then, the obtained results are further verified by the Poincaré map analysis, phase plot, and time evolution on dynamical behavior of the system, such as stability, bifurcation, and chaos. Thus excellent agrement between these results is found.

THE CONTROL OF STOCHASTIC COMPLEX DAMPED NONLINEAR SYSTEMS

2007 ◽  
Vol 18 (08) ◽  
pp. 1263-1275 ◽  
Author(s):  
QUN HE ◽  
YONG XU ◽  
GAMAL M. MAHMOUD ◽  
WEI XU
Keyword(s):  
Lyapunov Exponent ◽  
Nonlinear Systems ◽  
Time Evolution ◽  
Excellent Agreement ◽  
Random Noise ◽  
Dynamical Behavior ◽  
Random Phase ◽  
Phase Plot ◽  
Periodic Attractors ◽  
Map Analysis

The aim of this paper is to continue our investigations by studying complex damped nonlinear systems with random noise. The effect of random phase for these systems is examined. The interested system demonstrates unstable periodic attractors when the intensity of random noise equals zero, and we show that the unstable dynamical behavior will be stabilized as the intensity of random noise properly increases. The phase plot and the time evolution are carried out to confirm the obtained results of Poincaré map analysis and top Lyapunov exponent on the dynamical behavior of stability. Excellent agreement is found between these results.

TAMING CHAOTIC ARRAYS BY PARAMETRIC EXCITATION WITH PROPER RANDOM PHASE

2009 ◽  
Vol 20 (10) ◽  
pp. 1633-1643 ◽  
Author(s):  
KAI LEUNG YUNG ◽  
YOUMING LEI ◽  
YAN XU
Keyword(s):  
Lyapunov Exponent ◽  
Excellent Agreement ◽  
Bifurcation Analysis ◽  
Parametric Excitation ◽  
Stochastic Systems ◽  
Dynamical Behavior ◽  
Random Phase ◽  
Spatiotemporal Evolution ◽  
Linear Stochastic Systems ◽  
Control Stability

A weak harmonic parametric excitation with random phase has been introduced to tame chaotic arrays. It has been shown that when the amplitude of random phase properly increases, two different kinds of chaotic arrays, unsynchronized and synchronized, can be controlled by the criterion of top Lyapunov exponent. The Lyapunov exponent was computed based on Khasminskii's formulation and the extension of Wedig's algorithm for linear stochastic systems. In particular, it was found that with stronger coupling the synchronized chaotic arrays are more controllable than the unsynchronized ones. The bifurcation analysis, the spatiotemporal evolution, and the Poincaré map were carried out to confirm the results of the top Lyapunov exponent on the dynamical behavior of control stability. Excellent agreement was found between these results.

GENERATING CHAOTIC LIMIT CYCLES FOR A COMPLEX DUFFING–VAN DER POL SYSTEM USING A RANDOM PHASE

2005 ◽  
Vol 16 (09) ◽  
pp. 1437-1447 ◽  
Author(s):  
YONG XU ◽  
WEI XU ◽  
GAMAL M. MAHMOUD
Keyword(s):  
Lyapunov Exponent ◽  
Surface Analysis ◽  
Limit Cycles ◽  
Noise Intensity ◽  
Stochastic Systems ◽  
Random Noise ◽  
Random Phase ◽  
Deterministic System ◽  
Van Der Pol ◽  
Local Lyapunov Exponent

Stochastic forces or random noises have been greatly used in studying the control of chaos of random real systems, but little is reported for random complex systems. Chaotic limit cycles of a complex Duffing–Van der Pol system with a random excitation is studied. Generating chaos via adjusting the intensity of random phase is investigated. We consider the positive top Lyapunov exponent as a criterion of chaos for random dynamical systems. It is computed based on the Khasminskii's formulation and the extension of Wedig's algorithm for linear stochastic systems. We demonstrate the stable behavior of deterministic system when noise intensity is zero by means of the top (local) Lyapunov exponent. Poincaré surface analysis and phase plot are used to confirm our results. Later, random noise is used to generate chaos by adjusting the noise intensity to make the top (local) Lyapunov exponent changes from a negative sign to a positive one, and the Poincaré surface analysis is also applied to verify the obtained results and excellent agreement between these results is found.

THE ASYMPTOTIC BEHAVIOR OF SEMI-INVARIANTS FOR LINEAR STOCHASTIC SYSTEMS

2002 ◽  
Vol 02 (02) ◽  
pp. 281-294
Author(s):  
G. N. MILSTEIN
Keyword(s):  
Asymptotic Behavior ◽  
Lyapunov Exponent ◽  
Linear System ◽  
Characteristic Function ◽  
Stochastic Systems ◽  
Random Variable ◽  
Characteristic Functions ◽  
Linear Stochastic Systems ◽  
Moment Lyapunov Exponent ◽  
The Moment

The asymptotic behavior of semi-invariants of the random variable ln |X(t,x)|, where X(t,x) is a solution of a linear system of stochastic differential equations, is connected with the moment Lyapunov exponent g(p). Namely, it is obtained that the nth semi-invariant is asymptotically proportional to the time t with the coefficient of proportionality g(n)(0). The proof is based on the concept of analytic characteristic functions. It is also shown that the asymptotic behavior of the analytic characteristic function of ln |X(t,x)| in a neighborhood of the origin of the complex plane is controlled by the extension g(iz) of g(p).

scholarly journals Brownian Behavior in Coupled Chaotic Oscillators

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2503
Author(s):  
Francisco Javier Martín-Pasquín ◽  
Alexander N. Pisarchik
Keyword(s):  
Quantum Mechanics ◽  
Brownian Motion ◽  
Time Evolution ◽  
Phase Difference ◽  
Stochastic Systems ◽  
Phase Synchronization ◽  
Deterministic Model ◽  
Dynamical Behavior ◽  
Deterministic Models ◽  
Chaotic Oscillators

Since the dynamical behavior of chaotic and stochastic systems is very similar, it is sometimes difficult to determine the nature of the movement. One of the best-studied stochastic processes is Brownian motion, a random walk that accurately describes many phenomena that occur in nature, including quantum mechanics. In this paper, we propose an approach that allows us to analyze chaotic dynamics using the Langevin equation describing dynamics of the phase difference between identical coupled chaotic oscillators. The time evolution of this phase difference can be explained by the biased Brownian motion, which is accepted in quantum mechanics for modeling thermal phenomena. Using a deterministic model based on chaotic Rössler oscillators, we are able to reproduce a similar time evolution for the phase difference. We show how the phenomenon of intermittent phase synchronization can be explained in terms of both stochastic and deterministic models. In addition, the existence of phase multistability in the phase synchronization regime is demonstrated.

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