Lyapunov Exponent and Rotation Number of Two-Dimensional Linear Stochastic Systems with Small Diffusion

1988 ◽  
Vol 48 (2) ◽  
pp. 442-457 ◽  
Author(s):  
E. Pardoux ◽  
V. Wihstutz
Keyword(s):  
Lyapunov Exponent ◽  
Rotation Number ◽  
Stochastic Systems ◽  
Two Dimensional ◽  
Small Diffusion ◽  
Linear Stochastic Systems

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).

Rotation number and Lyapunov exponent in two-dimensional maps

1989 ◽  
Vol 34 (3) ◽  
pp. 366-377 ◽  
Author(s):  
A. Lambert ◽  
R. Lima ◽  
R.Vilela Mendes
Keyword(s):  
Lyapunov Exponent ◽  
Rotation Number ◽  
Two Dimensional

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.

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.

Lyapunov Exponents and Stochastic Stability of Two-Dimensional Parametrically Excited Random Systems

1993 ◽  
Vol 60 (3) ◽  
pp. 677-682 ◽  
Author(s):  
S. T. Ariaratnam ◽  
Wei-Chau Xie
Keyword(s):  
Lyapunov Exponent ◽  
Stochastic Stability ◽  
Stochastic Systems ◽  
Random Systems ◽  
Random Fluctuation ◽  
Largest Lyapunov Exponent ◽  
Random Disturbance ◽  
Two Dimensional ◽  
Parametrically Excited ◽  
The Largest Lyapunov Exponent

The variation of the largest Lyapunov exponent for two-dimensional parametrically excited stochastic systems is studied by a method of linear transformation. The sensitivity to random disturbance of systems undergoing bifurcation is investigated. Two commonly occurring examples in structural dynamics are considered, where the random fluctuation appears in the stiffness term or the damping term. The boundaries of almost-sure stochastic stability are determined by the vanishing of the largest Lyapunov exponent of the linearized system. The validity of the approximate results is checked by numerical simulation.

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