The maximal Lyapunov exponent for a three-dimensional system driven by white noise

Abstract In this paper, based on a perturbation method, the asymptotic expansions of the invariant measure and the maximal Lyapunov exponent for a three-dimensional system excited by a white noise are evaluated. All possible singular boundaries of the first or the second kind that exist in the one-dimensional phase diffusion process are considered and the results of the maximal Lyapunov exponent are obtained. In addition, the P-bifurcation behaviors are investigated.

[1]  Yu-Kweng Michael Lin,et al.  Probabilistic Structural Dynamics: Advanced Theory and Applications , 1967 .

[2]  James B. Seaborn,et al.  Hypergeometric Functions and Their Applications , 1991 .

[3]  R. Khas'minskii,et al.  Necessary and Sufficient Conditions for the Asymptotic Stability of Linear Stochastic Systems , 1967 .

[4]  Matjaž Perc,et al.  Singing of Neoconocephalus robustus as an example of deterministic chaos in insects , 2007, Journal of Biosciences.

[5]  M. Perc,et al.  Detecting chaos from a time series , 2005 .

[6]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[7]  K. Liew,et al.  The Maximal Lyapunov Exponent for a Three-Dimensional Stochastic System , 2004 .

[8]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[9]  X. Q. Wang,et al.  Parametric Resonance of a Two Degrees-of-Freedom System Induced by Bounded Noise , 2009 .

[10]  M. Perc.  The dynamics of human gait , 2005 .

[11]  Matjaz Perc,et al.  Nonlinearities in mating sounds of American crocodiles , 2009, Biosyst..

[12]  W. Xie.  Moment Lyapunov exponents of a two-dimensional system under both harmonic and white noise parametric excitations , 2006 .

[13]  H. Kantz,et al.  Nonlinear time series analysis , 1997 .

[14]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[15]  Desmond J. Higham,et al.  An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations , 2001, SIAM Rev..

[16]  K. Liew,et al.  Maximal Lyapunov exponent of a co-dimension two bifurcation system excited by a white noise , 2005 .

[17]  W. Xie.  Moment Lyapunov exponents of a two-dimensional system under combined harmonic and real noise excitations , 2007 .

[18]  Matjaž Perc,et al.  DETERMINISTIC CHAOS IN SOUNDS OF ASIAN CICADAS , 2006 .

[19]  M. Perc.  Nonlinear time series analysis of the human electrocardiogram , 2005 .

[20]  L. Arnold.  Random Dynamical Systems , 2003 .