On stationary probability density and maximal Lyapunov exponent of a co-dimension two bifurcation system subjected to parametric excitation by real noise

Abstract In this paper, the asymptotic expansions of the maximal Lyapunov exponents for a co-dimension two-bifurcation system which is on a three-dimensional center manifold and is excited parametrically by an ergodic real noise are evaluated. The real noise is an integrable function of an n-dimensional Ornstein–Uhlenbeck process. Based on a perturbation method, we examine almost all possible singular boundaries that exist in one-dimensional phase diffusion process. The comparisons between the analytical solutions and the numerical simulations are given. In addition, we also investigate the P-bifurcation behavior for the one-dimensional phase diffusion process. The result in this paper is a further extension of the work by Liew and Liu [1] .

[1]  N. Sri Namachchivaya,et al.  Maximal Lyapunov exponent and rotation numbers for two coupled oscillators driven by real noise , 1993 .

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

[3]  Volker Wihstutz,et al.  Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator and application , 1986 .

[4]  S. Ariaratnam,et al.  Lyapunov Exponents and Stochastic Stability of Coupled Linear Systems Under Real Noise Excitation , 1992 .

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

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

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

[8]  K. Liew,et al.  The Lyapunov exponent for a codimension two bifurcation system that is driven by a real noise , 2003 .

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

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

[11]  K. Nishioka.  On the stability of two-dimensional linear stochastic systems , 1976 .

[12]  M. Rosenstein,et al.  A practical method for calculating largest Lyapunov exponents from small data sets , 1993 .

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

[14]  Frank Kozin,et al.  Sample Stability of Second Order Linear Differential Equations with Wide Band Noise Coefficients , 1974 .

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

[16]  K. Liew,et al.  On the almost-sure stability condition for a co-dimension two-bifurcation system under the parametric excitation of a real noise , 2004 .

[17]  John L. Casti,et al.  Advanced theory and applications , 1982 .

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

[19]  R. Roy.  Stochastic averaging of oscillators excited by colored Gaussian processes , 1994 .

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

[21]  V. Wihstutz.  Perturbation Methods for Lyapunov Exponents , 1999 .

[22]  Daniel Liberzon,et al.  Spectral Analysis of Fokker--Planck and Related Operators Arising From Linear Stochastic Differential Equations , 2000, SIAM J. Control. Optim..

[23]  T. Caughey.  Nonlinear Theory of Random Vibrations , 1971 .

[24]  N. Sri Namachchivaya,et al.  Almost-sure asymptotic stability of a general four-dimensional system driven by real noise , 1994 .