Lyapunov exponents of systems with noise and fluctuating parameters

This paper deals with the problem of determination of Lyapunov exponents in dynamical systems with noise or fluctuating parameters. The method for identifying the character of motion in such systems is proposed. This approach is based on the phenomenon of complete synchronization in double-oscillator systems via diagonal, master-slave coupling between them. The idea of effective Lyapunov exponents is introduced for quantifying the local stability in the presence of noise. Examples of the method application and its comparison with bifurcation diagrams representing the system dynamics are demonstrated. Finally, the properties of the method are discussed.

[1]  Tomasz Kapitaniak,et al.  Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization , 2003 .

[2]  Andrzej Stefański,et al.  Estimation of the largest Lyapunov exponent in systems with impacts , 2000 .

[3]  G. Benettin,et al.  Kolmogorov Entropy and Numerical Experiments , 1976 .

[4]  Qishao Lu,et al.  A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impact-vibrating systems , 2006 .

[5]  M. Shinozuka,et al.  Simulation of Stochastic Processes by Spectral Representation , 1991 .

[6]  Homer Rahnejat,et al.  Multi-body dynamics : monitoring and simulation techniques , 1997 .

[7]  Karl Popp,et al.  Dynamics of oscillators with impact and friction , 1997 .

[8]  P. Müller.  Calculation of Lyapunov exponents for dynamic systems with discontinuities , 1995 .

[9]  V. I. Oseledec.  A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[10]  A. Wolf,et al.  13. Quantifying chaos with Lyapunov exponents , 1986 .

[11]  Iberê L. Caldas,et al.  Calculation of Lyapunov exponents in systems with impacts , 2004 .

[12]  M. Oestreich,et al.  Bifurcation and stability analysis for a non-smooth friction oscillator , 1996 .

[13]  James A. Yorke,et al.  Dynamics: Numerical Explorations , 1994 .

[14]  Tomasz Kapitaniak,et al.  Using Chaos Synchronization to Estimate the Largest Lyapunov Exponent of Nonsmooth Systems , 2000 .

[15]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory , 1980 .

[16]  I. Shimada,et al.  A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems , 1979 .

[17]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application , 1980 .

[18]  Tomasz Kapitaniak,et al.  Evaluation of the largest Lyapunov exponent in dynamical systems with time delay , 2005 .