Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization

A novel method of estimation of the largest Lyapunov exponent for discrete maps is introduced and evaluated for chosen examples of maps described by difference equations or generated from non-smooth dynamical systems. The method exploits the phenomenon of full synchronization of two identical discrete maps when one of them is disturbed. The presented results show that this method can be successfully applied both for discrete dynamical systems described by known difference equations and for discrete maps reconstructed from actual time series. Applications of the method for mechanical systems with discontinuities and examples of classical maps are presented. The comparison between the results obtained by means of the known algorithms and novel method is discussed. 2002 Elsevier Science Ltd. All rights reserved.

[1]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

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

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

[4]  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 .

[5]  Calculus of iterations and labor-capital core-periphery relative redistribution dynamics , 1994 .

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

[7]  T. Kapitaniak,et al.  MONOTONE SYNCHRONIZATION OF CHAOS , 1996 .

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

[9]  H. Fujisaka,et al.  Stability Theory of Synchronized Motion in Coupled-Oscillator Systems , 1983 .

[10]  Dmitry E. Postnov,et al.  SYNCHRONIZATION OF CHAOS , 1992 .

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

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

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

[14]  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 .