《3-lyapunov指数计算方法_wolf算法_》.pdf

Physica 16D (1985)285-317 North-Holland, Amsterdam DETERMINING LYAPUNOV EXPONENTS FROM A TIME SERIES Alan WOLF~-, Jack B. SWIFT, Harry L. SWINNEY and John A. VASTANO Department of Physics, Universityof Texas,Austin, Texas 78712, USA Received 18 October 1984 We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow. Contents convergence of nearby orbits in phase space. Since 1. Introduction nearby orbits correspond to nearly identical states, 2. The Lyapunov spectrum defined 3. Calculation of Lyapunov spectra from differential equations exponential orbital divergence means that systems 4. An approach to spectral estimation for experimental data whose initial differences we may not be able to 5. Spectral algorithm implementation* resolve will soon behave quite differently-predic- 6. Implementation details*