Article

Determining Lyapunov exponents from a time series

Department of Physics, University of Texas, Austin, Texas 78712, USA
Physica D Nonlinear Phenomena (Impact Factor: 1.67). 01/1985; DOI: 10.1016/0167-2789(85)90011-9

ABSTRACT 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.

1 Bookmark
 · 
330 Views
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: The phenomenon of chimera states in the systems of coupled, identical oscillators has attracted a great deal of recent theoretical and experimental interest. In such a state, different groups of oscillators can exhibit coexisting synchronous and incoherent behaviors despite homogeneous coupling. Here, considering the coupled pendula, we find another pattern, the so-called imperfect chimera state, which is characterized by a certain number of oscillators which escape from the synchronized chimera's cluster or behave differently than most of uncorrelated pendula. The escaped elements oscillate with different average frequencies (Poincare rotation number). We show that imperfect chimera can be realized in simple experiments with mechanical oscillators, namely Huygens clock. The mathematical model of our experiment shows that the observed chimera states are controlled by elementary dynamical equations derived from Newton's laws that are ubiquitous in many physical and engineering systems.
    Scientific Reports 09/2014; · 5.08 Impact Factor
  • Source
    Journal of Superconductivity and Novel Magnetism 09/2014; · 0.70 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In the present work, we aim to investigate the chaotic characteristics of measured temperature signals in the friction process, and further reveal the dynamic behavior of the friction system. Experiments are conducted on a reciprocating tribometer under different working conditions, and the contact temperature is measured by a thermocouple throughout the friction process. The phase trajectories and chaotic parameters are obtained based on the phase–space reconstruction of the temperature time series. The results show that the temperature signals acquired from different tests possess the same dynamic evolution law. As the time goes on, the phase trajectories follow the dynamic rule of “convergence–stability–divergence”. This evolution process corresponds to the stages of the “forming, keeping and disappearing” of the chaotic attractor. In the attractor forming stage, the correlation dimension increases gradually and the Lyapunov exponent varies from negative to positive. Then, both the correlation dimension and the Lyapunov exponent remain at a steady level in the attractor keeping stage. Finally, the correlation dimension decreases and the Lyapunov exponent varies to a negative value in the chaotic attractor disappearing stage. It makes it possible to identify friction states by analyzing the temperature time series.
    Wear 09/2014; 317(1-2):17-25. · 1.26 Impact Factor

Full-text (3 Sources)

View
170 Downloads
Available from
May 16, 2014