Lyapunov exponents

  Chaos arises from the exponential growth of infinitesimal perturbations, together with global folding mechanisms to guarantee boundedness of the solutions. This exponential instability is characterized by the spectrum of Lyapunov exponents [67]. If one assumes a local decomposition of the phase space into directions with different stretching or contraction rates, then the spectrum of exponents is the proper average of these local rates over the whole invariant set, and thus consists of as many exponents as there are space directions. The most prominent problem in time series analysis is that the physical phase space is unknown, and that instead the spectrum is computed in some embedding space. Thus the number of exponents depends on the reconstruction, and might be larger than in the physical phase space. Such additional exponents are called spurious, and there are several suggestions to either avoid them [68] or to identify them. Moreover, it is plausible that only as many exponents can be determined from a time series as are entering the Kaplan Yorke formula (see below). To give a simple example: Consider motion of a high-dimensional system on a stable limit cycle. The data cannot contain any information about the stability of this orbit against perturbations, as long as they are exactly on the limit cycle. For transients, the situation can be different, but then data are not distributed according to an invariant measure and the numerical values are thus difficult to interpret. Apart from these difficulties, there is one relevant positive feature: Lyapunov exponents are invariant under smooth transformations and are thus independent of the measurement function or the embedding procedure. They carry a dimension of an inverse time and have to be normalized to the sampling interval.

 
• The maximal exponent
• The Lyapunov spectrum