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Abstract
The literature about non‐linear dynamics offers a few recommendations, which sometimes are divergent, about the criteria to be used in order to select the optimal calculus parameters in the estimation of Lyapunov exponents by direct methods. These few recommendations are circumscribed to the analysis of chaotic systems. We have found no recommendation for the estimation of λ starting from the time series of classic systems. The reason for this is the interest in distinguishing variability due to a chaotic behavior of determinist dynamic systems of variability caused by white noise or linear stochastic processes, and less in the identification of non‐linear terms from the analysis of time series. In this study we have centered in the dependence of the Lyapunov exponent, obtained by means of direct estimation, of the initial distance and the time evolution. We have used generated series of chaotic systems and generated series of classic systems with varying complexity. To generate the series we have used the logistic map.
Highlights
Characterizing dynamical systems using the analysis of uni-dimensional time series has been a method widely developed since the 1980s, there are several questions that need some consideration, at least in the cases when the indicators are obtained from time series resulting from behavioral investigation
If the application of the discrete dynamical system for two close trajectories eventually leads us to separate points, the absolute value of the derivative off is greater than 1 when we evaluated at those trajectory points
The reason for this is the interest in distinguishing variability due to a chaotic behavior of determinist dynamic systems of variability caused by white noise or linear stochastic processes, and less in the identification of nonlinear terms from the analysis of time series
Summary
The discovery of chaotic behavior in deterministic dynamical systems has changed some philosophical aspects in the prevailing scientific paradigm and has opened new perspectives for the design and analysis of time series (Barnett and Choi, 1989; Casdagli, 1991; Casdagli et al, 1991; Sayers, 1991; Berliner, 1992; McCaffrey et al, 1992; Nychka et al, 1992; Gerr and Allen, 1993; Takens, 1993).In the 1980s, the breakthroughs in the analysis of time series based on the Qualitative Theory of Dynamical Systems have yielded a set of indexes. Characterizing dynamical systems using the analysis of uni-dimensional time series has been a method widely developed since the 1980s, there are several questions that need some consideration, at least in the cases when the indicators are obtained from time series resulting from behavioral investigation. In this type of investigation, like in most situations in real life, the data combine deterministic dynamics with noise of different nature and magnitude; in addition to this, in Psychology it is difficult to maintain the same observation situation for a long time and this leads to a reduction in the length of the series, the reliability of such indexes can be questionable. We have tried to provide answers to the questions arising from the calculation of dominant
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