Testing chaotic dynamics via Lyapunov exponents

Abstract

SUMMARYWe propose a new test to detect chaotic dynamics, based on the stability of the largest Lyapunov exponent from different sample sizes. This test is applied to the data used in the single-blind controlled competition tests for non-linearity and chaos that were generated by Barnett et al. (1997), as well as to several other chaotic series. The results suggest that the new test is particularly effective when compared to other stochastic alternatives (both linear and non-linear). For large sample sizes the power of the test is one, although for small sample sizes it diminishes occasionally.

A new test for chaos and determinism based on symbolic dynamics

“…These results (a) concur with the conclusions given in Shintani and Linton (2004), and (b) outperform those found with other statistical techniques (see Barnett et al, 1997;Matilla-García et al, 2004;Fernández-Rodríguez et al, 2005). In consequence, validity and utility of our testing procedures seems to be confirmed.…”

“…Therefore they are not testing for chaos. Shintani and Linton (2004) and Fernández-Rodríguez et al (2005) have developed new techniques in order to give a statistical framework to the Lyapunov exponents methods and, hence, to properly construct a test for chaos. Shintani and Linton (2004) and Fernández-Rodríguez et al (2005) correspond, respectively, to the latest improvements of the two main numerical approaches for computing the maximal Lyapunov exponent (namely, the tangent space method and the direct method).…”

“…Shintani and Linton (2004) and Fernández-Rodríguez et al (2005) have developed new techniques in order to give a statistical framework to the Lyapunov exponents methods and, hence, to properly construct a test for chaos. Shintani and Linton (2004) and Fernández-Rodríguez et al (2005) correspond, respectively, to the latest improvements of the two main numerical approaches for computing the maximal Lyapunov exponent (namely, the tangent space method and the direct method). 1 It is well-known, for both approaches, that the output of the Lyapunov-exponent based tests is sensitive to the choice of the parameters embedding dimension and delay (we refer the reader to Casdagli et al, 1991;Schreiber and Kantz, 1995 for detailed discussions).…”

A non-parametric independence test using permutation entropy

“…All these results indicate that the DJIA daily stock prices and the three exchange rate time series do not follow a random walk. As regarding volatility, measured via the proxy variables |R t | d , these results point out that, while rejection of the random walk hypothesis for the three exchange rate returns might be due to potential strong volatility clustering, this is not the case for the DJIA since independence cannot be rejected for absolute power 3 The exchange rate time series under study in this section have been recently analyzed looking for chaotic behavior in Fernández et al (2005); the same happens for the DJIA data set, see Shintani and Linton (2004 rather we take the number of order patterns in the observed series as a measure of its complexity. Although this methodology loses a certain amount of detailed information, some essential features of the dynamics are kept, among others, dependence or independence of the data generating process.…”

Market structure and the stability and volatility of electricity prices

“…If, however, the dynamics are not completely unfolded in reconstructed phase space, these invariants depend on the embedding dimension. Therefore, by increasing the embedding dimension enough, the dynamics are completely unfolded when an invariant stops changing (see Fernández-Rodríguez et al [15] for an example regarding the largest Lyapunov exponent and a statistical test for chaotic dynamics).…”