2.4 Short summaries of Papers [ii]

This paper presents a statistical framework based on a blockwise bootstrap procedure that tests for the presence of a positive Lyapunov exponent in an observed stochastic time series. Because a positive Lyapunov exponent is an operational definition of deterministic chaos if the dynamical system generating the time series is dissipative, the proposed framework is a test for deterministic chaos. Specifically, the null hypothesis Ho and the alternative hypothesis H1 are formulated as

where is the unknown parameter, i. e. the largest Lyapunov exponent. As can be seen, the dynamics are non-chaotic under the null hypothesis which is a weakness of this framework. The proposed framework was tested on the Henon (1976) map which is described by the following map:

then the sequence of points obtained by iteration of the mapping either diverges to infinity or tends to a strange attractor. When the initial point is

the sequence of points

tend to a strange attractor. Test samples were constructed by the following "observer" map:

where y is the noise level and is the measurement error. Thus, the "observed" time series, i. e. the test samples, was

where N = 200 and N = 1000.

The noise levels were -y = 0.05 and -y = 0.25, block sizes of 2, 4, 6, 8 and 10 were used in each test and 400 bootstrap values were calculated. The null hypothesis was rejected in all cases for a significance level of 0.025. This is in accordance with the true largest Lyapunov exponent, which in this case is

Prof. Mikael Bask