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A new test for chaos in deterministic systems

By Georg A. Gottwald and Ian Melbourne


<p>We describe a new test for determining whether a given deterministic dynamical system is chaotic or non-chaotic. In contrast to the usual method of computing the maximal Lyapunov exponent, our method is applied directly to the time-series data and does not require phase-space reconstruction. Moreover, the dimension of the dynamical system and the form of the underlying equations are irrelevant. The input is the time-series data and the output is 0 or 1, depending on whether the dynamics is non-chaotic or chaotic. The test is universally applicable to any deterministic dynamical system, in particular to ordinary and partial differential equations, and to maps. </p> <p> Our diagnostic is the real valued function <i>p(t)=∫φ(<b>x</b>s)cos(θ(s)) ds, </i> where φ is an observable on the underlying dynamics <b>x</b>(t) and <i>θ(t)&=ct+∫φ(<b>x</b>s)cos(θ(s)) ds.</i> The constant c > 0 is fixed arbitrarily. We define the mean-square displacement <i>M(t)</i> for <i>p(t)</i> and set K=lim<sub>t→∞</sub>\log<b><i>M</b>(t)</i>\log<i>t</i>. Using recent developments in ergodic theory, we argue that, typically, <i>K=0</i>, signifying non-chaotic dynamics, or <i>$K=1$</i>, signifying chaotic dynamics.</p

Year: 2004
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