A new test for chaos in deterministic systems
By Georg A. Gottwald, Ian Melbourne. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2002
623 citations recorded by the source corpus.
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. Our diagnostic is the real valued function p(t)= t 0 (x(s))cos(θ(s))ds, where φ is an observable on the underlying dynamicsx(t) and θ(t)=ct+ ∫ 0 t (x(s))ds. The constant c > 0 is fixed arbitrarily. We define the mean–square displacement M(t) for p(t) and set K=limt→∞logM(t)/logt. Using recent developments in ergodic theory, we argue that, typically, K=0, signifying non–chaotic dynamics, or K=1, signifying chaotic dynamics.
Subfields: Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/836b276bdacfa85cbe045db492945f8183daf85e
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Citations in the corpus
- On the Implementation of the 0-1 Test for Chaos (2009, methodology): “3 In [8, 11] it was shown that the results are valid for almost all observables φ.”
- The Lyapunov Characteristic Exponents and Their Computation (2008, methodology): “In the category of methods based on the analysis of a time series constructed by the coordinates of the orbit under study, one may list the frequency map analysis of Laskar [83, 86, 84, 85], the “0–1” test [64, 65], the method of the low-frequency spectral analysis [137, 81], the “patterns method” [”
- Testing for Chaos in Deterministic Systems with Noise (2004, background): “C D ] 1 5 O ct 2 00 4 TestingforChaosin Deterministic Systemswith Noise Georg A.Gottwald SchoolofM athsand Stats University ofSydney NSW 2006,Australia Ian M elbourne Departm entofM athsand Stats University ofSurrey Guildford GU2 7XH,UK 12 October,2004”
- Loss of Chaos in Combustion Noise as a Precursor of Impending Combustion Instability (2013, neutral): “A modified mean square displacement Dc may be defined to ensure better convergence properties but with the same asymptotic growth rate as [32]:”