Ian Melbourne

A new test for chaos in deterministic systems

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.

On the Implementation of the 0-1 Test for Chaos

In this paper we address practical aspects of the implementation of the 0–1 test for chaos in deterministic systems. In addition, we present a new formulation of the test which significantly increases its sensitivity. The test can be viewed as a method for distilling a binary quantity from the power spectrum. The implementation is guided by recent results from the theoretical justification of the test as well as by exploring better statistical methods for determining the binary quantities. We give several examples to illustrate the improvement.

Testing for chaos in deterministic systems with noise

Abstract Recently, we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal Lyapunov exponent. In this paper, we investigate the capability of the test to cope with moderate amounts of noisy data. Comparisons are made between an improved version of our test and both the “tangent space method” and “direct method” for computing the maximal Lyapunov exponent. The evidence of numerical experiments, ranging from the logistic map to an eight-dimensional Lorenz system of differential equations (the Lorenz 96 system), suggests that our method is superior to tangent space methods and that it compares very favourably with direct methods.

Almost Sure Invariance Principle for Nonuniformly Hyperbolic Systems

We prove an almost sure invariance principle that is valid for general classes of nonuniformly expanding and nonuniformly hyperbolic dynamical systems. Discrete time systems and flows are covered by this result. In particular, the result applies to the planar periodic Lorentz flow with finite horizon.Statistical limit laws such as the central limit theorem, the law of the iterated logarithm, and their functional versions, are immediate consequences.

Large deviations for nonuniformly hyperbolic systems

We obtain large deviation estimates for a large class of nonuniformly hyperbolic systems: namely those modelled by Young towers with summable decay of correlations. In the case of exponential decay of correlations, we obtain exponential large deviation estimates given by a rate function. In the case of polynomial decay of correlations, we obtain polynomial large deviation estimates, and exhibit examples where these estimates are essentially optimal. In contrast with many treatments of large deviations, our methods do not rely on thermodynamic formalism. Hence, for Holder observables we are able to obtain exponential estimates in situations where the space of equilibrium measures is not known to be a singleton, as well as polynomial estimates in situations where there is not a unique equilibrium measure.

Heteroclinic cycles involving periodic solutions in mode interactions with O(2) symmetry

In this paper we show that in O(2) symmetric systems, structurally stable, asymptoticallystable, heteroclinic cycles can be found which connect periodic solutions with steady states and periodic solutions with periodic solutions. These cycles are found in the third-order truncated normal forms of specific codimension two steady-state/Hopf and Hopf/Hopf mode interactions. We find these cycles using group-theoretic techniques; in particular, we look for certainpatterns in the lattice of isotropy subgroups. Once the pattern has been identified, the heteroclinic cycle can be constructed by decomposing the vector field on fixed-point subspaces into phase/amplitude equations (it is here that we use the assumption of normal form). The final proof of existence (and stability) relies on explicit calculations showing that certain eigenvalue restrictions can be satisfied.

The Lorenz Attractor is Mixing

We study a class of geometric Lorenz flows, introduced independently by Afraimovič, Bykov & Sil′nikov and by Guckenheimer & Williams, and give a verifiable condition for such flows to be mixing. As a consequence, we show that the classical Lorenz attractor is mixing.

A vector-valued almost sure invariance principle for hyperbolic dynamical systems

We prove an almost sure invariance principle (approximation by d-dimensional Brownian motion) for vector-valued Holder observables of large classes of nonuniformly hyperbolic dynamical systems. These systems include Axiom A dieomorphisms and flows as well as systems modelled by Young towers with moderate tail decay rates. In particular, the position variable of the planar periodic Lorentz gas with finite horizon approximates a 2-dimensional Brownian motion.

Asymptotic stability of heteroclinic cycles in systems with symmetry. II

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R 3 . Field and Swift, and Hofbauer, considered examples in R 4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique. In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R 4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R 4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

On the validity of the 0-1 test for chaos

In this paper, we present a theoretical justification of the 0–1 test for chaos. In particular, we show that with probability one, the test yields 0 for periodic and quasiperiodic dynamics, and 1 for sufficiently chaotic dynamics.