We describe a new test for determining whether a given deterministic dynamical system is chaotic or nonchaotic. 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 is 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 φ(x(s)) cos(θ(s))ds where φ is an observable on the underlying dynamics x(t) and θ(t) = ct + t 0 φ(x(s))ds. The constant c > 0 is fixed arbitrarily. We define the mean-square-displacement M(t) for p(t) and set K = lim t→∞ log M(t)/ log t. Using recent developments in ergodic theory, we argue that typically K = 0 signifying nonchaotic dynamics or K = 1 signifying chaotic dynamics.

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 to distill 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 to determine the binary quantities. We give several examples to illustrate the improvement.

Abstract. 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 Hölder 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.

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates L n of the transfer operator. This was previously an intractable problem.Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points.In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of n j=1 L j ) for the class of systems under consideration. In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for L n and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.This version includes minor corrections in Sections 10 and 11, and corresponding modifications of certain statements in Section 1. All main results are unaffected. In particular, Sections 2-9 are unchanged from the published version.

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" 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.

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.

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Consider an It\^{o} process $X$ satisfying the stochastic differential equation $dX=a(X)\,dt+b(X)\,dW$ where $a,b$ are smooth and $W$ is a multidimensional Brownian motion. Suppose that $W_n$ has smooth sample paths and that $W_n$ converges weakly to $W$. A central question in stochastic analysis is to understand the limiting behavior of solutions $X_n$ to the ordinary differential equation $dX_n=a(X_n)\,dt+b(X_n)\,dW_n$. The classical Wong--Zakai theorem gives sufficient conditions under which $X_n$ converges weakly to $X$ provided that the stochastic integral $\int b(X)\,dW$ is given the Stratonovich interpretation. The sufficient conditions are automatic in one dimension, but in higher dimensions the correct interpretation of $\int b(X)\,dW$ depends sensitively on how the smooth approximation $W_n$ is chosen. In applications, a natural class of smooth approximations arise by setting $W_n(t)=n^{-1/2}\int_0^{nt}v\circ\phi_s\,ds$ where $\phi_t$ is a flow (generated, e.g., by an ordinary differential equation) and $v$ is a mean zero observable. Under mild conditions on $\phi_t$, we give a definitive answer to the interpretation question for the stochastic integral $\int b(X)\,dW$. Our theory applies to Anosov or Axiom A flows $\phi_t$, as well as to a large class of nonuniformly hyperbolic flows (including the one defined by the well-known Lorenz equations) and our main results do not require any mixing assumptions on $\phi_t$. The methods used in this paper are a combination of rough path theory and smooth ergodic theory.Comment: Published at http://dx.doi.org/10.1214/14-AOP979 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org