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Ian Melbourne

Publications and source records attributed to Ian Melbourne.

70 records · Page 4Linked to original sources
arXiv2013

Weak Convergence to Stable Lévy Processes for Nonuniformly Hyperbolic Dynamical Systems

We consider weak invariance principles (functional limit theorems) in the domain of a stable law. A general result is obtained on lifting such limit laws from an induced dynamical system to the original system. An important class of examples covered by our result are Pomeau-Manneville intermittency maps, where convergence for the induced system is in the standard Skorohod J_1 topology. For the full system, convergence in the J_1 topology fails, but we prove convergence in the M_1 topology.

math.DS↗
arXiv2013

A Huygens principle for diffusion and anomalous diffusion in spatially extended systems

We present a universal view on diffusive behaviour in chaotic spatially extended systems for anisotropic and isotropic media. For anisotropic systems, strong chaos leads to diffusive behaviour (Brownian motion with drift) and weak chaos leads to superdiffusive behaviour (Lévy processes with drift). For isotropic systems, the drift term vanishes and strong chaos again leads to Brownian motion. We establish the existence of a nonlinear Huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behaviour in odd space dimensions.

math.DS↗
arXiv2011

First and higher order uniform dual ergodic theorems for dynamical systems with infinite measure

We generalize the proof of Karamata's Theorem by the method of approximation by polynomials to the operator case. As a consequence, we offer a simple proof of \emph{uniform dual ergodicity} for a very large class of dynamical systems with infinite measure, and we obtain bounds on the convergence rate. In many cases of interest, including the Pomeau-Manneville family of intermittency maps, the estimates obtained through real Tauberian remainder theory are very weak. Building on the techniques of complex Tauberian remainder theory, we develop a method that provides \emph{second (and higher) order asymptotics}. In the process, we derive a \emph{higher order Tauberian theorem} for scalar power series, which to our knowledge, has not previously been covered.

math.DS↗
arXiv2009

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.

nlin.CD↗
arXiv2009

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 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.

nlin.CD↗
arXiv2008

Existence and convergence properties of physical measures for certain dynamical systems with holes

We study two classes of dynamical systems with holes: expanding maps of the interval and Collet-Eckmann maps with singularities. In both cases, we prove that there is a natural absolutely continuous conditionally invariant measure $μ$ (a.c.c.i.m.) with the physical property that strictly positive Hölder continuous functions converge to the density of $μ$ under the renormalized dynamics of the system. In addition, we construct an invariant measure $ν$, supported on the Cantor set of points that never escape from the system, that is ergodic and enjoys exponential decay of correlations for Hölder observables. We show that $ν$ satisfies an equilibrium principle which implies that the escape rate formula, familiar to the thermodynamic formalism, holds outside the usual setting. In particular, it holds for Collet-Eckmann maps with holes, which are not uniformly hyperbolic and do not admit a finite Markov partition. We use a general framework of Young towers with holes and first prove results about the \accim and the invariant measure on the tower. Then we show how to transfer results to the original dynamical system. This approach can be expected to generalize to other dynamical systems than the two above classes.

math.DS↗
arXiv2008

Decay of correlations and invariance principles for dispersing billiards with cusps, and related planar billiard flows

Following recent work of Chernov, Markarian, and Zhang, it is known that the billiard map for dispersing billiards with zero angle cusps has slow decay of correlations with rate 1/n. Since the collisions inside a cusp occur in quick succession, it is reasonable to expect a much faster decay rate in continuous time. In this paper we prove that the flow is rapid mixing: correlations decay faster than any polynomial rate. A consequence is that the flow admits strong statistical properties such as the almost sure invariance principle, even though the billiard map does not. The techniques in this paper yield new results for other standard examples in planar billiards, including Bunimovich flowers and stadia.

math.DS↗
arXiv2006

Decay of correlations for slowly mixing flows

We show that polynomial decay of correlations is prevalent for a class of nonuniformly hyperbolic flows. These flows are the continuous time analogue of a class of nonuniformly hyperbolic diffeomorphisms for which Young proved polynomial decay of correlations. Roughly speaking, in situations where the decay rate $O(1/n^β)$ has previously been proved for diffeomorphisms, we establish the decay rate $O(1/t^β)$ for typical flows. Applications include certain classes of semidispersing billiards, as well as dispersing billiards with vanishing curvature. In addition, we obtain results for suspension flows with unbounded roof functions. This includes the planar periodic Lorentz flow with infinite horizon.

math.DS↗
arXiv2006

Dynamics on unbounded domains; co-solutions and inheritance of stability

We consider the dynamics of semiflows of patterns on unbounded domains that are equivariant under a noncompact group action. We exploit the unbounded nature of the domain in a setting where there is a strong `global' norm and a weak `local' norm. Relative equilibria whose group orbits are closed manifolds for a compact group action need not be closed in a noncompact setting; the closure of a group orbit of a solution can contain `co-solutions'. The main result of the paper is to show that co-solutions inherit stability in the sense that co-solutions of a Lyapunov stable pattern are also stable (but in a weaker sense). This means that the existence of a single group orbit of stable relative equilibria may force the existence of quite distinct group orbits of relative equilibria, and these are also stable. This is in contrast to the case for finite dimensional dynamical systems where group orbits of relative equilibria are typically isolated.

nlin.PS↗
arXiv2006

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 diffeomorphisms 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.

math.DS↗
arXiv2005

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.

math.DS↗
arXiv2004

Testing for Chaos in Deterministic Systems with Noise

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.

nlin.CD↗
arXiv2004

The Lorenz attractor is mixing

We study a class of geometric Lorenz flows, introduced independently by Afraimovic, 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.

math.DS↗
arXiv2003

A New Test for Chaos

We describe a new test for determining whether a given deterministic dynamical system is chaotic or nonchaotic. (This is an alternative to the usual approach of computing the largest Lyapunov exponent.) Our method is a 0-1 test for chaos (the output is a 0 signifying nonchaotic or a 1 signifying chaotic) and is independent of the dimension of the dynamical system. Moreover, the underlying equations need not be known. The test works equally well for continuous and discrete time. We give examples for an ordinary differential equation, a partial differential equation and for a map.

nlin.CD↗