Ian Melbourne
    Current research interests: smooth ergodic theory and dynamical systems; rates of mixing and statistical properties; nonuniform, singular, and partial hyperbolicity; applications to Lorentz gases and Lorenz-like attractors; infinite ergodic theory; compact and noncompact group extensions; testing for chaos; homogenization for deterministic systems (convergence of fast-slow systems to SDE limits); Huygen's principle for diffusion/superdiffusion in weakly chaotic spatially extended systems

    In the last few years, I have worked on problems in ergodic theory, looking at statistical and probabilistic aspects of dynamical systems. This includes systems with no symmetry, compact symmetry and noncompact symmetry. For example, Mike Field, Andrew Török and I show that hyperbolic flows are stably mixing with superpolynomial decay of correlations for open and dense sets of Cr flows, r at least 2.

    Using operator renewal theory, I have extended Dolgopyat's ideas on rapid mixing to a class of nonuniformly hyperbolic flows (including suspended Hénon attractors). An additional truncation step yields results for slowly mixing flows including optimal results for the infinite horizon planar periodic Lorentz gas with circular obstacles. Subsequent results with Péter Bálint deal with the general infinite horizon planar periodic Lorentz gas. (Still being written up.) Bálint and I also proved superpolynomial decay of correlations for the flow in the case of billiards with cusps.

    Dalia Terhesiu and I have extended the theory of operator renewal sequences to the infinite ergodic theory setting, and have obtained results on convergence of the transfer operator, and second order asymptotics, for a large class of systems with infinite measure, including intermittency maps and parabolic rational maps of the complex plane.

    Also with Terhesiu, as a byproduct of an approach that failed in the infinite case (so far), we have developed a simplified treatment of decay of correlations and statistical limit laws for systems modelled by Young towers in the finite measure case. We recover existing results for subexponential tails and obtain improved estimates for irregularly varying tails.

    Matthew Nicol and I have proved the almost sure invariance principle for nonuniformly hyperbolic flows for both scalar and vector valued observables. For example, particles in the finite horizon planar periodic Lorentz gas undergo approximately two-dimensional Brownian motion. Mark Holland and I prove analogous results for the Lorenz attractor.

    Roland Zweimüller and I have proved results on the weak invariance principle in the presence of a stable law, applying to systems such as Pomeau-Manneville intermittency maps. Convergence fails in the standard J1 Skorokhod topology but holds in the M1 Skorokhod topology.

    Georg Gottwald and I have developed a new 0-1 test for chaos. This test appears to have many advantages over the standard test of computing the maximal Lyapunov exponent. Click here for a comparison of the 0-1 test and the maximal Lyapunov exponent for the forced van der Pol oscillator. Here is an improved version of the 0-1 test that works well with moderately noisy data (and which seems to compare favourably with the maximal Lyapunov exponent). Further significant improvements have led to the (final?) implementation. The mathematical justification for why the test works is provided by the earlier work on ergodic theory mentioned above.

    Previous research interests: bifurcation theory and symmetry; equivariant dynamical systems; Ginzburg-Landau theory

    Previously, my research focused on bifurcation theory/dynamical systems with noncompact symmetry (usually Euclidean symmetry), with particular emphasis on the rigorous foundations of Ginzburg-Landau theory.
    Click here for a survey article on rigorous results concerning structure and universality of reduced equations in spatially-extended systems.

    As far as compact symmetry groups are concerned, I have worked on symmetries of chaotic attractors where the symmetry appears only on average, on robust heteroclinic cycles which provide an elementary mechanism for producing intermittent phenomena, and on bifurcation from relative periodic solutions and discrete rotating waves (periodic solutions with spatiotemporal symmetry).

    Click here and here for work of David Chan (Ph. D. student) on bifurcations from rotating waves in systems with spherical symmetry. Some important results for systems with E(3) symmetry remain to be written up.

    Preprints

    Publications since 2001

    Older publications

    Meetings 2012 onwards