Matthew Nicol

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.

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.

Some Curious Phenomena in Coupled Cell Networks

Abstract We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of cells and in lattices (and periodic arrays) of cells that cannot readily be explained by conventional symmetry considerations. We also show that different types of dynamics can coexist robustly in single solutions of systems of coupled identical cells. The examples include a three-cell system exhibiting equilibria, periodic, and quasiperiodic states in different cells; periodic 2n × 2n arrays of cells that generate 2n different patterns of synchrony from one symmetry-generated solution; and systems exhibiting multirhythms (periodic solutions with rationally related periods in different cells). Our theoretical results include the observation that reduced equations on a center manifold of a skew product system inherit a skew product form.

Symmetry of attractors and the Karhunen-Loegve decomposition

Recent fluid dynamics experiments [13, 10, 4] have shown that the symmetry of attractors can manifest itself through the existence of spatially regular patterns in the time average of an appropriate observable such as the intensity of transmitted light in the Faraday experiment. In this chapter we discuss how the symmetry of attractors can be detected numerically in solutions of symmetric PDEs and how symmetry considerations affect the appropriateness of a popular method for computing asymptotic dynamics in PDEs—the Karhunen-Loeve decomposition.

Euclidean extensions of dynamical systems

We consider special Euclidean (SE(n)) group extensions of dynamical systems and obtain results on the unboundedness and growth rates of trajectories for smooth extensions. The results depend on n and the base dynamics considered. For discrete dynamics on the base with a dense set of periodic points, we prove the unboundedness of trajectories for generic extensions provided n = 2 or n is odd. If in addition the base dynamics is Anosov, then generically trajectories are unbounded for all n, exhibit square root growth and converge in distribution to a non-degenerate standard n-dimensional normal distribution. For sufficiently smooth SE(2)-extensions of quasiperiodic flows, we prove that trajectories of the group extension are typically bounded in a probabilistic sense, but there is a dense set of base rotations for which extensions are typically unbounded in a topological sense. The results on unboundedness are generalized to SE(n) (n odd) and to extensions of quasiperiodic maps. We obtain these results by exploiting the fact that SE(n) has the semi-direct product structure Γ = Gn, where G is a compact connected Lie group and n is a normal Abelian subgroup of Γ. This means that our results also apply to extensions by this wider class of groups.

On the unfolding of a blowout bifurcation

Abstract Suppose a chaotic attractor A in an invariant subspace loses stability on varying a parameter. At the point of loss of stability, the most positive Lyapunov exponent of the natural measure on A crosses zero at what has been called a ‘blowout’ bifurcation. We introduce the notion of an essential basin of an attractor A . This is the set of points x such that accumulation points of the sequence of measures 1 n ∑n − 1 k = 0 δ f k (x) are supported on A . We characterise supercritical and subcritical scenarios according to whether the Lebesgue measure of the essential basin of A is positive or zero. We study a drift-diffusion model and a model class of piecewise linear mappings of the plane. In the supercritical case, we find examples where a Lyapunov exponent of the branch of attractors may be positive (‘hyperchaos’) or negative, depending purely on the dynamics far from the invariant subspace. For the mappings we find asymptotically linear scaling of Lyapunov exponents, average distance from the subspace and basin size on varying a parameter. We conjecture that these are general characteristics of blowout bifurcations.

Extreme value theory for non-uniformly expanding dynamical systems

We establish extreme value statistics for functions with multiple maxima and some degree of regularity on certain non-uniformly expanding dynamical systems. We also establish extreme value statistics for time-series of observations on discrete and continuous suspensions of certain non-uniformly expanding dynamical systems via a general lifting theorem. The main result is that a broad class of observations on these systems exhibit the same extreme value statistics as i.i.d processes with the same distribution function.

Extreme value theory and return time statistics for dispersing billiard maps and flows, Lozi maps and Lorenz-like maps

In this paper we establish extreme value statistics for observations on a class of hyperbolic systems: planar dispersing billiard maps and flows, Lozi maps and Lorenz-like maps. In particular we show that for time series arising from Hölder observations on these systems the successive maxima of the time series are distributed according to the corresponding extreme value distributions for independent identically distributed processes. These results imply an exponential law for the hitting and return time statistics of these dynamical systems.

On the Fine Structure of Stationary Measures in Systems which Contract-on-Average

Suppose {f1,...,fm} is a set of Lipschitz maps of ℝd. We form the iterated function system (IFS) by independently choosing the maps so that the map fi is chosen with probability pi (∑mi=1pi=1). We assume that the IFS contracts on average. We give an upper bound for the upper Hausdorff dimension of the invariant measure induced on ℝd and as a corollary show that the measure will be singular if the modulus of the entropy ∑ipi log pi is less than d times the modulus of the Lyapunov exponent of the system. Using a version of Shannons Theorem for random walks on semigroups we improve this estimate and show that it is actually attainable for certain cases of affine mappings of ℝ.