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Organizers |
The dynamics of finite data: hyperbolicity and distortion
by
Arno Berger
University of Alberta
Numerical data generated by real-world dynamical systems such as oceanographic models are typically finite. It is often not clear what qualitative and quantitative dynamical information can be deduced from such data in a rigorous manner. This talk will discuss two concepts that address this question: finite-time hyperbolicity and (scale) distortion of data sets.
Date received: November 3, 2008
Second largest eigenvalues and almost invariant structures for hyperbolic systems
by
Chris Bose
Department of Mathematics and Statistics, University of Victoria, Canada
Coauthors: Gary Froyland, University of New South Wales
We discuss the problem of constructing eigenfunctions with eigenvalues inside the unit disc for measurable, nonsingular and invertible transformations. For expanding one-dimensional maps the problem is well-understood in terms of spectral properties of the associated transfer operator restricted to a suitable subspace of L1 consisting of regular functions. For invertible maps we will show that one must instead EXPAND the domain of the transfer operator to obtain similar results. A construction using the so-called generalized baker's map shows this to be a feasible program and reveals some of the structure that will hold for such 'almost invariant functions' in the hyperbolic setting.
Date received: October 30, 2008
Ergodicity for partially hyperbolic diffeomorphisms
by
Keith Burns
Northwestern University
I will survey recent results which extend Hopf's method for proving ergodicity to a large class of partially hyperbolic diffeomorphisms.
Date received: November 3, 2008
Critical dimension of non-singular actions
by
Anthony Dooley
University of New South Wales
Let (X, B, m, T) be a non-singular ergodic dynamical system. We define the lower critical dimension of T to be the largest value of a ∈ [0, 1] such that [1/(na)] liminfSk=1n [(dm○T)/(dm)] > 0 a.e. and the upper critical dimension b in a similar way. These are invariants of metric equivalence. There is a natural weaker notion of equivalence, which we call Hurewicz equivalence which also preserves a and b. We shall discuss the classification of dynamical systems using these notions.
Date received: November 3, 2008
Coherent sets in flows and Perron-Frobenius cocycles
by
Gary Froyland
University of New South Wales
Coauthors: Simon Lloyd, Naratip Santitissadeekorn (UNSW), Anthony Quas (UVic)
We present an analysis of one-dimensional models of dynamical systems that possess ``coherent sets''; global structures that disperse more slowly than local trajectory separation. We introduce the notion of a Perron-Frobenius cocycle, give a characterisation of the Lyapunov spectrum of this cocycle, and present a strengthened version of the Multiplicative Ergodic Theorem. We describe applications to flows, including those arising from ocean models.
Date received: November 3, 2008
A cone exchange map
by
Arek Goetz
San Francisco State University
Coauthors: Peter Ashwin and Anthony Quas
A basic system to rotations on cones presents surprising challenges in determining the long term behavior of the orbits. We will report of our recent joint work with Ashwin and Quas. In particular we explain that a piecewise rotation with two half planes that is invertible has bounded rings and recurrent points. This is all despite the existence of an obvious finite invariant measure.
Date received: November 3, 2008
The geometric dimension of an equivalence relation and finite extensions of finite groups
by
Valentyn Golodets
University of New South Wales
Coauthors: A.H. Dooley
We say that the geometric dimension of a countable group G is equal to an integer n if any free Borel action of G on a standard Borel probability space (X, m) induces an equivalence relation EXG of geometrical dimension n in the sense of Gaboriau, or briefly geom-dim(EXG)=n.
Let G be as above and geom-dim(G)=n and let K be a finite extension of G. Does geom-dim(K)=n?
We prove that for any integer n, n ≥ 1, there exists a big enough class of group An such that if G belong to An then geom-dim(G)=n, and any finite extension K of G belongs to An too.
The important case n = 1 is considered more explicitly. We prove that A1 contains a big enough set of free products amenable groups. In particular, all free groups and all free finite products of finite groups belongs to A1.
We use some results and constructions from combinatorial group theory belonging to Karrass, Hanna Neumann, John Stallings and others, in a combination with methods of orbital equivalence theory.
Date received: November 3, 2008
Extreme behaviour in chaotic dynamical systems with suspension flow applications.
by
Mark Holland
University of Exeter, UK
Coauthors: M. Nicol, A. Torok.
We study extreme statistics for chaotic dynamical systems. That is, we investigate the long time range distribution of the maxima of observations on a typical solution trajectory. As applications we consider suspension flows over chaotic maps and mention preliminary results on extremal properties of the Lorenz equations.
Date received: October 30, 2008
Fast approximation of long term dynamical behaviour for continuous-time systems
by
Péter Koltai
Technische Universität München, Germany
Coauthors: Gary Froyland and Oliver Junge
Transfer operator methods are widely used in applications to determine long term dynamical behaviour. They are based on simulations of the dynamical system. However, for systems arising from ODEs, simulation is computationally very expensive. Instead of the associated transfer operator we propose to analyze the infinitesimal generator of the system. We develope theory and show numerical examples in the talk.
Date received: October 31, 2008
Large deviations and moderate deviations for slowly mixing dynamical systems
by
Ian Melbourne
University of Surrey, UK
We obtain results on large deviations for a large class of nonuniformly hyperbolic dynamical systems with polynomial decay of correlations 1/nb, b > 0. This includes systems modelled by Young towers with polynomial tails, extending recent work of M. Nicol and the author which assumed b > 1. As a byproduct of the proof, we obtain slightly stronger results even when b > 1. The results are sharp in the sense that there exist examples (such as Pomeau-Manneville intermittency maps) for which the obtained rates are best possible. In addition, we obtain results on moderate deviations.
Date received: October 23, 2008
Extreme value theory for dynamical systems
by
Matthew Nicol
University of Houston
Coauthors: Mark Holland (University of Exeter) and Andrew Torok (University of Houston)
We present results on extreme value theory (which concerns the distribution of successive maxima of a time-series of observations) for dynamical systems. The main result is that a broad class of observations on certain non-uniformly hyperbolic systems exhibit the same extreme value statistics as iid processes with the same distribution function.
Date received: October 30, 2008
Feedback control of unstable periodic orbits
by
Claire Postlethwaite
University of Auckland
It is often the case that the desired output from a system (be it experimental or numerical) is a periodic orbit or pattern which is unstable. In 1992 Pyragas introduced a method of time-delayed feedback control which can be used to stabilise such unstable solutions. This method has attracted much attention, as it has the advantages of being both non-invasive, and also requiring only a knowledge of the period of the orbit a priori.
I will present results on two examples. The first example involves stabilising unstable periodic orbits resulting from a subcritical Hopf bifurcation in the Lorenz equations. The second involves the stabilisation of periodic orbits with arbitrarily large period which have originated from a heteroclinic bifurcation.
Date received: October 19, 2008
Motion Estimation of Image Sequence Data in the Framework of the Frobenius-Perron Operator
by
Naratip Santitissadeekorn
University of New South Wales
In many areas of science, main sources of data are given in a form of image sequences or movies, which capture the underlying dynamics. To understand the transport behavior of these dynamical systems, it is mandatory to extract the velocity fields from image sequences.
Traditionally, the Particle Image Velocimetry (PIV) is used in fluid imagery to locally approximate velocity filed based on a cross-correlation technique. Suppose that an image seguence depicts a flow in such a way that the essential characteristics of the flow are captured. In particular, an image sequence may be viewed as the evolution of image pixel intensities under the Frobenius-Perron operator.
Then, a variational model based on the principle the infinitesimal generator of the Frobenius-Perron operator can be developed to globally estimate the velocity field.
Date received: October 30, 2008
Folded saddle-nodes: where canards meet hopf
by
Martin Wechselberger
School of Mathematics & Statistics, University of Sydney
Coauthors: Martin Krupa (Radboud Universiteit, Nijmegen, The Netherlands)
Folded saddle-nodes occur generically in one parameter families of singularly perturbed systems with two slow variables. We show that these folded singularities are the organising centres for two main delay phenomena in singular perturbation problems: canards and delayed Hopf bifurcations. We combine techniques from geometric singular perturbation theory -- the blow-up technique -- and from delayed Hopf bifurcation theory -- complex time paths analysis -- to analyse the flow near such folded saddle-nodes. In particular, we show the existence of canards as intersections of stable and unstable slow manifolds. Furthermore, we define a way in/way out function describing the maximal delay expected for generic solutions passing through a folded saddle-node singularity reminiscent of classical delayed Hopf bifurcation theory.
Date received: October 20, 2008