INVARIANT MEASURES AND INVARIANT MANIFOLDS FOR RANDOM DYNAMICAL SYSTEMS

Ludwig Arnold, Bremen

We first explain the concept of a random dynamical system (RDS) as a symbiosis of ergodic theory and smooth dynamics. We introduce invariant measures for RDS and discuss by way of examples how to find them. We then recall the celebrated multiplicative ergodic theorem of Oseledets which replaces linear algebra for RDS and makes local theory of nonlinear RDS possible. Random invariant manifolds which are at the heart of this local theory are constructed and dynamically characterized.

SYMBOLIC DYNAMICS AND PERIODIC ORBITS FOR THE CARDIOID BILLIARD

Arnd Bäcker, Ulm

By means of a binary symbolic dynamics the periodic orbits of the strongly chaotic cardioid billiard are studied. The corresponding partition is mapped to a topological well-ordered symbol plane. In the symbol plane the pruning front which seperates allowed and forbidden orbits is obtained from orbits running either into or through the cusp. We show that all periodic orbits correspond to maxima of the Lagrangian and give a list up to code length 15. Using periodic orbit averages the topological and metric entropy are calculated and compared with the values obtained by other methods and an analytical estimate ot the metric entropy. Furthermore the statistical properties of periodic orbits are investigated.

SCALING BEHAVIOUR OF PERIOD LENGTHS OF DYNAMICAL SYSTEMS WITH ROUNDOFF

Christian Beck, London

The orbits of chaotic discrete-time dynamical systems approach periodic orbits when they are are calculated by a computer of finite precision. The average period length usually scales with the machine precision, the scaling exponent being the so-called roundoff scaling exponent. Under certain assumptions it can be related to the Renyi dimensions of the attractor of the map. We also investigate probability distributions of period lengths obtained for various discretization schemes.

ROUND-OFF PATHOLOGIES AND RELATED STATISTICS

Michael Blank, Nice

When solving differential equations or other dynamical systems on a computer, the effect of finiteness (round-off) can sometimes be very drastic. For example, a stable periodic orbit may be replaced by one with a multiple of the true period. The problem is of special interest in the case of chaotic dynamical systems. We discuss on the one hand conditions under which chaotic properties survive discretization, and on another hand examples when, even in the limit of vanishing perturbation, a ”localization” phenomenon takes place: trajectories which should normally be dense remain confined to a small number of points. To investigate statistical aspects of the influence of the considered perturbations we introduce the notion of statistical probability of an event, which means the portion (among all discretizations from a given family), when the event takes place. These statistical probabilities are calculated analytically for the problem of stabilization of an unstable periodic trajectory, ergodicity phenomenon, and some others.

EXPLORING INVARIANT SETS AND INVARIANT MEASURES

Michael Dellnitz, Bayreuth, Martin Rumpf, Freiburg

We propose a method to explore invariant measures of dynamical systems. The method is based on numerical tools which directly compute invariant sets using a subdivision technique, and invariant measures by a discretization of the Frobenius-Perron operator. Finally appropriate visualization tools help to analyze the numerical results and to understand important aspects of the underlying dynamics. This fact will be illustrated for examples provided by the Lorenz system.

NUMERICAL BIFURCATION ANALYSIS: OJECTIVES, METHODS, CAPABILITY, LIMITATIONS

E. J. Doedel, Montreal

The objectives and some basic methods of numerical bifurcation analysis will be described. A number of different computational examples will be used to illustrate both the power and the limitations of these techniques. Some new directions will also be mentioned.

EVENTUALLY EXPANDING ONE DIMENSIONAL MAPS

Ale Jan Homburg, Berlin

A lot is known on the dynamics of smooth unimodal maps, where the derivative at the turning point vanishes. I discuss the dynamics of piecewise smooth unimodal maps with nowhere vanishing derivative. Necessary and sufficient conditions will be given for such maps to be eventualy expanding (f is eventually expanding if |Df^{N}| > 1 for some N > 0) and so to have an absolutely continuous invariant probability measure.

OBSERVABLE INVARIANT MEASURES AND THE VARIATIONAL PRINCIPLE

Gerhard Keller, Erlangen

We consider dynamical systems defined by (piecewise) smooth maps f : M → M. In the first part of this talk I shall explain (to the non-experts) why in many hyperbolic situations “observable” invariant measures are characterized by a variational principle involving entropies and the derivative of f. In the second part the possibility (and the difficulties) to extend these ideas to nonhyperbolic maps is discussed by looking more closely at some unimodal interval maps.

MINIMAL CENTER OF ATTRACTION OF MEASURABLE SYSTEMS AND THEIR DISCRETIZATIONS

P.E. Kloeden, Geelong

A generalization of the Birkhoff minimal center of attraction is introduced for discrete time dynamical systems that are generated by a single-valued or multi-valued masurable mapping. Its properties will be discussed, in particular it will be shown to be equal to the minimal closed set containingthe supports of the semi–invariant measures of the system. The effects of spatial discretization of the state space on it will also be considered.

COMPUTATIONAL ASPECTS OF RANDOM PERTURBATIONS OF DYNAMICAL SYSTEMS

Yuri Kifer, Jerusalem

I am going to review results on random perturbations of dynamical systems with some kind of hyperbolicity or expanding which provide a theoretical background for computations of parameters of dynamical systems since only parameters which are stable to random errors can be computed. I shall discuss also some of these results which provide direct approximate methods of computations of some of the parameters. Some questions connected with the speed of convergence will be discussed, as well.

BREAKDOWN OF INVARIANT TORI

Jens Lorenz, Albuquerque

I consider parameter dependent systems of ordinary differential equations with a branch of invariant tori. Typically, the tori will disappear (break) in certain parameter regions, and a great variety of bifurcations is possible. Given an invariant torus for a certain parameter value, how can one decide if the branch can be continued or not? In principle, the general perturbation theory of invariant manifolds of parameter dependent systems is applicable to this question. I use this theory in Fenichel’s form, which requires knowledge of so–called Lyapunov–type numbers. If phase locking occurs on the torus, these numbers can be determined in terms of Floquet exponents. If the flow on the torus is ergodic, however, it is not clear how to compute the number which measures the attractivity towards the torus. I will give a partial answer to this question.

THE STRUCTURE OF BASINS OF ATTRACTION AND THE OCCURRENCE OF WADA BASINS

Helena E. Nusse, Groningen

In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is nonempty. A point P is a boundary point of a basin B if every open neighborhood of P has a nonempty intersection with basin B and at least one other basin; the boundary of a basin is the set of all boundary points of that basin. When there are several basins, their boundaries can be thoroughly intermeshed. When there are three basins of attraction, is it possible that every boundary point of one basin is on the boundary of the two remaining basins? Is it possible that all three boundaries of these basins coincide? When this last situation occurs then the boundaries have a complicated structure. This phenomenon does occur naturally in simple dynamical systems. The purpose of this lecture is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms by using the basic notion of ”basin cell”. A basin cell is a trapping region generated by some well chosen periodic orbit and determines the structure of the corresponding basin. A basin B is a Wada basin if every boundary point of B is also on the boundary of at least two other basins. A qualitative result describing whether a basin is a Wada basin will be presented. Our theorem provides numerically verifiable conditions guaranteeing that a basin is a Wada basin.

STOCHASTIC STABILITY OF OSELEDETS SPACES

Gunter Ochs, Bremen

Lyapunov exponents (exponential growth rates) and Oseledets spaces (corresponding equivariant subspaces) of a matrix cocycle (product of random matrices) are in general not stable under perturbations. However, under certain types of stochastic perturbations stability of Lyapunov exponents can be ensured. We will show, that stability of Lyapunov exponents already implies stability of Oseledets spaces in a weak sense.

INVARIANT TORI IN DYNAMICAL SYSTEMS

Volker Reichelt, Aachen

Consider a dynamical system given as an autonomous ODE with an attractive invariant manifold M. Under certain assumptions concerning the flow within and towards the manifold M the existence of a perturbed invariant manifold for all C^{1}-close systems can be proven via graph transform. A discretized version of the graph transform is developed to compute invariant tori numerically. The influence of the discretization error will be examined and some examples will be given to discuss the performance of the algorithm.

THE INVERSE PROBLEM FOR PARTITIONED ITERATED FUNCTION SYSTEMS IN IMAGE COMPRESSION

Dietmar Saupe, Freiburg

In the late 1980’s Michael Barnsley proposed a method for encoding images which breaks away from all classical ’transform-quantize-entropy encode’ type of approaches. He suggested to use attractors of dynamical systems as approximations of image data. Thus, the code consists in a description of the dynamical system and the encoding becomes an inverse problem: for a given image find a system whose attractor matches the data. The solution to this problem was later given in the form of partitioned iterated function systems. The image is segmented into a set of ranges, and for each range a similar image portion (up to spatial scaling and affine intensity transformation), called domain, must be sought. We discuss the mathematical setup of this approach and continue with an overview of the many practical problems with the method: how should the image be segmented, where should one search for matching image portions, how should the intensity transformation be designed, and what methods are available for a speed up of the large search.

A FIXED POINT THEOREM BASED ON LYAPUNOV EXPONENTS

Bjoern Schmalfuß , Bremen

One of the most important theorems in mathematics is the well known Banach Fixed Point Theorem. We give an extension of this theorem for random dynamical systems based on Lyapunov exponents. In addition we discuss applications of this random fixed point theorem. These applications cover - Existence of a stationary solution and an invariant measure for (partial) stochastic differential equations.

- Existence of a random pitchfork bifurcation for a Stratonovich equation

du = (αu − u^{3} + g(u))dt + u ∘ dw, α ∈ R.

- Existence of random invariant manifolds for particular random dynamical systems.

PROBABILISTIC TECHNIQUES IN THE NUMERICAL ANALYSIS OF DYNAMICAL SYSTEMS

Andrew Stuart, Stanford

I will survey the existing theory of convergence for the numerical integration of dynamical sytems. This theory is only well-developed for the fixed time-step integration of dissipative dynamical systems, and for invariant sets with a hyperbolic structure. Thus a number of substantial gaps in the theory remain and I will highlight these. I will consider three such gaps and describe how probabilistic techniques might potentially be brought to bear to enlarge the theory in these cases. The three areas which I will describe are: (i) the use of adaptive time-steps, which yields a discontinuous dynamical system for the evolution of the approximate solution and the time-steps together; (ii) the effect of discretization on non-hyperbolic attractive invariant sets for which, typically, only upper semicontinuity with respect to perturbation may be established; (iii) the treatment of Hamiltonian systems where the question of how to interpret long-time simulations remains completely open. In the spirit of a workshop many of the ideas described will be embryonic and only loosely formulated.

A STATISTICAL METHOD FOR DETECTING CYCLES IN DISCRETE DYNAMICAL SYSTEMS

Jan Wenzelburger, Bielefeld

This paper introduces a statistical method for detecting cycles in discrete time dynamical systems. The continuous state space is replaced by a discrete one consisting of cells. Hashing is used to represent the cells in the computer’s memory. An algorithm for a two-parameter bifurcation analysis is presented which uses the statistical method to detect cycles in the discrete state space. The output of this analysis is a colored cartogram where parameter regions are marked according to the long–term behavior of the system. Moreover, the algorithm allows to compute basins of attraction of cycles.

NUMERICAL COMPUTATION OF THE INVARIANT MEASURE FOR CIRCLE MAPS

Bodo Werner, Hamburg

The determination of the density of an orientation preserving diffeomorphism of S^{1} is an ill posed problem — due to the phase locking phenomenon caused by rational rotation numbers. The discretization of the Perron–Frobenius operator by stochastic transition matrices M can be considered as a regularization. Of special interest is the Frobenius eigenvector and the spectrum of M and their relation to the continuous situation. Numerical experiments are performed for the Arnol’d family of circle maps

x ↦ x + λ + ε sin(2πx)

with special attention to parameter values close to Arnol’d tongues. It is demonstrated how a good approximation of the rotation number by rational numbers of small denominators can be utilized.