Workshop
"Invariant Measures and Invariant Manifolds"
ABSTRACTS
The same information in .dvi-file.
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 Baecker,
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 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 and Martin Rumpf,
Bayreuth / 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:
OBJECTIVES, 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.
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.
MINIMAL CENTER OF ATTRACTION
OF MEASURABLE SYSTEMS
AND THEIR DISCRETIZATIONS
Peter 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
containing the supports of the semi-invariant measures of the system.
The effects of spatial discretization of the state space on it will also
be considered.
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.
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=(alpha u-u^3+g(u))dt+u o dw, alpha Element 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 DISCRET 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+lambda+epsilon sin(2pi x)
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.
Further information about the workshop
DFG-Project Connecting Orbits (Bielefeld)
DFG-Schwerpunktprogramm Dynamik (Berlin)
Winfried Kleß, 96-05-28, 96-05-31
Thorsten Göke, 96-12-09
Department of Mathematics |
University of Bielefeld