In mathematics you don't understand things. You just get used to them.

In the middle 30's, Johnny was fascinated by the problem of hydrodynamical turbulence. It was then that he became aware of the mysteries underlying the subject of non-linear
     partial differential equations. His work, from the beginnings of the Second World War, concerns a study of the equations of hydrodynamics and the theory of shocks. The
     phenomena described by these non-linear equations are baffling analytically and defy even qualitative insight by present methods. Numerical work seemed to him the most
     promising way to obtain a feeling for the behaviour of such systems. This impelled him to study new possibilities of computation on electronic machines ...

The aim of the workshop is to discuss current topics related to the computation and modeling of dynamical systems, one of the favourite areas of John v. Neumann. As in all other areas, a high level of specialization characterizes the field of dynamical systems. One goal of the workshop is to connect people working on theoretical aspects of algorithms and computational complexity with those who find computational challenges emerging in various applications. The workshop's schedule is coordinated with the timetable of a larger conference, the central events will take place at the Hungarian Academy of Sciences on October 15th. Here the participants of the central conference will be joined by the participants of the workshops and the main theme will be a series a talks connected to various topics in the range of von Neumann's interest.
 
 

Global attractors for semiflows without uniqueness

Numerical methods for dynamical systems
on unbounded domains
 
 

Wolf-Jürgen Beyn, Bielefeld University
 
 

The longtime behaviour of solutions for evolution equations
on a spatially unbounded domain can differ significantly from that
on a bounded domain. Typical examples are fronts and pulses that travel
forever on the infinite line but usually die out on any finite
domain when they reach the boundary.

In the talk we review a general method for freezing such solutions
in a finite domain. We derive a partial differential algebraic equation
(PDAE) by which one can separate the evolution of the form of the solution
from the motion of its 'center'. The formulation can be used
for time integration of initial value problems
 as well as for the bifurcation analysis of certain patterns.
We show how
the method generalizes to infinite dimensional dynamical systems
that are equivariant under the action of a (generally noncompact) Lie group.
Several numerical issues of this PDAE approach will be discussed,
such as the choice of asymptotic boundary conditons and appropriate
time and space discretizations. Applications will be presented to
reaction-diffusion  systems that show travelling waves or spiral waves
on domains with one or two unbounded space directions.

Families of Elemental Periodic Orbits
of the Circular Restricted 3-Body Problem
 
 

Eusebius Doedel, Department of Computer Science

Concordia University, 1455 boulevard de Maisonneuve O.
Montreal, Quebec, H3G 1M8, CANADA
 

The N-body problem of Celestial Mechanics has been one of the most studied
problems in Mathematics.
Of particular interest is the study of its periodic solutions, to which
many people have made important contributions.
Renewed interest in the N-Body problem arises from the recent work of
Chenciner and Montgomery, who proved the existence of a surprising, planar
figure-8 periodic solution of the 3-body problem.
Such an orbit had earlier been predicted by C. Moore.
Sim\'o has found many more periodic orbits of this type, where N bodies
follow a single planar curve ("choreographies").
Some references to this remarkable work are listed in [1].
 

The  Circular Restricted 3-Body Problem (CR3BP) is a special limiting
case of the 3-Body Problem.  It describes the dynamics of a body with
negligible mass under the gravitational influence of two massive bodies,
called the primaries, where the primaries move in circular orbits about
their barycenter.
This problem has also received much attention in the literature (see [1]
for some references), as the study of its periodic orbits and their
stable and unstable manifolds is important in space-mission
design. For example, the Genesis mission, currently in a so-called
"Halo-orbit", was designed by Martin Lo and co-workers at JPL, using
concepts and techniques from Dynamical Systems Theory.
 

In this talk I will show how boundary value continuation techniques can
be very effective tools in the study of periodic solutions of the N-Body
Problem and, in particular, the CR3BP.
First I will show how the methods in [1,2] can be used to compute the
families of periodic orbits associated with the libration points
(equilibria) of the CR3BP, and further bifurcating families.
In addition, using extended boundary value systems, one can determine how
the bifurcation points depend on the mass-ratio parameter $\mu$, which
represents the ratio of the mass of the smaller body to the total mass.
Similarly, one can trace the dependence of homoclinic orbits on $\mu$.
This allows a rather complete classification of all primary families and
of some secondary and tertiary bifurcating families, for all values of
$\mu$. In the discussion of these rather extensive results (of which the
computer data fill several CDs!), I will concentrate on some of the most
interesting observations.
 
 

Various aspects of this work are done in cooperation with
Randy Paffenroth and Herb Keller (Caltech),
Don Dichmann (Torrance),
Jorge Gal\'an and colleagues (Sevilla),
A. Vanderbauwhede and Willy Govaerts (Gent),
and
Yuri Kuznetsov (Utrecht).

{[1]}
 E. J. Doedel, R. C. Paffenroth, H. B. Keller, D. J. Dichmann,
    J. Gal\'an, A. Vanderbauwhede, Continuation of periodic solutions in
    conservative systems with application to the 3-Body problem, Int. J.
    Bifurcation and Chaos, Volume 13, Number 6, 2003, (to appear).
    (http://cmvl.cs.concordia.ca/publications.html
{[2]}
 F. J. Mu\~noz-Almaraz, E. Freire, J. Gal\'an, E. J. Doedel,
    A. Vanderbauwhede, Continuation of periodic orbits in conservative
    and Hamiltonian systems, Physica D, 2003, (to appear).

{[3]}
 E. J. Doedel, R. C. Paffenroth, H. B. Keller, D. J. Dichmann,
    J. Gal\'an, A. Vanderbauwhede, Elemental Periodic Orbits associated
    with the Libration Points in the Circular Restricted 3-Body Problem,
    (in preparation).
 

Ghost solutions in nonlinear boundary-value problems

Gábor Domokos
Budapest University of Technology and Economics
Center for Applied Mathematics and Computational Physics

and

Philip Holmes
Princeton University
Program in Applied and Computational Mathematics

Numerical dynamics for delay differential equations
    as displayed in the work of Gyula Farkas

Barnabas M. Garay (Budapest University of Technology)

During his short career, GYULA FARKAS (born in Kaposvar, Hungary; 1972)
wrote 21 papers. He died in a car accident on the 27th of February, 2002,
three months after his 29th birthday,
the week before his PhD Diploma was handed to him. His PhD Thesis as well
as a great part of his publications are devoted to numerical dynamics of
retarded functional differential equations. This talk --- based on his
posthumous paper "Small delay inertial manifolds under numerics: a
numerical structural stability result" (J.Dyn.Diff.Eq. 14(2002), 549-588)
--- is a tribute to his memory.

Consider the delay differential equation

(1)     x'(t) = Ax(t) + f(x(t)) + g(x(t-c))

where A is an n by n constant matrix, c > 0 is a parameter, f,g : R^n --->
R^n are bounded two times continuously differentiable functions with
bounded derivatives. Let T_c(t) denote the solution semigroup generated by
the linear ordinary differential equation  x'(t) = Ax(t)  acting on the
space of continuous functions C([-c,0],R^n). Projection P_c : C([-c,0],R^n)
---> C([-c,0],R^n) defined by  P_c(\varphi)(s) = exp(As) \varphi(0) leads
to a T_c - invariant decomposition of C([-c,0],R^n), the phase space of (1).
By letting the delay parameter  c ---> 0 , it is elementary to show that
the spectral gap of this decomposition can be made arbitrarily large.

In a final analysis --- this implies that structural stability of the
limiting ordinary differential equation

(2)     x'(t) = Ax(t) + f(x(t)) + g(x(t))

is preserved by standard discretizations of the original delay equation (1),
for delay and stepsize sufficiently small. The why's and how's are the
essence of Gyula Farkas's paper as well as of the present talk: inertial
manifolds for the exact (explaining also the geometry behind Tonelli's
proof to Peano's existence theorem for ordinary differential equations) and
the discretized dynamics of (1). A brief survey on numerical structural
stability theory is also given.
 
 
 
 
 
 
 

 MatCont : a Matlab toolbox for dynamical systems

Speaker : W. Govaerts (Ghent, Belgium).
Address : Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 - S9,
B - 9000 Gent, Belgium.

E-mail : Willy.Govaerts@rug.ac.be

URL : allserv.rug.ac.be/~wgovarts

Co - authors : Yuri A. Kuznetsov (Utrecht, NL) and A. Dhooge (Gent,B).

MATCONT is a Matlab toolbox for the interactive, graphical, numerical study of parameterized
ODEs. It has the look - and - feel of  CONTENT [3] but is completely rewritten.
The current version is freely available at:
http://allserv.rug.ac.be/~ajdhooge
where also a slightly more general non - GUI version CL_MATCONT is available.
 

MATCONT provides dynamic simulation (all Matlab solvers are incorporated) and the
numerical continuation of curves of equilibria, limit points, Hopf points, limit cycles, and
flip, fold and Neimark Sacker bifurcations of limit cycles. It can detect and compute bifurcation
points, store computed curves, switch to new branches, compute symbolic derivatives etcetera.
 

In the case of limit cycles Matcont  discretizes the BVP exactly as in AUTO  [1]  and CONTENT [3],
i.e. by orthogonal collocation.
The systems that arise in this way are typically sparse and their sparsity increases with the number of
test intervals used in the discretization. In Matcont and CL_MATCONT the
sparsity of the linearized systems is exploited by using the Matlab sparse matrix routines
 

Matlab has many advantages but it has one big disadvantage :
it is usually much  slower than compiled code.
Fortunately, compiled code can  be introduced in Matlab.
Also, there is the Matlab Profiler for analyzing the code and measuring
 the time spent during every call and
in the execution of any line of the code. So it allows to decide where compilation is most useful; it also
allows to compare the details of various codes to make fair comparisons.

We discuss in particular the implementation in
Matcont of the algorithms in [2] for the computation of
flip, fold and Naimark - Sacker bifurcations of limit cycles. These use
a minimally extended system, i.e. we append only a scalar equation to the definition of limit cycles
in the case of flip and fold; we introduce an additional variable and append two equations in the
case of Neimark - Sacker.
These algorithms are not implemented in any other publicly available package.
AUTO97-00 [1] uses a fully extended system, i.e. the number of state variables is approximately doubled (flip and fold) or tripled (Naimark - Sacker).
 

References:

[1] E. J. Doedel, A. R. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. Sandstede and X. J. Wang,
AUTO97-AUTO2000 :
Continuation and Bifurcation Software for
Ordinary Differential Equations (with HomCont), User's Guide,
Concordia University, Montreal, Canada (1997-2000).
(http://indy.cs.concordia.ca).

[2] E. J. Doedel, W. Govaerts and Yu. A. Kuznetsov, Computation of Periodic Solution Bifurcations in ODEs using Bordered Systems, to appear in SIAM Journal on Numerical Analysis (2003)

[3 ]Yu. A. Kuznetsov and V. V. Levitin, CONTENT: Integrated Environment for analysis of dynamical systems. CWI, Amsterdam (1997): ftp://ftp.cwi.nl/pub/CONTENT

Title: Computing manifolds in dynamical systems.
Author(s): Michael E. Henderson
Affiliation(s): IBM Research
email address(es): mhender@watson.ibm.com
web   address(es): http://www.research.ibm.com/people/h/henderson/

Abstract: I will discuss an approach to computing two types of
"manifolds" that are found in dynamical
systems computations, and present several applications of the algorithm.
One dimensional manifolds can
be easily represented as points along a curve, with new points added at
one of the two endpoints. For surfaces
and higher dimensional manifolds this becomes an advancing front problem;
a problem which seems to have a simple
solution, which is notoriously difficult in the details. The idea is to
instead represent a manifold
as a set of points, together with balls in the tangent space of the
manifold at each point. The projection of
these balls onto the manifold gives a set of overlapping neighborhoods
which covers part of the manifold. A
continuation method for computation can be devised by finding a point
near the boundary of this union (actually a
point in the tangent space), projecting the point onto the manifold, and
adding the point and ball to the union.
This produces a well distributed set of points on the manifold, with
higher density where the curvature is larger.
It also reduces to pseudo-arclength continuation for a curve, and so is
in some sense the natural extension
of that method to higher dimensions.

Fixed points, periodic orbits, heteroclinic and homoclinic connections,
and many types of singular motions in
dynamical systems have been formulated as solutions of systems of
parameterized algebraic or two point boundary
value problems. (See the set of objects which can be computed with AUTO.)
These are sometimes called Implicitly
Defined Manifolds. At a regular point on such a manifold we can compute
the tangent space, estimate the curvature of
the manifold, and project a point in the tangent space onto the manifold,
and the algorithm described above can be used directly.

A second type of manifold - Invariant Manifolds, or more precisely the
image of a curve under a flow, are global
objects, and unlike the implicitly defined manifolds there is no local
way to project a point onto the manifold.
Instead,each point on the initial curve defines a trajectory of points on
the manifold. Moving along the trajectory
is straight-forward, and a little tensor analysis gives an expression for
the evolution of a local approximation to
the manifold along the trajectory. Points on this fattened trajectory are
the balls needed to compute the manifold.

In addition to some simple geometric examples, like the torus, this
algorithm has been used to compute periodic
orbits for a pair of torsionally couple pendula, equilibrium states of a
twisted rod, and invariant manifolds
in the Lorenz system.
 

  The Algorithms behind GAIO - Set Oriented
Numerics for Dynamical Systems

Oliver Junge    
 Univ. Paderborn
Institute of Mathematics

 Over the last few years a new approach
to the numerical treatment of dynamical
systems has been developed.  Using a
hierachical partitioning of the
interesting part of phase space, global
objects like invariant sets and invariant
manifolds can be efficiently computed.
In addition, these methods allow for
a natural discretization of an associated
transfer operator of the system.  An
analysis of parts of the spectrum of the
resulting matrix reveals a great deal of
the macroscopic dynamics of the underlying
system.

 In this talk we will give an overview over
the algorithms and related applications
and demonstrate a couple of example
computations using the associated software
package GAIO ("Global Analysis of Invariant
Objects").
 
 
 
 
 
 
 
 

 Equation-Free Multiscale Computation:  Enabling Microscopic
Simulators to Perform System Level Tasks.
 

Ioannis G. Kevrekidis
 

 Department of Chemical Egineering, PACM and Mathematics
               Princeton University
yannis@princeton.edu

I will present and discuss a framework for computer-aided multiscale analysis,

which enables models at a "fine" (microscopic/stochastic) level of description
to perform
modeling tasks at a "coarse" (macroscopic, systems) level.
These macroscopic modeling tasks, yielding information over long time
and large space scales, are accomplished through appropriately
initialized calls to the microscopic simulator for only short times
and small spatial domains: "patches" in macroscopic phase space-time.

Traditional modeling approaches start by deriving macroscopic evolution
equations (balances closed through constitutive
relations).  An arsenal of analytical and numerical techniques for the
efficient solution of such evolution equations (usually Partial
Differential Equations, PDEs) is then brought to bear on the problem.
Our equation-free (EF) approach, introduced in PNAS (2000) when
successful, can bypass the derivation of the macroscopic evolution
equations {it when these equations conceptually exist but are not
available in closed form}.

We discuss how the mathematics-assisted
development of a computational superstructure may enable alternative
descriptions of the problem physics (e.g. Lattice Boltzmann (LB),
Brownian Dynamics (BD), kinetic Monte Carlo (KMC) or
Molecular Dynamics (MD) microscopic simulators, executed over
relatively short time and space scales) to perform systems level tasks
(integration over relatively large time and space scales, "coarse"
bifurcation analysis, but also optimization and control tasks) directly.

In effect, the procedure constitutes a systems identification based,
"closure on demand" computational toolkit, bridging
microscopic/stochastic simulation with traditional continuum
scientific computation and numerical analysis.  We illustrate these
"numerical enabling technology" ideas through examples from chemical
kinetics (LB, KMC), rheology (Brownian Dynamics), homogenization and
the computation of "coarsely self-similar" solutions, and discuss
various features, limitations and potential extensions of the
approach.

           Averaging methods in climate models.

                Yuri Kifer (Hebrew University)

                      Abstract.

      K.Hasselmann suggested in 1978 to study climate-weather interactions
      in the framework of the averaging setup considering climate as a slow
      and weather as a fast (chaotic) motion. The averaging approach suggests
      to approximate the slow motion by a motion obtained by averaging in
      fast variables but Hasselmann suggested a better diffusion approximation
      of the averaging error. I shall discuss some recent rigorous results
      related to this approach.
 

Title: On the computation of invariant measures in random dynamical
systems

Author: Peter Kloeden

Affiliation:
Fachbereich Mathematik,
Johann Wolfgang  Goethe-University,
D 60054 Frankfurt am Main, Germany
 

email address: kloeden@math.uni-frankfurt.de

web   address: http://www.math.uni-frankfurt.de/~numerik/kloeden/
 

Invariant measures of dynamical systems generated e.g. by difference
equations can be computed by discretizing the originally continuum state
space, and replacing the action of the generator by the transition mechanism
of a Markov chain. In fact they are approximated by stationary vectors of
these Markov chains. Here we extend this well known approximation result
for deterministic autonomous difference equations and the underlying
algorithm
to the setting of random dynamical systems, i.e. dynamical systems on the
skew product of a probability space carrying the underlying stationary
stochasticity and the state space, a particular non-autonomous framework.
The systems are generated by difference equations driven by stationary
random processes modelled on a metric dynamical system. The approximation
algorithm involves spatial discretizations and the definition of
appropriate random Markov chains with stationary vectors converging to the
random invariant measure of the system. Extensions to deterministic
non-autonomous  difference eqautions will also be mentioned.

1. P. Imkeller and P. Kloeden, On the computation of invariant measures in
random dynamical systems, submittted

THE VARIATION OF FINANCIAL  PRICES:  SCALING AND FRACTALS
 

Benoit Mandelbrot

Do financial prices follow the coin tossing model or Brownian motion?
>From the mathematical viewpoint, this would have been convenient, but in
fact prices are very far from Brownian.  The are often discontinuous or
nearly so and records of their changes look non-stationary.  To model those
two departure from so-called "normality", the speaker has invoked since
1963 a principle of scale invariance, first fractal, then multifractal.
This principle has proven to be extremely effective.  The presentation will
begin by a sketch of the basic facts and close on some very recent results
 
 

Title:             Topological methods in rigorous numerics of dynamical systems
Author(s):         Marian Mrozek
Affiliation(s):    Department of Computer Science, Jagiellonian University, Kraków, Poland
email address(es): mrozek@ii.uj.edu.pl
web   address(es): http://www.ii.uj.edu.pl/~mrozek
 

Abstract
========

   One of the main goals of the theory of dynamical systems consists in predicting
the asymptotic behaviour of trajectories of the system. Despite of many
achievments of the theory, the applicability of the theory to concrete dynamical
systems is limited. The reason is that in many cases the verification
of the necessary assumptions in a concrete system is very difficult if not just
impossible. As a consequence, in many if not most applications, the use of computer
simulations remains the only available tool. However, by the very nature of computer,
such simulations cannot guarantee rigorous results in the mathematical sense. Moreover,
there are examples of dynamical phenomena, which are present in the finite discretizations
studied on the copmuter, but disappear in the system the discretizations are supposed to approximate.

   Nevertheless, it turns out that with the help of a computer at least some questions
in the theory of dynamical systems may be answered rigorously for some concrete problems.
The aim of the talk will be to present one of such approaches, based on topological invariants:
the Conley index and the fixed point index. On one hand, the invariants
may be used to give some criteria, which characterize the existence of some dynamical phenomena
such as periodic orbits, homo- and heteroclinic connections and chaotic invariant sets.
On the other hand, a computer may be used to rigorously compute the invariants for
a class of concrete problems.

   We will present a brief introduction to the Conley index theory for flows and maps.
Then we will show how the theory may be rigorously discretized in a way wich provides
algorithms for computing the invariants for the original problem. This will be achieved
with the help of multivalued maps, which arise naturally when a trajectory of a dynamical
system is computed togehter with the rigorous error bounds. The talk will be completed
with some examples of concrete problems, where the scheme was successfully carried out.
 

Model reduction for fluids, using proper orthogonal decomposition and
balanced truncation

Clancy Rowley

Many of the powerful tools of dynamical systems and control theory have
gone largely unused for fluid systems, because the governing equations
are so dynamically complex (high-dimensional and nonlinear).  This talk
addresses modeling and model-reduction techniques, which are used to
obtain low-order models tractable enough to be used for analysis and
control.

The method of Proper Orthogonal Decomposition (POD) and Galerkin
projection is a popular technique for obtaining low-dimensional models,
in which the governing equations are projected onto basis functions
consisting of the most energetic modes, determined from data from
simulations or experiments.  The method is tractable even for very
large systems such as fluids, but can produce very poor models, since
low-energy modes can be critically important to the dynamics.  Balanced
truncation is a method popular in the control theory community for
obtaining reduced-order realizations of stable, linear input-output
systems.  Unlike POD/Galerkin, this method has provable bounds on the
error introduced in the truncation, but is computationally intractable
for systems of very large dimension, such as fluids.

In this talk, we use ideas from the POD/Galerkin method to obtain
approximate balanced truncations which are computationally tractable
even for very large systems.  A central message is that the inner
product one uses in POD/Galerkin procedure plays a crucial role.  We
illustrate the methods by obtaining reduced-order models for a
compressible flow past a cavity, and an incompressible channel flow.
 

Title: Linear Programming, Dynamical Systems and Integral Geometry
Author: Michael Shub, IBM Watson Research Center

Abstract: Interior point methods in linear programming theory frequently
"follow" the "central path" which is a solution curve of the Newton
vectorfield associated to the logarithmic barrier function. Motivated by
the question of whether or not there is a strongly polynomial algorithm
for linear programming, we study the dynamics of these vectorfields and
the curvature of the solutions. With tools from integral geometry, we
prove that on the "average" the curvature of these solution curves is
"small". We study the behavior of these vector fields near the boundary.
The dynamics suggests that the curvature is even smaller. Small curvature
may help explain why straight line approximations to the solution curves
work well in practice.
This is joint work with Jean-Pierre Dedieu.

Title   : Robust normal forms for analytic vector fields

Speaker : Warwick Tucker (Uppsala, Sweden)

Address : Department of Mathematics, Uppsala University, Box 480,
          751 06 Uppsala, Sweden

E-mail  : warwick@math.uu.se

Web page: www.math.uu.se/~warwick

Abstract: The aim of this talk is to introduce a technique for describing
trajectories of systems of ordinary differential equations passing near
saddle-fixed points. In contrast to classical linearization techniques,
the normal form methods we present allow for perturbations of the
underlying vector fields. This robustness is vital when modeling systems
containing small uncertainties, and in the development of numerical ODE
solvers producing rigorous error bounds.

A special case of the techniques to be presented was successfully used to
prove that the Lorenz equations admit a strange attractor. We will use
this example as an illustration of the methods described during the talk.

References:

[1] W. Tucker, "A Rigorous ODE solver and Smale's 14th Problem"
    Found. of Comp. Math., 2:1, pp. 53-117, 2002.

[2] W. Tucker, "Robust normal forms for analytic vector fields"
    Submitted. (http://www.math.uu.se/~warwick/main/papers.html)
 
 

** Map of the inner city of Budapest. Guesthouse marked with BIG arrow.
Technical University campus indicated by hand-drawn circle.

** Detailed map of the TU campus, Guesthouse marked as #3, conference
site (location of talks) indicated by hand-drawn arrow.

**How to get from the airport to the Guesthouse or the Gellert Hotel

    Cabs at the airport are rather expensive (~USD 25) and tend to overcharge.
If you decide to take a cab, make sure that you agree on the rate with the driver in advance. They can charge
ONLY FLAT RATES, which depend on the district of Budapest where you drive.
    I recommend the Airport Minibus Service (~USD 10), especially for single travellers. This service is available
immediately after you emerge from the customs area. Normally you have to wait for 15-20 minutes, the minibuses
take a couple of travellers and drop them in sequence.
   In case  minibus a tip 10-15% to the driver is recommended,
   cab drivers mostly own their vehicle, so the tip is less in this case.
   The EXACT address of the guesthouse is in Hungarian:
   BME Professzori Vendégház
   Stoczek utca 5-7  (Martos kollégium)
   1111 Budapest
If you print this out and show it to the receptionist of the Minibus service or the cab driver, they will know.
The guesthouse is located on top of a university dormitory. The dorm is not very tidy, but do not get scared. After entering, turn
left immediately after the reception. You will find a closed glass door, press the intercom on the right hand side and you will
be connected to the reception of the guesthuse who will open the door for you. After that you enter an elevator, which brings
you to the top floor (7th), this is a non-stop elevator so there are only two buttons. After exiting the elevator you are in the
guesthouse. Telephone: +36-1-463-3939

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Multiple Parameter Continuation

Michael E. Henderson
IBM Research
P.O. Box 218, Yorktown Heights,   NY 10598
 
 
 

A new continuation method for computing implicitly defined manifolds of
arbitrary dimension will be described.
The method is based on a representation of the manifold as the projection
of a set of overlapping spherical balls,
with new points being added by choosing a point on the boundary of the
collection of balls. This will be shown
to be equivalent to  computing the vertices of a certain set of convex
polyhedra, which form a polyhedral decomposition
of the manifold, and which are a weighted Voronoi diagram for the centers
of the balls. This decomposition
can be used to generate a triangulation of the manifold similar to the
Delaunay triangulation.
Several applications will be described,  including the computation of a
manifold of periodic motions of a pair
of coupled pendula. The representation may also be used for manifolds
defined in other ways, provided that a method
can be found to project from the tangent space onto the manifold. Methods
for invariant manifolds of mappings and flows
will be presented.
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