## Numerical and Computational Challenges in Science
and Engineering

### Workshop on Computational Challenges in Dynamical Systems

December 3 - 7, 2001

### Speaker Abstracts

**Uri M. Ascher**, Univerity of British Columbia

*On advantages and limitations of structure preserving difference
schemes for differential equations*

Structure preserving numerical methods for time-dependent differential
systems have received a lot of attention recently. The challenge is
to find for a given dynamical system discretization methods which yield
discrete dynamical systems possessing ``important properties'' of their
continuous couterpart to a good accuracy even without closely reproducing
all exact system's features (especially without pointwise accuracy).

Occasionally, it is possible to analyze the discrete dynamical system
by transforming it and considering the passage of the transformed discrete
system to a nearby continuous limit. If the limit ``ghost differential
system'' behaves well then the approximate solution is qualitatively
good. But if not then by following these ``wrong'' features the discrete
dynamical system may become unstable, or worse, produce deceptively
looking, qualitatively wrong solutions.

We will discuss several examples of Hamiltonian systems, including
certain PDEs in one space variable and highly oscillatory ODEs.

**Dwight Barkley**,* *University of Warwick,
UK

** Dynamics in the Cylinder's Wake**

The first instability of the wake of a circular cylinder is to the
periodic two-dimensional von Karman vortex street. I have previously
carried out a numerical Floquet analysis of the vortex street to three-dimensional
perturbations, and calculated the thresholds, wavenumbers, and Floquet
modes of this secondary instability. I now present an instantaneous
-- temporally local -- stability analysis which shows the flow to be
linearly stable throughout the periodic cycle, a very surprising result
within the fluid dynamics community.

Time permitting I will discuss a simple numerical scheme for reducing
computation time in very large fluid (and other PDE) simulations by
about a factor of two with minimal programming effort.

**Sue Ann Campbell**, University of Waterloo

*Computing Higher Order Terms for Centre Manifolds for Delay Equations*

We present an algorithm for computing higher order terms for centre
manifolds for delay equations using the symbolic algebra language Maple.
The advantage of obtaining symbolic as opposed to numerical results
is that one can obtain expressions which can be used to prove analytical
results, such as where in parameter space Hopf bifurcations are supercritical.
The disadvantage is that the expressions obtained may be too large to
be useful. We will illustrate the algorithm on a model for metal cutting.

**Walter Craig**, McMaster University

*Traveling surface water waves*

This talk with describe an existence theory and a computational method
for traveling wave solution of the Euler quations for a fluid with a
free surface. The approach uses Zakharov's Hamiltonian form of the equations
of motion in an essential way, posing the equations of motion in surface
variables in terms of the Dirichlet-Neumann operator for the fluid domain.
Both two- and three-dimensional solutions are obtained. The existence
theory is in the case in which at least some surface tension is present,
and there is a close analogy with the resonant Lyapunov center theorem.
The numerical computations of nonlinear and quite steep surface water
waves are based on these surface variables, and use a surface spectral
method and a Taylor expansion of the Dirichlet-Neumann operator which
is an extension of the Hadamard variational formula for the Green's
function. A clear distinction between shallow water and deep water waves
is observed. This work is in collaboration with D. Nicholls.

**Eusebius Doedel**, Concordia University

*Continuation of Periodic Solutions in Conservative Systems with
Application to the Figure-8 orbit of Montgomery, Chenciner and Simo*

I will show how boundary value continuation software can be used to
compute families of periodic solutions of conservative systems. A very
simple example will be used to illustrate the main idea. I will also
show how the computational approach can be generalized to follow the
recently discovered figure-8 orbit of Montgomery, Chenciner, and Simo,
as the mass of one of the bodies is varied. The numerical results show,
among other things, that there exist continuous paths from the figure-8
orbit to periodic solutions of the restricted three-body problem.

This is joint work with Randy Paffenroth and Herb Keller at Caltech,
Andre Vanderbauwhede in Gent, and Jorge Galan in Sevilla.

**Mitrajit Dutta**, University of New Hampshire

** Robust Route to Unshadowability in Physical Systems**

In computer simulations of chaotic systems, small errors grow exponentially,
thereby questioning the validity of the numerical results. Fortunately,
shadowing techniques often show that the simulated trajectories are
close approximations to the true trajectories. However, chaotic attractors
containing unstable periodic orbits with different numbers of local
unstable directions are necessarily unshadowable. Here we argue that
this route to unshadowability should be common in coupled dynamical
systems. Our argument is robust in the sense that it does not require
any embedded invariant manifold or any exact condition like unidirectional
coupling. Thus, it should hold for real physical systems. Numerical
simulations of the forced, damped double pendulum are presented as an
example.

**Donald Estep**, Colorado State University

**Preservation of Invariant Rectangles under Discretization**

An important issue in the study of reaction-diffusion equations is
determining whether or not solutions blow-up. This carries over to numerical
solutions, in which case we have the additional concern of determining
if discretization either inhibits blowup or causes it to occur artificially.
The existence of invariant rectangles inside of which solutions remain
for all time is an important factor for addressing these issues in many
cases. After presenting some motivating examples, we discuss the preservation
of invariant rectangles under discretization in two ways. We construct
special numerical methods that preserve invariant rectangles exactly
and show how to use adaptive error control to preserve invariant rectangles
in an approximate sense.

**Jorge Galan,** University of Sevilla

*Bifurcations of relative equilibria and continuation of tori in
Hamiltonian systems with symmetries.*

Symmetries in Hamiltonian systems are very common and play a fundamental
role in the analysis of its bifurcation behavior. A solution that is
invariant under the action of the symmetry is called a relative equilibrium.
In this contribution we analyze by numerical continuation and a global
symmetry reduction the existence of "bridges" between subharmonic
bifurcations of relative equilibria. The constancy of the rotation number
along the bridge is the key to extend the analysis to arbitrary tori.

This work has been done in collaboration with E. Freire, F. J. Mu\~noz
Alamaraz, E. Doedel and A. Vanderbauwhede.

**Karin Gatermann**, Berlin and ORCCA, London

**Symbolic computations for chemical reaction systems**

Chemical reaction systems of mass action type are polynomial differential
equations. The structure og the polynomials is determined by graphs.
Because of the graph theoretic structure and the polynomial structure
special results are known. I will explain how the latttice suggests
Lyapunov functions and how the lattice may be investigated with Hermite
normal form. Secondly, I will explain the importance of deformed toric
varieties in the work by Clarke on stability of reaction networks and
Hopf bifurcation. Computational examples like the Calcium oscillation
will be given.

**Martin Golubitsky**, University of Houston

**Coupled Oscillators and Symmetry**

Models for a variety of physical and biological systems (such as animal
gaits and the beating pattern of the leech heart)

have the form of coupled networks of systems of differential equations.
One important feature of such networks is that they support solutions
in which some components vary synchronously and some vary with well
defined time lags (spatio-temporal symmetries).

In this lecture we build on the example of animal gaits and discuss
the mathematics of spatio-temporal symmetries and how these solutions
arise in coupled cell networks. We describe a global theorem that gives
necessary and sufficient conditions for a given network to support solutions
with given spatio-temporal symmetries.

We end with a discussion of the beating of the leech heart that when
interpreted naively seems to contradict the theory --- but in fact leads
to interesting mathematical questions and to notions of local symmetries
and approximately symmetric periodic solutions.

This research is joint with Luciano Buono, Marcus Pivato, and Ian Stewart.

**John Guckenheimer**, Cornell University

**The Forced van der Pol Equation Revisited**

The forced van der Pol equation is one of the classical examples in
dynamical system theory. Nonetheless, the bifurcations of this system
have not been thoroughly studied. This lecture will give describe new
results of Kathleen Hoffman, Warren Weckesser and myself that are based
upon investigation of the slow flow associated with the

van der Pol system. We produce a global bifurcation diagram for the
slow flow that encompasses the solutions of the system without canards.

**Michael
E. Henderson**, TJ Watson Research Center

*Multiple Parameter Continuation/Computing Implicitly Defined Manifolds*

Many structures of interest in dynamical systems can be written as
the solution of either an algebraic system or a two point boundary value
problem, which depends on a small number of parameters. Geometrically
these solutions are pieces of low dimensional manifolds embedded in
either spaces of dimension 2 up to around 100 (fixed points), or spaces
of dimension in the thousands (periodic motions, homoclinic and heteroclinic
trajectories). The boundaries of these pieces are singular sets, at
which several pieces meet.

Previous continuation methods have computed tilings of the manifold
(a union of non-overlapping neighborhoods). The main algorithmic difficulty
is how to maintain this representation when a new tile is added, i.e.
to ensure that the new tile does not overlap the old ones. Allgower
and Schmidt's simplicial continuation method does this by using sections
of a

mesh on the embedding space. In Keller's pseudo-arclength continuation
(for one dimensional manifolds) the tiles are intervals, and the overlap
is eliminated simply by trimming one side of the interval. In dimensions
2 and higher attempts have been made to use surface meshs as tilings,
but it has proven difficult to maintain the tiling - you can't just
trim one mesh cell against another and get compatible cells. (Rheinboldt
and Brodzik have developed an algorithm which avoids local, but not
global, overlap.) Simplicial continuation is not well suited to manifolds
embedded in high dimensional spaces, so we need to extend predictor-corrector
methods to higher dimensions.

I will describe a new predictor-corrector continuation method for computing
implicitly defined manifolds which uses a covering instead of a tiling.
Instead of using a mesh it represents the manifold as the union of overlapping
neighborhoods, each centered at a point on the manifold. To extend the
manifold, a new point is chosen from the boundary of this union

and a neighborhood of it is added to the collection. The problem of
overlapping neighborhoods is now moot, but is replaced by the seemingly
difficult task of finding a point on the boundary of a set of overlapping
neighborhoods. I will show that when the neighborhoods are spherical,
or nearly spherical, the boundary can be expressed very simply in terms
of a certain set of convex polyhedra. Furthermore, a boundary point
can be found in terms of the vertices of these polyhedra . The resulting
algorithm is very simple, and since the new point need only be near
the boundary, surprisingly robust.

**Tony Humphries, **University of Sussex

*Travelling Waves (TWs) in Lattice Differential Equations (LDEs)*

LDEs are differential-difference equations defined by a system of ODEs
coordinatized by a lattice. There is extensive theory and computation
of TW solutions of parabolic type PDEs, but little beyond existence
is known about TW solutions of spatially discrete analogues. Finding
TW solutions of LDEs requires the solution of functional differential
equation BVPs with both advanced and retarded terms. We will discuss
some of the issues involved in accurate numerical computation of these
solutions using collocation methods. We will also present numerically
computed TW solutions for several problems including both dissipative
(Nagumo) and conservative (FKT) examples.

Arieh Iserles, University of Cambridge

**Computational and dynamical aspects of double-bracket flows**

In this talk we report recent results and work in progress (joint with
Tony Bloch) on various aspects of double-bracket flows and their generalisations.
We present a computational algorithm, based on Magnus-type expansions
of the underlying Lie algebra action and discuss different geometric
and dynamical aspects of these flows. In particular we show that for
the classical double-bracket flow $Y'=[[N,Y],Y]$, where (without loss
of generality) $N$ is symmetric,

the distance $\|Y-N\|$ is minimised along the isospectral orbit by the
stable fixed point in all $p$-Schatten norms for $p>1$ (Brockett
alredy proved this for $p=2$, i.e. the Frobenius norm), as well as classifying
all symmetric gauges for which fixed points are optimal (in the above
sense) for $2\times 2$ matrices. We also consider generalised flows
of the form $Y'=[Y,[g(Y-N),Y]]$ and discuss their (nontrivial) dynamics.

**Angel Jorba**, Universitat de Barcelona

*Models for the dynamics of the Trojan asteroids*

Joint work with F. Gabern

In this talk we will focus on the dynamics near the Lagrangian points
of the Sun-Jupiter system. To try to account for the effect of Saturn,
we will develop a specific model based on the computation of a true
solution of the planar three-body problem for Sun, Jupiter and Saturn,
close to the real motion of these three bodies. Then, we will derive
the equations of motion of a fourth infinitesimal particle moving under
the attraction of these three masses. Using suitable coordinates, the
model will be written as a time-dependent perturbation of the well-known
spatial Restricted Three-Body Problem.

Next, we will discuss some techniques to study these four body models.
They are based on computing, up to high order, suitable normal forms
and first integrals. From these expansions, it is not difficult to derive
approximations to invariant tori (of dimensions 2, 3 and 4) as well
as bounds on the speed of diffusion on suitable domains.

**Oliver Junge**, University of Paderborn

*A rigorous computer assisted analysis of the global dynamics of
an infinite dimensional map*

We present a method to prove certain statements about the global dynamics
of an infinite dimensional map, namely the Kot-Schaffer growth-dispersal
model for plants. The method combines (rigorous) set-oriented numerical
tools, the Conley index theory and certain analytic considerations.
It not only allows for the detection of a certain dynamical behaviour
but also for a precise computation of the corresponding invariant sets
in phase space. We will give an outline of the method and exemplarily
show how to prove the existence of a heteroclinic orbit as well as of
a horseshoe. This

is joint work with M. Allili, S. Day and K. Mischaikow.

**Yannis G. Kevrekidis**, Princeton University

*``Coarse" Integration/Bifurcation Analysis via Microscopic
Simulators: micro-Galerkin methods *

I will present a time-stepper based approach to the ``coarse"
integration and stability/bifurcation analysis of distributed reacting
system models. The methods I will discuss are applicable to systems
for which the traditional modeling approach through macroscopic evolution
equations (usually partial differential equations, PDEs) is not possible
because the PDEs are not available in closed form.

If an alternative, microscopic (e.g. Monte Carlo or Lattice Boltzmann)
description of the physics is available, I will illustrate how the microscopic
simulator can be enabled (through a computational superstructure) to
perform certain

integration and numerical bifurcation analysis tasks at the coarse,
systems-level directly. This approach, when successful, can circumvent
the derivation of accurate, closed form, macroscopic PDE descriptions
of the system.

Facilitating, through such numerical "enabling technologies"
the direct, systems level analysis of microscopic process models may
advance our understanding and use of nonequilibrium systems.

**Eric Kostelich,** Dept. of Mathematics,
Arizona State University,

*Chaotic data analysis: Is it really any good?*

Much effort has been undertaken over the past two decades to elucidate
the dynamical properties of chaotic processes from time series measurements.
One example is the estimation of Lyapunov exponents from time delay
embeddings. However, the results are subject to large uncertainties.
This talk will argue that there are no obvious probability models for
errors that arise from artifacts of the embedding function or from poor
approximations of the natural measure of the underlying attractor. Consequently,
the construction of confidence intervals for dynamical quantities like
Lyapunov exponents does not seem feasible. Nevertheless, it may still
be possible to get a reasonable idea of the size of the

uncertainties associated with their computation from nonlinear time
series.

**Bernd Krauskopf**, University of Bristol

*Computing unstable manifolds in delay differential equations*

We explain and demonstrate a new numerical method for the computation
of 1D unstable manifolds of a saddle-type periodic orbit of a delay
differential equation. Its performance is illustrated with the example
of a semiconductor laser subject to phase conjugate optical feedback.

**Herb Kunze**, University of Guelph

*Using the Banach fixed point theorem to solve inverse problems
in differential and integral equations*

Banach's fixed point theorem plays a role in the (often approximate)
solution of many inverse problems in mathematics. The essential idea
is to express the problem in terms of a contractive map on a complete
metric space in order to use the fixed point theorem and some more recently
developed related results, most notably Barnsley's collage theorem.
For the inverse problem, one is given a solution, possibly an approximation
or an interpolation of experimental data, and then one seeks a map of
a particular form which has this ``target'' solution as a fixed point.

In this talk, the theoretical underpinnings of a method of solution
to some general inverse problems in differential and integral equations
is presented. In essence, collage-like theorems for the setting at hand
are developed. In each application, the end result is an algorithm which
produces a least-squares minimization problem. The construction and
solution of the minimization problem is computationally intensive. We
examine applications to initial value problems, two-point boundary value
problems, and Volterra and Fredholm integral equations of the second
kind.

Many of the parameter estimation techniques used by researchers in
differential equations can be viewed as inverse problems of the outlined
type; this work appears to be the first rigorous mathematical justification
of such ad hoc methods of error minimization.

**Rachel Kuske**, University of Minnesota

**Isolating the stochastic dynamics in models sensitive to noise**

Many systems which are sensitive to noise exhibit dynamical features
from both the underlying deterministic behavior and the stochastic elements.
Then the stochastic effects are obscured in this mix of dynamics. Several
different methods

have recently been applied to separate the "deterministic"
and "stochastic" dynamics. These approaches lead to simplified
approximate models which can be analyzed or simulated efficiently, providing
useful measures of the noise sensitivity. The methods combine projection
methods and the identification of important scaling relationships to
exploit features common to these systems, for example, presence of multiple
time scales, limited regions of strong noise-sensitivity, and resonances.
These approaches will be outlined in the context of two specific problems-
metastable interface dynamics and stochastic delay-differential equations.
Extensions to other problems will also be discussed.

**Gregory Lewis**, The Fields Institute

The numerical approximation of the normal form coefficients for a double
Hopf bifurcation.

I discuss an analysis of the primary bifurcations that occur in a mathematical
model of the differentially heated rotating fluid annulus. More specifically,
I present the analysis of the double Hopf bifurcations that occur along
the transition between axisymmetric steady solutions and non-axisymmetric
rotating waves. Center manifold reduction and normal

forms are used to deduce the local behaviour of the full system of partial
differential equations from a low-dimensional system of ordinary differential
equations.

Analytically, the coefficients of the normal form equations can be
found in terms of certain unknown functions, namely the steady axisymmetric
solution, the eigenfunctions and some Taylor coefficients of the center
manifold function. The unknown functions can be numerically approximated
from steady partial differential equations in two spatial dimensions.

Thus, a combination of analytical and numerical methods are used to
obtain numerical approximations of the coefficients of the normal form
equations. The numerical results are validated by comparison with experimental
observations.

**Andre Longtin**, University of Ottawa

*The challenges of memory effects in neurodynamical systems*

Memory effects due e.g. to finite propagation times in e.g optical
and physiological systems pose a great theoretical and computational
challenge. However, by accounting for the spatial propagation of activity,
delayed dynamics actually are a simplification over PDE's. We first
present recent work on reducing large scale ionic models of "bursting"
neural activity to simple two-dimensional dynamical systems with delay;
this delay is used to model the backpropagation of firing activity from
the cell's body to its dendrites and back. This reduction preserves
bifurcations in the full ionic model, enables an analytical study of
the bursting dynamics, and provides a drastic computational simplification
for real life problems where many such cells are coupled. We then discuss
work on delayed bistable systems which arise also in the context of
laser and neuron dynamics. We show that bistable dynamics with linear
delayed feedback can be understood in terms of the unfolding of a Takens-Bogdanov
bifurcation. We provide the two-dimensional normal form for this

infinite-dimensional system in terms of the original parameters; this
form again can be used to simplify computations. We also discuss a preliminary
computational analysis of the multistability of a cluster of such bistable
systems. Finally we describe a memory effect seen in excitable systems,
which leads to correlations between firing intervals. Modeled as a dynamic
threshold rather than a delay, it leads, in the presence of periodic
forcing, to phase locking and chaos. We derive the Lyapunov exponents
for this model, and discuss the difficult problems that arise in the
analysis of noise-induced crossings when such memory is present.

**Kurt Lust**, K.U.Leuven

*Bifurcation Analysis of Large-Scale Systems via Timesteppers.*

We will describe a numerical method to perform a bifurcation analysis
of certain large-scale dynamical systems by putting an additional routine
on top of an existing simulation code. A straightforward application
of this idea is the computation of limit cycles by the single shooting
method. However, the method can also be used to compute steady-state
solutions. In fact, one could even consider to use our algorithm to
study systems with unknown equations, as long as it is possible to construct
a time integrator for the system. (This is done in the presentation
of Prof. Dr. Y. Kevrekidis.) Even the timestepper is not essential:
one can also use the method to perform a bifurcation analysis of a high-dimensional
map. The main assumption made is that the Jacobian of the map - the
monodromy matrix for periodic solutions - has only few large eigenvalues
(where large means: close to or outside the unit circle in the complex
plane.)

All the above problems have one thing in common: they can be written
as a large nonlinear system of equations with one or more parameters.
After linearization, we obtain a bordered matrix with the Jacobian of
the map minus the identity matrix as the big upper left block, and rows
and columns corresponding to additional constraints such as a phase-
or pseudo arclength condition, and the parameters respectively. In this
presentation, we will present a method, inspired on the Recursive Projection
Method of Shroff and Keller, to solve these nonlinear systems efficiently,
simultaneously computing the dominant eigenvalues of the Jacobian of
the map.

We will also present some results for steady-state and periodic solutions
of PDE systems.

**Kristen Morris**, University of Waterloo

**Controller Design for Infinite-Dimensional Systems**

The dynamics of many physical problems are modelled by partial differential
equations. These models have an infinite-dimensional state-space. In
general, it is necessary to use a numerical approximation to simulate
the response of the system and to compute controllers for the system.
However, a scheme that yields good results when used for simulation
may be inappropriate for use in controller design. For instance, a scheme
for which the simulation results converge quickly may have a corresponding
controller sequence that converges very slowly or not at all. Criteria
appropriate for evaluation of an approximation scheme will be discussed.
These ideas will be applied to the problem of designing a controller
to solve a $\mathcal{H}^\infty$ disturbance-attenuation problem.

**Jim Murdock**, Iowa State University

*Finding preserved geometrical structures in dynamical systems
via normal forms.*

A system of differential equations in normal form near a rest point
is often observed to decouple, or partially decouple, revealing various
preserved geometrical structures such as fibrations and foliations.
(A foliation is {\it preserved} if two orbits beginning on the same
leaf share the same leaf at all times as they evolve; the leaves themselves
need not be invariant.) It is often possible to see, in advance of computing
the normal form, what these preserved structures are going to be. These
structures are often not sufficient to render the normalized system
analytically solvable, which raises the question: can these structures
nevertheless be exploited to advantage when carrying out numerical solution
of the equations?

**N. Sri Namachchivaya**, University
of Illinois at Urbana-Champaign

*Nonstandard Reduction of Noisy Mechanical Systems *

We develop rigorous methods to study one to one and one to two resonance
problems as a random perturbation of four-dimensional Hamiltonian systems.
The focus of this paper is the development of general techniques of
stochastic averaging of randomly-perturbed four-dimensional integrable
Hamiltonian systems with an elliptic fixed point, and certain nontrivial
(yet generic) resonance condition. Classical stochastic averaging makes
use of these integrable structures to identify a reduced diffusive model
on a space which encodes the structure of the fixed points.

The interest of this paper is when classical methods fail because the
original Hamiltonian system has resonances, which give rise to singularities
in the lower-dimensional description}. At these singularities, glueing
conditions will be derived, these glueing conditions completing the
specification of the dynamics of the reduced model. The general dimensional
reduction techniques developed here, consists of a sequence of averaging
procedures that are uniquely adapted to study noisy mechanical systems.

Using the above theory, we develop numerical algorithms to computationally
study the reduced stochastic models for mechanical systems. Whereas
the original system is 4-dimensional, the state space of the reduced
model is a 2-dimensional graph; thus we are interested in various relevant
partial differential equations on this graph (viz. the stationary and
time-dependent Fokker-Planck equations for transition densities and
stationary distributions, the Pontryagin-Witt equation for mean exit
times, and other Feynman-Kactype equations).

This research is joint with Seunggil Choi, Richard Sowers, and Lalit
Vedula.

**Israel
Ncube, York University**

We consider a network of three identical neurons with multiple signal
transmission delays. The model for such a network is a system of delay
differential equations. With the aid of the symbolic computation language
MAPLE, we derive the corresponding system of ordinary differential equations
describing the semiflow on the centre manifold. It is shown that

two cases of a single Hopf bifurcation occurs at the trivial fixed point
of the full nonlinear system of delay equations, primarily as a result
of the structure of the associated characteristic equation. These are
(i) the simple root Hopf, and (ii) the double root Hopf. This presentation
focusses on the second case, paying particular attention to possible
change of criticality of the bifurcation.

**Dan Offin,** Queens University

**Instability of Symmetric minimizing orbits for Hamiltonian systems**

We will report on recent work in the area of stability/instability
for families of periodic solutions determined by variational principles.
The focus of this talk will be on applications to Hamiltonian systems
with symmetry groups, and the instability of symmetric action minimizing
orbits. We shall discuss several examples in the area of celestial mechanics,
where the symmetry groups arise from assumptions made on the symmetry
class of the periodic orbit. The figure eight orbit of Chenciner and
Montgomery for the planar three body problem is an example, and will
be briefly described. Another orbit family recently discovered by Chenciner-Venturelli
, the Hip-Hop family, for the spatial four body problem, may be analyzed
with the techniqes used to prove our results. We shall show why this
family is generically hyperbolic on a reduced space (modulo symmetries).
The numerical challenges of finding the orbits, and providing a stability
analysis is thereby shown to be solved simultaneously, if the variational
principle can be implemented numerically.

**Bart Oldeman**, University of Bristol

*A numerical implementation of Lin's method for homoclinic branch
switching*

Joint work with Prof. Alan Champneys and Dr. Bernd Krauskopf.

We present a new numerical method for homoclinic branch switching in
AUTO/HomCont. This method transfers a $1$-homoclinic orbit to an $n$-homoclinic
orbit, where $n>1$. Applications include studying inclination and
orbit flip bifurcations, homoclinic-doubling cascades and Shil'nikov
bifurcations.

This allows us to explore routes to chaos in applicable models. It also
gives the possibility to reliably find multi-hump travelling waves in
applications such as the FitzHugh-Nagumo nerve-axon equations and a
5th order Hamiltonian KdV model for water waves. We used Sandstede's
model, a theoretical normal form like system of ordinary differential
equations, as a testbed for the algorithm and later successfully applied
it to the applications mentioned above. Even though this scheme is based
on Lin's method, a theoretical approach, it is very robust and more
reliable than traditional shooting methods, especially if the unstable
manifold has a dimension higher than one. We give a demonstration of
the implementation in AUTO2000.

**Hinke Osinga**, University of Bristol

*Computation and visualisation of two-dimensional global manifolds*

We will address the problem of computing a two-dimensional global stable
or unstable manifold of a hyperbolic equilibrium. We focus on the technique
to grow the manifold as a collection of levelsets that are defined as
the sets of points with equal geodesic distances to the equilibrium
(along the manifold). Even though this method has restrictions, so far
we did not encounter these in practice. For the purpose of demonstration,
we present a constructed example to explain

these limitations.

Already in a three-dimensional phase space it may be hard to visualise
the results. We will briefly show some new ideas that are useful when
the manifold has a complicated geometry, such as, for example, the stable
manifold of the origin in the Lorenz system.

This is joint work with Bernd Krauskopf (Bristol).

**Randy Paffenroth**, Caltech

*AUTO2000 and Continuation of Periodic Orbits Around Lagrange Points*

AUTO is a software for continuation and bifurcation problems in ordinary
differential equations originally written in 1980 and widely used in
the dynamical systems community. Recently a modernization of the program
has been undertaken to improve several of its capabilities and the result
of this effort is called AUTO2000. A key component of this modernization
is the Python language, which has been used extensively to improve the
user interface and provide extended functionality. In this talk we will
discuss the implementation of the parameter continuation algorithms
that underly the AUTO2000 software and demonstrate how they may be applied
to the gravitational N-body problem. Our computational focus will be
the families of periodic orbits which eminate from the Lagrange points
in the Circular Restricted Three Body problem.

This is joint work with Eusebius Doedel at Concordia University, Herb
Keller at Caltech, Andre Vanderbauwhede in Gent, and Jorge Galan in
Sevilla.

**Dirk Roose**, K.U.Leuven

*Computing periodic solutions and homoclinic orbits of Delay Differential
Equations using DDE-BIFTOOL*

At K.U.Leuven, we have developed a Matlab-based software package DDE-BIFTOOL
for the numerical bifurcation analysis of systems of delay differential
equations with fixed and/or state-dependent delays. We use a collocation
approach for the computation of periodic solutions and their stability.
We have extended this method to compute homoclinic and heteroclinic
orbits, using an appropriate choice of the boundary conditions. The
capabilities of DDE-BIFTOOL will be illustrated through the analysis
of some model problems, incl. travelling waves in delay partial differential
equations. This is joint work with Koen Engelborghs, Tatyana Luzyanina
and Giovanni Samaey.

**Andy Salinger**, Sandia National Labs

*Stability Analysis Algorithms for Large-Scale Applications*

Stability analysis algorithms have been developed to work with massively
parallel application codes. These algorithms

include an eigenvalue approximation algorithm (for linear stability
analysis) and a set of continuation algorithms for

tracking turning point (fold), pitchfork, and Hopf bifurcations. Since
our aim is to improve the computational design capability of established
engineering codes, we have at first chosen algorithms that are non-invasive
over those which are more robust. We will discuss the ramifications
of that decision.

With these algorithms we have analyzed several incompressible flow
problems, ranging from classical Rayleigh-Benard problems to CVD reactor
models to free surface flow manufacturing applications. The algorithms
are shown to scale to 3D flow systems with finite element discretizations
of millions of unknowns, run on hundreds of processors.

(Joint work with Rich Lehoucq, Roger Pawlowski, Louis Romero, John
Shadid, Ed Wilkes, and Nawaf Bou-Rabee.)

**Bjorn
Sandstede****,** Department of Mathematics, The Ohio State
University

*On the numerical computation of PDE spectra of travelling waves*

Travelling waves (in particular fronts, pulses and wave trains) are
interesting solutions to partial differential equations on one-dimensional
unbounded domains. These waves can be computed numerically in a robust
fashion as solutions to appropriate boundary-value problems. The issue
addressed in this talk is the computation of the spectrum of the PDE,
linearized about a travelling wave. One way of computing the spectrum
is to truncate the domain to a large interval, to impose boundary conditions
at the endpoints, and to then compute the spectrum of the resulting
PDE operator, for instance, by discretization. Of interest is then the
effect of the boundary conditions on the spectrum. We discuss this issue
and show that boundary conditions can change the spectrum quite dramatically.
We also outline an alternative approach to the computation of spectra
that relies on Evans-function techniques.

**Tim Sauer**, George Mason University

*Shadowing breakdown and large simulation errors*

In chaotic systems, small errors are magnified, bringing into question
almost any computer simulation of the system. Shadowing techniques show
that under some circumstances the simulated trajectory may be a close
approximation to a true trajectory. In the absence of shadowing, long
correct trajectories may not be computationally available, and there
are open questions about how accurately long-term averages on the underlying
space can be computed. We discuss some of

these questions. In particular we propose a scaling law for the bias
between the correct time average and the expected value of the computer-simulated
time average, as a function of one-step error.

**Samuel S.P. Shen**, University of Alberta

*Forced Evolution Equations as Asymmetric Dynamical Systems: Bifurcation,
Stability, and Collision of Uniform Solitons*

Let (L, M) be a Lax pair. A forced evolution equation (a partial differential
equation) is of the form

L_t + LM - ML = f(x).

This is an asymmetric dynamical system and here f(x) is a given forcing
function decaying to zero rapidly as |x| goes to infinity. Because the
forcing term breaks those symmetries associated with the unforced systems,
the traditional analytical methods, such as the inverse scattering method
and Backlund transform, do not work any more. So far, one can only use
numerical methods and asymptotic approximation to solve this type of
forced evolution equations. In this talk the forced Korteweg-de Vries
equation, forced cubic nonlinear Schrodinger equation and forced sine-Gordon
equation are discussed. A user-friendly software has been developed,
which solves these three types of forced evolution equations. The software
is based upon the semi-implicit spectral method and its algorithm does
not require artificial dispersion or dissipation terms, which are commonly
used in algorithms. This algorithm is very accurate and efficient. Some
conspicuous solution behavior of the forced systems will be shown, which
does not occur in unforced systems, such as the generation and collision
of uniform upstream advancing solitons in a channel flow of water over
a bump. Another interesting result is that a stationary forced Korteweg-de
Vries equation can have multiple solitary wave solutions. Bifurcation
diagrams have been found analytically for some specific types of forcings.
For this bifurcation problem, it will be demonstrated how to use our
software to find out which branch of solutions are stable.

References:

1. S.S.P. Shen, Forced solitary waves and hydraulic falls in two-layer
flows, J. Fluid Mech., 234, 583-612 (1992).

2. L. Gong and S.S.P. Shen, Multiple supercritical solitary wave solutions
of the stationary forced Korteweg-de Vries equation and their stability,
SIAM J. Appl. Math. 54, 1268-1290 (1994).

3. S.S.P. Shen, R.P. Manohar and L. Gong, Stability of the lower cusped
solitary waves, Phys. Fluids A 7, 2507-2509 (1995).

4. S.S.P. Shen, Energy distribution for waves in transcritical flows
over a bump, Wave Motion 23, 39-48(1996).

**John Stockie**, University of New Brunswick

*Parametric Resonance in Immersed Boundaries*

Resonance is often discussed in the context of damped mechanical systems
subjected to external, periodic forcing, wherein the system is stable,
but exhibits a peak in the response at a critical resonant frequency.
Instead, we consider systems that are subjected to _internal forcing_
via periodic variations in a parameter, thereby giving rise to very

different solution behaviour.

In particular, we examine the stability of fluid flows containing immersed,
elastic boundaries, where the flow is driven by periodic variations
in the elastic properties of a solid material. Such a system is a prototype
for active biological tissues such as heart muscle fibres immersed in
blood. Using Floquet theory, we derive an eigenvalue problem which can
be solved numerically to determine values of the forcing frequency and
fluid viscosity for which the system becomes unstable. We also describe
direct numerical simulations of the fluid-structure interaction that
are being performed to verify the existence of these parametric resonances.

This is joint work with R. Cortez (Tulane), C. Peskin (NYU) and D.
Varela (CalTech).

**Emily Stone**, Utah State University and Abe
Askari, The Boeing Company

**Nonlinear Models of Dynamics in Drilling**

In this talk I will discuss our current research on the chatter instability
in drilling. Chatter is a self-excited oscillation between the machine
tool and the workpiece that limits productivity of machining operations,
reduces the quality of the product and shortens machine tool life. Up
until recently all models of chatter have been linear, with delay effects

in the case of regenerative chatter. These models only partially explain
the instabilities observed in the machining process.

In aircraft manufacture drilling is a critical machining process: over
a million holes may be drilled in the creation of a commercial passenger
jet. To address the problem of chatter in drilling, we are developing
a suite of nonlinear models of metal cutting that can be merged with
finite element studies of drill vibration modes and informed by large
scale of

simulations of metal cutting operations. Typically, engineering studies
of chatter have restricted themselves to the question of linear stability
of a steady cutting solution; in addition to that we are studying the
effects of the nonlinear terms in the model on the resulting dynamics.
Ultimately contact will be made with laboratory results from experiments
conducted in Seattle and St. Louis, with the goal of directing tool
design and allowing machine operators to avoid chatter regimes in drilling.

**Andrew
Stuart,** Mathematics Institute, Warwick University

**Particles in a Random Velocity Field**

(joint work with Hersir Sigurgeirsson, SCCM, Stanford)

The aim of this work is to find a mathematical model for the motion
of particles in a turbulent velocity field, consistent with experimental
observations about particle distributions; and then to use this model
to study the effect of particle collisions on particle distributions.

We describe a mathematical model in which the velocity field is modelled
as a Gaussian random field, and the particles are assumed to move according
to Stokes' law. The velocity field may then be viewed as the solution
of a stochastic PDE. An algorithm for the time-integration of the coupled
stochastic PDE-ODE is described, including the handling of collisions.
Some analysis of the cost of collision detection is also presented.

The model is analyzed in the framework of random dynamical systems
and shown to be well-posed; in addition a random attractor is shown
to exist. By a combination of numerical simulation, exploiting the existence
of a random attractor, and some analysis when a natural scale separations
occur, the particle distributions are studied, with and without collisions.

**Edriss S. Titi**, University of California

*Postprocessing Galerkin Methods *

In this talk we will present a postprocessing procedure for the Galerkin
method which involves the use of an approximate inertial manifold to
model the high wavenumbers component of the solution in terms of the
low wavenumbers. This {\it postprocessing Galerkin method}, which is
much cheaper to implement computationally than the Nonlinear Galerkin
(NLG) Method, possess the same rate of convergence (accuracy) as the
simplest version of the NLG, which is more accurate than the standard
Galerkin method. Our results valid in the context of spectral and finite
element Galerkin methods and for many nonlinear parabolic equations
including the Navier-Stokes equations. We will also present some computational
study to support our analytical results.

This talk is based on joint works with Bosco Garcia-Archilla, Len Margolin,
Julia Novo and Shannon Wynne.

**Erik S. Van Vleck**, Colorado School of
Mines

*Computation of Spectral Intervals for Nonautonomous Linear Differential
Equations *

It is well known that the real parts of the eigenvalues of the coefficient
matrix in an autonomous linear differential equation determine the stability
properties of solutions. However, for nonautonomous linear differential
equations simple examples show that the eigenvalues of the coefficient
matrix function can give incorrect stability information. We consider
different definitions of spectrum for nonautonomous linear differential
equations and their uses. We review known perturbation results, derive
some consequences, and show relationships between different types of
spectrum with

an eye toward the impact on numerical approximation of spectral intervals.

Joint work with Luca Dieci.

**W. Yao, P. Yu and C. Essex**, University of
Western Ontario

**Competitive modes and their applications**

We introduce a technique, in analogy to modes in linear systems, that
we call competitive modes. This method helps in redicting chaotic parameter
regimes for a given system, or for creating highly complicated chaotic
systems. Necessary conditions for a system to be chaotic are proposed.
Examples will be given to show how competitive modes are used for these
purposes.

**James Yorke,** University of Maryland

*Ensemble weather forecasting: when good forecasts go bad*

Our University of Maryland Group is developing techniques to tell how
reliable forecasts are.

**Yuan Yuan**, University of Western Ontario

*A review of the computation of the simplest normal forms*

This is joint work with P. Yu.

This talk is intend to give a review on the computation of the simplest
normal forms (SNF) of differential equations. After a brief introduction,
attention will be focused on the development of efficient methods for
computing the SNF with the aid of computer algebra systems. A number
of singularities will be presented. ``Automatic'' symbolic computer
programs using Maple will be discussed, showing the high potential of
using the SNF to solve complex dynamical systems.

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