# SCIENTIFIC PROGRAMS AND ACTIVITIES

February 26, 2017

## Numerical and Computational Challenges in Science and Engineering

### Speaker Abstracts

 Uri M. Ascher Kirsten Morris Dwight Barkley Jim Murdock Sue Ann Campbell N. Sri Namachchivaya Walter Craig Israel Ncube Eusebius Doedel Dan Offin Mitrajit Dutta Bart Oldeman Donald Estep Hinke Osinga Jorge Galan Randy Paffenroth Karin Gatermann Dirk Roose Martin Golubitsky Andy Salinger Willy Govaerts Bjorn Sandstede John Guckenheimer Tim Sauer Michael E. Henderson Samuel S.P. Shen Tony Humphries John Stockie Arieh Iserles Emily Stone Angel Jorba Andrew Stuart Oliver Junge Edriss S. Titi Yannis G. Kevrekidis Erik S. Van Vleck Eric Kostelich Weiguang Yao Bernd Krauskopf James Yorke Herb Kunze Yuan Yuan Rachel Kuske Gregory Lewis Andre Longtin Kurt Lust

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.

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.

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.