c orbits in nonlinear systems vicijanbouwe/phd/connecting-orbits.pdf · the combination of...

CONNECTING ORBITS IN NONLINEAR SYSTEMS

VICI GRANT PROPOSAL

JAN BOUWE VAN DEN BERG

AUGUST 28, 2011

1 REGISTRATION FORM (BASIC DETAILS)1A DETAILS OF APPLICANT

Name: prof.dr. G.J.B. (Jan Bouwe) van den BergMale/female: maleAddress: Department of Mathematics, VU University Amsterdam

de Boelelaan 1081, 1081 HV AmsterdamPreference for correspondence in English: noTelephone: 020 5987686Cell phone: 06 42403561Fax: 020 5987653Email: [email protected]: http://www.math.vu.nl/ ˜janbouwe/Use of extension clause: no

1

2 Jan Bouwe van den Berg

1B TITLE OF RESEARCH PROPOSAL

Connecting Orbits in Nonlinear Systems

1C SUMMARY OF RESEARCH PROPOSAL

Patterns are all around us: our fingerprints, the red spots typical for measles, the stripes on zebras, andthe convection rolls in the atmosphere and oceans which shape our climate. It is a formidable challengeto understand and predict the large contrasts featured in these patterns. Much is known about how suchpatterns start to develop from a homogeneous state, but we are still unable to predict or analyze the featuresof fully developed patterns. These fascinating nonlinear problems require the integration of computationswith topology.

Numerical simulation produces clear pictures of the dynamics, but the information it provides is localand non-rigorous, since in scientific computing one selects an approximate version of the problem, andcomputes on this prototype. The fundamental novelty of the proposed research lies in developing topolog-ical tools that unequivocally link the outcome of such calculations with the true problem. My goal is tocreate topologically validated computational machinery for finding the pivotal objects of interest in largeamplitude pattern formation: the paths along which dynamical systems change from one state into another.

These connections play an organizing role. First, connecting orbits describe localized patterns such aspulses and boundary layers. Second, they act as building blocks from which more complicated, sometimeschaotic, patterns can be constructed. Third, the transitions between different patterns, and their emergencefrom a homogeneous background, are connections between structured and trivial equilibrium states.

Only an innovative combination of computational methods and topological techniques can provideresults that are both sufficiently detailed and robust, validating predictions beyond the scope of simulations.The outcomes are then fully justified, which is vital for our aim of making rock-solid predictions aboutconcrete models. This is the way forward in key evolution problems for finite and infinite dimensionaldynamical systems, such as pattern formation phenomena.

1D KEYWORDS

nonlinear dynamicsconnecting orbitscomputer-assisted calculationstopological techniquespattern formation

1E HOST INSTITUTION

VU University Amsterdam

1F NWO DIVISION

EW (Physical sciences)

1G NWO DIVISIONAL DISCIPLINE

EW 03: Mathematics

1H NWO DOMAIN

Beta

VICI proposal — Connecting Orbits in Nonlinear Systems 3

2 RESEARCH PROPOSAL

2A DESCRIPTION OF THE PROPOSED RESEARCH

CONNECTING ORBITS IN NONLINEAR SYSTEMS

I OBJECTIVES

I.1 INTRODUCTION

We recognize patterns everywhere. They appear in myriad shapes with bright contrasts, their structurechanges and they occasionally vanish, the fundamental question being: how does one type of behaviourdevelop into another? The paths along which such changes occur are connecting orbits in an underlyingnonlinear dynamical system. In their organizing role in pattern formation, connecting orbits are crucial forour understanding. On the one hand these connections are robust objects, while on the other hand they arevery sparse in phase space, making them difficult to pin down.

By their very nature, connections require a global (geometric and topological) approach. Ideally this iscoupled with numerical calculations, which offer clear pictures of the dynamics, but the information thenumerics provide is local (in phase and parameter space) and non-rigorous. Strikingly, today’s advances incomputer speed and algorithm development make it possible to utilize the power and robustness of topo-logical methods to rigorously validate computational results. The current practice in scientific computingis to select a finite dimensional (i.e. approximate) version of the problem, and compute on this prototype.The fundamental novelty of our approach lies in developing topological tools that unequivocally link theoutcome of such calculations with the true problem. My goal is to create

Topologically validated computational machinery for finding connecting orbitswhich are pivotal in global nonlinear dynamics and large amplitude patterns.

Patterns distinctly visualize the complexity of the dynamical systems that describe the evolution of gen-eral nonlinear processes. By Conley’s decomposition theorem [21], connecting orbits universally form thebackbone of any nonlinear dynamical system. In the special case of gradient (i.e. dissipative) systems,all dynamics consists of connecting orbits going from one equilibrium state to another. General nonlineardynamical systems are decomposed into “recurrent components” (e.g. periodic orbits, homoclinic loops,chaotic attractors) and “transient dynamics”: the orbits that evolve the system from one recurrent compo-nent to another [21]. The connecting orbit structure is thus essential for understanding the dynamics.

The topologically validated computational approach is stable under errors in parameter values, andprovides precise information (e.g. about shape). The connections can thus be used as building blocksfor understanding complicated dynamics, as ingredients for forcing theorems, and as seeds of topologi-cal information in Morse-Conley-Floer complexes. The combination of qualitative global analysis withquantitative information from topologically validated computations will establish a breakthrough. The re-sulting understanding of connecting orbit structures leads to novel insights in global evolutionary aspectsof dynamical systems in general, and pattern formation in particular.

I.2 TOPOLOGY AND COMPUTATION

The state-of-the-art mathematical description of pattern formation is largely restricted to situationswhere either smallness of the pattern (bifurcation methods, normal forms), or separation of scales, can beexploited, sometimes rigorously (geometric singular perturbation theory, oscillatory integrals, justificationof amplitude equations), often formally through matched asymptotic expansions. Those “asymptoticness”assumptions (e.g. separation of scales) are usually made to make the problem mathematically tractable,often without physical justification. Moreover, such an analysis is feasible only if the asymptotic problemsare sufficiently simple, limiting its applicability considerably.

The objective of this proposal is to go far beyond the perturbative setting. Two complementary tools


Pattern formation

The transition from normal heartbeat to ventricular fibrillation is a potentially lethal phenomenon.Rather than contracting in a coordinated fashion, electric signals and contraction waves travel throughthe ventricular muscle in a seemingly chaotic fashion (due to “re-entry” of the waves), and the heart failsto pump blood into the arteries. The crucial issue is: how does the dynamical system (representing thespatio-temporal behaviour of the heart) move from one state (normal heartbeat) to another (re-entry)?

3/31/09!

1!

Jan Bouwe!van den Berg!

VU University Amsterdam!

Pattern Formation!

Floer Homology!

Equilibria!

Connecting orbits!

Compute it!

: chain groups!

: boundary operator!

Two dimensional patterns in fluidflows, chemical reactions, desertvegetation and zebra markings.

This is a seminal and extremely complicated instance of a tran-sition between spatio-temporal patterns. We see such patterns andtransitions everywhere around us in science and daily life, inspiringresearchers in a diversity of fields: physics (e.g. crystal growth, tur-bulent flows), biology (e.g. neural activation patterns, animal skins)medicine (e.g. spiral waves in heart tissue), geology (e.g. curvedrock layers, desert formation). The science of pattern formationrevolves around finding common principles behind patterns fromdifferent contexts that look similar. In all these studies connectingorbits act as important centers of information for three reasons.

First, we often observe very localized patterns, such as lightpulses (solitons) and cell walls (lipid bi-layers). In the appropri-ate dynamical system setting these correspond to connecting or-bits; homoclinic if a homogeneous state is the backdrop for a localpattern, heteroclinic when it concerns the boundary between two different states.

Second, when there are localized patterns, one often observes a great variety of combinations of such“simple” patterns (e.g. multiple pulses, polycrystalline materials, many patches of vegetation). Under-standing these more complicated patterns thus requires two steps: analyzing the individual buildingblock patterns, as well as understanding the “rules” governing the combinatorics of the building blocks.

Third, many patterns are not static but evolving. Usually a system stays in one configuration for along time and then transits to another. For example, the heart beats regularly for years until it suddenlytransfers to a fibrillating state. More generally, spatial patterns may be invaded by energetically morefavorable ones. In particular, the initial emergence of a pattern from a pristine background state is such atransition. These temporal transitions between (spatially) different states are precisely what connectingorbits in dynamical systems describe. When the limiting states are spatially homogeneous (e.g. onespecies completely eradicating another), then these dynamical systems are relatively gentle ordinarydifferential equations (ODEs). When the spatio-temporal behaviour concerns transitions between trulyspatial patterns (periodic or otherwise), then the dynamical system, usually a partial differential equation(PDE), is set in an infinite dimensional space.

In summary, in pattern formation connections both form the patterns and organize their structure.

assist in extrapolating the results from an asymptotic analysis towards large amplitudes, namely numericalsimulations and topological/variational techniques. While the reliability of conclusions based on computersimulations is often hard to judge, topological techniques usually provide little qualitative informationbesides mere existence of solutions. In the best cases one can determine the fixed points of the vectorfieldand their linearization. This local behaviour near equilibria tells us only a very limited part of the story:global dynamics is out of reach of direct analysis except in some special low dimensional cases.

With the computational techniques in this proposal we will be able to probe the global dynamics ofthe vectorfield to a significantly deeper level, including in particular the connections between equilibria,and more generally between sets with recurrent dynamics (such as periodic orbits). Once we get ourhands on these connecting orbits, we can draw refined conclusions about the global dynamics, decideon chaoticity, etc. To achieve this we need topologically validated computations: the vast amount of


calculations necessitates computer assistance, while topological validation rigorously controls the errorsdue to discretization, truncation and rounding, so that spurious solutions are avoided.

There are two major components to this proposal. First, to develop a flexible topologically validatedcomputational method for rigorously establishing connecting orbits. This gives the global dynamics ofspecific orbits. While this is often important in its own right, it also raises a second fundamental question:what does this tell us about the global dynamics of the entire system? The answer lies in forcing techniques,either based on (analytical) gluing methods or on (topological) Morse-Conley-Floer theory (see page 6).

The timeliness of this undertaking is not rooted in present-day computer speed alone. The compu-tational approach at the center of this proposal is based on recent ideas in my work with Lessard andMischaikow on stationary (periodic) patterns [8, 9]. We have developed a computational scheme wherethe patterns are obtained as zeros of a nonlinear map in some (infinite dimensional) function space. This isthen translated into a fixed point problem. In a neighborhood of an approximate fixed point we prove thata Newton-like operator is contracting, and the uniform contraction rate allows us to control rigorously theerrors due to computational approximations that we inevitably make. Since the contraction mapping prin-ciple is an “analysis-style” fixed point theorem, this requires a major analytic effort. Indeed, the Banachfixed point theorem lies on the crossroads of analysis and topology, and we call this rigorous error controlthe topological validation step.

The work with Lessard and Mischaikow is just the beginning, showing that the ideas can be effectuated.Now we must combine them with more elaborate tools to address connecting orbits, which are analyticallymuch more elusive. With the novel ideas put forward in this proposal, the time is right for a first full scaletopological computational theory for connecting orbit problems, which is applicable to nonlinear systemsin pattern formation and beyond.

I.3 PRINCIPAL EXAMPLE

To illustrate the computational approach, the associated analytic estimates, and some topological as-pects, we present an example related to periodic orbits, which allows us to focus on the main ideas and tiethe principal ingredients together. The Swift-Hohenberg PDE

∂u∂ t

=�

✓∂ 2

∂x2 +1◆2

u+au�u3 (1)

is a paradigm in pattern formation science. When heating from below a fluid between two plates, a finitewavelength instability leads to the occurrence of so-called Rayleigh-Benard convection rolls. The profile u,which depends on space x and time t, describes the size of convection rolls, and the parameter a indicatesthe magnitude of the driving force (the temperature difference between the bottom and top). When consid-ering stationary patterns, this reduces to the ODE

u0000+2u00+(1�a)u+u3 = 0, (2)

interpreted as a dynamical system in four dimensional phase space. Below we discuss how computationand topology can be combined to prove that the “spatial” dynamics of (2) is chaotic. Taking advantage ofthe associated variational framework, we can prove that all these stationary profiles of (1) are dynamicallystable under physical perturbations [8, 10]. In particular, we infer that when the temperature difference ina Rayleigh-Benard cell is sufficiently large, then there are many different attractors, all corresponding togeometrically different velocity profiles.

Our approach only requires proving the existence of a single periodic solution, which then by Morsetheory arguments, based on an underlying variational principle that we have developed [10, 35], implieschaoticity of the Swift-Hohenberg ODE, schematically [8]:

)

chaos. (3)

The topological forcing theorem can be viewed as a counterpart for fourth order ODEs of the famous


Global topology and dynamics: Morse-Conley-Floer theory

Morse theory relates the topology of a closed manifold X with the critical points of a function F on X .This relation involves the Morse index µ(x) of critical points x, i.e., the dimension of the unstablemanifold of x (or the number of negative eigenvalues of the Hessian D2F (x)) under the gradient flow

dxdt

=�—F (x).

Let bk be the k-th Betti number of X (characterizing part of its topology), and let ck be the number ofcritical points of F with Morse index k. In the generic case (all critical points being nondegenerate)these sequences bk and ck must satisfy the Morse inequalities

m

Âk=0

(�1)m�kbk m

Âk=0

(�1)m�kck, for any m = 0,1,2, . . . .

This implies, among others, that the number of critical points of any function F on X is bounded belowby the sum of the Betti numbers, at least in the generic case (the general statement is only slightlyweaker). Another informative way to describe the Morse inequalities is as follows:

•

Âk=0

bktk =•

Âk=0

cktk� (1+ t)

•

Âk=0

qktk, for all t 2 R,

where qk � 0 is a lower bound on the number of connecting orbits of the gradient flow between acritical point of index k+ 1 and one of index k. We conclude that Morse theory provides informationabout the number of critical points and connecting orbits of a gradientsystem. For example, for the depicted (deformed) torus, which has Bettinumbers b0 = 1, b1 = 2, b2 = 1, the six critical points force the existenceof at least one connecting orbit (this is a minimal scenario; there happen tobe more in this particular picture).

The relation between critical points, connecting orbits and the (global)topology becomes even stronger, supplying more detailed information,when we consider the Morse-Floer homology. This algebraic topolog-ical invariant can be constructed even for strongly indefinite variationalproblems, which are characterized by the property that all critical pointshave infinite Morse index µ . These types of problems are commonplace in pattern formation (cf. §II.3and §II.4). In essence, Floer’s approach resolves this issue by defining a relative index µ rel.

The Morse-Floer homology intricately encodes information about critical points and connectingorbits for gradient flows. The Conley index is a topological generalization of Morse theory that isapplicable to arbitrary (non-gradient-like) dynamics, and it relates the flow on (the boundary of) iso-lating neighborhoods with the topology of their invariant sets. Morse-Conley-Floer theory forms a setof techniques for unveiling global topological information about invariant sets of the dynamics. Thetheory is strong on general statements about minimal scenarios, but it comes up short in providing pre-cise data for concrete dynamical systems. In this proposal, computational methods are combined withMorse-Conley-Floer theory to get the best of both worlds.

“period-3 implies chaos” result for interval maps [49]. The variational nature of the problem causes solu-tions of (2) to conserve the energy

E = u000u0 �12

u002 +u02 �a �1

2u2 +

14

u4 +(a �1)2

4.

To force chaos we need to find a single periodic solution with energy E = 0 and shape as depicted in (3).In the remainder of this section we focus on the topologically validated computational approach for

finding concrete orbits in dynamical systems, which is at the heart of the current proposal. Since in thisexample we are looking for a periodic solution, we apply a Fourier transformation. Let 2p

L be the a prioriunknown period of the solution that we want to find. Taking advantage of the left-right symmetry of the


problem, we write u asu(y) = a0 +2

•

Âk=1

ak cos(kLy). (4)

This reduces the ODE (2) to an infinite set of algebraic equations for the Fourier coefficients {ak}, wherea�k ⌘ ak:

gkdef=⇥L4k4

�2L2k2 +(1�a)⇤ak + Â

k1+k2+k3=kki2Z

ak1ak2ak3 = 0, for all k � 0.

The energy constraint E = 0 can be (re)written as

e def= 23/2L2

•

Âk=1

k2ak +

"a0 +2

•

Âk=1

ak

#2

�a +1 = 0.

With the notation x = (L,a0,a1,a2, . . .) for the unknowns and f (x) = (e,g0,g1,g2, . . .) for the equations,we are thus looking for a solution of the infinite set of equations f (x) = 0, and we have arrived at the stageof the analysis where the specifics of the original problem are no longer visible.

We define the finite dimensional truncations xF = (L,a0, · · · ,am�1,0,0,0, . . .), and the Galerkin projec-tion

fF(xF)def=

e(xF)

gF(xF)

�,

where gF denotes the first m components of g. Note that fF has both a finitely truncated domain and afinitely truncated co-domain. A numerical solution xF = (L, a0, a1, · · · , am�1,0,0,0, . . .) of fF(xF) ⇡ 0 isfound via standard iteration methods.

Crucially, we now need to add a topological validation step (see [24, 8]) to turn this into a mathemat-ically rigorous proof. We stress that not the interval arithmetic, although necessary and computationallytime consuming, but the analytic error estimates due to the finite dimensional reduction, are the crux of theproblem.

Let the (m+1)⇥(m+1) matrix JF be the numerically computed inverse of the derivative D fF(xF). Wedefine the “linear part” of gk as l k(L) = L4k4

�2L2k2 +(1�a). Using these we define the linear operatoron sequence spaces

A def=

2

6666664

JF 0 0 0 · · ·

0 l m(L)�1 0 0 · · ·

0 0 l m+1(L)�1 0 · · ·

0 0 0 l m+2(L)�1

......

.... . .

3

7777775,

which acts as an approximate inverse of the linear operator D f (xF). Finally, to lift the finite dimensionalcomputation to the infinite setting of our original problem, we consider the operator

T (x) def= x�A · f (x), (5)

which is reminiscent of a Newton operator. Via rigorous estimates on the remainder terms we show thatT is a contraction map on a small ball B around xF in an appropriate Banach space. The (unique) fixedpoint of T in B solves f (x) = 0.

More precisely, we use a weighted `•-norm (s > 1)kxk= sup{|L|, |a0|, |a1|,2s

|a2|,3s|a3|,4s

|a4|, . . .}.

The weights ensure that the Fourier coefficients decay algebraically for all sequences in the associatedBanach space, in accordance with the smoothness of solutions to (2). The ball of radius r centered at xF is

BxF (r)def= xF +

•

’k=�1

�

r|k|s

,r|k|s

�, (0s def

= 1).

We look for fixed points of T by showing that T is a contraction mapping on these balls. Indeed, theunique fixed point ex, that results from the Banach contraction mapping theorem, satisfies f (ex) = 0 and thuscorresponds via (4) to a periodic solution of the ODE (2).


For the operator T to be contracting, the radius of the ball BxF (r) should not be too small (larger thanthe distance between the approximate solution xF and the true solution ex ), but also not too large (otherwiseuniqueness is lost and the estimates will fail). Analytically, we need the estimates��[T (xF)� xF ]k

��Ck,

supw,w2B0(r)

��[DT (xF +w)w]k�� eCk(r),

where Ck � 0 are explicit constants and eCk(r) are explicit (fifth order) polynomials in r (with positivecoefficients). It turns out that for sufficiently large k, say k � M,

Ck = 0, and eCk(r) =✓

Mk

◆seCM(r).

This uniform estimate enables us [8] to reduce the problem of showing that T is a contraction to merelychecking a finite set of polynomial inequalities of the form

Pk(r)def=Ck + eCk(r)�

r|k|s

< 0 for k =�1,0,1, . . . ,M. (6)

Cumbersome if not impractical to check by hand, this is easily done using interval arithmetic on a computer.Applying the forcing result (3) then leads to the results about convection patterns discussed at the start ofthis section.

I.4 STATE-OF-THE-ART

Equilibria, periodic orbits, connecting orbits and more generally invariant manifolds are the fundamen-tal components through which much of the structure of the dynamics of nonlinear differential equations isexplained. There is a vast literature on simulation techniques for approximating these objects. Particularly,the last thirty years have witnessed a strong interest in developing numerical methods for connecting orbits[18, 17, 32, 47]. As mentioned in [26], most algorithms for computing heteroclinic or homoclinic orbitsreduce the question to numerically solving a boundary value problem on a finite interval, where the bound-ary conditions are given in terms of linear or higher order approximations of invariant manifolds near thesteady states.

This proposal builds on the continuing progress in scientific computing. On the other hand, the veracityof the numerical results can typically not be faithfully established, and we need to overcome this obstacle.The field of rigorous numerics (using interval arithmetic) has made a lasting impact in dynamical systems,by extracting coarse topological information from the systems, often revealing complicated dynamics.Particularly, the proof that the Lorenz attractor is chaotic [64, 53] was an essential breakthrough. In recentyears there has been remarkable progress in computational methods, fueled by both computer speed and thedevelopment of novel algorithms. Advances have been most rapid in the analysis of maps, i.e., dynamicalsystems with discrete time, which by their very nature are more receptive to computational techniques,see e.g. [1, 23]. In parallel, the underlying theory has significantly progressed, especially concerning thecombinatorial approach to Conley index computations [42].

In the last years a variety of authors have developed tools, combining interval arithmetic with topo-logical constructions, to prove the existence of homoclinic and heteroclinic solutions for specific ordinarydifferential equations. One effective approach uses advanced ODE integrators and interval arithmetic toinvestigate the images of rigorously integrated sets on a collection of Poincare sections, and derive re-sults from the topological information thus obtained through Conley index arguments (e.g. [65, 46, 2]).Other successful methods are based on the Melnikov function to detect intersections of stable and unstablemanifolds [39], and functional analytic arguments combined with spectral estimates [13].

The current proposal selects another strategy, which is most amenable to the problems originating frompattern formation (high or infinite dimensional, dependence on physical parameters), and which mostclosely matches the developments in scientific computing. Our approach, see §I.3, has been shown to workfor periodic solutions in [8, 22, 24], while some of the ideas go back to [68, 56]. Recently, we have refinedthe analytic estimates considerably, which enables us to solve more delicate problems (see Section I.3


for an overview and [8] for details), including periodic solutions in multiple dimensions [33] and delayequations [48]. Besides, we have shown how this method can be adapted to encompass rigorous parametercontinuation and branch following [9], so that we can track families of solutions. These developmentsform the ideal platform for taking the next leap and tackling the important and elusive connecting orbits.

The current proposal builds on the state of the art and introduces the following innovative features:

1. Novel rigorous combination of topologically validated computational techniques with (approximate)parametrizations of local stable and unstable manifolds. Since the topologically validated computa-tional machinery is rooted in a functional analytic framework, it is scalable and hence ideally suited forextension to infinite dimensional problems, such as connections in pattern forming PDE models.

2. Focus on pivotal solutions of continuous time dynamical systems, namely ordinary and partial differ-ential equations. This emphasis on continuous time problems is essential, since differential equationsare by far the most utilized type of mathematical models in physics, chemistry, biology, medicine andeconomy.

3. The topologically validated computational approach is robust under errors in physical parameter values,and provides precise quantitative (e.g. the shape) as well as qualitative (e.g. stability) information aboutthe connecting orbits.

4. Since topological validation leads to rigorous existence results, the connecting orbits can be used asnew seeds of information in Morse-Conley-Floer theory, leading to forcing theorems and insight in theglobal structure of the nonlinear dynamics.

II METHODOLOGY

The topologically validated computational technique described in §I.3 for periodic solutions, must beoverhauled for the purpose of finding connecting orbits. Let us outline the main issues, focussing onconnecting orbits for systems of coupled second order ODEs

d2~udx2 = Y(~u), (7)

where~u 2Rn, and Y : Rn!Rn is a smooth nonlinearity. Such systems are ubiquitous in reaction-diffusion

problems in chemistry and biology, and a plethora of patterns is found in such models. Reaction-diffusionprocesses are connected to morphogenesis in biology, as well as at the basis of animal coats and skinpigmentation [54, 51]. They are thus ideal for pilot studies, and we have implemented successfully themain ideas to demonstrate “proof of concept”, taking several shortcuts available for a test problem.

For the sake of clarity (both for exposition and to avoid technicalities) we assume the symmetry con-dition ~u0(0) = 0, i.e., we look for symmetric homoclinic orbits, corresponding to left-right symmetriclocalized patterns, which satisfy the asymptotic “boundary” conditions

limx!±•

~u(x) =~u• 2 Rn.

Let y def= (~u0,Y(~u)) be the vectorfield in 2n-dimensional phase space corresponding to (7). We assume that

the limit point p = (~u•,~0) 2 R2n is a hyperbolic equilibrium with a stable manifold W s(p) of dimension n(reversal symmetry implies the unstable manifold is also n-dimensional). Hence, a symmetric connectingorbit corresponds to a solution of the boundary value problem (L large)8

><

>:

~u00(x) = Y(~u(x)), in [0,L],~u0(0) = 0,(~u(L),~u0(L)) 2W s

loc(p).

(8)

Before furnishing a finite dimensional reduction necessary for a computational approach, we considerthe description of the stable manifold W s(p). One can obtain rigorous approximations of the stable mani-fold is a variety of ways, such as the topological construction in [70]. We make use of the parameterizationmethod for invariant manifolds, which facilitates efficient, high order approximations of local stable and


unstable manifolds. The theoretical developments of these parametrizations [14, 15, 16] have led to nu-merics for discrete time maps [41]. In this project we pioneer the computational algorithms for continuoustime flows, including rigorous control in the form of explicit and sufficiently sharp error estimates.

Let us assume the linearization Dy of the vectorfield at the equilibrium p has n distinct eigenvalues{l s

i}ni=1 with strictly negative real part, with {vs

i}ni=1 the associated eigenvectors. Let Ls be the n⇥ n

diagonal matrix having {l si} on the diagonal, and As be the 2n⇥ n matrix whose columns are the stable

eigenvectors {vsi}.

The parameterization method provides an injection P : Bd (0)⇢Rn!R2n, d > 0, such that P(0) = p,

DP(0) = As, and P (Bd (0)) ⇢W s(p), i.e., it parametrizes the n-dimensional local stable manifold of p.We express P as a power series

P(q) = Â|k|�0

ak q k , (9)

where k is a multi-index, and the coefficients ak are determined recursively through the requirement thatP satisfies the invariance equation y[P(q)] = DP(q)Lsq .

Writing P = (P(0),P(1)), problem (8) corresponds to finding a solution (q ,~u) 2 Rn⇥C2([0,L])n of

8><

>:

~u00(x) = Y(~u(x)), in [0,L],~u0(0) = 0,~u(L) = P(0)(q), ~u0(L) = P(1)(q).

Integrating the differential equation once from the left and once from the right we obtain the integralequation

~u(x) = f (q ,~u)(x) def= P(0)(q)+(x�L)

Z x

0Y(~u(s))ds+

Z L

x(s�L)Y(~u(s))ds.

Finally, appending the remaining boundary condition, one defines F : Rn⇥C([0,L])n

!Rn⇥C([0,L])n by

F(q ,~u) def=

P(1)(q)�

R L0 Y(~u(s))ds

f (q ,~u)�~u

!. (10)

Solutions of F(q ,~u) = 0 then correspond to symmetric homoclinic solutions of (7).For the periodic solutions in §I.3 it was suitable to use Fourier transformation and truncation of the series

to obtain a finite dimensional reduction. Here we need a different strategy, as the Fourier series convergetoo slowly, and we select an approach naturally adapted from scientific computing. We begin by choosinga mesh {xi = ih}m

i=0 of the interval [0,L] (h = L/m). Consider the subset Sh ⇢ C([0,L]) of piecewiselinear continuous functions with nodes at {xi}. One can identify Sh and Rm+1. Given v 2C([0,L]) definePhv = (v0,v1, . . . ,vm) 2 Sh by vi = v(xi), for i = 0, . . . ,m. We denote by uh = (Ph)n~u the component-wisepiecewise linear projection.

Choosing a parameterization order N 2 N, we let

PN(q) =⇣P(0)

N (q),P(1)N (q)

⌘def= Â

0|k|Nak q k .

The finite dimensional projection F(m,N) : Rn⇥ (Sh)n

! Rn⇥ (Sh)n of (10) is

F(m,N)(q ,uh) =

P(1)

N (q)�R L

0 Y(uh(s))ds

f (m,N)(q ,uh)�uh

!,

where each f (m,N)i (q ,uh), i = 1, . . . ,n, is given by

[ f (m,N)i (q ,uh)] j = [P(0)

N (q)]i +(x j �L)Z x j

0Yi(uh(s))ds+

Z L

x j

(s�L)Yi(uh(s))ds.

Analogous to the setup for periodic solutions, we have the full infinite dimensional problem F(q ,~u) = 0and a finite dimensional approximation F(m,N)(q ,uh) = 0 side by side. Analytic techniques are needed toestimate the defect between the two. For the integral equation component these estimates turn out to bemore tractable than for the truncated Fourier series, whereas the components involving the finite truncation


of the parametrization of the local stable manifold are much more involved, but they can in this setting beaccomplished using techniques from complex analysis [7].

With this outline of a topologically validated computational approach to connecting orbits as a basis, wenext describe four (interconnected) research problems which form the nuclei of PhD and Postdoc projects.They require a combination of techniques from both ODE and PDE theory, substantial computationalimplementation and algorithm development, as well as advances in their underlying foundational theory.As such, the subprojects will give ample opportunity for the PhD students and Postdocs involved to developtheir skills, learn new techniques, and make seminal contributions to dynamical systems theory and themathematics of pattern formation.

While the nature of exploratory research implies that one cannot foresee all obstacles or provide avery detailed workplan, we nevertheless outline the main ideas in each project. Furthermore, we indicateapplications to pattern formation problems throughout.

II.1 CONNECTIONS IN SYSTEMS OF ODES

The first subproject concerns developing efficient algorithms for topologically validated computationsof connecting orbits in general systems of ordinary differential equations. We have implemented the mainideas described above for a test problem [7]. However, the current approach is “quick-and-dirty”, takingfull advantage of the specifics of the problem. In the seminal steps towards a general method for systems ofODEs, it is crucial to monitor computational efficiency. We will capitalize on prevalent simulation methodsfrom scientific computing (cf. [27, 59]).

Goal: Develop a general framework for topologically validated computations of con-necting orbits in systems of nonlinear ODEs.

For the general system u0 = j(u) with u2Rn, an integral formulation is readily available: u(t) = u(0)+R t0 j(u(s))ds. By approximating u by piecewise linear functions, we may again discretize. Note that the

method is very flexible: higher degree splines and refined grid choices, both standard practice in scientificcomputing, are painlessly incorporated. Furthermore, in this integral formulation the errors accumulatedisproportionally at the right end of the interval, hence it is better to divide the interval into subintervalsand discretize the equivalent system u(t)�u(t i)�

R tti

j(u(s))ds = 0 for t 2 (t i,t i+1], i = 1, . . . ,`.The parametrization of general stable and unstable manifolds, as well as the associated required analytic

bounds, need careful scrutiny, aiming for estimates that are sufficiently sharp to satisfy computationalrequirements. There is a foundation of theoretical work to build on here [14, 15, 16]. For high dimensionalmanifolds it is not practical to parametrize the entire (un)stable manifold. Instead, parametrizations andapproximations of lower dimensional slow stable invariant manifolds will be needed. Namely, connectingorbits generically enter the equilibrium along the eigendirection corresponding to the leading eigenvalue,or the next-leading one in case of an orbit-flip configuration [40]. The parametrization estimates thus maybe restricted to those directions, with special care for the role of resonances between eigenvalues [14, 16].

We aim for a generally applicable (and available) code. Indeed, there are many problems where thequestion of existence of connecting orbits is an obstacle to the full analysis of the problem, since even inlow dimensional normal form systems the relevant solutions are often intractable analytically (e.g. [38]).Our approach will lead to a breakthrough. Transitions described by connecting orbits appear for examplein reaction-diffusion models in developmental biology [36], chemical reactions [37] and desert forma-tion [45]. One instance of a seminal application to pattern formation in fluid dynamics is the balancebetween hexagonal and stripe patterns [28]. For a modified version of the Swift-Hohenberg equation withan up-down symmetry breaking term, an asymptotic analysis was performed for patterns just above theonset of the instability of the trivial state. Transitions between stripes and hexagons were found to begoverned by the system of second order equations [28]

(4A00

1 + cA0

1 +b1A1 �b2A22 �3A3

1 �12A1A22 = 0,

A00

2 + cA0

1 +b1A2 �b2A1A2 �9A32 �6A2

1A2 = 0.(11)


Here A1 and A2 are the amplitudes of stripe and hexagonal patterns, c is the invasion speed of the pattern,and b1 and b2 are parameters measuring the distance to the onset of instability and the extent of symmetrybreaking, respectively. Starting with the case of co-existence (c = 0) we will apply the developed methodsto prove the existence of (moving) boundary layers between the two patterns.

A supplementary fundamental avenue that needs exploration is to incorporate symmetries inherent tomany prototypical pattern forming systems, which requires an adapted strategy to accomplish the necessarytopological robustness. For example, in (11) the case with zero propagation speed is Hamiltonian. Sucha variational symmetry, shared with many other pattern forming systems (e.g. Equation (2)), causes non-genericity of the dynamics. For periodic solutions it means that they come in families (hence the additionalrequirement E = 0 in §I.3), while for connecting orbits the intersection of stable and unstable manifolds isnontransversal. Our customary method demands transversal intersections, and we will use unfoldings andphase conditions (cf. [18]) to handle the nontransversality caused by the symmetries fundamental to manypattern forming systems.

Another generalization concerns connecting orbits between periodic orbits rather than between equilib-ria. In the context of differential equations one of the simplest and best understood means of “generating”chaotic dynamics is through solutions that are homoclinic to periodic orbits. There are two fundamentalprerequisites to obtain results on connections between periodic orbits. First, one must be able to rigor-ously and accurately identify periodic orbits. The method presented in §I.3 has proven effective for thistask [48, 8]. Second, one must be able to parameterize stable and unstable manifolds for the periodic or-bits. In principle there is no obstacle to the parameterization method being employed [16], and we intendto instigate an algorithmic implementation suitable for topological validation.

II.2 FAMILIES OF CONNECTIONS AND BUILDING BLOCKS

Establishing connecting orbits through topologically validated computations is just the first step. In thissubproject we examine their dynamically relevant properties.

Goal: Use connecting orbits as building blocks for complicated patterns.

The first step is to study the dependence on parameters. Most physical models have parameters andone needs to be able to investigate the dynamics over large ranges of parameter values. Thus, the abilityto efficiently and rigorously compute branches of solutions is highly desirable. For example, consider thedependence of the Swift-Hohenberg (2) on the parameter a , so that (after Fourier transforming) we needto solve f (x,a) = 0 for an interval of parameter values a , cf. §I.3. Under the assumption that Dx f (x,a)is nonsingular along the branch of zeros that we are computing, the Implicit Function Theorem impliesthat the branch of zeros can be viewed globally as the graph of a function of the parameter a . In animplementation, when computing on the finite dimensional reduction fF(xF ,a) = 0, we first compute anapproximate zero xF at a fixed value a = a0. Next, we compute a vector xF tangent to the solution graph:

Dx f (xF ,a0)xF +Da f (xF ,a0)⇡ 0.

Using the vectors xF and xF , we define the set of predictors by xa = xF +Da xF , where Da = a �a0 issmall. We now look for fixed points of Ta(x) = x�A· f (a,x) in balls Bxa (r) centered at these predictors.After developing the appropriate analytic bounds, the problem reduces to checking a finite number of poly-nomials inequalities of the form Pk(r, |Da |)< 0, k =�1,0, . . . ,M, cf. (6). The polynomials Pk now dependon both the distance r in Banach space and the distance |Da | in parameter space to the base point (xF ,a0).

Repeating this process, we can topologically validate the predictor-corrector algorithm to find parameterdependent families of solutions, which form continuous branches due to a pasting procedure that exploitsthe convenience of the polynomial inequalities, see [9]. The method is amenable to rigorous pseudo-arclength continuation as well [9, 43]. The main challenge in adapting this method to connecting orbits isconstructing algorithms for obtaining parameterized families of (un)stable manifolds. From the theoreticalpoint of view this has been described in [15], and we shall focus on the open question of a topologicallyvalidated computational implementation.


The next problem is capitalizing on the connecting orbits as building blocks for more complicatedpatterns. Namely, there are various results that concatenate connecting orbits to form new patterns. For ap-plying these “gluing” theorems, such as [25, 62, 60], transversality of the unstable and stable manifolds isthe vital property. Since the connecting solutions are obtained through a topological contraction principle,they are unique and, crucially, stable under perturbations. Intuitively it is thus clear that they must corre-spond to transversal intersections of stable and unstable manifolds — otherwise perturbations would easilyruin (uniqueness of) the solution — and due to the contraction mapping at the base, a proof using analytictechniques is within grasp. Such a mathematical proof will be a novel results both for periodic patternsand connecting orbits. Our approach thus puts us in a unique position, enabling the rigorous verification oftransversality criteria required in forcing and gluing theorems.

A related fundamental problem concerns the “physical” stability against perturbations when the con-nection is an equilibrium solution of a PDE. This is important because unstable solutions are unlikely tobe observed in experiments or show up in nature. This problem is connected to transversality, since non-transversality of the connecting orbit corresponds to the presence of a zero eigenvalue, the border betweenstability and instability. For self-adjoint problems this means stability cannot change along continuousbranches of transversal connecting orbits. The relation between the stability properties in de PDE and thespectrum of the matrix D fM(xF) in the finite dimensional reduction needs to be investigated: the “domi-nant” part of the linearization (which determines (in)stability) is expected to be captured by the eigenvaluesof the finite dimensional reduction, cf. [13].

Finally, it will be interesting to investigate the opposite of continuous branches, namely bifurcations. Wewill work towards a topologically validated computational technique for bifurcations based on a combina-tion of the contraction mapping ideas summarized in §I.3, together with simultaneously obtained informa-tion on the derivatives along the solution curve and with respect to parameters, so that general bifurcationresults (cf. [19]) can be invoked with explicit analytic bounds. Such methods would be a breakthroughfor periodic patterns of ODEs/PDEs (cf. [69]), while applications to connecting orbits, with their zoo ofpossible bifurcations [40], are the subsequent natural step.

II.3 CONNECTING ORBITS IN PARTIAL DIFFERENTIAL EQUATIONS

Connecting orbits in PDEs describe transitions between patterns. The main challenge, as compared toODEs, is the infinite dimensional phase space.

Goal: Compute and validate connecting orbits in infinite dimensional systems.

We will focus initially on connecting orbits in a purposefully chosen paradigm, the nonlinear Cauchy-Riemann equations ⇢

pt +(qx �Hp(p,q,x)) = 0,qt � (px +Hq(p,q,x)) = 0, (12)

where H(p,q,x) 2C2(R2⇥S1) is a planar non-autonomous Hamiltonian. The stationary problem reduces

the PDE to the Hamilton equations. Equation (12) is a classical strongly indefinite PDE [31], with linksto pattern formation, and there remain plenty of open questions. It admits a variational formulation as the(formal) gradient flow of the functional

F (p,q) =Z 1

0

⇥pqx �H(p,q,x)

⇤dx, (13)

bringing the possibilities of Morse-Floer homology into play to explore the dynamic and topological con-sequences of connecting orbits, once they have been found. The strongly indefinite character exemplifiesthe infinite dimensional challenges. Results will be new already for stationary patterns, to which our ODEmethods should be extendable in a relatively direct manner, though the main aim of this subproject is todevelop widely applicable topologically validated computational techniques for connecting orbits in PDEs.

First, to find the t-independent limit states (x-periodic equilibria (ep(x),eq(x))), the method as summa-rized in §I.3 is pertinent. It should be noted that it would be the first time these techniques [8, 34] are used


x

q

p

+

!

Figure 1: Left: strands of a braid. Right: a positive and a negative crossing.

for strongly indefinite problems. However, the constructions do not directly use the finiteness of the Morseindex and we are confident that our methods will extend to the strongly indefinite setting. Indeed, the con-traction rate of operator T in (5) depends on the eigenvalues of A�1

⇡ DF(xF)⇡ —2F (ep,eq): heuristically,the larger their magnitude is, the stronger the contraction is. Thus, eigenvalues with large magnitude donot adversely effect our rigorous computational methods.

To make use of the critical points, their relative index µ rel, as defined in the context of Floer theory, mustbe computed in order to determine the dimension of the intersection of stable and unstable manifolds. Ourapproach to resolve relative indices is based on two related observations. First, our methods for establishingequilibria are based on a finite dimensional computation, and using regularity of the solutions to bound theerrors induced by the truncation. Second, the relative index is equivalent to the Fredholm index of thelinearized equation (or, more generally, a homotopy of linear operators), hence there exist a priori boundson the set of eigenvalues which cross zero as a result of the spectral flow. Combining these argumentssuggests that, with appropriate global estimates, the relative index can be determined via an absolute indexfor the corresponding finite dimensional representation.

Having acquired detailed information on the equilibria, attention turns to the connections between them.To adjust the topologically validated computation method for connecting orbits from the ODE to the PDEsetting, one may Fourier transform (12) in the x variable to obtain an infinite set of ODEs. This does notchange the main issue of looking for connecting orbits in an infinite dimensional space, but it highlights theanalogy with the ODE problem. The infinite set of differential equations needs to be finitely truncated todo computations, and this brings fresh challenges to the PDE setting. First, analytic estimates are neededto control the defect introduced by this truncation. Since the bounds on the truncation errors are closelyrelated to the regularity of solutions, we do not expect insurmountable obstacles, as all solutions of (12)are smooth.

The second pressing problem is the infinite dimensionality of the (un)stable manifold. The parametriza-tion method works in general Banach spaces [14], but under the assumption that the underlying dynamicsis a flow, hence it is not directly applicable to indefinite problems. Even when one can overcome this,parametrizing infinite dimensional manifolds is impractical from a computational point of view. However,since orbits enter the equilibria along the leading eigenvalue direction, there is no need to approximatethe entire stable manifold: the parametrization techniques should be “localized” along these leading direc-tions. Here we may capitalize on the fact that the theory [14] is applicable to (finite dimensional) invariantsubmanifolds. Once again, an important aspect will be to bring the theory and computational algorithmstogether through explicit analytic error estimates.

The prototype problem (12) has a special structure, which enables us to harness the discovered connect-ing orbits in a remarkable way. Since (12) is the formal gradient flow of (13), we may consider the Floerhomology of maximal invariant sets. Therefore, both equilibria and connecting orbits obtained throughtopologically validated computations, can be used to force the existence of other solutions. The crucialtopological forcing structure is expressed by braids: consider the evolving strands

b k(·)def=�(x,qk(·,x), pk

x(·,x))�� x 2 [0,1]

,

where {(qk(t,x), pk(t,x))}Kk=1 are solutions of (12). As illustrated in Figure 1, these solution form braids

with both positive and negative crossings. It turns out that the maximum principle implies that along


the evolution the braid can only change positive crossings into negative crossings. As a consequence,braid classes are isolating neighborhoods and we may consider the Floer homology of a braid class [6].Since we have independent methods for calculating this invariant [6], it is possible to combine that globaltopological information with the local information of rigorously computed equilibria and connections toforce a multitude of braided solutions of (12). Somewhat surprisingly, there is a connection with mixingprocesses of viscous fluids, which are elegantly captured [12] by the t-independent solutions of the Cauchy-Riemann equation (12). Finding connecting orbits for (12) may thus reveal, through Floer homology,information about those mixing solutions.

Finally, the developed methods will be applicable to more general systems. The first step is to replacethe Hamiltonian vectorfield in (12) by a general one. Such equations, although no longer gradient systems,have Poincare-Bendixson like behaviour [30]. The structure of the invariant set is thus still relativelysimple. Nevertheless, it means that t-periodic orbits come into play, and the resulting connections betweenthese pose new computational and analytic challenges. This has a geometric-algebraic counterpart in theaccompanying necessary adaptation of the construction of Floer homology in the spirit of Morse-Botthomology [11].

Other application areas of the developed techniques include connecting orbits in lattice dynamics anddelay-differential equations (cf. [50, 48]), as well as spiral waves in reaction diffusion systems, whichin suitable coordinates may be described as connecting orbits to periodic stationary patterns (modulatedwaves) in nonlinear PDEs without well-defined initial value problems [61].

II.4 MODULATED TRAVELLING WAVES

The final subproject concerns propagating disturbances in PDEs, a problem which in the last few yearshas taken central stage in pattern formation [28, 29]. Modulated (non-uniformly) translating structures areobserved in a variety of systems, leading to stripes, (hexagonal) spots, and spirals [4, 61]. These appearin fluid dynamics (discussed below) and solid combustion models [3], whereas two-dimensional spiralwaves, which can described by a very similar mathematical setting, appear in chemical reactions (e.g. theBelousov-Zhabotinsky reaction) [67], in the oxidation of carbon-monoxide on platinum surfaces [55], andin cardiac tissue [66].

Existence of periodically modulated travelling waves has been proved in perturbative, small ampli-tude settings [20], but fully nonlinear methods are presently unavailable, and we outline below a noveltechnique.

Goal: Use Morse-Floer homology to investigate the propagation of periodically modu-lated travelling waves.

Consider, for definiteness (one easily generalizes to higher order equations, systems of equations, dif-ferent nonlinearities, etc), the fourth order equation (incorporating both Swift-Hohenberg and extendedFisher-Kolmogorov models) [63, 57]

ut =�g uxxxx +buxx +u�u3, g > 0, b 2 R. (14a)When one perturbs the unstable homogeneous state u ⌘ 0, travelling structures are observed that leavebehind a stationary periodic pattern, see Figure 2. In particular, these are not standard travelling waves (i.e.u(x, t) = u(x� ct)), but rather modulated travelling waves characterized by the shift-periodicity

u(t +T,x) = u(t,x�L) for all t,x 2 R, (14b)u(t,x)!U

±

(x) as x !±•. (14c)Here U

±

are L-periodic stationary solutions, and c = LT is the velocity of the connecting front. If U

±

arenot both homogeneous states this is not a uniformly travelling wave.

Introduce new variables (s,y) by u(t,x) = v(ct � x,x) = v(s,y). Then (14b) is equivalent to the period-icity v(s,y+L) = v(s,y), while (14c) is equivalent to lims!±• v(s,y) = U

⌥

(y). In the new variables theoriginal PDE (14a) transforms into

g (vssss �4vsssy +6vssyy �4vsyyy + vyyyy)�b(vss �2vsy + vyy)+ cvs = v� v3. (15)


0 50 100 150

−1

−0.5

0

0.5

1

Figure 2: Left: modulated travelling wave of (14a) for g = 1, b =�

13 . Right: sketch of the corresponding

connecting orbit in infinite dimensional phase space between an equilibrium and a periodic state.

We may thus consider the space of L-periodic functions in y, and look for a connecting orbit of (15) in thetime-variable s between the cycles U+ and U

�

.The PDE (15) does not have a well-posed initial value problem in s. On the other hand, the functional

L =Z L

0

hg2(v

2yy �6v2

sy �8vssyvs +2vsssvs � v2ss)+

b2 (v

2y � v2

s )+14(v

2�1)2

idy

is a Lyapunov functional: dLds =�c

R L0 v2

s dy 0. The setting is very similar to that of spatial dynamics [44],but with a variational twist.

While the techniques that will emerge from the approach in §II.3 are eventually applicable to this type ofmodulated travelling wave problem, in this subproject we take a different, original route to understand theconnecting orbit structure. It has the advantage of disclosing a collection of connections simultaneously.The approach is based on knowledge of the periodic stationary limit states, which can be assessed usingthe techniques from §I.3, and the fact that the system is variational, hence the natural domain for Floerhomology. Actually, it makes for a nonstandard, innovative application of Floer homology theory, sincewe intend to formulate results about connecting orbits, using known information about critical points,rather than the usual exclusive focus on equilibria.

The main goal of this subproject is therefore to develop a Floer homology in this setting, to explore theexistence of connecting orbits and thus to prove the existence of modulated travelling waves. As in all Floerhomology constructions, we need to obtain bounds on solutions, so that we can employ compactness toextract converging sequences (these bounds should descend from standard ones for the original parabolicequation (14a)). Furthermore, as also alluded to in §II.3, we need to understand the spectra of criticalpoints and relate them to the Fredholm index of connecting paths.

Finally, to extract information about connecting orbits corresponding to periodically modulated travel-ling waves, we need to combine the global topological data bestowed by Floer homology, with an analysisof the critical points (stationary periodic patterns). There are two roughly complementary approaches:1. Local: when U+ is close to U

�

(e.g. they appear in a saddle-node bifurcation, cf. [58]) one can use abifurcation analysis to calculate the homology (in an isolating neighborhood of U

±

) to find connectingorbits.

2. Global: continuation away from bifurcation points can be accomplished using a Floer homology ana-logue of the Conley index techniques from [52] (cf. [22]), as long as the bifurcation diagram of sta-tionary solutions is known. For example, for b >

p

8g the bifurcation diagram of (14a) is completelyunderstood [5], while in other situations we will use topologically validated continuation computationson the stationary solutions.

As generalizations one should subsequently consider problems in several spatial dimensions, i.e., prob-lems in which a more-dimensional periodic pattern (hexagonal for example) is generated behind the front.


III CONCISE PLAN OF WORK

III.1 RESEARCH PLAN

The research team is envisaged to comprise the PI, two Postdocs and two PhD students. Of the foursubprojects listed in §II, the first one, developing the method for general systems of ODEs (§II.1) is suitablefor a PhD student: he/she can start by working out several applications (as indicated in the main text) anddevelop the method in steps, with implementation issues being a major component, and a smaller amountof time spent on theoretical aspects. These will be developed in part in collaboration with the first Postdoc,who will work on adapting the method to connections in infinite dimensional settings (PDEs). This requiresmore advanced analytic abilities and is therefore most suitable for a Postdoc. While we will initially focuson Cauchy-Riemann equations (§II.3) there are plenty of additional challenging avenues to explore, aspresented in the main text. The second PhD student will be working on the development of the topologicalvalidation of the quantitative and qualitative information about the computed connecting orbits (§II.2).Theoretical development and implementation for test problems will go hand in hand, making it particularlysuitable for a PhD project. The last subproject, described in detail in §II.4, requires a wide perspective,and it is an excellent project for a Postdoc, with concrete initial steps towards subgoals, but also withextensive possibilities for generalizations. While all subprojects have well-defined goals, collaborationbetween Postdocs and PhD students is important, as expertises, especially regarding algorithmic aspects,should be shared for optimal results.

team member taskPI Coordination, collaboration, coaching, overall responsibilityPhD student 1 Connections in systems of ODEs §II.1PhD student 2 Families of connections and building blocks §II.2Postdoc 1 Connecting orbits in PDEs §II.3Postdoc 2 Modulated travelling waves §II.4

III.2 RESEARCH GROUP

The research team will be hosted by the Department of Mathematics within the Faculty of Sciencesof the VU University Amsterdam. Mathematical Analysis is one of the focus areas of the Departmentof Mathematics, illustrating its capacity to support this research project. The members of the nonlineardynamics group relevant to the project are

area of expertiseprof. J.B. van den Berg dynamical systems & applicationsprof. R.C. van der Vorst topological and geometric methodsprof. J. Hulshof partial differential equationsdr. B. Rink lattice dynamicsdr. R. Planque variational methods, mathematical biologydr. F. Pasquotto symplectic geometry, Hamiltonian dynamics

with five PhD students currently working in the group.The research fits in well with the national mathematics cluster on Nonlinear Dynamics of Natural Sys-

tems (NDNS+), the NWO-funded nationwide collaboration in dynamical systems, of which the PI is thedirector and in which the nonlinear dynamics group plays an important role. Interaction with the groupsof Doelman (Leiden; reaction-diffusion systems), Peletier (Eindhoven; variational techniques), Steven-son/Homburg (University of Amsterdam; numerics/bifurcations) and Frank (CWI; scientific computing)is expected to benefit the project. Internationally, collaborations will be continued and extended withS.B. Angenent (Wisconsin), S. Day (William & Mary), R. Ghrist (Pennsylvania), K. Mischaikow (Rut-gers), J.-P. Lessard (BCAM, Bilbao) and J.F. Williams (SFU, Vancouver). The latter two collaboratorsspecialize in numerical algorithms and scientific computing, while the others have valuable expertise intheoretical aspects of the proposal.


2B RESEARCH IMPACT – OPTIONAL, BUT WHEN FILLED OUT THE RESULTS OF THE AS-SESSMENT OF THE RESEARCH IMPACT, WHETHER POSITIVE OR NEGATIVE, WILL BETAKEN INTO ACCOUNT IN THE OVERALL RATING OF THE APPLICATION.

Direct societal, technological or industrial implications are not part of the proposal.

2C NUMBER OF WORDS USED

7990 words in Section 2A.12 words in Section 2B.

2D ANY OTHER IMPORTANT REMARKS WITH REGARD TO THIS APPLICATION

—

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