Invariant Manifolds and Global Bifurcations

John GuckenheimerDepartment of Mathematics, Cornell University, Ithaca, NY 14853, USA.

Bernd Krauskopf and Hinke M. OsingaDepartment of Mathematics, The University of Auckland,

Private Bag 92019, Auckland 1142, New Zealand

Björn SandstedeDivision of Applied Mathematics, Brown University,

182 George Street, Providence, RI 02912, USA(Dated: January 2015)

Invariant manifolds are key objects in describing how trajectories partition the phase spaces ofa dynamical system. Examples include stable, unstable and center manifolds of equilibria andperiodic orbits, quasiperiodic invariant tori and slow manifolds of systems with multiple timescales.Changes in these objects and their intersections with variation of system parameters give rise toglobal bifurcations. Bifurcation manifolds in the parameter spaces of multi-parameter families ofdynamical systems also play a prominent role in dynamical systems theory. Much progress has beenmade in developing theory and computational methods for invariant manifolds during the past 25years. This article highlights some of these achievements and remaining open problems.

Computer investigations of dynamical systems havebecome a indispensable tool throughout the sciences.These studies often focus upon the geometry of thephase space of the system. Based upon the conceptsof genericity and transversality, dynamical systems the-ory describes typical behaviors. These descriptions in-volve invariant manifolds of dimension larger than one,such as the stable and unstable manifolds of equilibriumpoints and periodic orbits. Tangency of pairs of invari-ant manifolds has been shown to be a key ingredient insome types of global bifurcations in a system. This briefsurvey describes a few examples of this phenomenon.It highlights numerical methods that identify invariantmanifolds and locate their intersections. The examplescenter around aspects of the FitzHugh-Nagumo equationthat has become a prototype for studying traveling wavesin dynamical systems described by partial differentialequations.

I. INTRODUCTION

Dynamical systems theory embodies a geometric viewof solutions to ordinary differential equations of theform

ẋ = f(x, η), x ∈ Rn, η ∈ R`,

where Rn is the phase space and R` the parameterspace. In creating the subject, Poincaré emphasizedthe planar case n = 2 and generic properties that aretypical among the set of all such equations. He de-scribed their phase portraits, which show how solutiontrajectories partition R2. Stable and unstable manifoldsof saddles are key entities in the phase portrait of a

generic planar system. Each is a pair of trajectoriesthat approach the saddle as t→ ±∞.

Figure 1 displays phase portraits of the FitzHugh-Nagumo vector field [1], given by{

v̇ = w + v − v3

3 ,ẇ = −ε (a v + b+ cw), (1)

for three different values of the system parameter ε andfixed suitable choices of a,b and c. There are three equi-libria in Fig. 1(a)–(c): the upper-left equilibrium is asink, the lower-right equilibrium is a source, and themiddle eqilibrium is a saddle, denoted p; moreover thereis also an outer stable periodic orbit throughout. Panel(b) shows the situation when there is a homoclinic orbitΓ0, which is simultaneously in the stable and unstablemanifold of p. At this homoclinic bifurcation, an unsta-ble periodic orbit Γ emerges from the homoclinic orbitΓ0 as ε is decreased. The periodic orbit Γ supplants thestable manifold of p as the boundary between the twoattractor basins: points near the source are in the basinof attraction of the sink in Fig. 1(a), and they are in thebasin of attraction of the outer stable periodic orbit inFig. 1(c).

This example illustrates the role of invariant man-ifolds and their intersections in organizing the phaseportraits of dynamical systems. The stable and unsta-ble manifolds of a planar saddle are easy to find: each isformed from just two trajectories that can be computedwith standard initial value solvers. However, the geom-etry and the numerical analysis quickly become muchmore complicated when multiple timescales are involvedor the dimension of the system increases.

The limit ε = 0 of the FitzHugh-Nagumo vector fieldis singular with a whole curve of equilibrium points.

2

2 1 0 1 21

0.5

0

0.5

1(a)

u

v

p

2 1 0 1 21

0.5

0

0.5

1(b)

u

v

p

Γ0

2 1 0 1 21

0.5

0

0.5

1(c)

u

v

p

Γ

FIG. 1. Phase portraits near a homoclinic bifurcation of the FitzHugh-Nagumo vector field (1) for (a, b, c) = (1.0, 0.05, 1.2);panel (a) for ε = 0.38 is before, panel (b) for ε = 0.375149 is approximately at, and panel (c) for ε = 0.37 is after thehomoclinic bifurcation. Shown are equilibria (black dots), the stable manifold (blue curves) of the saddle p, the unstablemanifold (red curves) of p, and periodic orbits (green curves).

With the change of timescale t 7→ t ε, the resultingslow-fast singularly perturbed system is a differential-algebraic equation (DAE) in the limit ε = 0. Whenb = c = 0, the system reduces to the Van der Polequation [2] whose relaxation oscillations have inspiredmuch of the development of singular perturbation the-ory for dynamical systems with multiple timescales. Afundamental aspect of the subject is the presence of in-variant slow manifolds along which trajectories evolveon the slow timescale. ‘Stiff’ numerical methods havebeen developed to compute trajectories along attract-ing slow manifolds more efficiently than is done withexplicit ‘non-stiff’ methods. However, trajectories maycome to places where they leave an attracting slow man-ifold, and the stiff methods no longer are the ones ofchoice. Geometric, analytic and numerical methods areall needed in order to develop a full understanding ofthe dynamics in these circumstances.

Vector fields in dimensions larger than two exhibita vastly larger range of phenomena than planar vec-tor fields. Beginning in the 1950’s with the work ofKolmogorov [3], KAM theory has shown that invarianttori are quite common in both conservative and dissi-pative dynamical systems. Enormous effort has goneinto studying chaotic dynamics since Smale’s discov-ery in 1960 of the geometric example called the horse-shoe [4, 5]. As important as both invariant tori andchaotic dynamics are in dynamical systems theory, nei-ther is discussed here. Our emphasis is upon invariantmanifolds that arise as either stable or unstable man-ifolds of equilibrium points of a vector field or as slowmanifolds of a system with multiple timescales. Newcomputational methods have been developed to visual-ize these objects, and new theory has been developed toexplain their role in organizing the dynamics of systems.As in the FitzHugh-Nagumo example, non-transversal

intersections of invariant manifolds can be regarded asglobal bifurcations that separate parameter regions ofa system with different qualitative behaviors. The de-tection of these phenomena has been important in un-derstanding puzzling observations that were difficult toexplain in other ways.

Beyond bifurcations, there are circumstances inwhich non-generic dynamical behavior is important inapplications. As an example, we discuss traveling-waveprofiles for infinite dimensional dynamical systems de-fined by partial differential equations (PDEs). Thetraveling waves are solutions of an equation with theproperty that they translate spatially in time. Thesespatial profiles of associated traveling waves are foundas homoclinic orbits of a reduction of the PDE to anordinary differential equation. They arise, for exam-ple, in the context of the Hodgkin–Huxley model [6]of action potentials for nerve cells. This model is oneof the landmark achievements of 20th century biology,and it motivated significant developments in dynamicalsystems theory, including the example discussed here.One version of the FitzHugh-Nagumo model is a PDEthat has been used to study propagation of such actionpotentials along nerves.

We have chosen to organize this brief overview of de-velopments in this area by means of three examples thatbuild upon the FitzHugh-Nagumo vector field intro-duced above. The first example, an inclination-flip bi-furcation, illustrates some of the complexity that occurswith homoclinic bifurcations in three-dimensional vec-tor fields. The second example introduces slow-fast sys-tems with two slow and one fast variable. Here, foldedsingularities are a new phenomenon that gives rise tosurprising dynamical phenomena such as mixed-modeoscillations. Finally, we study the traveling-wave pro-files of the FitzHugh-Nagumo PDE. Interspersed with

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the examples are sections that provide minimal back-ground material for establishing the mathematical set-ting of our discussion. Following the examples we givea brief overview of some of the numerical methods usedin this work.

II. BACKGROUND

Manifolds are defined as locally Euclidean topologi-cal spaces. The manifolds discussed in this paper aresubmanifolds of the state spaces and parameter spacesof dynamical systems. Submanifolds of the state spaceare invariant if they are unions of trajectories. We alsoconsider submanifolds with boundary that are locallyinvariant : trajectories enter or leave the submanifoldonly through its boundary. In topology, submanifoldsare often defined implicitly as the set of solutions to asystem of equations. In contrast, the invariant mani-folds of dynamical systems such as stable manifolds arefrequently defined by asymptotic properties of trajec-tories as t → ±∞. Consequently, theoretical questionsconcerning the existence and smoothness of invariantmanifolds of dynamical systems are subtle, and the de-velopment of numerical algorithms for computing themis hardly straightforward. Each type of invariant man-ifold presents its own set of issues: we give examplesthat illustrate current research in this area.

Basic theory of manifolds can be viewed as a general-ization of linear algebra. The implicit function theoremgives conditions that guarantee that the set of solutionsS to a system of m equations g(x) = 0 in Rn form amanifold of dimension n−m, namely, the derivative Dgmust have maximal rank m at all points of S. The in-teger m is the codimension of S and the null space ofDg(x) is the tangent space of S at x ∈ S. Two subman-ifolds S1 and S2 are transverse if their tangent spacesspan Rn. Transverse intersections of submanifolds areagain submanifolds. Manifolds can also be defined bycoordinate charts, atlases and transition functions thatglue together coordinate charts on their overlaps. Nu-merically, continuation methods based upon the implicitfunction theorem have become a standard tool for com-puting one-dimensional manifolds. These methods arebased on the observation that the curve S defined bya regular system of n − 1 equations g(x) = 0 in Rnis a trajectory of vector fields that are tangent to thenull space of Dg on S. Methods for higher-dimensionalmanifolds are far less common and their development isan active area of research; see, for example, Ref. [7].

III. HOMOCLINIC ORBITS IN HIGHERDIMENSIONS

The homoclinic bifurcation of the FitzHugh-Nagumomodel (1) shown in Fig. 1 is typical of planar vector

fields, where a single periodic orbit bifurcates from thehomoclinic orbit and its stability depends on the rela-tive strengths of the two real eigenvalues of the equi-librium involved. At a generic codimension-one homo-clinic bifurcation of an equilibrium the dimensions ofthe stable and unstable manifolds necessarily add up tothe dimension of the phase space. Hence, in the planethey are both one-dimensional objects, and they havebranches which coincide at the homoclinic bifurcation.

In higher dimensions this is no longer the case: atleast one of the two invariant manifolds is of dimensionlarger than one and, at a homoclinic bifurcation, thestable and unstable manifolds of the equilibrium do notcoincide, instead intersecting in a single trajectory —the homoclinic orbit Γ0. The behaviors associated withhomoclinic orbits depend upon the types and magni-tudes of the eigenvalues of the equilibrium (through thesaddle quantity that determines the stability of nearbyperiodic orbits), as well as twisting of the flow aroundthe homoclinic orbit. Already in R3, the case we dis-cuss in this paper, the dynamics near a homoclinic orbitmay be very complicated and surprising. The overalldynamics is organized by invariant surfaces, in partic-ular, by two-dimensional stable manifolds of equilibriaand saddle periodic orbits.

The classical example of Shilnikov [8, 9] considers asaddle focus p of a vector field in R3 with a homoclinicorbit Γ0, where one branch of the one-dimensional un-stable manifold Wu(p) lies in the two-dimensional sta-ble manifold W s(p) and, hence, spirals back into p.When the saddle quantity is negative so that Γ0 is at-tracting, then a single stable periodic orbit bifurcatesfrom Γ0. However, when the saddle quantity is positiveand Γ0 is not attracting then there exists a chaotic in-variant set of saddle type near Γ0; or, equivalently, thereare Smale horseshoes in a suitable Poincaré section.This celebrated result by Shilnikov shows that chaoticdynamics can be located by finding a codimension-onehomoclinic bifurcation in R3; here, an important ingre-dient is the spiraling nature of the flow near the saddlefocus p due to the existence of complex-conjugate eigen-values. As a result, the stable manifold W s(p), whenfollowed backwards along the homoclinic orbit Γ0, formsa helix with infinitely many twists as it returns to p; seealso Ref. [10].

Homoclinic bifurcations in R3 to saddle points p withtwo real stable eigenvalues are also typical and canfound in many applications. One can ask if and whenchaotic dynamics are found near such a homoclinic bi-furcation, as in the Shilnikov case. The crucial geo-metric ingredient to answer this question lies again inhow the two-dimensional stable manifold W s(p) twistswhen it returns back to p along Γ0. Under suitablegenericity conditions, W s(p) accumulates on the one-dimensional strong stable manifold W ss(p) ⊂W s(p) atthe homoclinic bifurcation. Near Γ0 the surface W

s(p)either forms a cylinder, which is orientable, or a Möbius

4

strip, which is nonorientable. In both cases a single pe-riodic orbit bifurcates from Γ0 that is either orientable(has two positive Floquet multipliers) or nonorientable(has two negative Floquet multipliers); depending onthe eigenvalues of p, the bifurcating periodic orbit maybe attracting, of saddle type or repelling. In short, onedoes not find chaotic dynamics near a codimension-onehomoclinic bifurcation to a real saddle in R3.

However, it turns out that chaotic dynamics canbe found near codimension-two homoclinic bifurca-tions called flip bifurcations, where the stable mani-fold W s(p) changes from orientable to nonorientable.This happens when one of the genericity conditions ofa codimension-one homoclinic bifurcation is no longersatisfied. The theory of flip bifurcations is reviewed inRef. [11], where further references can be found. Thereare two types, called inclination flip and orbit flip bi-furcations, and they come in three cases each, denotedA, B and C, as defined by conditions on the eigenval-ues of p. Importantly, case C features the existenceof a chaotic saddle. Flip bifurcations have been foundin a number of systems, including the Hindmarsh-Rosemodel of a class of neuronal cells [12], a Van der Pol-Duffing model [13], and in reaction-diffusion systemswith nonlocal coupling [14]. Finding a flip bifurcationin a given system not only requires the detection ofthe homoclinic orbit Γ0, but also the determination ofwhether W s(p) is orientable or not. The capability ofdetecting flip bifurcations, via the formulation of well-defined test functions (that use the adjoint of the vectorfield), has been incorporated into the Homcont [15] partof the package AUTO [16]; see also Sec. VI.

A. Inclination flip bifurcation of type A

We now show how the stable manifold W s(p) at a ho-moclinic orbit can suddenly change from being a cylin-der to being a Möbius strip. To this end we consideran inclination flip of type A, which can be found andstudied conveniently[17] in the model vector fieldẋ = a x+ b y − a x2 + (µ̃− α z) (2− 3x)x+ δ z,ẏ = b x+ a y − 32 b x

2 − 32 a x y − (µ̃− α z) 2y − δ z,ż = c z + µx+ γ x z + αβ (x2 (1− x)− y2),

(2)which was constructed and introduced in Ref. [18] tofeature different kinds of codimension-two homoclinicbifurcations in an accessible way. The origin 0 is anequilibrium of (2) and, for the choice of parameters

a = −0.05, b = 1.05, c = −1.2, α = 0,β = 1, γ = 0, δ = 0, µ = 0, µ̃ = 0,

(3)

with α = αA ≈ 0.860183 there is a homoclinic orbit Γ0to 0 that satisfies all the conditions of a codimension-two inclination flip bifurcation IF of type A. When the

parameter α is varied from α = αA, the homoclinic or-bit Γ0 persists. However, it changes from an orientablehomoclinic orbit for α < αA, denoted Ho, to a nonori-entable (or twisted) homoclinic orbit for α > αA, de-noted Ht.

Figure 2 illustrates how the two-dimensional stablemanifoldW s(0), when followed along the homoclinic or-bit Γ0, returns to the origin 0. In this figure the (x, y, z)-space of (2) has been transformed so that the eigen-vectors of this saddle are the coordinate axes; hence,the one-dimensional unstable manifold Wu(0) is tan-gent at 0 to the vertical axis and the two-dimensionalstable manifoldW s(0) is tangent to the horizontal planethrough 0. On W s(0) we also show the strong sta-ble manifold W ss(q) and a weak trajectory ωs− tangentto the weak stable eigenvector. Note that there is asecond equilibrium q, which is a saddle focus, and itsone-dimensional stable manifold W s(q) is also shown inFig. 2.

The organization of phase space by W s(0) at themoment of homoclinic bifurcation is presented inFig. 2(a1), (b) and (c1). To illustrate the orientabilityof W s(0), this surface is divided along the homoclinicorbit Γ0 and ω

s− into a solid part and a transparent part.

In panels (a1) and (c1), when it is followed (backward intime) along Γ0, the stable manifold W

s(0) accumulateson the strong stable manifold W ss(0), meaning thatit satisfies the genericity conditions of a codimension-one homoclinic bifurcation. In Fig. 2(a1) the solid halfreturns on the solid side and the transparent half re-turns on the transparent side. Here W s(0) forms an ori-entable surface, namely a cylinder, and we are dealingwith an orientable homoclinic bifurcation Ho. Noticethat the cylinder surrounds the secondary equilibriumq and its one-dimensional stable manifold W s(q). InFig. 2(c1), on the other hand, the solid half of W s(0)returns back along Γ0 on the side of the transparenthalf, and vice versa, so that W s(0) forms a Möbiusstrip and we are dealing with a nonorientable homo-clinic bifurcation Ht. Notice further that, when W

s(0)is nonorientable, it is a much more complicated surfacein R3; in particular, W s(0) now accumulates on thecurve W s(q).

The transition between the two cases Ho and Ht ofcodimension one takes place at the codimension-two in-clination flip bifurcation IF shown in Fig. 2(b). Here,the surface W s(0) does not close up along W ss(0), butinstead aligns along the orbit ωs−, that is, it returnstangent to the weak stable eigendirection. Hence, thesurface W s(0) is neither orientable nor nonorientablebut ‘in between’ the two cases.

In order to understand the properties of W s(0) at Hoand Ht it is very helpful to consider its intersection set

Ŵ s(0) of W s(0) with a sufficiently large sphere thatcontains Γ0 in its interior. Figure 2(a2) and (c2) showstereographic projections of such a sphere, and panels

5

(a1)

Ho

(a2)

Ho

(a3)

H0

(b)

IF

(c1)

Ht

(c2)

Ht

(c3)

Ht

FIG. 2. Transition along a curve of homoclinic bifurcation through an inclination flip IF of type A of (2). Shown are W s(0)(blue surface and curves), Wu(0) (red curve), Γ0 (red curve), W

ss(q) (cyan curve and dots), and the weak trajectory ωs−(light blue curve and dots) at the orientable homoclinic bifurcation Ho for α = 0.7 in row (a), at the inclination flip IF forα = 0.860183 in (b), and at an nonorientable homoclinic bifurcation Ht for α = 1.0 in row (c); the other parameters areas in (3). Panels (a1), (b) and (c1) show the situation in R3; intersection sets of invariant objects with a sufficiently largesphere are shown in stereographic projection in panels (a2) and (c2), and are sketched in panels (a3) and (c3), at Ho andHt, respectively. Images from Ref. [17]. c©2013 Society for Industrial and Applied Mathematics. Reprinted with permission.All rights reserved.

(a3) and (c3) are respective topological sketches. At Hothe set Ŵ s(0) consists of a single curve whose two endpoints connect up to the curve at the intersection points

Ŵ ss− and Ŵss+ of W

ss(0) with the sphere; see Fig. 2(a2)

and (a3). The resulting two closed curves (one on eachside of the sphere) are the intersection set of the cylin-der formed by W s(0) along Γ0. What the intersectionset of the stable manifold W s(0) with the sphere looks

6

like when W s(0) forms a Möbius strip containing Γ0is less obvious and probably somewhat surprising. AsFig. 2(c2) and (c3) show, at a nonorientable homoclinic

bifurcation Ht the intersection set Ŵs(0) consists of

a single closed curve with two arcs that connect the

points Ŵ ss− and Ŵss+ in a spiraling fashion to Ŵ

s−(q)

and Ŵ s+(q), respectively.The associated two-parameter unfoldings of the

codimension-two homoclinic flip bifurcations of type Acan be found in Ref. [17]. The study of how W s(0) orga-nizes the phase space near inclination flip bifurcations oftype B is ongoing; it involves bifurcating periodic orbitsof saddle type and their stable and unstable manifolds,which may be orientable or nonorientable. Finding thestructure of invariant manifolds for the most compli-cated type C of inclination flip bifurcations, involvingsaddle hyperbolic sets with infinitely many saddle peri-odic orbits, remains an interesting challenge.

IV. SLOW-FAST SYSTEMS AND THEIRINVARIANT MANIFOLDS

The FitzHugh–Nagumo vector field (1) is an exam-ple of a system with multiple timescales when the pa-rameter ε is small. Many aspects of the behavior ofsuch slow-fast systems, particularly in dimensions threeand higher, have only recently become better under-stood through developments in geometric singular per-turbation theory [19]. Here, we highlight the analysisof folded singularities in systems with two slow and onefast variables as an example of the essential role of in-variant manifolds in dynamical systems.

Slow-fast systems are written in their slow timescaleas {

ε x′ = f(x, y, η, ε),y′ = g(x, y, η, ε),

(4)

where x ∈ Rk are the fast variables, y ∈ R(n−k) are theslow variables, ε is the ratio of timescales and η ∈ R`are other system parameters. The critical manifold isthe set of solutions of the equation f(x, y, η, 0) = 0, andy′ = g(x, y, η, 0) defines the slow flow as a differential-algebraic equation (DAE) when restricted to the criticalmanifold. Where Dxf is regular, the implicit functiontheorem gives x = h(y, η) on the critical manifold andthe DAE reduces to an ODE. Furthermore, where Dxfis hyperbolic, stable manifold theory [20] guarantees theexistence of locally invariant slow manifolds close to thecritical manifold for small ε > 0. Points on the criticalmanifold where Dxf is singular are folds and the slowflow of the critical manifold is no longer defined. Wherefolds are simple, the slow flow can be desingularized atthe expense of changing the direction of time on sheetsof the critical manifold where det(Dxf) < 0. In the full

system with ε > 0, trajectories that approach a simplefold ‘jump’ along the fast direction.

Consider now three-dimensional systems with twoslow variables. In these systems, the critical manifold isa two-dimensional surface with attracting and repellingsheets. Trajectories that flow from an attracting sheetto a repelling sheet are canard orbits that play a dra-matic role in the dynamics. Because repelling slow man-ifolds are unstable on the fast timescale, the slow-timeevolution near these manifolds seems to be discontin-uous as trajectories on either side turn away abruptly.Canard orbits appear near folded singularities, pointson the fold curve where the desingularized system hasan equilibrium.

Benôıt [21] analyzed the intersections of the attract-ing and repelling slow manifolds at folded saddles, prov-ing that invariant extensions of the manifolds intersecttransversally along canard orbits with an angle thatis O(ε). In the singular limit ε = 0, the stable manifoldof a folded saddle separates trajectories on the attract-ing slow manifold that flow all the way to the fold curveand then jump, from trajectories that turn away fromthe fold before reaching it. When ε > 0, some trajecto-ries immediately adjacent to the stable manifold formcanard orbits that flow onto the repelling slow manifoldbefore jumping. The separation of trajectories alongthe canard orbits is abrupt and creates the stretchingthat is characteristic of chaotic invariant sets. Indeed,Haiduc [22] proved that this mechanism explains thelandmark results of Littlewood [23–25] on the forcedVan der Pol equation [2] that demonstrated the exis-tence of chaotic dynamics in an explicit dissipative sys-tem for the first time.

The geometry that is associated with folded nodesis even more complicated and surprising than that offolded saddles. Benôıt showed that the attracting andrepelling slow manifolds twist as they approach a foldednode, creating multiple canard orbits in the process.Benôıt [26] and Wechselberger [27] analyzed the amountof twisting that occurs and the bifurcations that pro-duce increasing numbers of canard orbits. The twist-ing is manifest in small-amplitude oscillations of trajec-tories that flow past a folded node. When these tra-jectories have a global return to the region around thefolded node, they give examples of mixed-mode oscilla-tions (MMOs).

A. The self-coupled FitzHugh–Nagumo equation

Recently, there has also been increasing interest inMMOs that are observed in the context of neural mod-els. As an example of an MMO that is organized bya folded-node singularity, we consider the FitzHugh–Nagumo equation with synaptic coupling back to itselfas a model of single-cell dynamics that is influenced by

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external electrical signals due to connections to othercells. This model three-dimensional system

v̇ = h− v3 − v + 1

2− γ s v,

ḣ = −ε (2h+ 2.6 v),ṡ = βH(v) (1− s)− ε δ s,

(5)

was introduced by Wechselberger [27]. Here, the thirdvariable, denoted s, describes the synaptic coupling,which occurs through voltage v. The parameter γ isthe coupling strength. The dynamics of s consists ofan activation term, determined by the parameter β,and a deactivation term, controlled by the decay rate δ.Activation is only occurring in the active phase, whenv > 0, as indicated by the Heaviside function H(v); inthe silent phase, when v < 0, the synaptic coupling sdecays on the same timescale as the gating variable h.The presence of the Heaviside function greatly simpli-fies the analysis of the silent phase, for which system (5)is a slow-fast system with v the fast and h and s theslow variables.

Figure 3(a)–(b) illustrates the response of system (5)for β = 0.035, γ = 0.5, δ = 0.565 and ε = 0.015.For these parameters, there exists a stable MMO peri-odic orbit Γ5 that exhibits five small-amplitude oscilla-tions, which constitute subthreshold oscillations in thesilent phase, followed by one large action potential; itsv-time series is shown in panel (a). Since H(v) = 0in the subthreshold regime, the structure of slow man-ifolds for system (5) is independent of β and can beanalyzed separately. Slow manifolds of system (5) havealso been studied in Ref. [28]. The critical manifoldS of system (5) is a cubic surface and a folded nodeat (v, h, s) ≈ (−0.4900, 0.6176, 0.2797) exists relativelyfar away from the cusp point at (v, h, s) = (0, 12 , 1), onthe side of the fold curve with smallest v. Figure 3(b)shows how the intersections between the attracting andrepelling slow manifolds of system (5) organize the sub-threshold oscillations near the folded node. These slowmanifolds were computed with the method explainedin Sec. VI. The repelling slow manifold Srε comprisesthe family of orbit segments that start in the planeΣ := {s = 0.2797} and end on the line Lr := {(v, h, s) ∈S | v = 0} = {(0, 12 , s)}. Similarly, the attracting slowmanifold Saε comprises the family of orbit segments thatstart on the line La := {(v, h, s) ∈ S | h = −6} and endin Σ. The slow manifolds Srε and S

aε intersect in canard

orbits, two of which, namely, ξ4 and ξ5, are highlightedin Fig. 3(b). These canards make four and five small-amplitude oscillations, respectively. As can be seen inFig. 3(b), the value of β is such that the periodic orbitΓ5 lands (approximately) on S

aε in between the two ca-

nard orbits ξ4 and ξ5, which determines the signatureof this MMO.

One advantage of our procedure for computing in-tersections between attracting and repelling slow man-

(a)

time

v

800 1000 1200 1400 1600 1800

-1

-0.5

0

0.5

1

(b)

hs

v

Lr

Srε Saε ξ4 ξ5

Γ5

0 0.015 0.03 0.045

2

4

6

−2 −1 0−2

−1

0

1

2

−2 −1 0−2

−1

0

1

2

−2 −1 0−2

−1

0

1

2

−2 −1 0−2

−1

0

1

2

ε

|| · ||2

(c)(d1)

(d1)

v

s

(d2)

(d2)

v

s

(d3)

(d3)

v

s

(d4)

(d4)

v

s

FIG. 3. (a) Mixed-mode oscillation of system (5) for γ =0.5, δ = 0.565, ε = 0.015 and β = 0.035. (b) Associatedattracting and repelling slow manifolds, Saε (red surface)and Srε (blue surface); also shown are the two canard orbitsξ4 (magenta curve) and ξ5 (orange curve); reproduced fromRef. [28]. (c) Continuation in ε of the canard orbit ξ5 (whileassuming H(v) ≡ 0); (d1)–(d4) Projections of ξ5 onto the(s, v)-plane at the correspondingly labeled points along thebranch.

8

ifolds is that we can continue such canard orbits in asystem parameter. A particularly interesting parame-ter is the timescale ratio ε; see also Ref. [29]. Geomet-ric singular perturbation theory predicts the existenceand characterization of slow manifolds and canard or-bits provided ε is small enough. The numerical meth-ods, on the other hand, work for a large range of valuesof ε that extends well beyond the known theory. Wehave found that these computations yield new predic-tions about the nature of different canard orbits.

Figure 3(c) illustrates such a numerical explorationwith the continuation in ε of the canard orbit ξ5 frompanel (b). We plot the L2-norm of the continued ca-nard orbit ξ5; the insets (d1)–(d4) show projectionsonto the (s, v)-plane of four selected canard orbits alongthe branch. When ε is decreased from ε = 0.015,we find that ξ5 accumulates onto the strong canard atε = 0, as predicted by the theory. In the other direc-tion, as ε is increased, an interesting transition occursat ε ≈ 0.0305, which is close to where the branch hasa minimum in the L2-norm: the canard orbits changefrom having s > 0 decreasing, as shown in Fig. 3(b),to having s < 0 increasing; the canard orbits past thistransition, including those shown in Fig. 3(d1)–(d4), allsatisfy s < 0. We disregarded the activation term in theequation for s during this continuation; notice that therestriction v ≤ 0 is not satisfied everywhere along thesecomputed canard orbits, so that their interpretation inthe context of MMOs is not straightforward. The con-tinuation branch undergoes three folds, at ε ≈ 0.0385,ε ≈ 0.0363 and ε ≈ 0.0412, respectively; panels (d1)and (d2) show two coexisting canard orbits for ε = 0.037on either side of the first fold, and panels (d3) and (d4)show coexisting canard orbits for ε = 0.04 on eitherside of the third fold. Note that the transition acrossthe third fold has the effect that ξ5 transforms into acanard orbit with only four oscillations; such transi-tions have been observed in other systems as well [29].Figure 3(d3) indicates that any trajectory of system (5)that starts on Saε near a solution on the branch segmentin between the second and third fold would exhibit onlyone small-amplitude oscillation before producing (pos-sibly more than) one action potential (when v becomespositive).

V. TRAVELING WAVES OF PDES

In many applications, localized traveling waves playan important role: they may, for instance, represent ac-tion potentials that propagate in a neuronal axon, lightblips that travel through an optical fiber, or solitary wa-ter waves in a channel. Instead of describing the mostgeneral type of PDE models, we focus here on systemsof reaction-diffusion equations of the form

ut = Duxx + f(u), (6)

where x ∈ R, u ∈ X = C0(R,Rn), and D is a non-negative diagonal diffusion matrix. Traveling waves aresolutions of the form

u(x, t) = v(x− ct), (7)

where v = v(z) describes the profile, and c is the se-lected wave speed. Substituting this ansatz into (6), wesee that traveling-wave profiles satisfy the ODE

Dvzz + cvz + f(v) = 0, (8)

where the wave speed c enters as a free parameter. Wecan rewrite (8) as the first-order system

Vz = F (V, c) (9)

and use dynamical-systems methods to analyze it. Iff(0) = 0, we can seek localized traveling waves of (6)with profiles v(z) that converge to zero exponentiallyas |z| → ∞. Localized traveling waves correspond,therefore, to homoclinic orbits V (z) of (9) that lie inthe intersection of the stable and unstable manifoldsof the equilibrium V = 0. Without additional struc-ture in the underlying ODE, homoclinic orbits arise ascodimension-one phenomena: the wave speed c suppliesa free parameter, which suggests that localized travelingwaves arise for a discrete set of wave speeds c. Underappropriate genericity conditions, Melnikov theory [11]shows that the stable and unstable manifolds of V = 0will unfold transversally along a homoclinic orbit V∗(z)upon changing c near the selected wave speed c∗, pro-vided

M :=

∫R〈ψ(z), Fc(V∗(z))〉dz 6= 0, (10)

where ψ(z) is the unique nontrivial bounded solution ofthe adjoint variational equation

Wz = −FV (V∗(z))∗W.

Localized traveling waves of (6), which are also re-ferred to as pulses, can be found as homoclinic orbitsof the associated traveling-wave ODE (9). Construct-ing homoclinic orbits is a challenging problem that canoften be addressed only through numerical computa-tion. However, when an additional slow-fast structureis present, geometric singular perturbation theory pro-vides a very effective tool to construct pulses. Once apulse has been identified, homoclinic bifurcation theorycan be used to study whether this localized travelingwave can give rise to multi-pulse solutions, which aretraveling waves with profiles that resemble several well-separated copies of the original pulse; these travel atwave speeds close to that of the original pulse.

Once the existence of a pulse v∗(z) with wave speedc∗ has been shown, one may want to determine whetherthe resulting solution u(x, t) = v∗(x − c∗t) is stable as

9

(a)

z

z

(b)

σ(L∗)

!λ

"λ

FIG. 4. (a) Stationary solutions of the PDE (11) are one-parameter families that depend on the location of their cen-ter of mass. (b) The spectrum σ(L∗) of a stable localizedtraveling wave.

a solution of the original PDE (6). A typical approachconsists of transforming (6) into the moving coordinateframe (z, t) = (x− c∗t, t) to get

ut = Duzz + c∗uz + f(u), (11)

which admits the stationary solution u(z, t) = v∗(z).Linearizing this PDE about v∗, we obtain the linearizedoperator

L∗u := (D∂2zz + c∗∂z + fu(v∗(z)))u. (12)

The operator L∗ can be viewed as an unbounded,densely defined, closed operator on X = C0unif : its spec-trum on this space provides the necessary informationthat can be used to prove linear and nonlinear asymp-totic stability of the traveling wave u(z, t) = v∗(z).We now review some key features of the spectrum ofL. First, λ = 0 always belongs to the spectrum sinceL∗v′∗(z) = 0, and the derivative of the pulse, therefore,provides an eigenfunction associated with λ = 0. Fig-ure 4(a) illustrates the one-parameter family v∗(· − p)of stationary solutions that is provided by a pulse v∗(z)via translation of the center of mass to any locationp ∈ R. Second, since the pulse profile v∗(z) convergesto zero as |z| → ∞, it can be shown that any elementin the spectrum of the asymptotic operator

L0u := (D∂2z + c∗∂z + fu(0))u (13)

that is associated with the rest state u = 0 also liesin the spectrum of L∗. The spectrum of L0 can bedetermined via Fourier transform: indeed, the spectrumS0 of L0 is given by

S0 = {λ ∈ C | for some k ∈ R,det[−Dk2 + i k c∗ + fu(0)− λ

]= 0}.

(14)

This set consists of curve segments in the complexplane. If S0 intersects the open right-half plane, thepulse will be unstable. Therefore, we assume from nowon that S0 lies in the open left-half plane (as the bor-der case where the spectrum of the rest state touchesthe imaginary axis will result in bifurcations[30, 31]):

in this case, the spectrum of L∗ in the closed right-halfplane consists of discrete isolated eigenvalues of finitemultiplicity, as is illustrated in Fig. 4(b).

We say that the pulse v∗ is spectrally stable if S0lies in the open left-half plane, the eigenvalue λ = 0of L∗ is simple, and L∗ has no other eigenvalues withpositive real part. A typical stability result consists ofthe statement that spectral stability of L∗ implies non-linear stability with asymptotic phase of the traveling-wave family {v∗(· − p); p ∈ R}. This result reducesthe question of nonlinear stability to studying spectralstability of L∗. If we take the view that the set S0 can,in principle, be calculated case by case, as it involvesonly an algebraic problem, then it remains to (i) findconditions that guarantee that λ = 0 is simple and (ii)identify any other unstable eigenvalues of L∗.

To analyze (i), we note that the equation L∗v = 0 isequivalent to solving the variational equation

Vz = FV (V∗(z), c∗)V

of the traveling-wave ODE (9) around the homoclinicorbit V∗(z) associated with the pulse v∗(z). In par-ticular, the condition that the null space of L∗ is onedimensional (λ = 0 has geometric multiplicity one) isequivalent to the condition that the tangent spaces ofthe stable and unstable manifolds at V∗(z) intersect inthe one-dimensional space spanned by V ′∗(z). Further-more, if the geometric multiplicity of λ = 0 is one, thenits algebraic multiplicity will be one if, and only if, theMelnikov integral M defined in (10) is not zero: in-deed, it can be shown that the adjoint solution ψ(z) isrelated to the adjoint eigenfunction of the adjoint opera-tor L∗∗. This result provides an interesting link betweenthe traveling-wave ODE and stability properties of thePDE linearization.

Regarding property (ii), we can write the eigenvalueproblem

L∗u = λu

as an equivalent system of linear ODEs of the form

Vz = FV (V∗(z), c∗)V + λBV (15)

with parameter λ. A complex number λ is an isolatedeigenvalue of L∗ if, and only if, (15) has a nonzero lo-calized solution, that is, a ‘homoclinic orbit’. In otherwords, if we denote by Es(λ) and Eu(λ) the linear sub-spaces of initial conditions of (15) at z = 0 that convergeto zero as z → ∞ and z → −∞, respectively, then weneed that these subspaces have a nontrivial intersection.Thus, choosing bases in these subspaces and calculatingtheir determinant, we see that λ is an eigenvalue if, andonly if, this determinant, a Wronskian of appropriatesolutions of (15), vanishes. This determinant, viewedas a function D(λ) of λ, is referred to as the Evansfunction [32]: it is analytic in λ for λ to the right of S0,

10

and its roots correspond to the sought eigenvalues; infact, the multiplicity of roots of D(λ) agrees with thealgebraic multiplicity of λ viewed as an eigenvalue ofL∗.

A. Traveling waves of the FitzHugh–Nagumoequation

To conclude this review, we illustrate the importanceof invariant manifolds in both the theoretical and nu-merical analysis of dynamical systems by consideringa model that exhibits all types of invariant manifoldsdiscussed in this paper. More specifically, we considertraveling waves of a FitzHugh–Nagumo model, whichhave spatial profiles that are homoclinic solutions of thethree-dimensional vector field

ε ẋ1 = x2,

ε ẋ2 =15 [s x2−x1 (x1 − 1) ( 110 − x1) + y − p

],

ẏ = 1s (x1 − y).

(16)

The geometry of the homoclinic orbits of this systemis organized by its invariant manifolds. Study of thisproblem motivated Jones and Kopell [33] to formulatea general result, the Exchange Lemma, that was used toprove existence of a homoclinic orbit of (16) for particu-lar wave speeds given by the parameter s. However, thehomoclinic orbit was computed accurately only recently,via intersections of several different types of invariantmanifolds [34].

System (16) is a slow-fast vector field with one slowvariable y and two fast variables x1 and x2. The criticalmanifold S, defined by {x2 = 0, y = x1 (x1 − 1) (0.1 −x1) + p}, is one dimensional and splits into left, middleand right branches, denoted Sl, Sm, and Sr, respec-tively: the inner branch Sm consists of sources and thetwo outer branches Sl and Sr are saddle equilibria ofthe layer equations. System (16) has an equilibrium qthat lies on the critical manifold and additionally solvesy = x1. The stability of q depends upon both p and s.Figure 5(a) shows a bifurcation diagram in the (p, s)-plane. The equilibrium q undergoes a Hopf bifurcationalong the U-shaped curve and homoclinic orbits to qare found along the C-shaped curve, where q ∈ Sl.

Approximations of the homoclinic orbits for smallε > 0 can be pieced together from the singular limit.Beginning at q ∈ Sl, the first segment of the homoclinicorbit follows the unstable manifold of q in the layerequations to the layer equilibrium on Sr with the same(x1, x2)-coordinates as q. As described by the ExchangeLemma, the trajectory then turns and follows Sr to avalue of the slow variable y where there is a connectingorbit that returns from Sr to Sl; the connecting orbitthen follows Sl back to q.

00.2

0.40.6

0.8

−0.1−0.05

00.05

0.1−0.1

0

0.1

0.2(b)

y

x2 x1

Sl Sm

Srq•

−0.2 −0.1 0 0.1 0.2 0.3 0.4 0.50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

(a)

s

p

Hopf

Hom

I

FIG. 5. (a) Bifurcation diagram of traveling waves for sys-tem (16) in the (p, s)-plane consisting of a U-shaped (blue)curve of Hopf bifurcations and a C-shaped (red) curve ofhomoclinic bifurcations; the dashed curves are their singu-lar limit as ε→ 0. (b) Traveling-wave homoclinic orbit (redcurve) to the saddle q with slow segments near the criticalmanifold (blue curve).

Fenichel proved that the critical manifold branchesSl and Sr perturb to locally invariant slow manifoldsfor small ε > 0, along with their stable and unstablemanifolds [20]. Hence, the slow-fast decomposition ofthe homoclinic orbits persists when ε > 0; however, asa heteroclinic connection between saddles, it occurs forparameters that lie on a curve in the (p, s)-plane. It isdifficult to compute the homoclinic orbits because therelevant slow manifolds are saddle like in the fast di-rections. Numerical solutions of initial value problemsthat start on or close to these manifolds can only fol-low them for times that are O(1) with respect to thefast timescale. Guckenheimer and Kuehn [34] devel-oped a two-point boundary value problem and associ-ated solver that locates these manifolds. The directionsof its stable and unstable manifolds were estimated aswell, yielding initial conditions for computing trajecto-ries on these manifolds with initial value solvers; seeSec. VI for the details of these calculations. This ap-

11

proach allows for the computation of the whole homo-clinic orbit as a composite of its slow and fast segments,each computed separately and matched together at therespective endpoints. The full homoclinic orbit to q isillustrated in Fig. 5(b).

Champneys et al. noted that the ‘CU’ bifurcationdiagram shown in Fig. 5(a) is puzzling [35]. They re-port that the curve of homoclinic bifurcations appearsto end without contacting another more degenerate bi-furcation. Further analysis of invariant manifolds re-solved this enigma [34]. The clue to the discrepancyis that q may undergo a subcritical Hopf bifurcation,after which it no longer lies on Sl. There is a family ofperiodic orbits emerging from q which bounds its sta-ble manifold. A consequence is that there is no wayfor trajectories following the slow manifold associatedwith Sl to reach q for parameters that are close to theHopf curve. The stable manifold of q and the unstablemanifold of the saddle slow manifold do not intersect.However, as the parameters move farther from the Hopfcurve, the stable manifold of the equilibrium begins tospiral around the periodic orbit created at the Hopf bi-furcation. It then passes through a tangency with theunstable manifold of the slow manifold Sl followed bytransversal intersections of the two manifolds; see Fig. 3in Ref. [34]. The tangency of these manifolds can beregarded as another codimension-one bifurcation thatoccurs along a curve in the (p, s)-plane. Because thistangency is independent of the connection from q to thesaddle slow manifold associated with Sr, the associatedtwo bifurcation curves cross transversally, intersectingat the end of the C-curve of homoclinic bifurcations. Werefer to Ref. [34] for more detail on the exponentiallysmall scales found in the folding of the C-curve.

This example illustrates that invariant manifolds ofdifferent kinds and their intersections play a prominentrole in shaping the dynamics of slow-fast vector fields.Homoclinic tangency of stable and unstable manifoldsof periodic orbits has long been a focus of the analy-sis of horseshoes and their bifurcations, but the phe-nomenology associated with the homoclinic orbits ofthe FitzHugh–Nagumo model have been a new develop-ment. Similarly, intersections between a repelling slowmanifold and the unstable manifold of a saddle equilib-rium are important in mixed-mode oscillations of theKoper model [13, 36], and the tangency of these man-ifolds demarcates part of a parameter-space boundaryfor these complex oscillations; see also Ref. [19]. In abroader context, we know relatively little about globalreturns of systems with multiple timescales; i.e., dy-namics that lead to recurrence of trajectories to spe-cific regions of a phase space following large excursionsfrom these regions. We even lack a sharp formulation ofmathematical problems and conjectures that generalizethe observations made in examples of three-dimensionalslow-fast systems.

VI. NUMERICAL METHODS FORMANIFOLDS

Equilibria, periodic orbits and their local bifurcationscan be found with standard dynamical systems soft-ware such as Auto [16], MatCont [37] and CoCo[38]. Here periodic orbits are computed as solutions toa boundary value problem (BVP) with periodic bound-ary conditions and an appropriate phase condition.More generally, the integrated boundary value solversof the above packages locate an orbit segment u(t) witht ∈ [0, 1] that satisfies the time-rescaled equation

u̇ = T f(u),

subject to specified boundary conditions, where T isthe integration time (which may be negative) associatedwith the normalized orbit segment u. The solution ofthe BVP is found with the method of collocation as apiecewise polynomial over a specified mesh. A first peri-odic solution can be constructed near a Hopf bifurcationor, when it is stable, found by numerical integration.

An approximate homoclinic or heteroclinic orbit canbe found and continued as an orbit segment whose endpoints lie in the stable and unstable linear eigenspacesnear the respective equilibrium; one speaks of projec-tion boundary conditions [39]. To find an initial orbitsegment u satisfying this BVP one can consider a peri-odic orbit of high period, or perform what is known asa homotopy step as implemented in the toolbox Hom-Cont [15] that is part of the package Auto; also sup-plied are test functions that allow the user to identifycertain codimension-two global bifurcations, includinginclination and orbit flips.

Several methods have been developed for computinginvariant manifolds of dimension higher than one, withemphasis on two-dimensional stable and unstable man-ifolds of equilibria in R3; see the survey Ref. [40]. Weconcentrate here on the general idea to select a regionof interest and define the two-dimensional (invariant)manifold in this region as a one-parameter family of or-bit segments, defined by a suitable BVP. A review ofthis approach can be found in Ref. [41]; for more gen-eral background information on continuation methodssee Ref. [42].

Restricting our discussion to three-dimensional sys-tems for simplicity, we consider an orbit segment u withone end point on a one-dimensional curve (for example,a line) and the other on a two-dimensional surface (forexample a planar section). This two-point boundaryvalue problem setup is very flexible, and the boundaryconditions on either end point can be formulated implic-itly; for example, one can also consider orbit segmentsof a fixed integration time or specified fixed arclength;see Refs. [41] and [40].

Figure 6(a) shows the computational setup for the re-pelling slow manifold of system (5) in Fig. 3(b). Here,

12

(a)!!!!!!

Lr

F

Sr

Σ

u

hs

v

(b)

"vs

Z

Σ

Γ

p#

##$vu

W s(Γ)

W u(p)

Q+

Q−

FIG. 6. Illustration of BVP setup for computing families oforbit segments. (a) An initial orbit segment u of system (5)with u(0) ∈ Lr ⊂ Sr and u(1) ∈ Σ; varying u(0) along Lrproduces the repelling slow manifold Srε ; reproduced fromRef. [28]. (b) Lin’s method setup for a connection betweenan equilibrium p and a periodic orbit Γ, consisting of twoorbit segments, Q− from p and Q+ from Γ, that end in asection Σ along a specified Lin direction Z; from Ref. [43,Fig. 1(a)]. c©2008 IOP Publishing & London MathematicalSociety. Reproduced with permission. All rights reserved.

u(0) is restricted to the line denoted Lr ⊂ Sr withv = 0, and u(1) is restricted to the plane Σ that is per-pendicular to the fold curve F and contains the folded-node singularity. To obtain such an orbit segment weperform two homotopy steps [28]. The orbit segmentu shown in Fig. 6(a) with associated integration timeT is an isolated solution of a solution family that isparametrized by the point u(0) on the line Lr = Lr(θ).Continuation in the parameter θ then produces the re-pelling slow manifold Srε as a surface.

In order to find more complicated connecting orbitsit may be useful to split the orbit into several segments,each to be computed with a BVP solver. In particu-

lar, this approach is used in implementations of whatis known as Lin’s method [44, 45]. The underlying ideais to compute pairs of orbit segments (with associatedintegration times) in such a way that their end pointsare constrained to lie along a specified vector directionon a common surface defining one of the boundary con-ditions for these segments. Lin’s method has been im-plemented for the detection of multipulse homoclinicorbits [46], and for so-called EtoP connections betweenequilibria and periodic orbits [43] and PtoP connectionsbetween periodic orbits [47].

Figure 6(b) shows the Lin’s method setup for thecomputation of a codimension-one EtoP connection be-tween a saddle equilibrium p and a saddle periodic or-bit Γ. The orbit segment Q− starts from a point on theunstable eigenvector vu of p and ends in the sectionΣ; similarly the orbit segment Q+ starts from Σ to apoint on the vector vs in the unstable bundle of Γ. Thetwo end points of Q− and Q+ in Σ lie along the Lindirection Z, which can be chosen freely provided a mildgenericity condition is satisfied. The signed differencebetween the two end points along the fixed directionZ is a well-defined test function that is referred to asthe Lin gap; continuation in a system parameter, whilekeeping Z fixed, can then be used to find a zero of theLin gap, which corresponds to the sought connectingorbit.

A similar strategy was used in calculating the homo-clinic orbit of system (16), but a customized boundaryvalue solver was developed to compute the slow seg-ments of this orbit. The homoclinic orbit is split intofour segments, two that follow the slow manifolds Srand Sl, one that connects the equilibrium q to Sr anda fourth that connects Sr to Sl. The connection fromq to Sr is the part of the homoclinic orbit that existsfor only a discrete value of the wave-speed parameter s.The first step in finding the homoclinic orbit is to com-pute this segment by a shooting algorithm that usesinitial conditions on the linear approximation of theone-dimensional manifold Wu(q), for different values ofthe wave speed s. The value of s for which this con-nection is found is then fixed in the remainder of thecalculations of the homoclinic orbit. Note that, sinceSr is normally hyperbolic, it is easy to determine whichdirection Wu(q) turns as it approaches Sr.

The next step in finding the homoclinic orbit is tocalculate accurate approximations to Sr and Sl with aboundary value solver. Figure 7(a) illustrates the setupof the two-point boundary value problem used to calcu-late Sr. In making the calculation well conditioned, itis important to choose boundary conditions that makea large angle with the vector field in an entire regionof the boundary manifold. Thus, instead of just com-puting the segment along the slow manifold where thedirection of the vector field changes rapidly, the bound-ary conditions are located transverse to the stable andunstable manifolds of Sr, as illustrated by the line seg-

13

−0.5

0

0.5

1

−0.05

0

0.05

0.1−0.02

0

0.02

0.04

0.06

0.08

0.1

(a)

y

x2x1

Br

Bl

Sr

(b)

Sr

Sl

y

x2

x1

Σ

Σ

FIG. 7. (a) Boundary value problem setup from Bl to Br forthe computation of slow manifold Sr of saddle type of (16)for p = 0, s = 1.2463 and ε = 0.001. (b) Associated hete-roclinic connection of slow manifolds: ten blue trajectoriesstarting along Sr are computed forward to a cross-section Σdefined by x1 = 0.5, and ten red trajectories starting on Slare computed backward to Σ. It is apparent In Σ that theunstable manifold of Sr intersects the stable manifold of Sl

transversally.

ment Br and rectangle Bl in Fig. 7(a). The (time)length of the trajectory is chosen so that the algorithmcomputes longer segments of Sr and Sl than those thatare part of the homoclinic orbit.

The directions of the stable and unstable manifoldsalong Sr and Sl were estimated by the linearization ofthe fast (layer) equations along the slow manifold; aninitial value solver was used to ‘sweep’ out the man-ifolds by computing sets of trajectories whose initialconditions were constructed from the linearization. Fig-ure 7(b) shows some of these trajectories as well as the

surfaces of the stable and unstable manifolds interpo-lated from these. These integrations were terminatedon a common plane to illustrate the transversality oftheir intersection along the fast segment of the hete-roclinic orbit that connects Sr to Sl. The ExchangeLemma describes important aspects of the geometry ofthis connection.

The matching conditions of the four segments of thehomoclinic orbit are more or less automatic from thenormal hyperbolicity of the slow manifolds. The slowsegments of the homoclinic orbit orbit must lie expo-nentially close to the slow manifolds Sr to Sl, so theycannot be distinguished from these manifolds numeri-cally. Thus, the errors associated with using approxi-mate boundary conditions that place the endpoints ofthe segments on the stable and unstable manifolds of Srto Sl cannot be resolved without heroic computations ofextreme precision. The continuity of the invariant man-ifolds with respect to perturbations of the vector fieldand their transversal intersections on the boundaries ofthe cross-sections used in the phase space extended withthe parameter s make us confident that the numericallycomputed trajectory is an excellent approximation tothe homoclinic orbit.

VII. DISCUSSION

This short review attempted to highlight some recentexamples of how the study of global invariant mani-folds and their bifurcations can help unravel the overalldynamics of a given system. The examples are by nomeans exhaustive, but we hope that they convey theusefulness of this approach and the associated advancednumerical methods. Indeed, invariant manifolds of dif-ferent kinds are also key ingredients in the dynamics ofnumerous other systems, and quite a number of chal-lenges remain. We mention a few of them briefly.

• The study of invariant manifolds near the morecomplicated cases B and C of codimension-twohomoclinic flip orbits is the subject of ongoing re-search; here also (possibly infinitely many) peri-odic orbits of saddle type play a role as well.

• The study of invariant manifolds near heterocliniccycles or chains involving equilibria and periodicorbits is a promising direction for future research.

• The examples in this paper all involve invariantmanifolds of dimensions one and two in three-dimensional vector fields. There is a robust lit-erature on identifying attracting slow manifoldsin systems of chemical reactions, especially in thecontext of combustion [48], and this appears to bean area ready for further development. Comput-ing stable manifolds of low codimension in high-

14

dimensional systems is one strategy to identify at-tractor basin boundaries in high-dimensional sys-tems [49].

• Higher-dimensional compact invariant objects,such as invariant tori, are also of great interest inmany areas of application, and computing themand their stable and unstable manifolds is an ac-tive area of research [50–53] not touched upon inthis review.

• Dynamical systems with symmetries, conservedquantities or network structure appear in manyapplications. Invariant manifolds are perhapseven more important as key ingredients in the

analysis of such systems, but adjustments of thenumerical methods to take account of these struc-tures is needed [54]. Beyond manifolds, more ge-ometric objects with singularities arise in thesesettings. Group theoretical methods are a power-ful tool for the analysis of systems with symme-try [55].

Acknowledgments: Guckenheimer gratefully ac-knowledges support from the National Science Founda-tion through grant DMS 1006272. Sandstede gratefullyacknowledges support from the National Science Foun-dation through grant DMS 1409742.

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