-
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.
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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
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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
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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
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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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