ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY
PERTURBED SYSTEM ARISING FROM THE STUDY OF CRYSTALLINE
GRAIN BOUNDARIES
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
Abstract. Mathematically, the problem considered here is that of heteroclinic connections for a
system of two second order differential equations of gradient type, in which a small parameter ǫ
conveys a singular perturbation. The motivation comes from a multi-order-parameter phase field
model developed by Braun et al [5] and [23] for the description of crystalline interphase boundaries.
In this, the smallness of ǫ is related to large anisotropy. The existence of such a heteroclinic, and
its dependence on ǫ, is proved. In addition, its stability is established within the context of the
associated evolutionary phase field equations.
1. Introduction
1.1. The model and prior results. The physical context is that of crystals existing in several
phases, and the general goal is to study the structure of interphase boundaries. The modeling
framework uses order parameters as both microscopic and mesoscopic descriptors of the state of
the material, with dynamics given by the gradient flow of a free energy functional involving the
order parameters and their spatial gradients. Properties of the crystalline structure lead naturally
to models based on several order parameters, which involve serious mathematical challenges not
found in Cahn-Hilliard and Allen-Cahn equations and related single-order-parameter models. One
new element is the set of invariants reflecting the crystallography of the material. This combined
with the usual lack of convexity when there are phase transitions makes the problems rich but
difficult. The geometry of the free energy function generally is quite intricate with many critical
points, and the dependence of the surface energy on orientation is typically not convex.
The model we will be describing in this section, and which forms the basis of the analysis to follow,
is that in [5] and [23]. It grew out of efforts in overcoming the ad hoc approach which has been
employed in single-order-parameter models to represent anisotropic interfaces. This model employs
an energy functional which is intimately related to the crystalline lattice and is formulated in terms
of physically based order parameters. The gradient energy terms are sums of squares of derivatives
with coefficients which reflect the anisotropy. What results is a continuous (diffuse) description
of an interface. One way to model anisotropic interfacial properties in a single-order-parameter
diffuse interface theory is to allow the gradient energy coefficients and the mobility coefficient to
depend on the spatial gradient of the order parameter, which is affected by the orientation of
the interface. In this way the surface energy and the kinetic coefficient can be assigned a given
anisotropy. While this approach allows a great deal of flexibility, it is also somewhat ad hoc.
Another approach involves introducing anisotropy through generalized gradient energy terms that
include higher order derivatives; this approach can also be difficult to justify on theoretical grounds.
On the other hand, the use of continuum models based on an underlying lattice such as we
are considering, have the advantage that the anisotropy appears in a natural way, and correctly
incorporates the crystal symmetries that are present.
N. D. A. Supported by grants from the University of North Texas and the University of Athens. C. S. supported
by I.K.Y..
1
2
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
Our description, in this section, of the crystalline framework and the resulting phase field model,
will be brief; full details can be found in [5]. We begin with a textbook physical description of
surface tension [19], since that concept is directly related to our construction of a free energy
functional below. Surface tension in its most well known form can be thought of either as a force
or (in the present case) as the energy that is required to generate or maintain an interface. On the
microscopic level, surface tension can be related to the number of broken bonds, which is exactly
what distinguishes atoms on the surface from those in the interior. Planar cuts of the crystal at
different orientations have different numbers of broken bonds per unit of area, and this induces
anisotropic surface tension.
In the present paper we will be dealing with 3D lattices, and with a type of crystal known as
FCC (face-centered cubic). Quite analogous problems and methods hold for other structures; for
example in the case of crystals of the hexagon-closely-packed (HCP) type the analogy is almost
complete [16]. The FCC is a periodic arrangement of atoms whose unit cell is a cube with atoms
occupying its corners and the centers of its faces. We easily count 12 closest neighbors to each
atom. We can
planar
identify 3 distinguished
cuts corresponding to the normal directions n =
1
1
1
1
1
√ ,√ ,√
, n = (1, 0, 0), n = 0, √2 , √2 (Fig. 1). It is a simple exercise to count broken
3
3
3
bonds for each of these orientations. We find respectively 3, 4, and 5 broken bonds per unit area.
n = (1, 0, 0)
x3
x2
n = (0,
x1
n=(
1
3
1
2
,
,
1
3
1
2
,
)
1
3
)
Figure 1. The distinguished planar cuts.
Each unit cell in the FCC contains the equivalent of 4 whole atoms and so a tetrahedron can be
associated with it (Fig. 2). Each numbered point of such a tetrahedron can serve as the origin of a
primitive cubic Bravais lattice. The FCC lattice then is decomposed into 4 numbered sublattices.
Following [5], in our work we will consider, as a specific illustrative prototype, the alloy Cu3 Au, so
that each lattice site is occupied by either a Cu or Au atom. Continuing this emphasis on illustrative
special prototypes, we focus on boundary regions between two phases, one of them termed “ordered”
(the Cu and Au atoms are as illustrated in Fig. 2) and the other totally disordered (the assignment
of any given site in the unit cell to Cu or Au is random, subject to the ratio of Cu to Au being the
same as in the ordered state).
There will be other occasions as well, during out treatment, when restrictions to special cases
are made. The reasons will be mainly to simplify the rigorous analysis. In fact a full mathematical
treatment of interphase boundaries in this multiple order parameter phase field scenario appears
presently to be an unreasonable goal, at least without a wealth of obfuscating details.
We will focus, at least at first, on interphase boundaries representing order-disorder transitions.
The order parameters In the ordered form the copper atoms occupy the centers of the faces and
the gold atoms the vertices. Four numbers ρ1 , ρ2 , ρ3 , ρ4 are defined, when ordering is imperfect, to
be the fraction of atoms on each primitive cubic sublattice which are Cu. When ordering is perfect,
copper represents 34 of the total. Hence for the ordered Cu3 Au state,
ρ1 (ord) = 0, ρ2 (ord) = ρ3 (ord) = ρ4 (ord) = 1,
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
2
3
4
1
x3
x2
x1
Cu
Au
Figure 2. A unit cell of the FCC lattice, and the tetrahedron whose corners serves
to number the four primitive cubic sublattices. The lower schematic indicates a way
to visualize the number of Cu and Au atoms assigned to each unit cell.
while for the disordered state
3
.
4
In our treatment, the order parameters ρi are taken to vary continuously within the order-disorder
transition region. The equations we will be dealing with are written in terms of the alternative
variables X, Y, Z, W , defined as linear combinations of the ρ’s:
ρ1 (dis) = ρ2 (dis) = ρ3 (dis) = ρ4 (dis) =
X = 41 (ρ1 + ρ2 − ρ3 − ρ4 ),
Y = 14 (ρ1 − ρ2 + ρ3 − ρ4 ),
Z = 14 (ρ1 − ρ2 − ρ3 + ρ4 ),
W = 14 (ρ1 + ρ2 + ρ3 + ρ4 ).
(1)
The intuition behind this transformation is that the new order parameters are more amenable
to continuizing [18]. The first three are nonconserved order parameters and the fourth, W , is
conserved, since it represents the total density of copper in the crystal. It will be taken fixed (in a
more complete model, W would be taken as fixed not pointwise, but only on the average [23]). In
what follows we choose to redefine the variables (X, Y, Z, W ) as multiples of the previous ones by
a fixed number. To avoid new notation, we continue to use the previous symbols (X, Y, Z, W ) for
the new variables. The disordered state now corresponds to
X = Y = Z = 0,
and the ordered state to
X = Y = Z = 1.
The free energy Since W is held fixed, the free energy function used in [5] depends only on
X, Y, Z and their gradients:
Z
[Q(∇X, ∇Y, ∇Z) + F (X, Y, Z)] dξ1 dξ2 dξ3 ,
(2)
J(X, Y, Z) =
V
4
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
where the space coordinates are (ξ1 , ξ2 , ξ3 ) and V is the volume occupied by the sample. Here Q is
a quadratic form and F is a fourth degree polynomial which is positive except at its several global
minima, including (0, 0, 0) and (1, 1, 1). The bulk free energy F must conform to certain crystalline
symmetries; if it is restricted to be a fourth degree polynomial, its general form is that given below
in (3).
The function Q represents the influence on the free energy of the differences between the order
parameters at a point and those at nearest neighbors. It is generally the case that this contribution
depends on the orientation of the line between these nearby points, and this dependence is a source
of anisotropy. The simplest type of quadratic form Q which accounts for anisotropy and which
satisfies other symmetry conditions is of the form Q = AQ1 + BQ2 , where Qi are simple sums
of squares of derivatives, Q1 being isotropic, and Q2 being a prototypical anisotropic quadratic
expression. The ratio B/A then will be taken as a measure of the degree of anisotropy of the free
energy.
As a result, we obtain the following for F and Q:
(3) F (X, Y, Z) = a2 (X 2 + Y 2 + Z 2 ) + a3 XY Z + a41 (X 4 + Y 4 + Z 4 ) + a42 (X 2 Y 2 + X 2 Z 2 + Y 2 Z 2 ),
Q1 =
(4)
Q2 =
1
2
1
2
∂X
∂ξ1
∂X
∂ξ2
2
2
+
+
∂Y
∂ξ2
∂X
∂ξ3
2
2
+
+
∂Z
∂ξ3
∂Y
∂ξ1
2 2
,
+
∂Y
∂ξ3
2
+
∂Z
∂ξ1
2
+
∂Z
∂ξ2
2 .
The approach here will be to assume dynamics governed by a gradient flow with respect to J, and
examine the nature of the interface between grains of ordered and disordered material. In general,
these two states will enjoy different bulk free energies, and the interface will migrate. However, the
motion depends on the values of the coefficients in (3), which in turn depend on the temperature.
The simplest situation is when the two bulk values of F are the same, and we assume that the
temperature is chosen so that this is the case, and that the two equilibria are (0, 0, 0) and (1, 1, 1),
representing the disordered and ordered state respectively. In this case a calculation shows that
a2 = 2, a3 = −12, a41 = a42 = 1, so that F (0, 0, 0) = F (1, 1, 1) = 0. The truncation to fourth
degree is discussed in [5] and the extension to sixth degree in [23]. The inclusion of cubic terms is
sufficient for the existence of first-order transitions [15]. The basis of the free energy is a lattice
model along the lines of bonds discussed above but complicated further by (1) the presence of two
species, and (2) the entropy term that has to be added (that can be ignored only at Oo Kelvin).
The phase field equations The governing evolution PDE’s are given by the L2 gradient flow of
this functional:




X
X
∂
(5)
τ  Y  = L  Y  − ∇F (X, Y, Z),
∂t
Z
Z
where L is a diagonal matrix of second degree elliptic operators in the space variables, ∇ denotes
the gradient with respect to the variables (X, Y, Z), and τ is a dimensionless relaxation time.
Although we have described the model in terms of the ordering of a special binary alloy, it serves
partially as a pattern for the treatment of other multiphase materials.
The fundamental paper [5], in addition to the derivation of the model, contains a bifurcation
analysis of the uniform steady states, numerical and formal asymptotic analyses of plane wave
solutions for large anisotropy ratios A/B ≡ ǫ−2 , and numerical calculation of the Wulff shapes.
This paper together with [23] are our basic references. Other related work is cited in these references.
We remark that the parameter ǫ should not be confused with the usual epsilon appearing before the
gradient in the Allen-Cahn and Cahn-Hilliard equations, where it has an entirely different meaning.
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
Grain boundaries as planar solutions Plane waves in the direction n and velocity V are
solutions of (5) of the form
(6)
X = x(n·(ξ1 , ξ2 , ξ3 ) − V t) = x(s), Y = y(s), Z = z(s),
with boundary conditions x(−∞) = y(−∞) = z(−∞) = 0, x(∞) = y(∞) = z(∞) = 1. They
represent planar interfaces with normal n separating an ordered state from a disordered state. The
functions x = (x, y, z) satisfy (derivatives are with respect to s)
−V x′ = Λx′′ − ∇F (x, y, z),
(7)
where Λ is a diagonal matrix whose elements are linear functions of A and B and quadratic functions
of n.
However, recall that the temperature has been chosen so that the coefficients of F are such that
F has equal depth wells at the equilibria of the order disorder transition: F (0, 0, 0) = F (1, 1, 1).
We see by taking the scalar product of (7) with x′ and integrating that V = 0.
1.2. Symmetries. Simplifications to the system (7) can be made by seeking only those profiles
which satisfy certain symmetry constraints. Doing so reduces the dimensionality of the model to
two, thereby making the mathematics more tractable, while retaining features characteristic of
higher-dimensional models. Of course the direction n must be chosen so that the resulting profile
satisfies those constraints, and the function F and the matrix Λ have to be compatible with them
as well.
One such possible constraint is the restriction of the order parameters to the plane Y = Z
(hence y = z). In the crystal, this is tied to the symmetry between two sites on the elementary
tetrahedron in Fig. 1. Note from (1) that Y = Z ⇒ ρ3 = ρ4 and that the two symmetric sites
share the same ξ1 coordinate. Therefore if we take n so that n2 = n3 , the symmetry Y = Z should
be preserved through the transition. This is indeed true. To exhibit the resulting system, we define
our anisotropy parameter
p
(8)
ǫ = B/A,
the reduced free energy
(9)
g(x, r) = F
1
1
x, √ r, √ r
2
2
(r is a modified order parameter), and the angle α by n = (cos α, √12 sin α, √12 sin α), 0 ≤ α ≤ π.
Then (7) is reduced to
(10)
(cos2 α + ǫ2 sin2 α)x′′ − gx (x, r) = 0, x(−∞) = r(−∞) = 0,
√
2
2
′′ − g (x, r) = 0, x(∞) = 1, r(∞) =
2,
(ǫ2 cos2 α + 1+ǫ
sin
α)r
r
2
As can be easily verified, the gradient flow (5) that determines the full dynamics of the phase
field model leaves invariant the subspace of functions
(11)
S = {(X(ξ1 , ξ2 , ξ3 ), Y (ξ1 , ξ2 , ξ3 ), Z(ξ1 , ξ2 , ξ3 )) : Y = Z} .
Note that the special cases α = 0 and α = π/2 correspond to the distinguished cuts (1, 0, 0) and
(0, √12 , √12 ) already mentioned above. When ǫ ≪ 1, the problem for the profile at α = 0 or α = π/2
is a singular perturbation problem from a limit profile at ǫ = 0, “singular” because passing to the
limit ǫ = 0 reduces the order of the system (11) from 4 to 2. In the case α = 0, the problem is
particularly difficult because of a degeneracy (irregularity) in the limit profile which does not allow
for the application of the Fenichel theory and its variants.
6
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
1.3. Statement of results. In the present paper we study the orientation α = π2 corresponding
to grain boundaries parallel to the second distinguished cut in Fig. 1. We establish existence and
stability of the connection between the ordered and disordered states in the invariant subspace S,
i.e. we do this for solutions of (10). The case α = 0 is very different both in results and methods
of proof. We refer to [1], [9].
More precisely, we prove the following theorems (see the remarks before (12)):
Theorem A (The heteroclinic orbit) If ǫ > 0 is sufficiently small, then there exists a solution
(xǫ (s), rǫ (s)) of (12) such that:
xǫ = x0 + x̃ǫ , rǫ = r0 + r̃ǫ with x̃ǫ , r̃ǫ ∈ H 2 (R) ∩ C 2 (R),
and
where
x̃ǫ , r̃ǫ ⊥ (x′0 , r0′ ) (in L2 (R) × L2 (R)),
k(x̃ǫ , r̃ǫ )k1,ǫ = O(ǫ2 ),
2
2
k(x̃ǫ , r̃ǫ )k21,ǫ = ǫ2 x̃′ǫ L2 + r̃ǫ′ L2 + kx̃ǫ k2L2 + kr̃ǫ k2L2 .
Here (x0 (s), r0 (s)) is the solution of (10) for ǫ = 0, defined below in Sec. 2.2.
Theorem B (Stability) Let (xǫ , rǫ ) be the heteroclinic orbit of Theorem A. Then if ǫ > 0 is
sufficiently small, the spectrum of the linearized operator Lǫ about (xǫ , rǫ ) has the following form:
Zero is a simple eigenvalue at the bottom of the spectrum and the rest of the spectrum is contained
in [C, ∞), where C is a positive constant independent of ǫ.
As we have mentioned before, existence for (10), α = π2 , can be treated via Fenichel’s invariant
manifold theory [8]. For this choice of α, (10) is equivalent, by simple stretching of variables, to
(12)
ǫ2 x′′ = gx (x, r),
r ′′ = gr (x, r)
This system is Hamiltonian and for ǫ = 0 it has a heteroclinic connection (x0 (s), r0 (s))), a
standard fact. Fenichel’s machinery, as presented for example in Jones [14], allows the reduction
of the problem on a two-dimensional center-like manifold, and by using the Hamiltonian structure
one can establish the existence of a heteroclinic connection for small ǫ > 0. In Sec. 5, we give a
self-contained presentation of the existence of the heteroclinic, using Fenichel’s geometric singular
perturbation theory. This section can be read independently from the rest of the paper.
The main approach to the question of existence in this paper is different from Fenichel’s; moreover
we establish the stability of the connection, relative to the time variable t appearing in (5). Except
for Sec. 5, our procedure is based on the study of the linearized operator of the parabolic problem
(5) about the zero order approximation to the heteroclinic, and thus it deals with an infinite
dimensional phase space and is more functional analytic. The main effort is to establish a theorem
on the spectrum of this operator. Then this result is utilized in Sec. 3, where the existence of the
heteroclinic is established fairly quickly via a contraction mapping argument. Finally in Sec. 4 we
establish stability of the heteroclinic via the theorem in Sec. 2 and a simple perturbation argument.
2. The spectrum
2.1. Preliminaries. The boundary value problem (12) to be considered is
(13)
ǫ2 x′′ = gx (x, r),
r ′′ = gr (x, r),
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
(14)
(15)
x = x(s), r = r(s), s ∈ R,
′
=
d
,
ds
x(−∞) = r(−∞) = 0, x(∞) = 1, r(∞) =
√
2.
For future reference we record here the most relevant facts about the function g. From (9), (3),
3
g(x, r) = 2(x2 + r 2 ) − 6xr 2 + x2 r 2 + r 4 + x4 ,
4
√
with equal minima at (0, 0) and (1, 2). Therefore
(16)
(17)
gx = 4x − 6r 2 + 2xr 2 + 4x3 , gr = r(4 − 12x + 2x2 + 3r 2 );
(18)
gxx = 4 + 2r 2 + 12x2 ≥ 4,
gxr = −12r + 4xr,
grr = 4 − 12x + 2x2 + 9r 2 .
Furthermore,
2x + 2x3
,
3−x
which can be√inverted for x ∈ [0, 3) to obtain x = χ(r), where χ : R→[0, 3) is smooth, even, and
χ(0) = 0, χ( 2) = 1, χ′ (0) = 0.
gx = 0 ⇔ r 2 =
(19)
2.2. The ǫ = 0 approximation. Setting ǫ = 0 in the first equation of (13) and using the expectation that x(s) ∈ [0, 1], we get x = χ(r).
Substituting into the second equation, we have
r ′′ − gr (χ(r), r) = 0.
(20)
We look for a solution to this equation such that r(−∞) = 0, r(+∞) =
Setting
(21)
2.
G(r) = g(χ(r), r), r ∈ R,
we have
(22)
√
G′ (r) = gx (χ(r), r)χ′ (r) + gr (χ(r), r) = gr (χ(r), r).
Thus (20) can be written as
(23)
r ′′ − G′ (r) = 0.
√
It can be checked from (17), (19) that as r ranges from 0 √
to 2, the function G′ (r) = gr (χ(r), r)
starts positive and then becomes negative. Also G(0) = G( 2) = 0.
√
It is known [4, 11] that in the phase plane there is a heteroclinic orbit connecting (0, 0)) to ( 2, 0).
This furnishes a solution to (20) with the desired behavior at infinity, unique up to translation. We
denote it by r0 (s) and call it “the base heteroclinic”.
√
We also know that r0′ (s) > 0 for s ∈ R, hence √
0 < r0 (s) < 2. We set x0 (s) = χ(r0 (s)). Then
x0 (−∞) = χ(r0 (−∞)) = χ(0) = 0; x0 (∞) = χ( 2) = 1. Hence the pair (x0 , r0 ) is a solution of
(13)–(15) for ǫ = 0. Furthermore, we know that√r0 (s) and r0′ (s) approach their limits exponentially
as s→ ± ∞. This follows from G′′ (0) > 0, G′′ ( 2) > 0 via linear theory.
From (23) we then have that r0′′ converges exponentially as |s|→∞ to 0. Also r0′′′ = r0′ G′′ (r0 ),
which implies that r0′′′ converges exponentially to 0.
8
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
Function spaces and the bistable operator In this paper the notation L2 , H 2 , C 2 will denote
the corresponding spaces of functions defined on the whole real line R: L2 (R), etc.
From the exponential convergence, we have that r0′ , r0′′ , r0′′′ ∈ L2 , which in turn gives that
r0′ ∈ H 2 . Similarly, since x0 = χ(r0 ), we get that x′0 ∈ H 2 .
We also note that for r ∈ R,
(24)
G′′ (r) = gxr (χ(r), r)χ′ (r) + grr (χ(r), r) = −
[gxr (χ(r), r)]2
+ grr (χ(r), r).
gxx (χ(r), r)
Set
.
q(s) = G′′ (r0 (s)), s ∈ R.
(25)
We define the operator B by
Bh = −h′′ + qh with D(B) = H 2 ,
(26)
which is unbounded and self-adjoint on L2 . It is called the bistable operator. The following facts
are well known:
• The essential spectrum of B is contained in [C1 , ∞), where C1 > 0.
• The smallest eigenvalue of B is 0, which is simple (we know that r0′ ∈ ker(B)).
Hence by the variational characterization of the eigenvalues of self adjoint operators (see [20]) we
have
Z ∞
′ 2
(h ) + qh2 ds ≥ C2 khk2L2 for all h ∈ H 2 , h ⊥ r0′ (in L2 ).
(27)
−∞
2.3. The operator Lǫ . We now consider the operator Lǫ , obtained by linearizing (13) about
(x0 , r0 ). We have
h1
−ǫ2 h′′1 + gxx (x0 , r0 )h1 + gxr (x0 , r0 )h2
,
(28)
Lǫ
=
h2
−h′′2 + gxr (x0 , r0 )h1 + grr (x0 , r0 )h2
with Lǫ : D(Lǫ ) −→ L2 × L2 , D(Lǫ ) = H 2 × H 2 . It is an unbounded self -adjoint operator in
L2 × L2 .
√
We recall that (x0 (s), r0 (s))→(0, 0) as s→ − ∞ and (x0 (s), r0 (s))→(1, 2) as s→ + ∞ (in all
cases, exponentially); and that (see (18))
(29)
.
gxx (0, 0)
√ 0, grr (0,√0) = 4, √
√ = 4, gxr (0, 0) =
gxx (1, 2) = 20, gxr (1, 2) = −8 2, grr (1, 2) = 12.
We write
(30)
so that
(31)
Then
(32)
Dǫ =
ǫ2 0
0 1
N (−∞) =
, N (s) =
4 0
0 4
gxx (x0 , r0 ) gxr (x0 , r0 )
gxr (x0 , r0 ) grr (x0 , r0 )
, N (+∞) =
,
√ 20 √ −8 2
.
−8 2 12
Lǫ h = −Dǫ h′′ + N (s)h, h = (h1 , h2 )⊥ .
In what follows, we make use of the operator
(33)
L̃ǫ h = −Dǫ h′′ + Ñ (s)h,
where Ñ (s) = N (−∞) if s ≤ 0; Ñ (s) = N (∞) if s > 0. We can easily show that
(34)
(L̃ǫ h, h)L2 ×L2 ≥ C3 khk2L2 ×L2 ∀h ∈ H 2 × H 2 ,
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
with C3 > 0 independent of ǫ > 0. Hence
(35)
σess (L̃ǫ ) ⊂ σ(L̃ǫ ) ⊂ [C3 , ∞).
Also (Lǫ − L̃ǫ )h = (N (s) − Ñ (s))h, where (N (s) − Ñ (s))→0 in the matrix norm exponentially as
|s|→∞.
It can be shown that the operator
2
2
2
2
(Lǫ − L̃ǫ )L̃−1
ǫ : L × L −→ L × L
is compact. In other words, (Lǫ − L̃ǫ ) is relatively L̃ǫ -compact. From this it follows that
(36)
σess (Lǫ ) = σess (L̃ǫ ) ⊂ [C3 , ∞)
( [13, p. 140]).
Occasionally we will be dropping the subscript ǫ and write L instead of Lǫ .
Throughout this paper, all the inner products are in the sense of LR2 or L2 × L2 , unless specified
∞
otherwise. The L2 × L2 inner product is defined by (u, v)L2 ×L2 = −∞ (u1 v1 + u2 , v2 ) ds, where
u = (u1 , u2 ), v = (v1 , v2 ) ∈ L2 × L2 Furthermore, all the constants Ci will be independent of ǫ > 0.
2.4. The operators K1ǫ (λ), K2ǫ (λ), and the reduction to a single equation with bistable
structure. The eigenvalues λ of L and their corresponding eigenfunctions (h1 , h2 ) ∈ H 2 × H 2
satisfy
(37)
−ǫ2 h′′1 (s) + gxx (x0 (s), r0 (s))h1 (s) + gxr (x0 (s), r0 (s))h2 (s) = λh1 (s),
(38)
−h′′2 (s) + gxr (x0 (s), r0 (s))h1 (s) + grr (x0 (s), r0 (s))h2 (s) = λh2 (s),
(39)
kh1 k2L2 + kh2 k2L2 = 1.
From now on for simplicity we will not explicitly write s and write instead gxx in the place of
gxx (x0 (s), r0 (s)), etc. Also since we are interested in the small eigenvalues of L, we can assume
without loss of generality that λ ≤ 2.
From (37) we have
(40)
−ǫ2 h′′1 + (gxx − λ)h1 = −gxr h2 .
We know that gxx ≥ 4 and since λ ≤ 2, we have gxx −λ ≥ 2. Hence for each ǫ > 0, λ ≤ 2, h2 ∈ L2 ,
there exists a unique solution h1 ∈ H 2 of the above equation.
Hence we can define a linear operator K1ǫ (λ) : L2 →L2 by
.
(41)
h1 = K1ǫ (λ)h2
for ǫ > 0, λ ≤ 2. We also define another linear operator K2ǫ (λ) : H 2 →H 2 for ǫ > 0, λ ≤ 2 by
writing
gxr
h2 + ǫ2 K2ǫ (λ)h2 ,
(42)
h1 = K1ǫ (λ)h2 = −
gxx − λ
.
gxr
ǫ (λ)h
where K2ǫ (λ)h2 = ǫ12 gxx
h
+
K
2 .
1
−λ 2
Observe that the first term on the right side of (42) is the solution h1 of (40) when ǫ = 0.
Substituting (42) into (38), we get
gxr
′′
2 ǫ
(43)
−h2 + gxr −
h2 + ǫ K2 (λ)h2 + grr h2 = λh2 ,
gxx − λ
which we rewrite in the form
2
2
gxr
gxr
′′
h2 = λ 1 +
h2 − ǫ2 gxr K2ǫ (λ)h2 .
(44)
−h2 + grr −
gxx
gxx (gxx − λ)
10
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
Hence the eigenvalue problem (37)–(39) is equivalent to
(45)
(46)
h1 = K1ǫ (λ)h2 ,
2
gxr
g2
h2 − ǫ2 gxr K2ǫ (λ)h2 ,
−h′′2 + grr − xr h2 = λ 1 +
gxx
gxx (gxx − λ)
kh1 k2L2 + kh2 k2L2 = 1.
(47)
Observe that on the left of (46), we have the bistable operator B in (26).
2.5. H 2 estimates.
Lemma 1. For every λ, λ1 , λ2 ≤ 2 and every ǫ > 0, the following inequalities hold:
(48)
kK1ǫ (λ)f kL2 ≤ C4 kf kL2 ∀f ∈ L2 ,
(49)
kK2ǫ (λ)f kL2 ≤ C5 kf kH 2 ∀f ∈ H 2 ,
(50)
k(K2ǫ (λ1 ) − K2ǫ (λ2 ))f kL2 ≤ C6 |λ1 − λ2 | kf kH 2 ∀f ∈ H 2 ,
where C4 , C5 , C6 are independent of λ, λ1 , λ2 , ǫ and f .
Proof Let f ∈ L2 , λ ≤ 2 and ǫ > 0, and set X = K1ǫ (λ)f ∈ H 2 . Then by the definition of the
linear operator K1ǫ (λ), we have
−ǫ2 X ′′ + (gxx − λ)X = −gxr f,
(51)
from which we obtain
(52)
−ǫ
2
Z
∞
′′
X X ds +
−∞
Z
∞
−∞
2
(gxx − λ)X ds = −
Z
∞
gxr f X ds,
−∞
and by integration by parts, which is valid since X ∈ H 2 , we get
Z ∞
Z ∞
Z ∞
2
2
′ 2
gxr f X ds.
(53)
ǫ
(gxx − λ)X ds = −
(X ) ds +
−∞
X ′ (s)→0
−∞
H 2 ).
−∞
(we also used that X(s),
as |s|→∞, since X ∈
Now using that gxx − λ ≥ 2 and
that gxr is bounded, we have by the Cauchy-Schwartz inequality that
Z ∞
|f | |X| ds ≤ sup |gxr | kf kL2 kXkL2 .
(54)
2kXk2L2 ≤ sup |gxr |
s
s∈R
−∞
Hence kXkL2 ≤ C4 kf kL2 . This proves (48).
Now let f ∈ H 2 . By the definition of K2ǫ (λ), we have that
−ǫ2 u′′ + (gxx − λ)u = −gxr f,
is equivalent to
u=−
gxr
f + ǫ2 K2ǫ (λ)f.
gxx − λ
Set X = K2ǫ (λ)f ; then
′′
gxr
gxr
2
2
2
−ǫ −
+ (gxx − λ) −
f +ǫ X
f + ǫ X = −gxr f,
gxx − λ
gxx − λ
which simplifies to
2
′′
−ǫ X + (gxx − λ)X = −
gxr
f
gxx − λ
′′
.
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
By working as before, we get
kXkL2
′′ gxr
1
f ≤ 2.
2
gxx − λ
L
Taking into account that gxx − λ ≥ 2 and that the functions gxx and gxr of s are bounded along
with their first and second derivatives, we can estimate the right hand side of the above inequality,
and get kXkL2 ≤ C5 kf kH 2 . This proves (49).
Set X1 = K2ǫ (λ1 )f, X2 = K2ǫ (λ2 )f . Then
′′
gxr f
2 ′′
, i = 1, 2,
−ǫ Xi + (gxx − λi )Xi = −
gxx − λi
and so
(55)
2
′′
−ǫ (X1 − X2 ) + (gxx − λ1 )(X1 − X2 ) = (λ1 − λ2 )X2 −
gxr f
gxr f
−
gxx − λ1 gxx − λ2
′′
.
Now by (49), we have
kX2 kL2 = kK2ǫ (λ2 )f kL2 ≤ C5 kf kH 2 ,
and so by working as in the proof of (48) and of (49) (to estimate the norm of the last term of
(55)), we get
′′ gxr f
gxr f
≤ C6 |λ1 − λ2 | kf kH 2 .
2kX1 − X2 kL2 ≤ C5 |λ1 − λ2 | kf kH 2 + −
gxx − λ1 gxx − λ2 L2
This proves (50). The proof of Lemma 1 is complete. ˜
2
xr
Lemma 2. Denote, as always, q = grr − ggxx
. If −h′′ + qh = f, h ∈ H 2 , h ⊥ r0′ (in L2 ) and
f ∈ L2 , then
khkH 2 ≤ C7 kf kL2 ,
with C7 independent of f .
Proof Working as in the proof of Lemma 1, we get
Z ∞
Z ∞
′ 2
2
f h ds ≤ kf kL2 khkL2 .
(h ) + qh ds =
−∞
−∞
Utilizing (27), we have
C2 khk2L2 ≤ kf kL2 khkL2 .
Hence
khkL2 ≤ C2−1 kf kL2 .
(56)
Also,
kh′′ kL2 = kqh − f kL2 ≤ kqk∞ khkL2 + kf kL2 ≤ kqk∞ C2−1 + 1 kf kL2 .
Obviously −2hh′′ ≤ h2 + (h′′ )2 , and
Z
(57)
2
∞
−∞
(h′ )2 ds ≤ khk2L2 + kh′′ k2L2 .
From (56), (57),we obtain that
which proves Lemma 2. ˜
khkH 2 ≤ C7 kf kL2 ,
12
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
2.6. A priori estimates.
Proposition 1. Suppose that (λ, h) is a solution to (37)-(39). Then there exist positive constants
ǫ0 , λ0 < min {2, C3 }, M0 , a0 , independent of ǫ, λ, h, such that if 0 < ǫ < ǫ0 and |λ| < λ0 , then the
following estimates hold:
(i) |λ| ≤ M0 ǫ2 ,
(ii) kpkH 2 ≤ M0 ǫ2 , |a| ≥ a0 , where
r0′
, p ⊥ r0′ (in L2 ).
kr0′ kL2
h2 = ay + p, y =
(For simplicity we do not write the dependence of λ, a, p on ǫ).
Proof Throughout the proof, Ci > 0 will denote constants independent of ǫ, λ, h. Substitute
2
h2 = ay + p in (46) and set K̃2ǫ (λ) := gxr K2ǫ (λ), σ := 1 + gxx (ggxr
, with |σ(s)| ≤ kσk∞ , s ∈ R
xx −λ)
(kσk∞ independent of λ since gxx − λ ≥ 2). Then (46) takes the following form (recall that
−y ′′ + qy = 0):
−p′′ + qp = λaσy − ǫ2 aK̃2ǫ (λ)y + λσp − ǫ2 K̃2ǫ (λ)p.
(58)
By Lemma 1, we have
(59)
kK̃2ǫ (λ)f kL2 ≤ C8 kf kH 2 ,
(60)
kK̃2ǫ (λ1 )f − K̃2ǫ (λ2 )f kL2 ≤ C8 |λ1 − λ2 | kf kH 2
for every f ∈ H 2 , λ, λ1 , λ2 ≤ 2 and ǫ > 0 (C8 independent of λ, λ1 , λ2 , ǫ, f ). Multiplying (58) by
y, integrating, and using that (−p′′ + qp, y)L2 = (p, −y ′′ + qy)L2 = 0 gives
λa(σy, y) − ǫ2 a K̃2ǫ (λ)y, y + λ(σp, y) − ǫ2 K̃2ǫ (λ)p, y = 0.
Since the function σ satisfies σ > 1, kykL2 = 1, via (59) we have
(61)
|λ||a| ≤ ǫ2 |a|C8 kykH 2 + |λ| kσk∞ kpkL2 + ǫ2 C8 kpkH 2 .
Applying Lemma 2 to (58) gives
kpkH 2 ≤ C7 kλaσy − ǫ2 aK̃2ǫ (λ)y + λσp − ǫ2 K̃2ǫ (λ)pkL2 ≤
≤ C7 |λ| |a| kσk∞ + ǫ2 |a|C8 kykH 2 + |λ| kσk∞ kpkL2 + ǫ2 C8 kpkH 2 ,
or in a more compact form,
kpkH 2 ≤ C9 |λ| |a| + ǫ2 |a| + |λ| kpkH 2 + ǫ2 kpkH 2 .
Choosing ǫ0 , λ0 such that C9 (λ0 + ǫ20 ) < 21 , we get
kpkH 2 ≤ 2C9 (|λ| |a| + ǫ2 |a|).
(62)
Now using Lemma 1, we have kh1 kL2 = kK1ǫ (λ)h2 kL2 ≤ C4 kh2 kL2 , so by (47) we have kh2 k2L2 ≥
1
C10 > 0 for C10 = 1+C
2 . Also
4
(63)
1 ≥ kh2 k2L2 = |a|2 + kpk2L2 ≥ C10 > 0.
In particular, |a| ≤ 1. Thus from (62) we have
(64)
kpkH 2 ≤ 2C9 (|λ| + ǫ2 ).
Further reducing ǫ0 , λ0 so that C9 (λ0 + ǫ20 ) <
(63) gives |a|2 ≥ 21 C10 .
√
C10
4 ,
we get from (64) that kpk2H 2 ≤ 12 C10 . This and
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
Returning to (61) and using the above, we have
p
|λ| C10 /2 ≤ C8 ǫ2 kykH 2 + |λ| kσk∞ kpkL2 + ǫ2 C8 kpkH 2
≤ C8 ǫ2 kykH 2 + |λ| kσk∞ + C8 ǫ2 2C9 (|λ| + ǫ2 ) ≤
≤ C11 (ǫ2 + (|λ| + ǫ2 )2 ) = C11 (ǫ2 + |λ|2 + ǫ4 + 2|λ|ǫ2 ).
p
Choosing even smaller λ0 such that C11 λ0 < 21 C10 /2, we get
|λ| ≤ C12 (ǫ2 + ǫ4 + 2|λ|ǫ2 ) ≤ C12 (2ǫ2 + 2|λ|ǫ2 ) ≤ C13 ǫ2 .
Finally utilizing the estimate in (64), we obtain kpkH 2 ≤ C14 ǫ2 .
p
This completes the proof of Proposition 1 with M0 = max {C13 , C14 }, a0 = C10 /2. ˜
2.7. Existence, uniqueness, and simplicity of critical eigenvalues. Following Nishiura [17],
we call the eigenvalues λǫ with the property limǫ→0 λǫ = 0 critical.
Proposition 2. The eigenvalue problem (37)-(39) has the following property: There exists ǫ1 < ǫ0
such that for 0 < ǫ < ǫ1 , there is a critical eigenvalue λ(ǫ) satisfying |λ(ǫ)| = O(ǫ2 ). Furthermore
λ(ǫ) is simple and it is the unique eigenvalue on the interval [−λ0 , λ0 ], where ǫ0 and λ0 are as in
Prop. 1
Proof
I. Existence of a critical eigenvalue With y as in Prop. 1, we search for a solution of (45)-(47)
of the form h2 = y + p, p ⊥ y. Substituting in (46) gives (recall that −y ′′ + qy = 0)
−p′′ + qp = λσy − ǫ2 K̃2ǫ (λ)y + λσp − ǫ2 K̃2ǫ (λ)p.
(65)
Next, we will be defining a map T : (λ, p) −→ (λ̂, p̂) for
(λ, p) ∈ S = {(λ, p) ∈ R × H 2 : |λ| ≤ M1 ǫ2 , kpkH 2 ≤ M2 ǫ2 , p ⊥ y (in L2 )}
(Mi constants independent of ǫ and to be determined later). S is a closed subset of the Banach
space R × H 2 equipped with the norm k(λ, p)k = |λ| + kpkH 2 .
−p̂′′ + q p̂ = λ̂σy − ǫ2 K̃2ǫ (λ)y + λσp − ǫ2 K̃2ǫ (λ)p,
ǫ2 K̃2ǫ (λ)y − λσp + ǫ2 K̃2ǫ (λ)p , y
.
λ̂ =
(σy, y)
(66)
(67)
Notice that (67) implies that the right side of (66) is L2 -orthogonal to y. So for (λ, p) ∈ S, there
is a unique (λ̂, p̂) ∈ R × H 2 , p̂ ⊥ y satisfying (66); hence the map T is well defined. Observe that
if (λ, p) ∈ S is a fixed point of T , then λ and h2 = y + p 6= 0 (since p ⊥ y) satisfy (46).
Using (59) and the C-S inequality (recall (σy, y) ≥ 1 and kykL2 = 1),
|λ̂| ≤ ǫ2 C8 kykH 2 + |λ|kσk∞ kpkL2 + ǫ2 C8 kpkH 2 ≤ ǫ2 C8 kykH 2 + (M1 M2 kσk∞ + M2 C8 )ǫ2 .
Choosing
(68)
M1 > max {C8 kykH 2 + 1, M0 },
(69)
M2 > max {C7 (M1 kσk∞ + C8 kykH 2 + 1), M0 /a0 },
)
(
r
λ0
−1/2
,
0 < ǫ1 < min (M1 M2 kσk∞ + C8 M2 )
, ǫ0 ,
M1
(70)
we get that |λ̂| ≤ M1 ǫ2 .
14
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
Also by Lemma 2 and (59),
kp̂kH 2 ≤ C7 kλ̂σy − ǫ2 K̃2ǫ (λ)y + λσp − ǫ2 K̃2ǫ (λ)pkL2
≤ C7 M1 ǫ2 kσk∞ + ǫ2 C8 kykH 2 + M1 ǫ2 kσk∞ M2 ǫ2 + ǫ2 C8 M2 ǫ2 ≤
(70) ≤ C7 ǫ2 (M1 kσk∞ + C8 kykH 2 + 1) ≤ (by (69)) M2 ǫ2 .
Hence T : S −→ S.
The next step is to show that T is a contraction on the closed set S. Let (λ1 , p1 ), (λ2 , p2 ) ∈ S;
then
|λ̂1 − λ̂2 | ≤ ǫ2 K̃2ǫ (λ1 )y − K̃2ǫ (λ2 )y 2 + K̃2ǫ (λ1 )p1 − K̃2ǫ (λ2 )p2 2 + kσk∞ kλ1 p1 − λ2 p2 kL2 .
L
L
Using (59), (60):
ǫ
ǫ
ǫ
K̃2 (λ1 )p1 − K̃2 (λ2 )p2 2 ≤ K̃2 (λ1 )(p1 − p2 ) 2 +
L
L
+ K̃2ǫ (λ1 )p2 − K̃2ǫ (λ2 )p2 2 ≤ C8 kp1 −p2 kH 2 +C8 |λ1 −λ2 | kp2 kH 2 ≤ C8 kp1 −p2 kH 2 +C8 M2 ǫ2 |λ1 −λ2 |.
L
Also
Hence
(71)
Also
kλ1 p1 − λ2 p2 kL2 ≤ |λ1 | kp1 − p2 kL2 + |λ1 − λ2 | kp2 kL2 ≤
≤ M1 ǫ2 kp1 − p2 kL2 + M2 ǫ2 |λ1 − λ2 |.
|λ̂1 − λ̂2 | = O(ǫ2 ) (|λ1 − λ2 | + kp1 − p2 kH 2 ) .
− (p̂1 − p̂2 )′′ +q (p̂1 − p̂2 ) = λ̂1 − λ̂2 σy−ǫ2 K̃2ǫ (λ1 )y − K̃2ǫ (λ2 )y +σ(λ1 p1 −λ2 p2 )−ǫ2 K̃2ǫ (λ1 )p1 − K̃2ǫ
By Lemma 2,
kp̂1 −p̂2 kH 2 ≤ C7 k λ̂1 − λ̂2 σy−ǫ2 K̃2ǫ (λ1 )y − K̃2ǫ (λ2 )y +σ(λ1 p1 −λ2 p2 )−ǫ2 K̃2ǫ (λ1 )p1 − K̃2ǫ (λ2 )p2 k
Working as before and using (71), we get
kp̂1 − p̂2 kH 2 = O(ǫ2 ) (|λ1 − λ2 | + kp1 − p2 kH 2 ) .
So combining the inequality above with (71) gives
|λ̂1 − λ̂2 | + kp̂1 − p̂2 kH 2 = O(ǫ2 ) (|λ1 − λ2 | + kp1 − p2 kH 2 ) .
Choosing ǫ1 > 0 sufficiently small, we see that if ǫ < ǫ1 , then T is a contraction. Applying the
Banach fixed point theorem, we have that there exists a unique (λ, p) ∈ S such that T (λ, p) = (λ, p),
that is λ̂ = λ and p̂ = p. So by (66) and (67), we get
−p′′ + qp = λσy − ǫ2 K̃2ǫ (λ)y + λσp − ǫ2 K̃2ǫ (λ)p.
Obviously λ, (h1 , h2 ) 6= 0 with h1 = K1ǫ (λ)h2 satisfy (45) and (46) and we can assume that (47) is
also satisfied. Hence we found an eigenvalue λ with |λ| ≤ M1 ǫ2 corresponding to an eigenfunction
(h1 , h2 ).
II. Uniqueness and simplicity of the critical eigenvalue
Suppose that for some ǫ < ǫ1 we have an eigenvalue λ̄ (not necessarily λ̄ 6= λ) such that |λ̄| ≤ λ0 ).
Then by Proposition 1,
|λ̄| ≤ M0 ǫ2 , kp̄kH 2 ≤ M0 ǫ2 , |ā| ≥ a0 ,
where h̄2 = āy + p̄, y ⊥ p̄ (again we suppress the dependence of λ, λ̄, h̄2 , p̄, ā on ǫ) with (h̄1 , h̄2 ) a
normalized eigenvector. Using (68)–(70), we obtain from the above that
p̄ |λ̄| ≤ M1 ǫ2 , ≤ M2 ǫ2 .
ā H 2
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
thus λ̄, āp̄ is a fixed point of T in S. Hence λ̄ = λ and
h̄1 =
K1ǫ (λ̄)h̄2
=
p̄
ā
= p. That is h̄2 = ā(y + p) = ãh2 and
ãK1ǫ (λ)h2
= ãh1 ,
where ã depends on ǫ > 0 and is due to the normalization of (h1 , h2 ). Since (h1 , h2 ) and (h̄1 , h̄2 )
are normalized, we get that |ã| = 1. Hence if ǫ < ǫ1 , λ is the unique eigenvalue in [−λ0 , λ0 ] and it
is simple. The proof of Proposition 2 is complete. ˜
2.8. The lower bound of the spectrum.
Lemma 3. Let λ < 0 be an eigenvalue with corresponding L2 normalized eigenfunction (h1 , h2 ).
Then kh2 kH 2 ≤ C16 , where C16 is independent of λ, ǫ.
Proof Let λ < 0 be an eigenvalue of (37) – (39) with corresponding eigenfunction (h1 , h2 ). From
(38),
2
2
2
2
+ grr
h1 + h22 ≤
h′′2 + λh2 = (gxr h1 + grr h2 )2 ≤ gxr
2
2
≤ sup gxr
+ grr
h21 + h22 .
s
2 + g 2 . Then via (39) and integration by parts, we have
Let C15 = sups gxr
rr
Z ∞
Z ∞
Z ∞
2
′′ 2
2
2
h2 ds + λ
h2 ds − 2λ
h′2 ds ≤ C15 .
−∞
−∞
−∞
Since λ < 0, we obtain from the inequality above
Z ∞
Z ∞
Z
′′ 2
′′ 2
h2 ds ≤ C15 ⇒
h2 ds +
−∞
−∞
∞
−∞
h22 ds ≤ C15 + 1.
Also from the identity −2h2 h′′2 ≤ (h′′2 )2 + h22 , we get
Z ∞
2
h′2 ds ≤ C15 + 1.
2
Hence kh2 kH 2 ≤
q
−∞
3
2 (C15
+ 1) = C16 . The proof of Lemma 3 is complete. ˜
Proposition 3. Let ǫ1 , λ0 be as in Proposition 2. There is 0 < ǫ2 < ǫ1 such that for 0 < ǫ < ǫ2 ,
there are no eigenvalues λ with λ ≤ −λ0 .
Proof Suppose that there is an eigenvalue λ with λ ≤ −λ0 . by (46),
2
gxr
′′
−h2 + qh2 = λ 1 +
h2 − ǫ2 gxr K2ǫ (λ)h2 .
gxx (gxx − λ)
R∞ We know that −∞ (h′ )2 + qh2 ds ≥ 0 ∀h ∈ H 2 , since the spectrum of the bistable operator
Bh = −h′′ + qh is positive. Also we recall that λ ≤ −λ0 . Multiplying both sides of the above
equation by h2 and integrating by parts gives:
λ0 kh2 k2L2 ≤ ǫ2 |(gxr K2ǫ (λ)h2 , h2 )| ≤ ǫ2 sup |gxr | kK2ǫ (λ)h2 kL2 kh2 kL2 ≤
≤ǫ
2
sup |gxr | kK2ǫ (λ)h2 kL2
s
s
2
≤ ǫ sup |gxr |C8 kh2 kH 2 ,
s
where we used that kh2 kL2 ≤ 1 and (49). Now using Lemma 3, we get kh2 kL2 ≤ C17 ǫ2 .
By (48):
kh1 kL2 = kK1ǫ (λ)h2 kL2 ≤ C4 kh2 kL2 ≤ C18 ǫ2 .
Obviously if ǫ < ǫ2 , then these estimates contradict that
kh1 k2L2 + kh2 k2L2 = 1.
The proof of Proposition 3 is complete. ˜
16
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
2.9. The basic result on the spectrum on Lǫ .
Theorem 1. If ǫ > 0 is sufficiently small, then the spectrum of Lǫ (c.f. (28)) has the following
form:
(i) At the bottom of the spectrum, there is a simple eigenvalue λ1 (ǫ) = O(ǫ2 ). Furthermore the
L2 × L2 normalized eigenfunction (hǫ1 , hǫ2 ) corresponding to λ1 (ǫ) satisfies
hǫ2 = aǫ
r0′
+ pǫ , pǫ ⊥ r0′ (in L2 ),
kr0′ kL2
aǫ ∈ R, pǫ ∈ H 2 , with |aǫ | ≥ a0 > 0, kpǫ kH 2 = O(ǫ2 ).
(ii) The rest of the spectrum is contained in [λ0 , ∞), where a0 , λ0 are positive constants independent of ǫ.
Proof of Theorem 1 By Proposition 2, if ǫ < ǫ2 , there exists a unique eigenvalue λ(ǫ) ∈ [−λ0 , λ0 ].
Moreover λ(ǫ) = O(ǫ2 ) and λ is simple. Also the corresponding h2 satisfies the required estimate.
Furthermore by Proposition 3, we have that there is no eigenvalue in (−∞, −λ0 ]. Hence λ is the
only eigenvalue in (−∞, λ0 ] and consequently it is the principal eigenvalue. Denote it by λ1 . By
the results in Section 2.1, we know that the essential spectrum of Lǫ is contained in [C3 , ∞), where
C3 > 0 is independent of ǫ, and λ0 < C3 .
The proof of Theorem 1 is complete. ˜
3. The heteroclinic
We seek a solution u = (x, r) of (13) of the form u = u0 + ũ, where u0 = (x0 , r0 ) and ũ = (x̃, r̃) ∈
H 2 × H 2 , ũ ⊥ u′0 in L2 . Obviously, u has the required behavior (15) at infinity. It remains to be
shown that u satisfies (13), from where we also get that u ∈ C 2 × C 2 .
We have
(72)
Lǫ ũ = N (ũ) + E,
where
(73)
and
N (ũ) = −
gx (x̃ + x0 , r̃ + r0 ) − gx (x0 , r0 ) − gxx (x0 , r0 )x̃ − gxr (x0 , r0 )r̃
gr (x̃ + x0 , r̃ + r0 ) − gr (x0 , r0 ) − gxr (x0 , r0 )x̃ − grr (x0 , r0 )r̃
x′′0
E=ǫ
.
0
The existence of a solution ũ of (72) follows from the following propositions (see Theorem 2 in
Sec. 3.1 below).
2
Proposition 4. If ǫ is sufficiently small, there exists a pair (c, ũ) with c ∈ R and ũ ∈ H 2 ×H 2 , ũ ⊥
u′0 , such that
(74)
Lũ = −cu′0 + N (ũ) + E, kũk1,ǫ = O(ǫ2 ),
where k(x, r)k21,ǫ = ǫ2 kx′ k2L2 + kr ′ k2L2 + kxk2L2 + krk2L2 .
Proposition 5. Let (c, ũ) be as in Proposition 4. Then c = 0.
In the remainder of the paper, it is assumed that 0 < ǫ < ǫ2 .
Lemma 4. For ǫ sufficiently small, we have
(i) h − kr′ ak 2 u′0 0 L
(ii) |(h, u′0 )| ≥
L2 ×L2
1 ′
kr
2 0 kL2
≤ C21 ǫ2 ,
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
where h = (h1 , h2 ) is the normalized eigenfunction corresponding to λ1 (recall h2 = ay + p, y =
r0′
kr ′ k 2 ) and u0 = (x0 , r0 ). C21 is independent of ǫ (recall that λ1 , h, a depend on ǫ).
0 L
Proof
(i) h −
a
a
a
′
′
′
≤ h1 − ′
u0 x0 + h2 − ′
r0 =
′
kr0 kL2
kr0 kL2
kr0 kL2
2
2
L2
L2
L ×L
ǫ
|a| a
ǫ
′
′
′
ǫ
=
K1 (λ1 )(ay + p) − kr ′ k 2 x0 2 +kh2 −aykL2 ≤ kK1 (λ1 )pkL2 + kr ′ k 2 K1 (λ1 )r0 − x0 L2 +kpkL2 .
0 L
0 L
L
By (48), the estimates kpkH 2 = O(ǫ2 ), |a| ≤ 1 and the definition of K2ǫ (λ), this is
λ1 gxr r0′
ǫ
g
g
xr
xr
2
2
′
2
ǫ
′
′
′
2
≤ C19 ǫ +C20 − gxx − λ1 r0 + ǫ K2 (λ1 )r0 + gxx r0 2 ≤ C19 ǫ +C20 gxx (gxx − λ1 ) 2 +ǫ C20 K2 (λ1 )r0 L2 ≤ C21
L
L
where we used that |λ1 | = O(ǫ2 ) and Lemma 1.
(ii) It follows trivially from (i) via khkL2 ×L2 = 1.
Lemma 5. For ǫ sufficiently small, for every w such that w ∈ H 2 × H 2 , w ⊥ h, the following
estimates hold:
(i) (Lǫ w, w) ≥ λ0 kwk2L2 ,
(ii) (Lǫ w, w) ≥ C23 kwk21,ǫ ,
where λ0 is as in Proposition 1, and C23 > 0 is independent of ǫ and w.
Proof (i) is obvious from the variational characterization of the eigenvalues of self-adjoint operators.
(ii) Let w = (w1 , w2 ) ∈ H 2 × H 2 , w ⊥ h, and 0 < C < 1. Then
(Lǫ w, w) − Ckwk21,ǫ = (1 − C)(Lǫ w, w) + C (Lǫ w, w) − kwk21,ǫ
≥ (1 − C)λ0 kwk2L2 + C
Z
∞
−∞
(gxx − 1)w12 + 2gxr w1 w2 + (grr − 1)w22 ds ≥
≥ (1 − C)λ0 kwk2L2 − CC22 kwk2L2 .
Choosing C sufficiently small completes the proof of the lemma with C23 := C. ˜.
Lemma 6. Let Ui ∈ H 1 × H 1 , kUi k1,ǫ ≤ M ǫ2 , i = 1, 2, M > 1, ǫ < 1. Then kN (Ui )kL2 ×L2 ≤
C24 M 3 ǫ3 and
kN (U1 ) − N (U2 )kL2 ×L2 ≤ C24 M 2 ǫδ kU1 − U2 kL2 ×L2 ,
where C24 , δ > 0 are independent of ǫ, M, Ui .
Proof Let U = (x, r) ∈ H 1 × H 1 with kU k1,ǫ ≤ M ǫ2 . Then kxkH 1 ≤ M ǫ, krkH 1 ≤ M ǫ2 , kxkL2 ≤
M ǫ2 , krkL2 ≤ M ǫ2 .
From (17), (18),(73),
2
6r − 2xr 2 − 4xrr0 − 2x0 r 2 − 4x3 − 12x2 x0
N (U ) =
.
12xr − 2x2 r − 2x2 r0 − 4xx0 r − 3r 3 − 9r 2 r0
We recall the Gagliardo-Nirenberg inequality [12]: if u ∈ H 1 , then kukLp ≤ C kukθH 1 kuk1−θ
for
L2
1
1
p ≥ 2, 0 ≤ θ ≤ 1, θ ≥ 2 − p . We estimate a typical term to explain the procedure.
Consider the term xr 2 in the expression of N (U ).
1
1/2
kxr 2 kL2 ≤ kx2 kL2 kr 4 kL2 2 = kxkL4 krk2L8 ≤
1/4
3/4
3/4
5/4
≤ C kxkH 1 kxkL2 krkH 1 krkL2 ≤ CM 3 ǫ23/4 .
18
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
The rest of the terms can be estimated analogously. Let Ui = (xi , ri ) ∈ H 1 × H 1 be such that
kUi k1,ǫ ≤ M ǫ2 , i = 1, 2. Then kxi k∞ ≤ CM ǫ3/2 and kri k∞ ≤ CM ǫ2 . We will treat a typical term
in N (U1 ) − N (U2 ) and leave the rest to the reader.
kx2 r22 − x1 r12 kL2 ≤ kx2 r22 − x2 r12 kL2 + kx2 r12 − x1 r12 kL2 ≤
≤ kx2 k∞ k(−r2 +r1 )(−r2 −r1 )kL2 +kr12 k∞ kx2 −x1 kL2 ≤ 2M 2 C 2 ǫ3 kr2 −r1 kL2 +C 2 M 2 ǫ4 kx2 −x1 kL2 .
˜
Proof of Proposition 4
I. The transformation ũ→w
For ũ ∈ L2 × L2 , ũ ⊥ u′0 , set
(75)
a
′
u h.
w = ũ − ũ, h − ′
kr0 kL2 0
By (i) of Lemma 4, the right hand side is a small perturbation of the identity. Hence if ǫ is
sufficiently small, the above linear transformation is invertible and ũ = Q(w). Also note that we
have the equivalence w ⊥ h ⇔ ũ ⊥ u′0 . Furthermore if ǫ is sufficiently small, then
kQ(w1 ) − Q(w2 )kL2 ×L2 ≤ C25 kw1 − w2 kL2 ×L2
and kQ(w)k1,ǫ ≤ C25 kwk1,ǫ , for every w1 , w2 ∈ L2 × L2 , w ∈ H 1 × H 1 , w1 , w2 , w ⊥ h, where C25 > 1
is a constant independent of ǫ, w, w1 , w2 .
II. The w-equation. Substituting (75) into (74) gives
(76)
Lw = −cu′0 + N (Q(w)) + E − (Q(w), h)λ1 h.
We define a function C : H 2 × H 2 →R by
1
(N (Q(W )) + E − λ1 Q(W ), h)
(77)
C(W ) = ′
(u0 , h)
(from (ii) of Lemma 4, |(u′0 , h)| ≥ 12 kr0′ kL2 ). We seek W ∈ H 2 × H 2 , W ⊥ h such that
(78)
LW = −C(W )u′0 + N (Q(W )) + E − (Q(W ), h)λ1 h.
We can define an operator
T : (H 2 × H 2 ) ∩ h⊥ →(H 2 × H 2 ) ∩ h⊥
by T W = Ŵ , where Ŵ is the unique solution in (H 2 × H 2 ) ∩ h⊥ of
(79)
LŴ = −C(W )u′0 + N (Q(W )) + E − (Q(W ), h)λ1 h.
This map is well defined via (77) and Lemma 5(i).
Let S = {W ∈ (H 2 × H 2 ) ∩ h⊥ : kW k1,ǫ ≤ M3 ǫ2 }, where M3 > 1 is to be chosen. Let W ∈ S.
Then
3
M33 ǫ3 + kx′′0 kL2 ǫ2 + M1 ǫ2 C25 M3 ǫ2 .
|C(W )| ≤ C26 kN (Q(W )) + E − λ1 Q(W )kL2 ×L2 ≤ C26 C24 C25
From (79) via (ii) of Lemma 5 and the above, we have
3
kT W k1,ǫ ≤ C27 C24 C25
M33 ǫ3 + kx′′0 kL2 ǫ2 + M1 C25 M3 ǫ4 .
By choosing M3 sufficiently large and ǫ sufficiently small, we obtain that kT W k1,ǫ ≤ M3 ǫ2 , i.e.
T S ⊂ S.
Now let W1 , W2 ∈ S; we have
|C(W1 ) − C(W2 )| ≤ C26 kN (Q(W1 )) − N (Q(W2 ))kL2 ×L2 + C26 M1 ǫ2 kQ(W1 ) − Q(W2 )kL2 ×L2 ≤
2
2
M32 ǫδ + M1 ǫ2 C25 kW1 −W2 k1,ǫ .
M32 ǫδ + M1 ǫ2 kQ(W1 )−Q(W2 )kL2 ×L2 ≤ C26 C24 C25
≤ C26 C24 C25
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
and
2
M32 ǫδ + M1 ǫ2 C25 kW1 − W2 k1,ǫ .
kT W1 − T W2 k1,ǫ ≤ C28 C24 C25
If ǫ is sufficiently small, T is a contraction on the closed set S and T S ⊂ S. Thus there exists
W ∈ S such that T W = W . Hence we obtain a pair C = C(W ) ∈ R and W ∈ H 2 × H 2 , W ⊥
h, kW k1,ǫ ≤ M3 ǫ2 , that satisfies (76). Obviously C = C(W ) and ũ = Q(W ) satisfy (74) and
ũ ∈ H 2 × H 2 , ũ ⊥ u′0 , kũk1,ǫ ≤ C25 M3 ǫ2 .
This completes the proof of Proposition 4. ˜
Remark 1 From (74) we see that ũ ∈ C 2 × C 2 .
Proof of Proposition 5
2
Set u = u0 + ũ = (x, r) ∈ C 2 × C 2 (recall
√ that ũ depends′ on ǫ ′and kũk1,ǫ = O(ǫ )). Then
x(s)→0, r(s)→0 as s→ − ∞, x(s)→1, r(s)→ 2 as s→∞ and x (s), r (s)→0 as |s|→∞. Rewriting
(74) in a more suitable form, we have
2 ′′
ǫ x − gx (x, r)
= cu′0 .
r ′′ − gr (x, r)
Now taking the inner product of both sides with u′ gives
Z ∞
(ǫ2 x′′ x′ + r ′′ r ′ − gx (x, r)x′ − gr (x, r)r ′ )ds = c(u′0 , u′0 + ũ′ ),
−∞
Z
∞
−∞
i.e.
d
ds
ǫ2 ′ 2 1 ′ 2
(x ) + (r ) − g(x, r) ds = c ku′0 k2L2 ×L2 + (u′0 , ũ′ ) ,
2
2
√
c ku′0 k2L2 ×L2 + (u′0 , ũ′ ) = g(0, 0) − g(1, 2) = 0.
From this, we assert that for sufficiently small ǫ we have c = 0. Indeed, since kũk1,ǫ ≤ C25 M3 ǫ2 ,
we have that kũ′ kL2 ×L2 ≤ C25 M3 ǫ, and so
ku′0 k2L2 ×L2 + (u′0 , ũ′ ) ≥ ku′0 k2L2 ×L2 − ku′0 kL2 ×L2 kũ′ kL2 ×L2 ≥ ku′0 kL2 ×L2 ku′0 kL2 ×L2 − C25 M3 ǫ > 0
if ǫ is sufficiently small. This completes the proof of Proposition 5. ˜
3.1. The existence theorem. We have therefore proved the following theorem:
Theorem 2. If ǫ > 0 is sufficiently small, then there exists a solution (xǫ , rǫ ) of (13) such that:
xǫ = x0 + x̃ǫ , rǫ = r0 + r̃ǫ with x̃ǫ , r̃ǫ ∈ H 2 ∩ C 2 ,
(x̃ǫ , r̃ǫ ) ⊥ (x′0 , r0′ ) (inL2 × L2 ),
and
k(x̃ǫ , r̃ǫ )k1,ǫ = O(ǫ2 ),
where
2
2
k(x, r)k21,ǫ = ǫ2 x′ L2 + r ′ L2 + kxk2L2 + krk2L2 .
20
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
4. The stability of the heteroclinic
We consider a family of functions (xǫ (s), rǫ (s)) ∈ C 2 × C 2 , ǫ ≥ 0, satisfying
(H1) (xǫ (s), rǫ (s)) continuous in ǫ uniformly for s ∈ R with (x0 (s), r0 (s)) as in Sec. 2.2,
lim|s|→∞ xǫ , lim|s|→∞ rǫ exist, and kx′ǫ k∞ , krǫ′ k∞ , kx′′ǫ kL2 , krǫ′′ kL2 / bounded uniformly in ǫ.
In this section we assume that gxx , gxr , grr are evaluated at (xǫ , rǫ ) satisfying (H1). We define
(using the same notation) the operator Lǫ as in (28) but now with (xǫ , rǫ ) in the place of (x0 , r0 ).
From the asymptotic behavior
C of gxx, gxr, grr, as |s|→∞ it follows as in Sec. 2.3 that for small
ǫ > 0 we have σess (Lǫ ) ⊂ 23 , ∞ .
We now consider the eigenvalue problem Lǫ h = λh, h = (h1 , h2 ), khkL2 ×L2 = 1. We can perform
steps (40) – (55) without any change1. However in order to continue, we need the following lemma
whose proof is an immediate consequence of (H1), Th. 3.1, pg. 482 in [6], and (27).
Lemma 7. Set q(s, ǫ) = grr −
2
gxr
gxx
and define the operator B(ǫ) in L2 via
B(ǫ)h = −h′′ + q(s, ǫ)h, with D(B(ǫ)) = H 2 .
Then if ǫ ≥ 0 is sufficiently small, the essential spectrum of B(ǫ) is contained in [ C21 , ∞) and
the smallest eigenvalue µ1 (ǫ) of B(ǫ) is simple, µ1 (ǫ)→0 and the corresponding L2 normalized
eigenfunction yǫ satisfies yǫ →y in H 2 (y as in Sec. 2.6).
Using this, we can prove the analogs of the results in Secs. 2.2, 2.7, 2.8 and obtain (using the
same notation):
Theorem 3. If ǫ > 0 is sufficiently small and (xǫ , rǫ ) satisfies (H1), then the spectrum of Lǫ
has the following form: At the bottom of the spectrum, there is a simple eigenvalue λ1 (ǫ) with
λ1 (ǫ) = O(ǫ2 + µ1 (ǫ)). Furthermore, the normalized eigenfunction (hǫ1 , hǫ2 ) corresponding to λ1 (ǫ)
satisfies
hǫ2 = aǫ yǫ + pǫ , pǫ ⊥ yǫ (in L2 ),
aǫ ∈ R, pǫ ∈ H 2 with |aǫ | ≥ a0 , kpǫ kH 2 = O(ǫ2 + µ1 (ǫ))
(a0 , λ0 are positive constants independent of ǫ, but depending on the family (xǫ , rǫ )).
The following corollary implies the stability of the heteroclinic orbit of (13) obtained in Sec. 3.
Corollary 1. Let (xǫ , rǫ ) be the heteroclinic orbit of (13) given in Theorem 2 above. Then if ǫ > 0
is sufficiently small, the spectrum of the linearized operator Lǫ about (xǫ , rǫ ) has the following form:
Zero is a simple eigenvalue at the bottom of the spectrum and the rest of the spectrum is contained
in [C29 , ∞) where C29 > 0 (and as always with constants, independent of ǫ).
Proof It is easy to show that (xǫ , rǫ ) satisfies (H1) via (72) and that zero is an eigenvalue of Lǫ
′
with normalized eigenfunction kuuǫ′ k , where uǫ = (xǫ , rǫ ). Now the corollary follows by Theorem 3.
ǫ
˜
5. The geometric singular perturbation approach
We study system (13)
(80)
ǫ2 x′′ = gx (x, r),
r ′′ = gr (x, r).
This can be viewed as a singular perturbation of the ǫ = 0 system
(81)
0 = gx (x, r),
r ′′ = gr (x, r),
1except now instead of the boundedness of the derivatives of g , g , we use (H1).
xr
xx
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
which may be called the reduced problem.
In the following we will be using well known work of Fenichel [8] to obtain an invariant manifold
Mǫ for (80), ǫ > 0, which depends in a smooth way on ǫ, the smoothness extended all the way to
ǫ = 0. In particular, the limǫ→0 Mǫ = M0 is meaningful and the set M0 is given by (a compact
piece of)
(82)
M0 = {(x, r) : gx (x, r) = 0}.
For applying this theory, there are two hypotheses that have to be verified:
(1) The set M0 should have a degree of smoothness.
(2) The set M0 should be normally hyperbolic.
After verifying these hypotheses and thus obtaining the invariant manifold Mǫ , we proceed to
examine the problem of the existence of the heteroclinic connection stated in Thm. 2: There exists
(x(s), r(s)) solution to (80), satisfying
√
(83)
lim (x(s), r(s)) = (0, 0), lim (x(s), r(s)) = (1, 2).
s→−∞
s→∞
The benefit of this kind of result is twofold. First, we achieve the reduction of problem (80) on
the manifold Mǫ , a lower dimensional object. Second, the regular dependence on ǫ all the way to
ǫ = 0 allows, in principle, for a regular perturbation argument that takes advantage of the existence
of such heteroclinic connections for system (81).
The perturbation of a heteroclinic connection in general does not render a heteroclinic connection.
However for the problem at hand, the Hamiltonian structure of (80) plays a crucial role and provides
a heteroclinic connection, thus a solution of (83). We note that the Hamiltonian structure of (13)
goes hand in hand with the gradient structure of (5).
We now present the details. For Fenichel’s work, we refer to [8, 21, 22, 24] and the article by
Jones in [14]. For studying (80) in regions where x varies rapidly, we shall sometimes use the (fast)
independent variable
(84)
d
s
.
ζ= , ˙=
ǫ
dζ
In this variable we obtain the system equivalent (when ǫ > 0) to (80)
ẍ = gx (x, r),
r̈ = ǫ2 gr (x, r).
(85)
The corresponding systems of equations are obtained by setting u1 = x, u2 = ǫx′ , u3 = r, u4 = r ′ .
Thus (80), (81), (85) become
(86)
ǫu′1
ǫu′2
u′3
u′4
=
=
=
=
u2
gx (u1 , u3 )
u4
gr (u1 , u3 ),
(87)
0
0
u′3
u′4
=
=
=
=
u2
gx (u1 , u3 )
u4
gr (u1 , u3 ),
(88)
u̇1
u̇2
u̇3
u̇4
=
=
=
=
u2
gx (u1 , u3 )
ǫu4
ǫgr (u1 , u3 ),
22
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
and
(89)
u̇1
u̇2
u̇3
u̇4
=
=
=
=
u2
gx (u1 , u3 )
ẍ = gx (x, r)
is equivalent to
0
r̈ = 0.
0,
Now M0 is defined as the set of equilibria of (89),
M0 = {(u1 , u2 , u3 , u4 ) |u2 = 0, gx (u1 , u3 ) = 0}
(or equivalently by (19)) = {(u1 , u2 , u3 , u4 ) |u2 = 0, u1 = χ(u3 )} ,
where we utilized the fact that gxx 6= 0. We can represent M0 as the graph of a smooth function
(u1 , u2 ) = h0 (u3 , u4 ), where h0 (u3 , u4 ) = (χ(u3 ), 0). (We observe that actually χ ∈ C ∞ ).
We now check the normal hyperbolicity of M0 in the context of (89). Thus we consider the
linearized operator about a point on M0 , namely


0 1 0 0
 gxx 0 gxr 0 

(90)
A=
 0 0 0 0 .
0 0 0 0
Its eigenvalues are determined by
λ2 (λ2 − gxx ) = 0,
√
from which we find that λ = 0 (double) and λ = ± gxx . Recall that gxx ≥ 4 (by (18)). Thus
M0 is normally hyperbolic, i.e. there are only the two zero eigenvalues on the imaginary axis. We
remark that the matrix A is simplectic, a reflection of the Hamiltonian character of (13) (to be
exploited later), and so its spectrum should not come as a surprise. The following theorem (see
[22]) can therefore be applied:
(91)
Theorem (Fenichel) Under the hypotheses of C ∞ smoothness and normal hyperbolicity for M0 ,
for a given compact subset K of the (u3 , u4 ) plane and a given m > 0, there is a function h(u3 , u4 , ǫ)
defined on K, and an ǫ0 = ǫ0 (m, K) so that for 0 < ǫ < ǫ0 the graph
(92)
Mǫ = { (u1 , u2 , u3 , u4 )| (u1 , u2 ) = h(u3 , u4 , ǫ)}
is locally invariant under (88). Furthermore
(93)
and
(94)
h : K × [0, ǫ0 ) −→ R2 is C m , jointly in (u3 , u4 , ǫ),
h(u3 , u4 , 0) = h0 (u3 , u4 ).
Here ǫ0 > 0 is a generally small number, and h0 was defined previously in relation to
√ M0 .
The manifold Mǫ is center-like for (88), (86). Note that (0, 0, 0, 0) and (1, 0, 2, 0) are still
equilibria of (88) for small ǫ > 0, and thus they lie in Mǫ . Linearizing (88) and evaluating at
(0, 0, 0, 0), we find


0 1 0 0
 4 0 0 0 

(95)
Aǫ (0, 0, 0, 0) = 
 0 0 0 ǫ ,
0 0 4ǫ 0
with eigenvalues determined by (λ2 − 4)(λ2 − 4ǫ2 ) = 0, hence λ1 = 2, λ2 = −2, λ3 = 2ǫ, λ4 = −2ǫ.
The first two eigenvalues correspond to motion normal to Mǫ , and the latter two correspond to
motion on Mǫ . The dynamics within Mǫ therefore has a saddle point at the origin with one
dimensional stable and unstable manifolds Wǫs (0, 0, 0, 0), Wǫu (0, 0, 0, 0) (in Mǫ ).
ANALYSIS OF THE HETEROCLINIC CONNECTION IN A SINGULARLY PERTURBED SYSTEM ARISING FROM THE STUDY OF C
√
These same calculations hold at the other critical point (1, 0, 2, 0), and thus it is a saddle with
respect to the dynamics on Mǫ .
√
Our strategy for proving the existence of a heteroclinic orbit connecting (0, 0, 0, 0) and (1, 0, 2, 0)
is based on showing that on Mǫ the unstable manifold of one of the equilibria meets the stable
manifold of the other, and thus, since they are one-dimensional, they have to coincide.
We will be establishing the following
Theorem C
For each ǫ sufficiently
√ small in (0, ǫ0 ), there is a heteroclinic orbit of (86) connecting the critical
points (0, 0, 0, 0), (1, 0, 2, 0) which lies on Mǫ , and hence is O(ǫ) away from M0 . Moreover along
the orbit we have the relations
u1 = χ(u3 ) + O(ǫ)
u2 = ǫχ′ (u3 )u4 + O(ǫ2 ).
(96)
Proof We begin by deriving the equations on Mǫ . By (92) we have
u1 = χ(u3 ) + ǫh1 (u3 , u4 , ǫ),
u2 = ǫh2 (u3 , u4 , ǫ).
(97)
From the invariance of Mǫ we obtain
(98)
∂h1 ′
∂h1 ′
ǫh2 (u3 , u4 , ǫ) = u2 = ǫu′1 = ǫ χ′ (u3 )u′3 + ǫ
u3 + ǫ
u4 .
∂u3
∂u4
Thus (96) follows.
The flow on Mǫ is given by
(99)
u′3 = u4
u′4 = gr (χ(u3 ) + O(ǫ), u3 ).
In the (u3 , u4 ) plane, let
1
,
Cǫ = (u3 , u4 ) : χ(u3 ) + O(ǫ) =
2
where O(ǫ) refers to the corresponding term in (99). When
r ǫ = 0, there
is a heteroclinic orbit
(r0 , r0′ ) of (99), intersecting C0 at a point (u03 , u04 ) = √12 , 2G √12
(see Sec. 2.2). We shall
choose K to be a large compact connected set in the (u3 , u4 ) plane with the above
√ mentioned
heteroclinic orbit in its interior. When ǫ > 0 is sufficiently small, then (0, 0) and ( 2, 0) remain
saddles of (99)√(since equilibria of (99) give equilibria of (86)). Also their unstable/stable manifolds
−
+
+
Wǫu (0, 0)/Wǫs ( 2, 0) depend smoothly on ǫ ≥ 0 and intersect Cǫ at (u−
3,ǫ , u4,ǫ )/(u3,ǫ , u4,ǫ ) with
uji,ǫ →u0i as ǫ→0, i = 3, 4, j = −, +. This is a consequence of the implicit function theorem. Since
−
(u−
3,ǫ , u4,ǫ ) is a solution of
(100)
1
χ(u3 ) + O(ǫ) = ,
2
u4 = hu (u3 , ǫ)
where (u3 , hu (u3 , ǫ)), u3 ∈ (u03 − δ, u03 + δ) (δ > 0 a small number independent of ǫ ≥ 0) is a
+
parametrization of Wǫu (0, 0), ǫ ≥ 0. The same holds for (u+
3,ǫ , u4,ǫ ).
24
N.D. ALIKAKOS, P.C. FIFE, G. FUSCO, AND C. SOURDIS
To conclude, we will be utilizing the Hamiltonian structure of (86). Notice that (86) can be
written in the form
(101)
ǫu′1 =
∂H
∂u2
u′3 =
∂H
∂u4
∂H
ǫu′2 = − ∂u
1
∂H
,
u′4 = − ∂u
3
where
1
H(u1 , u3 , u2 , u4 ) = (u22 + u24 ) − g(u1 , u3 ).
2
Thus the solutions of (101) lie on level sets√ of H. Let now Uǫ− (s), Uǫ+ (s) be solutions of
±
, u±
(86) parameterizing Wǫu (0, 0, 0, 0) and Wǫs (1, 0, 2, 0) with Uǫ± (0) = ( 21 , u±
2,ǫ
3,ǫ , u4,ǫ ) ∈ Mǫ , i.e.
√
±
±
s
u
u±
Uǫ± (s) = (u±
1 (s),
2 (s), u3 (s), u4 (s)). Recall that Wǫ (0, 0, 0, 0) \ Wǫ (1, 0, 2, 0) is the image of
√
Wǫu (0, 0) \ Wǫ2 ( 2, 0) on Mǫ .
+
−
+
We claim that u−
i,ǫ = ui,ǫ , i = 2, 3, 4, from which it will follow that Uǫ ≡ Uǫ . Notice that we
want to determine three variables, although (96) with (100) furnishes only two
√ equations. The
third equation is provided by the Hamiltonian, via the fact that (0, 0) and (1, 2) are minima of
equal energy for g (by design of (3)). Thus
√
H(0, 0, 0, 0) = H(1, 0, 2, 0) = 0.
(102)
±
±
Collecting all the equations together, we have that (u±
2,ǫ , u3,ǫ , u4,ǫ ) solve
(103)
1
2
− (χ(u3 ) + ǫh1 (u3 , u4 , ǫ)) = 0,
u2 − (ǫχ′ (u3 )u4 + O(ǫ2 )) = 0,
g( 12 , u3 ) − 21 (u22 + u24 ) = 0.
But via the implicit function theorem, for ǫ > 0 sufficiently small (103) has a unique solution in a
neighborhood of (0, u03 , u04 ), i.e. Uǫ− (0) = Uǫ+ (0). Thus
−
Uǫ (s), s ≤ 0,
Uǫ (s) =
Uǫ+ (s) s ≥ 0,
is the required heteroclinic orbit of (86). This completes the proof of Theorem C. ˜
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N.D. Alikakos, Department of Mathematics, University of North Texas and University of Athens
E-mail address: nalikako@earthlink.net
P.C. Fife, Department of Mathematics, University of Utah, UT 84112, USA
E-mail address: fife@math.utah.edu
G. Fusco, Univ. degli Studi dell’Aquila, L’Aquila, Italy
E-mail address: fusco@mailhost.univaq.it
C. Sourdis, Mathematics Department, University of Athens, Athens, Greece
E-mail address: sour@dei.gr