1991 Society for Industrial and Applied Mathematics
007
SIAM J. MATH. ANAL.
Vol. 22, No. 6, pp. 1631-1650, November 1991
THE EXISTENCE OF INFINITELY MANY TRAVELING FRONT AND
BACK WAVES IN THE FITZHUGH-NAGUMO EQUATIONS*
BO DENGt
Abstract. Consideration is given to the FitzHugh-Nagumo equations of bistable type. The existence of
traveling front and back waves with any finite number of pulses is proved. The speed of such a multiple
pulse wave is characterized by its number of pulses: the more pulses it has, the slower it travels. Traveling
impulse and traveling train solutions are also found. These traveling waves arise from the bifurcation of a
doubly twisted front-back wave loop. The method is based on the theory of heteroclinic loop bifurcation,
the geometric theory of singular perturbation and the Melnikov method.
Key words, traveling wave, twisted heteroclinic loop, singular perturbation, Melnikov integral
AMS(MOS) subject
classifications, primary 35K57"
secondary 34B99, 34C28, 34C45, 34D15
1. Introduction. Consider the FitzHugh-Nagumo equations
(1.1)
vt
v,, + f(v)- w,
w,
e(v- yw),
where f(v)=v(v-a)(1-v) is a cubic polynomial and 0<a<1/2, e>0, and y>0 are
parameters. A solution (v, w)(x, t) that is bounded over x R and R is called a
traveling wave if it is a function of one variable and there is a constant c so that
(v, w)(x, t) (v, w)(x + ct).
The simplest traveling waves might be constant, or steady-state solutions. Depending on the value of y, there are one, two, or three steady states (of. Fig. 1.1), which
are the intersections of the nullclines w =f(v) and v yw. In this paper, however, we
are interested in the case when y is greater than the critical value Yl := Vmax/fmax and
there are three steady states. We restrict our attention further to the leftmost and
rightmost stable states, which we denote by al and a2 in Fig. 1.1, respectively, and
FIG. 1.1. 3/0 is chosen so that al and a are symmetric with respect to the inflection point (/)inf,finf) of the
cubic curve w =f(v). The thickly drawn segments g and 2 on w --f(v) do not contain the extreme points but
contains a =0 and the intersect point {w= wz} f) w=f( v)} and gz contains
they are long enough so that
a2 and (1, O) as interior points, respectively.
Received by the editors September 28,1989; accepted for publication (in revised form) October 30,1990.
t Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 685880323.
1631
1632
Bo DENG
study nonconstant traveling waves connecting al and a2. The various types of connecting
waves considered here are as follows. A traveling wave (v, w)(x + ct) is said to be a
traveling front if
lim (v, w)(x + ct)
al
and
lim (v,
w)(x + ct) a,
exist for all x. Likewise, by a traveling back wave we mean the same limits exist except
that the first limit is a2 while the second limit is a A traveling wave solution is said
to be an impulse of a if
lim (v, w)(’r)
a
An impulse of a2 is analogously defined. Last, a traveling wave solution is said to be
a traveling train if (v, w)(’) is periodic in -.
We further characterize the waves of the same type according to their numbers
of pulses contained. To be precise, choose and fix a neighborhood of each al and a:
for the equations. A traveling wave has a pulse from al if there is a closed interval
’/’0’l’’/’l such that (v, w)() arises from the chosen vicinity of al, enters into the
vicinity of a2, and only afterwards falls back to the vicinity of al as increases from
’o to 1 (cf. Fig. 1.2). A pulse from a2 is defined similarly. A front (back, respectively)
wave is called k-front (k-back, respectively) wave if it has k pulses from al. A front
(back, respectively) without a pulse is referred to as a simple front (back, respectively).
An impulse of ai is called a k-impulse if it has k pulses from ai. A traveling train is
-
pulse
(a)
voru
x
(b)
FIG. 1.2. (a) One pulse (thickly drawn curve) forms when the traveling wave jumps from a given
neighborhood of a into a given neighborhood of a and only afterwards drops into the neighborhood of a
again over an interval " [%, h].
(b) When the traveling speed c > O, a 1-front here moves to the left with time.
EXISTENCE OF INFINITELY MANY TRAVELING WAVES
1633
called a k-train if within the minimum period of the periodic traveling wave there are
k pulses. An impulse or a traveling train is simple if there is only one pulse. We
emphasize that the comparison among fronts or other types of waves in terms of the
number of pulses makes sense only if the neighborhoods of al and a2 are fixed, but
they are allowed to vary with parameters.
The profiles of a given traveling wave are simply the graphs of v and w over the
real line (cf. Fig. 1.2). Thus, a traveling wave moves to the left with time if the traveling
velocity c is positive. Likewise, it travels to the right if c < 0. Also, it is trivial to verify
that if (v, w)(x+ct) is a traveling wave then (, )(x+(-c)t):=(v, w)(-x+ct) is
another traveling wave solution, traveling in the opposite direction. For this reason,
we only consider the existence of leftward traveling waves (i.e., with c > 0) from now
on.
By smoothness in this paper we mean differentiability of as many times as needed.
Our main result is the following theorem.
THEOREM 1.1. Let 0<a<1/2 be fixed in the FitzHugh-Nagumo equations (1.1).
There exists a small eo and two smooth functions y(e) and 6(e), 0 <- e <-Co, such that
the following is satisfied for all 0 < e < Co.
(a) On the relevant y, c)-parameter space there are two smooth curves c Ci, o( y)
defined on the interval IT-y(e)[ < 6(e) and i= 1,2 (Ci,o here and all other curves below
are smoothly parametrized by e also, but e is usually suppressed for simplicity) such that
(1.1) has a simple front wave of speed cl,o(y) and a simple back wave of speed C2,o(y).
(b) There is a sequence {Cl,k(Y)}_l of smooth curves of the left half interval
0 < y(e) y < 6(e) such that (1.1) has a k-front wave of speed el,k(Y) for every k 1,
2,... and O< y(e)-y<6(e). Similarly, there is a sequence {C2,k(Y)}=l of smooth
curves of the right half interval 0 < y y( e < 6( e such that (1.1) has a k-back wave of
speed C2,k(Y) for every k= 1, 2,... and 0< y-y(e)<6(e).
(c) There is a smooth curve c,(y) of the left half interval 0< y(e)- y< 6(e) such
that 1.1 has a simple impulse wave of a with speed c1,(y) for every 0 < y( e 3’ < 3(e ).
Similarly, there is a smooth curve c2,(y) of the right half interval 0< y-y(e)< 6(e)
such that (1.1) has a simple impulse wave of a2 with speed e2,( y) for every 0< y- y(e) <
().
(d) The simple front and back wave curves Ci.o( y) intersect transversely at y( e ). The
intersectionpoint, (y(e), c(e)), is smooth in e. At e =0, (y(0), c(0))= (9/(2-a)(1-2a),
(1-2a)/x/) := (Yo, Co). The slopes of c,o( y) satisfy C,o(Yo) =0 and C,o(Yo) <0, respectively, at e O. Moreover, for fixed e > O, 3’ and
1, 2, the sequence { Ci, k (3’)} is monotone
decreasing in k O, 1, 2,... and converges to the corresponding impulse curve c,( y) as
k--> oo. Furthermore, every Cl.k curve is asymptotically tangent to the cl,o curve from the
left of y(e) as 3’ -> Y(e)- and, similarly, every C2,k curve is asymptotically tangent to the
C2,o curve from the right of y(e) as y--> y(e) +.
(e) There is a neighborhood of (y(e), c(e)) in the (y, c)-parameter space for each
0 < e < eo such that (1.1) has a simple traveling train solution for some speed c if and
only if y, c) is in this neighborhood and lies below the curve Cl, LJ c2, LJ y(e), c(e)). See
Fig. 1.3.
It would be of mathematical interest to assume that the recovering variable w also
diffuses slightly along the spatial line. This leads to the consideration of the following
system of reaction diffusion equations"
(1.2)
vt
v,,, + f( v)
w,
wt
Kw,,, + e( v
yw).
This is the same as the FitzHugh-Nagumo equations (1.1) except for a small diffusion
term W,x with I1 << 1 in the w-equation. It is easy to see that the steady states of (1.2),
1634
Bo DENG
FIG. 1.3. The bifurcation diagram of Theorem 1.1. Qx and Q2 here are the Melnikov functions associated
with the simple front wave and the simple back wave, respectively. The simple front (back, respectively) wave
curve cl, (C2,o, respectively) is the level set of Qx--0 (Q2 0, respectively).
in particular, the steady states al and a2, are the same as those of (1.1) and all the
definitions for traveling waves of different types considered above can be directly
extended for (1.2). As we will see later, the following theorem is a direct consequence
of Theorem 1.1 in the context of singular perturbation.
THEOREM 1.2. Let 0< a < 1/2 be fixed in (1.2). There are small constants eo and Ko
and smooth functions y(e, ), c(e, ), and 6(e, ) of 0 <- e <- eo and [1 <- o such that
for 0 < e < eo and [] <= o all the conclusions of Theorem 1.1 are satisfied when y(e, ),
c(e, ), and 8(e, ) are substituted for y(e), c(e), and 8(e), respectively. Moreover,
y(O, )= To and c(O, )= Co, the same constants as in Theorem 1.1. In this case, of
course, all the curves Ci.k depend smoothly on as well.
The FitzHugh-Nagumo equations have been studied extensively for the last two
decades. This system of reaction-diffusion equations is a qualitative model for several
applications including nerve conduction (Hodgkin and Huxley (1952), FitzHugh
(1961), and Nagumo, Arimoto, and Yoshizawa (1963)), neuronal interactions at the
population level (Wilson and Cowan (1972)), chemical and biochemical reaction
(Ortoleva and Ross (1975)), as well as electronic transmission lines (Nagumo, Arimoto,
and Yoshizawa (1963)). The references quoted here only reflect the author’s limited
understanding on this subject. For the case when y =0, Hastings (1974), (1976) and
Casten, Cohen, and Lagerstrom (1975) have studied the existence of impulse and
traveling train solutions. Rinzel and Keller (1973) have studied the same problem
except that the function f is replaced by a piecewise continuous function f=
H(v-a)-v with H to be the Heaviside step function. For the case when e, a, y are
all small, Hastings (1982) and Evans, Fenichel, and Feroe (1982) have studied the
existence of impulse solutions of double pulses and traveling trains of multiple pulses.
Their results are closely related to the saddle-focus homoclinic bifurcation theorem
by Sil’nikov (1970). Feroe (1982) and, more recently, Wang (1988I) have also studied
this type of phenomenon with the piecewise linear function f For large y, Carpenter
(1977) has studied the existence of traveling front and back waves as well as impulse
and traveling train solutions through a constructive singular perturbation approach.
Rinzel and Terman (1982) have also considered the same problem for the piecewise
EXISTENCE OF INFINITELY MANY TRAVELING WAVES
1635
linear case. Indeed, as pointed out by one of the referees of this paper, the existence
of traveling waves for (1.1) has been the subject of many other researchers including
Aronson and Weinberger (1975), Conley (1975), Greenberg (1973), Keener (1980),
Langer (1980), McKean (1970), Pauwelussen (1980), Rauch and Smoller (1978), and
probably many more. With only a few exceptions, all the traveling wave solutions
investigated so far can be characterized as simple wave solutions in our terminology.
The multiple pulse front and back waves obtained here have not previously been
proved to exist, nor investigated numerically.
The proof of the theorem is based on three important theories in dynamical
systems, namely, the bifurcation theory of a doubly twisted heteroclinic loop by Deng
(1991), the geometric singular perturbation theory by Fenichel (1979), and the Melnikov
method. Although the idea of using the Melnikov integral together with singular
perturbation theory can be found in Kokubu, Nishiura, and Oka (1988) and Lin (1989),
our singular perturbation approach is quite different. Whereas other researchers emphasize the "singular" aspects of the problems which inevitably lead to techniques like
asymptotic expansion, matching principle, etc., we only need to address the "regular"
aspects of the singular perturbation problems. This point of view is taken from Fenichel
(1979) which asserts that a singular perturbation problem is essentially a regular
perturbation problem in terms of invariant manifold theory, in particular, the center
manifold theory. Extending his idea via invariant manifold theory to connecting orbits,
we naturally see some connections between the singular perturbation theory and the
Melnikov method.
This paper is organized as follows. In 2, we will state the heteroclinic loop
bifurcation theory from Deng (1991) from which the proof of our main result Theorem
1.1 will be derived. The remaining sections are devoted to verifying all the conditions
of that theorem. Specifically, we will introduce the Melnikov function (which was
called the separation function by Kokubu, Nishiura, and Oka (1988)) in 3. In 4-8,
all the nondegenerate conditions of Theorem 2.1 will be verified. At first sight, these
conditions may appear next to impossible to check. However, since they are all generic
with respect to the existence of a heteroclinic loop, it is not too surprising to see that
the existence of the twisted loop, which only requires our extended Melnikov method,
indeed contains enough information for its genericity. In 9, Theorem 1.2 will be
proved based on the singular perturbation method and the proof of Theorem 1.1.
In Theorem 1. l(e) it appears that the neighborhood around (y(e), c(e)) where
the traveling train solutions may occur, as well as the lengths of those curves for
traveling front, back and impulse waves, depend on the parameter e. In fact, they can
be made uniform for all small e > 0. Unfortunately, we cannot show this fact in this
paper. It requires some nontrivial modification and generalization of our Theorem 2.1
to singularly perturbed systems. The same comment also applies to Theorem 1.2 with
respect to e > 0 and [[ << 1.
This paper was originally inspired by the work of Rinzel and Terman (1982).
David Terman helped me understand their work correctly. This made it possible for
me to find the twist structure of the front-back wave loop at the bifurcation point
(yo, c0) through their numerical bifurcation diagram for the simple front, simple back,
and simple impulse waves (cf. Fig. 1.3). Finally, let us mention the other equally
important motivation that lies beyond the scope of this paper. We would like to
eventually prove that all the multiple pulse front and back waves found here are stable
with respect to the PDEs (1.1) and (1.2). The fact that the steady states al, a2, the
simple front, simple back, simple impulse, and simple train solutions near the bifurcation point (%, Co) are all stable is well known. (See, e.g., Rinzel and Terman (1982)
1636
Bo DENG
and the references therein. See also Wang (1988II) for the stability problem of multiple
pulse impulses for the piecewise linear case.)
2. The bifurcatioa of a twistel heterocliaic lool. Recall that a traveling wave (v, w)
is a function of one variable Thus, if we let u :- v’, where the prime denotes the
derivative in ’, then (v, u, w) satisfies the following system of first-order ordinary
differential equations:
.
(2.1)
v’:u, u’:cu-f(v)+w,
w’:e-(v-yw).
For e > 0, it is trivial to check that this system has three equilibrium points when y > y
(of. Fig. 1.1). They are those points (v, u, w) satisfying that u 0 and (v, w) equals to
the steady states of (1.1) discussed in the Introduction. For this reason and for simplicity,
we will denote throughout the equilibria with u 0, (v, w)= ai just by ai alone. Also,
as the counterpart of traveling front of (1.1), a solution of (2.1) is call a heteroclinic
orbit from al to a2 if
lim (v, u, W)(’r)=al
and
lim (t, U, w)(’r)--a2.
A heteroclinic orbit from a2 to al is defined analogously. On the other hand, a solution
of (2.1) is called a homoclinic orbit to al if
lim (v, u, w)(’) a.
A similar definition applies to a homoelinic orbit to a2. Note that a heteroclinic orbit
of (2.1) from al to a2 (respectively, from a2 to al) gives rise to a traveling front
(respectively, back) wave of (1.1) while a homoclinie orbit of (2.1) gives rise to an
impulse wave of (1.1). In a similar way as in the previous section, we can define
k-heteroclinic, k-homoclinic, and k-periodic orbits with respect to some neighborhoods
of al and a2 for (2.1). We leave this to the reader. Therefore, our strategy to prove
Theorem 1.1 is to prove the same theorem except that (1.1) is replaced by (2.1) and
the traveling fronts, etc., are replaced by heteroclinic orbits, etc., respectively. To do
this, we will apply a theorem on the bifurcations of a twisted heteroclinie loop from
Deng (1991) (el. Theorem B of Deng (1991)). Although that theorem as well as the
result on singular perturbation by Fenichel and the method of Melnikov integral are
available for any finite-dimensional system, we will treat them only in 3 here for
simplicity. Also, we need to warn the reader in advance that the theorem we are about
to state is the time-reversed version of Theorem B of Deng (1991). Note that upon
time reversal, the stable manifold becomes the unstable manifold and vice versa. Also,
a heteroelinic orbit from a to a2 becomes a heteroclinic orbit from a2 to a, and so
on. But a homoclinic orbit of al remains the same.
To state the theorem we begin with its hypotheses (2.2a)-(2.2e). Since the bifurcation problem to be discussed is of codimension two, we will only include the so-called
relevant parameter a(a =(y, c) in our case) in a vector field X := X(x), where x
and a 2. We certainly allow other parameters (say e in our case) to be included in
the vector field but we will usually suppress them unless otherwise indicated.
(2.2a)
The relative expansion of a. Let X denote a family of vector fields in 3
parametrized by a relevant parameter a in E. Suppose a a(a), i= 1, 2
are hyperbolic equilibrium points of X for all a and are relatively expansive
in the sense that the eigenvalues A of the linearization DX’(ai) satisfy
AI<A:<O<A3 and A3+A2>O.
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EXISTENCE OF INFINITELY MANY TRAVELING WAVES
Of course, A here also depends on ai for i= 1, 2. The eigenvalues A2 and
3 are called the principal stable and the principal unstable eigenvalues,
respectively. Their eigenvectors are thus referred to as the principal stable
and the principal unstable eigenvectors, respectively (see (1.2) of Deng
(1991)).
(2.2b)
The nondegeneracy of a heteroclinic loop. There is a parameter value ao so
that the equation x’=X(x) has a nondegenerate heteroclinic loop.
Specifically, there exists a simple heteroclinic orbit z* from al to a2 and a
simple heteroclinic orbit z* from a2 to al at the same parameter ao. Moreover,
by nondegeneracy we mean that the following two conditions are satisfied.
First, z*(’) is asymptotically tangent to the principal stable eigenvector of
-c,
at, j # as ’- / and the principal unstable eigenvector of ai as
respectively. Second, the following strong inclination conditions hold:
(2.2b’)
lim
T,.,(, W]
--
Ta, W’ + Ta, W.s,
where i, j 1, 2 and # j, Tp W means the tangent space of a given manifold
W at a base point pc W. Also, W, W’, W is the standard notation for
the stable, unstable, and strong stable manifolds of ai, respectively. They
are two-dimensional, one-dimensional, and one-dimensional, respectively,
in this case (see (1.5) and (1.7) of Deng (1991)).
(2.2c)
The double twist of a nondegenerate heteroclinic loop. Let
nondegenerate heteroclinic loop. Let
ei
lim
z*(’)-__a_
+
,.
Zl*
and
z* form a
z*(’)-a
be the unit principal unstable and stable eigenvectors along which the
heteroclinic orbit z* comes from a and goes towards at, respectively, then
and ej- point to opposite sides of TzT W} at r--oo and
respectively. Here, i, j 1, 2, ij (cf. Fig. 2.1 and see (1.9) and Definition
1.1 of Deng (1991)).
e-
Remark. The definition of twisted heteroclinic orbit has much in common with
that of twisted homoclinic orbit. The geometric notion of twisted homoclinic orbit was
given by Deng (1989) and Chow, Deng, and Fiedler (1990), based on the author’s
strong A-lemma. It was inspired by a work of Yanagida (1987).
w7
FIG. 2.1. A nondegenerate and doubly twisted heteroclinic loop in
sides of W.
3.
e
and
e. point
to the opposite
1638
)o
DENG
(2.2d)
The continuation of z*. There exist two curves 0-hetl and 0-het2 in the
parameter space a E 2 which intersect at ao transversely so that when
c E 0-heti there is a simple heteroclinic orbit zi, from a to aj and z, is
the continuation of z* in the sense that zi, z* and zi, (r) are continuous
in r and a (see (1.1a) of Deng (1991)).
(2.2e)
The transverse crossing of the stable and unstable manifolds along z*. Let
Qi(a) be the Melnikov function defined in the next section; then the gradient
vectors VQl(co) and VQ2(ao) are linearly independent (see (1.10b) of Deng
(1991)). We remark that since the discussions of Melnikov functions in 3
is independent of what follows, we may certainly find out the precise
definition before continuing.
THEOREM 2.1. Suppose conditions (2.2a)-(2.2e) are satisfied. Then the following
holds.
(a) There is a sequence {k-hetl}=l of smooth curves in such that for every k 1,
2,... the equation x’= X(x) has a k-heteroclinic orbit from a to a2 if and only if
a k-hetl. Similarly, there is a sequence {k-het2}=l of smooth curves in 2 such that
for every k 1, 2,... there is a k-heteroclinic orbit from a to al if and only if a k-het2.
(b) There is a smooth curve hom for each i= 1, 2 such that there is a simple
homoclinic orbit of a if and only if a homi.
(c) The curve O-heti is simply the level set of Q O, and it is divided by the other
O-hetj curve into two parts. Let O-hetbe the half of the O-heti curve that points to the
gradient direction of VQj (cf Fig. 2.2). Then all the curves {k-hetl}_l together with
0-het
Q=O
FIG. 2.2. The bifurcation diagram of Theorem 2.1. The doubly twisted heteroclinic loop is drawn in R 2.
0-hetf point to the same side of the 0-heti curve which is the level set Qi 0 and Qi is the Melnikov
function for the primary connection from ai to aj when a ao.
V Qi and
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EXISTENCE OF INFINITELY MANY TRAVELING WAVES
are in the sector bounded by 0 -het/l and 0-het and converge to ao asymptotically
tangent to O-het-. Moreover, {k-ketl}k=l lies between the 0-het / and hOml curves and
converges to the hom curve monotonely in k-1, 2,... (cf Fig. 2.2). Analogous result
holds for the sequence {h-het2}=l and the hOmE curve.
(d) Let A be the sector bounded by the homoclinic curves hOml and homE. Then
there is a simple periodic solution of the vector field X if and only if s A.
Proof. The theorem is the same as Theorem B of Deng (1991) provided that the
vector field X is replaced by its time-reversed vector field -X Thus the proof is
complete. What follows is meant to help the reader pin down the parallel comparisons
between the Melnikov functions from condition (2.2e) and from that theorem, respectively. First, the Melnikov function Q here is essentially a positive constant multiple
--(1)
--(1)/
of Qi
there. Second, the condition (1.10b) of Theorem B is replaced by OQ/Os
>0
which was really what we used in the proof of Theorem B of Deng (1991). Note that
the 0-heti curve here is the si-axis there, which is the level set Q)= 0, and that the
positive direction of si in Theorem B, which is the same as 0-het- here, was chosen
(1)lOs O. This positive derivative in turn is equivalent to
in correspondence with OQi/
--(1)
that VQI1) and O-het point to the same side of the level set Q
=0. Also, the linear
independence of V,lr(1) and V,2r(l is equivalent to the transverse intersection of O-hetl
and 0-het2 at So.
homl
.
3. The Melnikov method. Let X(x) be a sufficiently smooth vector field in 3
with parameter s. Let a and 2 be two equilibrium points having a two-dimensional
stable manifold W (ai, s) and a one-dimensional unstable manifold W" (ai, s), respectively. In what follows we will write W or W(ag) interchangeably for the stable
manifold and so on for simplicity, provided that there is no confusion involved. Suppose
at some So there is a heteroclinic orbit z* from a to a2 (an identical consideration
can also be given to a2 to al connections). Then it must be z* WU(al) WS(a2) at
So. We would like to know how the heteroclinic connection z* changes with the
parameter. In many applications, the Melnikov method presented below is very useful
for attacking this problem.
Naturally, we would like to examine how the "signed distance" between the stable
and unstable manifolds changes with the parameter. To implement this intuitive idea,
we choose and fix a point Zo* z* from the orbit and a two-dimensional plane 5; which
is perpendicular to z*’ and through Zo*. The intersection E (q W is necessarily a curve,
whereas E f3 W is just a point for every s near So. Choose and fix a vector e on X
that is perpendicular to the curve E 0 W at Zo* and So. Let be a straight line that
goes through the point pU(s):= El’) W"(al, s) and that is of the direction of e. Then
must intersect the stable manifold Efq W(aE, s) at a unique point p(s) for s
sufficiently close to So. p"(s) and p(s) can be chosen differentiable and satisfying
p(so) pU(so)= Zo*. Now, there must be a smooth function Q(s) such that
,
(3.1)
pS(s)-p"(s)
Q(s)e or Q(s)= (p(s)-pU(s)). e/llell
=.
See Fig. 3.1. The function Q(s) serves what we called the "signed distance" between
W(a,) and W"(al) above. We will call it the Melnikov function (or the separation
function by Kokubu, Nishiura, and Oka (1988)). We are interested in the solutions of
Q(s) 0 since that precisely gives rise to those parameters at which there is a heteroclinic orbit from a to aE.
Several modifications can be made in the construction of Q(s) above. The
requirements that E be perpendicular to z* and that e be perpendicular to E f-) W are
not necessary. For instance, take the case where X is the same as above but e is replaced
1640
Bo DENG
,.
FIG. 3.1. The long dashed curve represents the intersection curve of the stable manifold of a at the
The Melnikov function Q(a) is defined as Q(a)=
point a=a and the plane
bifurcation
(p(
-p"( ))e/Ilell 2,
,
by another vector that is just transverse to the stable manifold curve E WS(a2, ao)
let
be the line through p U(a) that is parallel to
at Zo*. Similarly, let
be the corresponding intersection point lEWS(a2, a), and let Q(a)=
/ll ll = be the Melnikov function. Then it is easy to see from Fig. 3.1
that there is a nonzero constant/3 independent of a so that
Q( a flQ( a + o(l()- p()l).
In fact,/3 Ilellcos 0/ll ll and/3 is positive (respectively, negative) if e and point to
the same (respectively, opposite) side of W(a2)f-)E, where 0 is the angle between e
and (cf. Fig. 3.1). It is clear that if we want to solve the equation Q(a)=0 by the
implicit function theorem it is important to know the behavior of the partial derivatives
of Q at the bifurcation point a ao. But, this alternative definition Q simply says that
both functions are essentially the same in the sense that
(3.2)
oQ( ao)
Oa
fl
oQ( ao)
Oa
The same conclusion also holds true if we relax the choice of to be a plane transverse,
instead of perpendicular, to the heteroclinic orbit z*. We also remark that the definition
of the Melnikov function above is not necessarily just restricted to vector fields in R 3.
It can be easily extended to stable and unstable manifolds of any finite dimensions
under the condition that they are in general position along Zo*. Moreover, in the case
of nonhyperbolic equilibrium points, we can analogously define the Melnikov function
between WU(al, a) and WCS(a2, a). Another extension we will need later is the "signed
distance" between points of a center unstable manifold WCU(al, a) and a center stable
manifold WCS(a2, a) whose dimensions satisfy, for our consideration only, dim Wcs=
dim Wu= 2 and dim W c= 1. Let and e be the same as above. Then f-) WCU(al, a)
is a curve too. Suppose this curve is parametrized by w e [-1, 1] so that w--0 always
corresponds to the intersection point
W"(al, a). Let pU(a, w) be a given point
from El) W and pS(a, w) be the corresponding point of If’) f’) Wcs, where is the
line through pU(a, w) with direction e. Define
,
q(a, w)- (pS(, w)_ pU(, w)), e/Ilell
which represents a differential along the e direction from W to Ws. Of course, we
have q(c,0)=Q(a), where Q(c) is the Melnikov function between W"(al) and
WS(a2). The purpose to introduce this function q(a, s) is to relate the condition
Oq(ao, O)
(3.3)
Ow
0
1641
EXISTENCE OF INFINITELY MANY TRAVELING WAVES
to the transverse intersection of WCU(al,
at ceo. The other useful property is
ao) and WCS(a2, ao) along the connection z*
aQ(ceo) oq(ceo, o)
(3.4)
ace
ace
which will be used later in computing the derivative of q(., 0).
Now, let us return to Theorem 2.1, in particular, the choice of the Melnikov
functions Ql(ce) and Q2(ce) from the hypothesis (2.2e). Note that we explicitly talked
about the directions V Qi(ce) which determine the bifurcation directions of those
multiple pulse heteroclinic orbits in our main theorem. But, on the other hand, we
have the freedom of choosing either e or -e, which is also transverse to W (a, ceo) f] E,
in the definition of the Melnikov function Q(ce). Thus, from now on we will specify
the direction e. To this end, recall the strong inclination limit (2.2b’). From that
condition, we can easily conclude that the principal stable unit eigenvector e+ is
transverse to the stable manifold WS(a2, ceo) near a. Thus, e+ defines an orientation
for WS(a2, ceo). Now, in the definition of Ql(a), choose a vector e which points to the
same side of WS(a2) at Zo* as el+ does up to the flow homotopy. Indeed, when Zo* is
+
sufficiently close to a, we can simply let e=el.
And, as mentioned earlier, Q(ce) can
be chosen to be (or essentially to be) a positive constant multiple of ,eo() in the proof
of Theorem B of Deng (1991), or (5.41), (5.43) of Chow, Deng, and Terman (1990).
So much for the theoretical aspect of the Melnikov function Q(ce). When an
application comes, what really matters is the so-called Melnikov integral which provides
us with a computable formula for the derivative aQ(ceo)/Oce. We introduce this integral
below.
Without loss of generality, let E and e be perpendicular to z* and E fq W
respectively. Let the orbit z*(r) be parametrized so that z*(0)= Zo* and z*(’) satisfies
the equation x’= Xo(x). Consider the variational equation y’= DXo(z*(r))y along
z* and its adjoint equation y’=-(DXo(z*(r))ry. Then there is a unique bounded
solution p(r), ’e, of the adjoint equation with the initial condition (0)=e (see,
e.g., Palmer (1984)). It is well known that
,
a,
(-o)=-
q()"
ox"o(z*())a, a,
where Q(ce)= (p$(ce)_pu(ce)). e/llell= (see Holmes (1980), Palmer (1984), or Guckenheimer and Holmes (1983)). In particular, when the vector field is two-dimensional
and e is chosen to be of the orthogonal vector (-z*’(0),z*’(0)) of z*’(0) where
z*’(0) := (z*’(0), z*’(0)) is the component form, it can be directly checked that
p(’) =exp
tr DXo(z*(s)) ds
(-z*’(r), z*’(r)),
where tr A means the trace of a given square matrix A. See, e.g., Melnikov (1964),
Holmes (1980), and Palmer (1984). Note that e here is uniquely determined (cf. Fig.
3.2). In summary, we have
(3.5)
exp
tr DXo(z*(s)) ds
(-z*’(r), z*’())
ace
1642
BO DENG
el
FIG. 3.2. The unique choice
of the vector e.
which is referred to as the Melnikov integral. Last, let us remark that in light of (3.2)
we will sometimes slightly abuse the notation by writing oQ(ao)/Oa as the same
Melnikov integral as above provided that the directions e and point to the same side
of the stable manifold. A justification for this is based on the statement of Theorem
2.1 that only the signs of the derivatives of a Melnikov function really count.
4. Proof of conditions (2.2a, d). Beginning with this section, we will show that the
hypotheses (2.2a-e) of Theorem 2.1 are satisfied for the reduced FitzHugh-Nagumo
equation (2.1). It is straightforward to see that the condition (2.2d) for the continuation
of the simple heteroclinic orbits is superfluous. Indeed, it is implied by the existence
of the simple heteroclinic orbits z* at ao from condition (2.2b) and the linear independence of V Ql(aO) and V Q2(ao). The reason to include it in the statement of
Theorem 2.1 is simply for a convenient parallel comparison between that theorem
and Theorem B of Deng (1991). Thus, the condition (2.2d) may now be removed from
our checklist.
To show condition (2.2a), recall the equilibria ai
the linearization of (2.1) at a
V’=U, U’=cU-f’(vi)V+W,
(vi, 0, w) from 1 and consider
W’=e--(V-yW).
C
The corresponding characteristic equation is
A(A,
When e
e)=A(c-A)(eY+A)-f’(
v)
/
+;t +-=0.
\c
C
0, it is straightforward to check that since f’(v)< 0 for i= 1, 2 (cf. Fig. 1.1),
A(A, 0)=0 has roots
(4.1)
A1
c -x/c 2 -4f’(vi)
2
<A=0<A3
c + x/c 2- 4f’(vi)
2
Since A(O, e) -f’(vi)ey/c + e/c > 0 for e > 0 and 0A(0, 0)/0A -f’(v) > 0, the second
root A must move to the left of the origin while A1 and A3 stay uniformly away from
the origin for small e>0. Hence Al(e)<A2(e)<0<A3(e) and A3(e)+A(e)>0 for
small e > 0 by continuity. This proves condition (2.2a) that the equilibrium points are
relatively expansive by definition.
5. Proof of condition (2.2b). The methods used in this section include the geometric
theory of singular perturbations by Fenichel (1979) and the Melnikov method discussed
above. It is a rather long section but it contains all the information we will need for
verifying the remaining conditions (2.2c-e).
EXISTENCE OF INFINITELY MANY TRAVELING WAVES
1643
Let us begin with the singular perturbation. The singular parameter for (2.1) is e.
When e 0, it becomes
v’= u, u’= cu-f(v)+ w, w’=0.
(5.1a)
Note that the variable w can be regarded as a parameter and the entire cubic curve
w =f(v) on the plane u 0 consists of equilibrium points of (5.1a). Let i be a bounded,
connected and closed segment on the cubic curve w =f(v) which contains ai and the
intersection point {w wj} fq {w =f(v)} but does not contain any of the extreme points
of the cubic curve (cf. Fig. 1.1). It is easy to check that the linearization at
is
,
V’= U,
U’=cU-f’(v)V+ W, W’=O,
where v are those points that {w=f(v)}c
and thus f’(v)<0 since
does not
contain any extreme point. Similar to (4.1), the roots for the characteristic equation
are the same as Aj in (4.1) except that f’(vi) there is now replaced by f’(v). It is also
straightforward to check that the corresponding eigenvectors are
(5.1b)
Vl
v2
v3
v)
A3
0
See Fig. 5.1. Since A and A are strictly nonzero, the eigendirections vl and v are
normal to g. Therefore, gi is normally hyperbolic according to Fenichel (1979). Thus,
by Theorem 9.1 of Fenichel (1979), there exists a global center-stable manifold WS(i)
and a global center-unstable manifold WU(gi). These manifolds are smooth in e for
[el<< 1. Moreover, when e>0, W is precisely the stable manifold WS(a, e) of a
while the unstable manifold W (a, e) of a is the unstable fiber, called ff by Fenichel
(1979), through ai. WU(a, e) is also smooth in e for lel<< 1. Furthermore, all the
invariant manifolds above depend smoothly on all other parameters. Thus, our strategy
now becomes to show the existence of a connection, i.e., W (a, e) f’) WS() 0, when
e 0 and to show the continuation of that connection for e > 0 by the Melnikov method.
FIG. 5.1. The strange loop at e =0. The two vertical heavy curves are c and 2, respectively. At the
critical parameter (% c)= (yo, Co), Zl* lies on the plane {w=0} while z*z lies on the plane {w= w2}, connecting
g and g2. At the appropriately perturbed point (% c)= y(e), c(e)) with e > O, z* becomes a connection from
a to a2, and it connects opposite sides of W( ), which is the stable manifoM of a after the perturbation
e > O. Similarly, z*2 connects opposite sides of W(’2) when (y, c) (y(e), c(e)) and e > O.
1644
so
DENG
The existence of the desired connections when e 0 is given in the Appendix.
The useful properties are summarized as follows. There is a connection Zl* (Vl*, u*, 0)
from al=0 to the intersection point {w=0}VI c2--(1,0,0 for (5.1a) when C=Co=
(1 2a)/x/ for all y and Vl* > 0, Ul* > 0. For the same Co, when y To 9/(2-1)(1 2a)
there is a connection z* (v2*, u2*, w2) from a2 (v2, 0, w) back to the intersection
point {w= Wz}fq 1 for (5.1a) with u* <0, where v= yoW, w2=f(v) and (yo, Co) is
the same as in Theorem 1.1(d).
Next, let us show that these two connections can be continued for parameters
near (e, y, c)= (0, yo, Co) via the Melnikov method. We consider this question for the
Zl* connection first. Let Zo* be an arbitrarily chosen point from Zl*, and let Z be the
corresponding plane perpendicular to z* at Zo*. Without loss of generality, let Zl*(0) Zo*
and Zl*(r)= (v*, u*, 0)(r) up to time translation along the solution. Since w in (5.1a)
can be regarded as a parameter, the center-stable manifold WS(2)f3 Z of 2 on Z
can be parametrized by w. Let e be the vector (-Ul*’(0), v*’(0), 0) (which is labeled as
el in Fig. 5.1) on {w=0} as discussed in the last section for vector field in 2; then e
must be transverse to WS(2) near Zo* since it points forwards (cf. Fig. 5.1). Let
Q(e, y, c) be the corresponding Melnikov function for W"(a, e) and WCS(2) (we
will see later in 7 that the direction e is indeed consistent with the requirement of
3 that it points to the same side of WS(a2) as e+ does). On the other hand, as
mentioned earlier, since w can be viewed as a parameter for the two-dimensional system
v’= u, u’= cu-f(v)+ w,
(5.2)
we can define a Melnikov function Q(c, w) for the connection (v*, u*) at the parameters
w 0 and c Co with respect to the corresponding straight line Z fq {w 0} and the
same direction e on {w =0}. It is easy to see that (o/oC)Ql(O, yo, Co)=(o/oc)Q(Co, 0).
Thus, by (3.2) and the Melnikov integral (3.5) we have, up to a positive constant
multiple,
(0, To, Co)=OC
(5.3a)
f
exp (-cr)(-u*’(’r), v*’(7")) (0,
u* (’r)) dr
exp (-cr)u* (r) 2 dr < O.
Therefore, by the implicit function theorem, the solutions of Q1--0 can be expressed
as a function of c cl,O(y, e), or c C,o(y) for short, near (0, Yo, Co). Moreover, an
identical argument yields
Q---2 (0, To, Co) O.
(5.3b)
oy
Therefore,
C,o(Yo, 0)
(5.3c)
0y
0
or
C,o(3o)
0
by the implicit function theorem. Similarly, define the other Melnikov function
Qz(e, y, c) for WU(az, e) and WS(’) with the same kind of choices of Z and e except
that (v*, u*, 0) above is replaced by (v2*, u*, w2), where w is a constant satisfying
v yoW2, w =f(v). By the same argument,
(5.4a)
oQ___2 (0, yo, Co)
Oc
f
exp (-cr)u*(r) 2 d’r < O.
EXISTENCE OF INFINITELY MANY TRAVELING WAVES
1645
The only difference is that oQ/o/(0, Vo, Co)=OQ/o w (Co, w2)" ow/03,<o. Indeed,
since w2 is strictly decreasing in y (cf. Fig. 1.1) and, by the Melnikov integral (3.5),
(Co, w2)=-
for
I_
exp (-c7.)(-u’2’(7.), v*’(7.)). (0, 1) d7.
f-oo
exp (-c7")u*(7") dT" > 0
u2* < 0, we have
oQz
(5.4b)
(0, yo, Co) < O.
Thus, by the implicit function theorem the solution function c
of Q2 0 satisfies
0C2.o(3’o, 0) < 0
(5.4c)
03’
C.o(y, e), or C2,o(y),
C.o(3,o) < 0.
or
This and (5.3c) show that cl,o and c2, 0 intersect transversely near (3’0, Co) for small e.
More precisely, the C:,o curve crosses the cl,o curve transversely from above into below
as 3’ increases through 3"0. Let the intersection be (3’(e), c(e)). Then at (3’(e), c(e))
with e > 0 there exists a heteroclinic loop connecting a and a_.
Next, to show the heteroclinic orbit z* of (5.1a) converges to a: along the principal
stable eigenvector of a we only need to show that z* is not contained in the strong
stable manifold WSS(a2, e) of a2 which is the stable fiber, ff, through a2 on the
Since in limit e 0
center-stable manifold of
lies on {w w} while Zl* lies on
and
for
because
0<a<1/2 and w2=f(v2), vz=
{w=0}
3"o=9/(2-a)(1-2a)#O
w2>0
from
e > 0 by continuity. Similar
for
small
still
stays away
)’oWE 0,
z* uniformly
to
the
connection.
arguments apply
z*
Last, to show the strong inclination property (2.2b’) along Zl* we show first that
the center-stable and the center-unstable manifolds of (5.1a) intersect transversely
along Zl* at the limit e 0 and then we will relate this transverse intersection to the
strong inclination property for e > 0. Recall that 1 and 2 lie on {u--0} and can be
parametrized by w so that when w-0, c--Co there is the connection z* from al to
2[q{w=0}. Recall the definition of the differential function q(e, % c, w) along the
forward pointing direction e from 3, where e is the same as in the definition of the
Melnikov function Q above in this section (cf. Fig. 5.1). As mentioned earlier in 3,
we need to show
.
s
Oq(O,
3’, Co,
OW
O)
SO
in order to prove the transverse intersection of WCS(2) and WU() along z*. However,
by treating (5.1a) as a two-dimensional system (5.2) again, it is easy to see that the
differential q between W and W for the three-dimensional system (5.1a) at e =0
is precisely the Melnikov function Q(c, w) of (5.2). Thus, Oq(O, 3",co, O)/Ow=
oQ(Co, O)/ow. Hence, by (3.2) and the Melnikov integral (3.5) we have
oq(O, % Co, O)
Ow
(5.5a)
[ exp (-c7")(-u*’(7"), v*’(7"))
=-/_
exp (-c7.)u *(7.)dT"<0
(0, 1) aT"
1646
since
BO DENG
u*(z) > O. Similarly, for the other connection z2* from a2 back to 1, we can show
a(O %
o, W2)
Ow
[ exp (-c’r)u*(’r) da" > 0
since u2*(r) < 0. This shows that by definition WU(i) and WCS(j) intersect transversely
and the corresponding strong inclination property is satisfied by the strong A-lemma
of Deng (1989), (1990). By "strong inclination property" for nonhyperbolic system we
mean that the same limit as (2.2b’) exists except that W}, W’, and W in the formula
are understood as WCS(aj), ’(ai), and g(ai), respectively. Because the strong inclination property is generic, that is, it persists for those small perturbations of the vector
field along which there exists the continuation of the connection z*, the strong
inclination property (2.2b’) also holds true for sufficiently small e > 0. This completes
the proof for condition (2.2b).
Remark. The nonzeroness of the Melnikov integrals in (5.3a), (5.4a), (5.5a), and
(5.5b) have also been derived by Lin (1989).
6. Proof of condition (2.2c). Recall that the connection Zl* is twisted if the principal
eigenvectors e2 and el+ along which the other connection z2* comes from a_ and goes
to al, respectively, point to opposite sides of the stable manifold WS(a:) of a: along
as above, let us examine the limiting case e 0 first. Note
Zl*. Using the same strategy
that in limit e 0, el+ is of the direction of the center eigenvector v: of al and e is of
the direction of the unstable tangent fiber -v3 of a2 (cf. Fig. 5.1). Thus, it suffices to
show v2 of al and -v3 of a_ point to opposite sides of the center stable manifold WCS(a)
To show this, recall the differential function q from the previous section and
of
the property (5.5a). Oq/Ow < 0 implies that WCU(l) and WCS() must split in such a
way that when w >0, WCS(2) lies behind WCU(l) near z* (cf. Fig. 5.1). Thus, v2 of
al and v3 of the intersection point 2 fq { w 0} point to the same side of WCS(2). Since
v3 of 2 fq {w 0} and -v3 of a point to opposite sides of WS(2), the desired result
is proved. Similar arguments show that z2* is twisted. This proves condition (2.2c).
a.
7. Proof of condition (2.2e). Recall from (5.3a, b) that when (e, y, c)= (0, 3/0, Co),
oQ/oc < o, OQ1/O’y 0. Also recall from (5.4a, b) that oQ2/oc < O, oQ2/oy < 0. It
obviously follows that V Q1 and V Q2 are linearly independent for small e > 0, where
the gradient operator V is taken with respect to the relevant parameters 3’ and c. Last,
do
from the proof of the twist of z/* above, it is easy to see that the vectors ei and
1,
point to the same side of WS(a) (cf. Fig. 5.1). Hence, the Melnikov function Q,
2, satisfies the specific requirement with respect to the direction ei in 3. This proves
e-
condition (2.2e).
Remark. Back in 1980, Langer proved that the stable and unstable manifolds cross
transversely with the velocity parameter c for small e > 0. This is also implied by the
nonzeroness of oQ/oc, which also indicates the direction in which the transversal
crossing takes place.
8. Proof of Theorem 1.1. As discussed in 2, to prove Theorem 1.1 we only need
to show that Theorem 2.1 is applicable to the reduced FitzHugh-Nagumo equation
(2.1). We have shown in 4-7 above that the hypotheses of Theorem 2.1 are all
satisfied. Thus, it is only left to determine the bifurcation directions for the curves of
the multiple pulse front and back waves by Theorem 2.1(c). Again, recall from (5.3a, b)
and (5.4a, b), or from the previous section, that oQ/oc < O, oQ1/oT O, and oQ2/oc < O,
EXISTENCE OF INFINITELY MANY TRAVELING WAVES
1647
oQ:/oy<O at (e, 7, c) (0, 7o, Co). That is, VQ1 and VQ2 point downward at (3, c)=
(3,(e), c(e)) on the (3’, c)-plane for small e _->0. Therefore, all the interesting bifurcations take place in the southwest sector bounded by Q1 0 and Q: 0 in the (3,, c)-plane.
I-1
This completes the proof.
9. Proof of Theorem 1.2. To prove this theorem it suffices to show that the reduced
traveling wave equations when restricted to the center manifold with respect to the
singular parameter K are "almost" the same as the FitzHugh-Nagumo equations (1.1).
To begin with, recast (1.2) into the traveling wave equations with z x + ct:
(9.1)
v’=u, u’=cu-f(v)+w, w’=x, Kx’=cx-e(v-3"w).
Treating as a singular parameter and writing this equation in terms of the fast time
:= z/ variable, we have
fi=(cu-f(v)+w), =r,x, 2,=cx-e(v-3"w),
0 the equilibrium points
where the dot means the derivative in t. Note that when
of (9.2) consist of the entire three-dimensional manifold:
(9.2)
f)=u,
o :=
x=-(v-,w
c
The linearization of (9.2) at g0 when
0 is I? t)= I/ 0 and J eX. It has only
one nonzero eigenvalue c > 0 and the corresponding eigenvector (0, 0, 0, 1) is normal
to the manifold go. For simplicity, let g’o also denote a sufficiently large, connected,
and compact set in what follows. Therefore, in the context of Fenichel (1979) the
invariant manifold go is normally hyperbolic. Also by Theorem 9.1 of Fenichel (1979)
again there is a center manifold
:= {x= h(v, u, w, ,)}
in a neighborhood of g’o for all
I[ << 1, and it is a smooth continuation of g’o, namely,
h(v, u, w, O)
=e-e- (v- 3"w).
C
Here, the other parameters are suppressed from the expression of h for simplicity. It
is very important to note from (9.2) that when e =0, {x =0} is always an invariant
manifold regardless of the parameter This implies that the function h can be chosen
so that
.
(9.3)
h(v, u,
w,K)=e--(v-3"w)+O(e).
C
According to the singular perturbation theory of Fenichel (1979) this manifold gK is
invariant for both the slow and fast equations (9.1) and (9.2) for all I]<< 1. Now,
recasting (9.1) on the center manifold g’K yields
(9.4)
v’=u, u’=cu-f(v)+w,
w’=e-(v-3"w)+O(er,),
C
because of w’= w h(v, u, w, ) and the estimate (9.3). Note that this is exactly the
same FitzHugh-Nagumo equation (1.1) except for a perturbation term O(eK) to the
w equation. Now, it is easy to see that all the analysis for (2.1) in the previous sections
applies to (9.4) as well. More specifically, e is the singular parameter, 3" and c are the
relevant parameters. The additional parameter
represents a trivial direction of
1648
Bo DENG
perturbation. That is to say, it will not change any qualitative structure of the system
with respect to the heteroclinic loop bifurcation of Theorem 2.1, neither the singular
perturbation structure in terms of e nor the Melnikov method. This completes the
proof.
Appendix. The result presented below is taken from McKean (1970) and Casten,
Cohen, and Lagerstrom (1975). Consider (5.1a). Since w’=0, we can treat w as a
parameter. For fmin < W0 <fmax, there are three roots for -f(v)+ Wo 0. Denote them
by 1 </32 </33 which implicitly depend on Wo (cf. Fig. A). Then (5.1a) is equivalent to
du
since dr=dv/u. Now, it is straightforward to check that u=A(v-fll)(3-v) with
A ---t-1/x/, C--A(fll + f13--2f12) is a polynomial solution going through fll and/3. In
particular, when w0=0, /31 =0, /32 a, /33 1, and A 1/v/ we obtain a connection
with a positive speed cl,0 (1 2a)/x/. Denote the corresponding solution by (v*, Ul*);
then 1 > v* > 0, u* > 0. This implies that the connection is from al to 2 f’l {w 0} since
v*’= u* >0. On the other hand, choosing A =-1/x/, we obtain another connecting
orbit (v*2,u*) for C2,0=A(1+3--22). But, in this case u*<0 since
thus v*’<0. This implies that the connection is from 2{w Wo} back to
{ w Wo}. Thus, it is only left to determine whether the corresponding speed c is positive
and equal to 1.o at some 3’ in order to obtain a "loop." Because the cubic curve
w=f(v) is symmetric with respect to the inflection point (tinf,finf):--((l+a)/3,
(1 + a)(1-2a)(2-1)/27) of w=f(v), we can choose (v, w) from the curve w=f(v)
to be the point that is symmetric to the origin (0, 0) with respect to the inflection point.
Because of this symmetry C2,o=--(fll+3--2fl2)/x/=(1--2a)/v/=Cl,o at Wo w2. A
direct computation yields yo:=V2/W2=9/(2-a)(1-2a), which is the same as in
Theorem 1.1(d). Certainly, for this parameter yo, the connection (v2*, u2*) is from a2
back to 1 f-) { w w2}.
FIG. A. /31 < f12 < f13 are the roots
of wo-f(v)=O.
REFERENCES
D. G. ARONSON AND H. F. WEINBERGER, Nonlinear diffusion in population genetics, combustion and nerve
propagation, in Proc. Tulane Program in Partial Differential Equations and Related Topics, Lecture
Notes in Math. 446, Springer-Verlag, Berlin, 1975, pp. 5-49.
EXISTENCE OF INFINITELY MANY TRAVELING WAVES
1649
G. A. CARPENTER, A geometric approach to singular perturbation problems with applications to nerve impulse
equations, J. Differential Equations, 23 (1977), pp. 335-367.
R. H. CASTEN, H. COHEN, AND P. LAGERSTROM, Perturbation analysis of an approximation to HodgkinHuxley theory, Quart. Appl. Math., 32 (1975), pp. 365-402.
S.-N. CHOW, B. DENG, AND B. FIEDLER, Homoclinic bifurcation at resonant eigenvalues, J. Dynamical
Systems and Differential Equations, 2 (1990), pp. 177-244.
S.-N. CHOW, B. DENG, AND D. TERMAN, The bifurcation of a homoclinic and periodic orbits from two
heteroclinic orbits, SIAM J. Math. Anal., 21 (1990), pp. 179-204.
C. CONLEY, On traveling wave solutions of nonlinear diffusion equations, Tech. Report 1492, Mathematics
Research Center, University of Wisconsin, Madison, WI, 1975.
B. DENG, The bifurcations of countable connections from a twisted heteroclinic loop, SIAM J. Math. Anal.,
22 (1991), pp. 653-679.
, Homoclinic bifurcations with nonhyperbolic equilibria, SIAM J. Math. Anal., 21 (1990), pp. 693-720.
, The Sil’ nikov problem, exponential expansion, strong A-lemma, C-linearization and homoclinic bifurcation, J. Differential Equations, 79 (1989), pp. 189-231.
J. W. EVANS, N. FENICHEL, AND J. A. FEROE, Double impulse solutions in nerve axon equations, SIAM J.
Appl. Math., 42 (1982), pp. 219-234.
N. FENICHEL, Geometric singular perturbation theory for ordinary differential equations, J. Differential
Equations, 31 (1979), pp. 53-98.
J. A. FEROE, Existence and stability of multiple impulse solutions of a nerve equation, SIAM J. Appl. Math.,
42 (1982), pp. 219-234.
R. FITZHUGH, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J.,
(1961), pp. 445-466.
J. M. GREENBERG, A note on Nagumo equation, Quart. J. Math. (Oxford), 24 (1973), pp. 307-314.
J. GUCKENHEIMER AND P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector
Fields, Springer-Verlag, New York, 1983.
S. P. HASTINGS, The existence of periodic solutions to Nagumo’s equation, Quart. J. Math., 25 (1974),
pp. 368-378.
, On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations, Quart. J.
Math., 27 (1976), pp. 123-134.
, Single and multiple pulse waves for the FitzHugh-Nagumo equations, SIAM J. Appl. Math., 42 (1982),
pp. 247-260.
A. L. HODGKIN AND A. F. HUXLEY, A quantitative description of membrane current and its application to
conduction and excitation in nerve, J. Physiol. (London), 117, 500 (1952).
P. HOLMES, Averaging and chaotic motions in forced oscillations, SIAM J. Appl. Math., 38 (1980), pp. 65-80.
H. KOKUBU, Y. NISHIURA, AND H. OKA, Heteroclinic and homoclinic bifurcation in bistable reaction diffusion
systems, preprint KSU/ICS 88-08, 1988.
R. LANGER, Existence of homoclinic travelling wave solutions to the FitzHugh-Nagumo equations, Ph.D.
thesis, Northeastern University, 1980.
X.-B. LIN, Heteroclinic bifurcation and singular perturbed boundary value problems, preprint, 1989.
H. P. MCKEAN, JR., Nagumo’s equation, Adv. in Math., 4 (1970), pp. 209-223.
V. K. MELNIKOV, On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc.,
12 (1964), pp. 1-57.
J. S. NAGUMO, S. ARIMOTO, AND S. YOSHIZAWA, An active pulse transmission line simulating nerve axon,
Proc. IRE, 50 (1963), pp. 2061-2070.
P. ORTOLEVA AND J. ROSS, Theory of propagation of discontinuities in kinetic systems with multiple time
scales: fronts, front multiplicity, and pulses, J. Chem. Phys., 63 (1975), pp. 3398-3408.
K. J. PALMER, Exponential dichotomies and transversal homoclinic points, J. Differential Equations, 55 (1984),
pp. 225-265.
J. P. PAUWELUSSEN, Heteroclinic waves of the FitzHugh-Nagumo equations, preprint, 1980.
J. RAUCH AND J. SMOLLER, Qualitative theory of FitzHugh-Nagumo equations, Adv. in Math., 27 (1978),
pp. 12-44.
J. RINZEL AND J. B. KELLER, Traveling wave solutions of a nerve conduction equation, Biophys. J., 12 (1973),
pp. 1313-1337.
J. RINZEL AND D. TERMAN, Propagation phenomena in a bistable reaction-diffusion system, SIAM J. Appl.
Math., 42 (1982), pp. 1111-1137.
L. P. SIL’NIKOV, A contribution to the problem of the structure of an extended neighborhood of a rough
equilibrium state of saddle-focus type, Math. USSR Sb., 10 (1970), pp. 91-102.
W.-P. WANG, Multiple impulse solutions to McKean’s caricature of the nerve equation. I--Existence, Comm.
Pure. Appl. Math., 41 (1988), pp. 71-103.
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Bo OENG
W.-P. WANG, Multiple impulse solutions to McKean’s caricature of the nerve equation. II--Stability, Comm.
Pure. Appl. Math., 41 (1988), pp. 997-1025.
n. R. WILSON AND J. D. COWAN, Excitatory and inhibitory interactions in localized populations of model
neurons, Biophys. J., 12 (1972), pp. 1-24.
E. YANAGIDA, Branching of double pulse solutions from single pulse solutions in nerve axon equations,
J. Differential Equations, 66 (1987), pp. 243-262.