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Journal of Computational and Applied Mathematics 103 (1999) 287–295

Internal layer oscillations in FitzHugh–Nagumo equation
YoonMee Ham ∗
Department of Mathematics, Kyonggi University, Suwon 442-760, South Korea
Received 6 April 1998; received in revised form 28 September 1998

Abstract
A controller-propagator system with a FitzHugh–Nagumo equation can be reduced to a free boundary problem when a
layer parameter ” is equal to zero. We shall show the existence of solutions and the occurence of a Hopf bifurcation for
c 1999 Elsevier Science B.V. All rights reserved.
this free boundary problem as the controlling parameter  varies.
AMS classication: primary 35R35; 35B32; secondary 35B25; 35K22; 35K57; 58F14; 58F22
Keywords: FitzHugh–Nagumo equation; Internal layer; Free boundary problem; Hopf bifurcation

1. Introduction
The FitzHugh–Nagumo equations provide a model for the conduction of action potentials along
unmyelinated nerve ber in [2, 7]:
”ut = ”2 uxx + f(u) − v;
vt = Dvxx + g(u; v);

(1)

where f(u) = u(1 − u)(u − a) for 0 ¡ a ¡ 21 and g(u; v) = u −
v for some constant
: The diusivity
”2 is a small positive constant and the parameter  controls the rate between u and v (see [8]).
The reaction terms satisfy the bistable condition, i.e., the nullcline f(u) − v = 0 is the triple-valued
function of u which are called h+ (v); h− (v) and h0 (v) dened on the interval (vmin ; vmax ) where
!
!
√
√
1 + a + 1 − a + a2
1 + a − 1 − a + a2
and vmax :=f
:
vmin :=f
3
3
Also, the nullclines of f(u) − v = 0 and g(u; v) = 0 must have three intersection points as shown in

Fig. 1.
∗

E-mail: ymham@kuic.kyonggi.ac.kr.

c 1999 Elsevier Science B.V. All rights reserved.
0377-0427/99/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 2 6 4 - 7

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Y.M. Ham / Journal of Computational and Applied Mathematics 103 (1999) 287–295

Fig. 1. Nullclines f(u) − v = 0 and g = 0 for the bistable system (1).

When the diusion constant ” is very small, the existence and uniqueness of solutions for problem
(1) are examined in [8]. The occurrence of a Hopf bifurcation in (1) as  varies was shown for a very
small ” in [5, 6]. In [8, 9], they showed there is a pitchfork bifurcation for problem (1) at some 
such that the front wave loses the stability and begins to move, it travels along the innite line which
means that a set of traveling front and back waves with nonzero velocities corresponds to breathing

solutions with innte period. When the diusion constant ” is exactly zero in (1), problem (1) can
be reduced to a free boundary problem (see [1]). The well posedness of the free boundary problem
was shown in [5] using the xed-point theorem and [3] using the semigroup theory. However, the
existence of a Hopf bifurcation in a free boundary problem for the FitzHugh–Nagumo equation has
not been shown yet. In [3], the authors considered McKean type of the reaction terms such that the
pseudo inverses h± (v) are linear functions with the slope −1. And using the Green’s function and
implicit function theorem, they had shown that the Hopf bifurcation for this problem exists at some
critical point : In this paper, we shall consider a free boundary problem of (1) and shall show the
occurrence of a Hopf bifurcation as the parameter  varies. We assume that problem (1) satises
the Neumann boundary conditions at x = 0 and 1. Let ” be exactly zero in (1), then (1) will be
reduced to the following free boundary problem:
vt = Dvxx + g(h+ (v); v); 0 ¡ x ¡ s(t); t ¿ 0;
vt = Dvxx + g(h− (v); v); s(t) ¡ x ¡ 1; t ¿ 0;
vx (0; t) = 0 = vx (1; t); t ¿ 0;
1
s′ (t) =
(v(s(t); t); t); t ¿ 0;


(2)

where s(t) is the free boundary and its velocity is a function (·). This velocity function can be
calculated by the existence theory of the travelling wave solutions (see [1]). Since the pseudo-inverses
of f are dened only on the interval (vmin ; vmax );
(·) must be normalized by
h+ (r) − 2h0 (r) + h− (r)
:
(r) = − √
(vmax − r)(r − vmin )
In order to show the Hopf bifurcation as the parameter  varies, we need to examine the eigenvalues for (2). However, since the reaction terms which are the pseudo-inverses of f are not linear,

Y.M. Ham / Journal of Computational and Applied Mathematics 103 (1999) 287–295

289

Fig. 2. The linear approximations of h± (v) at the point  which are u− = h− () + (h− )′ () v and u+ = h+ () + (h+ )′ () v:

the approximation of the pseudo-inverse is the essential step. There are many linear approximations
for the pseudo-inverse functions, but we need to obtain the linear approximation function with the
same slope. In the next section, we shall linearize the pseudo-inverse functions which have the same

slope and then obtain the free boundary problem.

2. The approximation and regularization
In this section, we shall linearize the pseudo-inverse h± (v); h0 (v) of f at some point in (vmin ; vmax )
(Fig. 2). The velocity of the free boundary s(t) must be zero when (·) = 0 and thus there exists a
point  in (vmin ; vmax ) such that h+ () − 2h0 () + h− () = 0; which means that f(u) −  = 0 has the
triple values of u. Let h+ () = ur ; h0 () = u0 and h− () = ul satisfying ul ¡ u0 ¡ ur and ur + ul = 2u0 .
This implies that u0 = 13 (1 + a): Since 0 ¡ a ¡ 12 ; it follows that ur and u0 both are positive constants
and ul is a negative constant. Furthermore, note that
(h+ )′ () =

1
1
= (h− )′ () = ′
f′ (ur )
f (ul )

which is a negative constant. We now consider the reaction term g by g(u; v) = u − (v + k) where
 k = ul − (h+ )′ () · : By the bistable condition,  must satisfy
vmax

(h+ )′ () v

max

+ u r − ul

¡

1
vmin
¡ 0 ′
:

(h ) () (vmin − ) + u0 − ul + (h+ )′ () 

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Y.M. Ham / Journal of Computational and Applied Mathematics 103 (1999) 287–295

The following lemma gives an approximation of problem (1)

Lemma 2.1. The problem of (1) is approximated by
vt = Dvxx − v + (h+ )′ ()v + (ur − ul )H (x − s(t)) in
± ;
2(h+ )′ () − 2(h0 )′ ()
C(v(s(t); t)); t ¿ 0;
s′ (t) = −

vx (0; t) = vx (1; t) = 0; t ¿ 0;
s(0) = s0 ; 0 ¡ s0 ¡ 1;

(3)

where
C(r) = √

r−
(vmax − r)(r − vmin )

and
+ = {(x; t) ∈ (0; 1) × (0; ∞): 0 ¡ x ¡ s(t)};

− = {(x; t) ∈ (0; 1)×(0; ∞): s(t) ¡ x ¡ 1}. Here,
H (·) is a Heaviside step function.
Proof. Since the function h± (v) is approximated by h± () + (h± )′ ()(v − ) and (h+ )′ () = (h− )′ ();
the velocity of the free boundary can be obtained. The reaction terms are reduced such that g(h+ ()+
(h+ )′ ()(v−); v) = h+ ()+(h+ )′ ()(v−)−(v+k) for 0 ¡ x ¡ s(t) and g(h− ()+(h− )′ ()(v− ); v)
= h− () + (h− )′ ()(v − )−(v + k) for s(t) ¡ x ¡ 1. Noting the fact that (h+ )′ () = (h− )′ () and
using the Heaviside function. Hence we obtain that vt = Dvxx −v+(h+ )′ ()v+(h+ ()−h− ()) H (x −
s(t)).
The well posedness of solutions of (3) can be obtained by a similar argument which was used
in the proof of Theorem 2.7 in [5, 3]. Hence, we shall examine an existence of stationary solutions
and the occurrence of a Hopf bifurcation in the next section.
We now adopt several notations from [3] in order to show a Hopf bifurcation for the problem
(3) as  varies:
We may assume the nite diusion constant D = 1 by a rescaling of t in (3). For brevity, we
let (h+ )′ () = − b1 and (h0 )′ () = b0 ¿ 0 with b0 ¿ 0 and b1 ¿ 0: Let G : [0; 1]2 →R be a Green’s
function of the dierential operator A:= −d 2 =dx2 ++b1 satisfying the Neumann boundary conditions
and let the domain of A be
D(A) = {v ∈ H 2; 2 ((0; 1)): vx (0) = vx (1) = 0}:
We dene a function g : [0; 1]2 →R by
g(x; s):= (ur − ul )

Z

s

1

G(x; y) dy = A−1 ((ur − ul )H (· − s))(x)

Y.M. Ham / Journal of Computational and Applied Mathematics 103 (1999) 287–295

291

and
: [0; 1]→R by
(s):= g(s; s): If we use a transformation
u(t)(x):= v(x; t) − g(x; s(t));
then problem (3) can be written by an abstract evolution equation
d
e s) = 2(b0 + b1 ) f(u; s)

(u; s) + A(u;
dt


(R)

(u; s)(0) = (u(0); s(0)) = (u0 ; s0 );

e is a 2 × 2 matrix whose (1,1)-entry is the operator A and all the others are zero. The
where A
nonlinear forcing term f is

f(u; s) =

(ur −ul ) C(u(t)(s(t)) +
(s(t)))G(x; s(t))
C(u(t)(s(t)) +
(s(t)))

!

:

The authors in [3] proved the well posedness of solutions applying the semigroup theory using
e and obtained that f : W ∩ X
e  →X
e is continuously
domains of fractional powers  ∈ ( 43 ; 1] of A and A
dierentiable where
W := {(u; s) ∈ C 1 ([0; 1]) × (0; 1): u(s) +
(s) ∈ I } ⊂open C 1 ([0; 1]) × R;
X  := D(A );

e  );
Xe  := D(A

Xe  = X  × R:

3. Stationary solutions and a Hopf bifurcation
3.1. Stationary solutions
The stationary problem, corresponding to (R), is given by
2(b0 + b1 )
(ur − ul ) · C(u∗ (s∗ ) +
(s∗ )) G(·; s∗ );

2(b0 + b1 )
0=
C(u∗ (s∗ ) +
(s∗ ));

′
′
u∗ (0) = 0 = u∗ (1):
Au∗ =

e ∩ W . For  6= 0; this system is equivalent to the pair of equations
for (u∗ ; s∗ ) ∈ D(A)
u∗ = 0; C(
(s∗ )) = 0:

(4)

We thus obtain:
Proposition 3.1. If (ur −ul )=vmax ¡ +b1 ¡ (ur −ul )= then (R) has a unique
solution (0; s∗ ) for all  6= 0 with s∗ ∈ (0; 1). The linearization of f at (0; s∗ ) is
Df(0; s∗ )(û; ŝ) =

27(b0 + b1 )
(û(s∗ ) +
′ (s∗ )ŝ)((ur −ul )G(s∗ ; s∗ ); 1):
 (1−a + a2 )3=2

stationary

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Y.M. Ham / Journal of Computational and Applied Mathematics 103 (1999) 287–295

The pair (0; s∗ ) corresponds to a unique steady state (v∗ ; s∗ ) of (F) for  6= 0 with
v∗ (x) = g(x; s∗ ) :
Proof. Since C(z) = 0 if and only if z = , we have
(s) =  and thus it will suce to show the
existence of s∗ ∈ (0; 1): We now dene
T (s) :=
(s) − :

Then T (0) = (ur −ul )=(+b1 )− ¿ 0; T (1) = − ¡ 0 and T ′ (s∗ ) =
′ (s∗ ) ¡ 0: Therefore, there exist
a unique value s∗ in (0; 1) such that
(s∗ ) = . Furthermore,
C ′ (
(s∗ )) =

1
27
=
¿0
vmax −  2(1 − a + a2 )3=2

since vmax −  = −vmin . Using the theorem in [3], we obtain the corresponding steady state (v∗ ; s∗ )
for (F).
3.2. A Hopf bifurcation
We now introduce a new parameter  = (27(b0 + b1 ))=( (1−a + a2 )3=2 ) and show that there is
a Hopf bifurcation from the curve  7→ (0; s∗ ) of steady states. We now state some denitions and
theorems for the Hopf bifurcation theory.
Denition 3.2. Under the assumptions of Proposition 3.1, dene (for
(Xe  ; Xe ) as

3
4

¡ 61) the operator B ∈ L

 (1−a + a2 )3=2
Df(0; s∗ ):
27(b0 + b1 )
We then dene (0; s∗ ) to be a Hopf point for (R) if there exists an 0 ¿ 0 and a C 1 -curve
(− + ∗ ; ∗ +  ) 7→ ((); ()) ∈ C ×Xe
B :=

0

0

C

(YC denotes the complexication of the real space Y ) of eigendata for −Ae + B such that
(i) (−Ae + B)(()) = ()(); (−Ae + B)(()) = () ();
(ii) (∗ ) = i with  ¿ 0;
(iii) Re () 6= 0 for all  ∈ (−Ae + ∗ B)\{±i};
(iv) Re ′ (∗ ) 6= 0 (transversality).

We now have to check (R) for Hopf points. In order to do this we need to solve the eigenvalue
problem
e s) + B(u; s) = (u; s);
−A(u;

which is equivalent to

(A + )u =  · (ur −ul )(
′ (s∗ )s + u(s∗ )) · G(·; s∗ );
s =  · (
′ (s∗ )s + u(s∗ )):

(5)

Y.M. Ham / Journal of Computational and Applied Mathematics 103 (1999) 287–295

293

We let v := u−(ur −ul ) · G(·; s∗ ), this system is equivalent to the weak system of equations
(A + )v = −(ur −ul ) · s∗ s
 s =  · (
′ (s∗ ) s + (u −u ) · G(s∗ ; s∗ ) + v(s∗ )):
r

(6)

l

As a result, we can see that it suces to nd a unique, purely imaginary eigenvalue  = i of
(5) with  ¿ 0 for some ∗ in order for (0; s∗ ; ∗ ) to be a Hopf point.
Theorem 3.3. Assume that for ∗ ∈ R\{0} ; the operator −Ae + ∗ B has a unique pair {±i} of
purely imaginary eigenvalues. Then (0; s∗ ; ∗ ) is a Hopf point for (R).

Proof. We assume without loss of generality that  ¿ 0, and ∗ is the (normalized) eigenfunction
of −Ae + ∗ B with eigenvalue i. We have to show that (∗ ; i) can be extended to a C 1 -curve
 7→ ((); ()) of eigendata for −Ae + B with ′ (∗ ) 6= 0.
For this let ∗ = ( 0 ; s0 ) ∈ D(A) × R. First, we see that s0 6= 0 and hence, without loss of generality,
we assume s0 = 1. Then E( 0 ; i; ∗ ) = 0 by (5), where
E : D(A)C × C × R → XC × C;

E(u; ; ) := ((A + )u− · (ur − ul ) (
′ (s∗ ) + u(s∗ ))G(·; s∗ );  −  · (
′ (s∗ ) + u(s∗ ))):
The equation E(u; ; ) = 0 is equivalent to  being an eigenvalue of −Ae + B with eigenfunction
(u; 1). We need to show two things; that E is C 1 and the transversality condition holds. The proof
of the rst one is similar to [3] and thus have only the transversality condition, Re ′ (∗ ) 6= 0 holds.
Let () = ( (); 1). Implicit dierentiation of E( (); (); ) = 0 implies that
D(u; ) E( 0 ; i; ∗ )( ′ (∗ ); ′ (∗ )) = (
′ (s∗ ) +
This means that the function û :=

′

0 (s

∗ )) · (G(·; s∗ ); 1):

(∗ ) and ˆ := ′ (∗ ) satisfy the equations

(A + i) û − ∗ û(s∗ )G(·; s∗ ) + ˆ

0

= (
′ (s∗ ) +

0 (s

∗ ))G(·; s∗ )

(7)

and
−∗ û(s∗ ) + ˆ =
′ (s∗ ) +
Putting (8) into (7) and using
(A + i)û + ˆ

1

0 (s
1

∗ ):

:=

(8)
0 −G(·; s

∗ ), as before, we obtain

= 0:

(9)

From (6) and (8), we have
ˆ
i
û(s∗ ) = ∗ − 2 :

∗

(10)

Multiplying û by (9) and integrating, we obtain
2i 

Z

0

1

û

1

ˆ
= ∗ − û(s∗ )


(11)

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Y.M. Ham / Journal of Computational and Applied Mathematics 103 (1999) 287–295

and
Z

1

0

|A1=2 û|2 + i

Z

0

1

|û|2 + ˆ

Z

1
1 û = 0:

(12)

0

Compare this equation with Eq. (11) and then substitute (10), then we get
2i

Z

0

1

|A1=2 û|2 − 22

Z

1

|û|2 − i

0

ˆ
= 0;
(∗ )2

which implies that
Re ˆ = 2(∗ )2
Hence
in [3],
passes

Z

0

1

|A1=2 û|2 ¿ 0:

the transversality condition holds for all ∗ ¿ 0. Therefore, by the Hopf-bifurcation theorem
there exists a family of periodic solutions which bifurcates from the stationary solution as 
∗ .

Finally, we will show that, whenever (R) admits a stationary solution, there is a unique ∗ ¿ 0
such that (0; s∗ ; ∗ ) is a Hopf point, and thus ∗ is the origin of a branch of nontrivial periodic
orbits.
Theorem 3.4. There exists a unique; purely imaginary eigenvalue  = i of (5) with  ¿ 0 for a
unique critical point ∗ ¿ 0 in order for (0; s∗ ; ∗ ) to be a Hopf point.
Proof. We only need to show that the function (u; ; ) 7→ E(u; i; ) has a unique zero with  ¿ 0
and  ¿ 0. This means solving system (6) with  = i and s = 1;
(A + i)v = −(ur −ul ) · s∗
i =  · (
′ (s∗ ) + (u −u ) · G(s∗ ; s∗ ) + v(s∗ )):
r

(13)

l

Since
′ (s∗ ) + (ur − ul )G(s∗ ; s∗ ) ¿ 0 and
′ (s∗ ) ¡ 0, a unique solution (; ∗ ) of (13) for  ¿ 0
and ∗ ¿ 0 exists which can be easily proved by the similar method used in [8].
The following theorem is what we have shown for the FitzHugh–Nagumo equation in a free
boundary problem:
Theorem 3.5. Assume that 0 ¡ =(ur −ul ) ¡ 1=( + b1 ); so that (R) and (F); respectively; has a
unique stationary solution (0; s∗ ); respectively (v∗ ; s∗ ); for all  ¿ 0. Then there exists a unique
∗ ¿ 0 such that the linearization −Ae+ ∗ B has a purely imaginary pair of eigenvalues. The point
(0; s∗ ; ∗ ) is then a Hopf point for (R) and there exists a C 0 -curve of nontrivial periodic orbits
for (R) and (F); respectively; bifurcating from (0; s∗ ; ∗ ) and (v∗ ; s∗ ; ∗ ); respectively.

Y.M. Ham / Journal of Computational and Applied Mathematics 103 (1999) 287–295

295

Acknowledgements
The author wishes to acknowledge the nancial support of the Korea Research Foundation made in
the program year in 1997. The present work was supported by the Basic Science Research Institute
Program, Ministry of Education, 1997, Project No. BSRI-97-1436.
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