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 this free boundary problem as the controlling parameter varies. c1999 Elsevier Science B.V. All rights reserved.
AMS classication: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=”2uxx+f(u)−v;
vt=Dvxx+g(u; v);
(1)
where f(u) =u(1−u)(u−a) for 0¡ a ¡1
2 andg(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) dened on the interval (v
min; vmax) where
vmin:=f
1 +a−√1−a+a2
3
!
and vmax:=f
1 +a+√1−a+a2
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: [email protected].
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) asvaries 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 innite line which means that a set of traveling front and back waves with nonzero velocities corresponds to breathing solutions with innte 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) satises 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;
s′(t) =1
(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 dened only on the interval (vmin; vmax); (·) must be normalized by
(r) = −h
+(r)−2h0(r) +h−
(r)
√
(vmax−r)(r−vmin)
:
Fig. 2. The linear approximations ofh±(v) at the pointwhich areu−=h−() + (h−)′()vandu+=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 (v
min; 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+() =u
r; h0() =u0 andh−() =ul satisfying ul¡ u0¡ ur andur+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
f′(u r)
= (h−)′() = 1
f′(u l)
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+ur−ul
¡1
¡
vmin
(h0)′() (v
min−) +u0−ul+ (h+)′()
C(r) =√ r−
the velocity of the free boundary can be obtained. The reaction terms are reduced such thatg(h+()+
(h+)′()(v
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:
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
dt(u; s) +Ae(u; s) =
2(b0+b1)
f(u; s)
(u; s)(0) = (u(0); s(0)) = (u0; s0);
(R)
where Ae is a 2×2 matrix whose (1,1)-entry is the operator A and all the others are zero. The 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 domains of fractional powers ∈(3
4;1] of A and Ae and obtained that f:W∩Xe
→Xe is continuously
dierentiable where
W:={(u; s)∈C1([0;1])
×(0;1): u(s) +(s)∈I} ⊂open C1([0;1])×R; X:=D(A); Xe:=D(Ae); Xe=X
×R:
3. Stationary solutions and a Hopf bifurcation
3.1. Stationary solutions
The stationary problem, corresponding to (R), is given by
Au∗=2(b0+b1)
(ur−ul)·C(u∗(s∗) +(s∗)) G(·; s∗);
0=2(b0+b1)
C(u∗(s∗) +(s∗)); u∗′(0) = 0 =u∗′(1):
for (u∗; s∗)∈D(Ae)∩W. For 6= 0; this system is equivalent to the pair of equations
u∗= 0; C((s∗)) = 0: (4)
We thus obtain:
Proposition 3.1. If (ur−ul)=vmax¡ +b1¡(ur−ul)= then (R) has a unique stationary
solution (0; s∗) for all 6= 0 with s∗ ∈(0;1). The linearization of f at (0; s∗) is
Df(0; s∗)( ˆu;sˆ) = 27(b0+b1)
(1−a+a2)3=2 ( ˆu(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 denitions and theorems for the Hopf bifurcation theory.
Denition 3.2. Under the assumptions of Proposition 3.1, dene (for 3
4¡ 61) the operator B∈L
(Xe;Xe) as
B:=(1−a+a
2)3=2
27(b0+b1)
Df(0; s∗):
We then dene (0; s∗) to be a Hopf point for (R) if there exists an 0¿0 and a C1-curve
(−0+∗; ∗+0)7→((); ())∈C×XeC
(YC denotes the complexication 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
−Ae(u; s) +B(u; s) =(u; s);
which is equivalent to (A+)u=·(ur−ul)(
′
(s∗)s+u(s∗))·G(·; s∗);
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+ (ur−ul)·G(s∗; s∗) +v(s∗)):
(6)
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 C1-curve (u;1). We need to show two things; that E is C1 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.
Multiplying ˆu by (9) and integrating, we obtain
0
Hence the transversality condition holds for all ∗¿0. Therefore, by the Hopf-bifurcation theorem in [3], there exists a family of periodic solutions which bifurcates from the stationary solution as
passes ∗.
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∗) + (ur−ul)·G(s∗; s∗) +v(s∗)):
(13)
Since ′(s∗) + (u
r−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 C0-curve of nontrivial periodic orbits
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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