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Travelling wave solutions near isolated double-pulse solitary waves of nerve axon equations
Volume 121, number 8,9
PHYSICS LETTERS A
18 May 1987
TRAVELLING WAVE SOLUTIONS NEAR ISOLATED DOUBLE-PULSE SOLITARY WAVES OF NERVE AXON EQUATIONS
Paul GLENDINNING MathematicsInstitute, University of Warwick, Coventry CV4 7AL, UK
Received 16 February 1987; accepted for publication 25 March 1987
It is known that a countable number of double-pulse solitary waves can be associated with a single-pulse solitary wave in reaction diffusion equations such as the FitzHugh—Nagumo equations of nerve conduction. Even when the conditions for this occur do not hold, a double-pulse solution can branch from a single-pulse wave. We describe how this happens, and show that double-pulse travelling waves exist over very small regions of the bifurcation parameters.
Simple models of the Hodgkin—Huxley equations of impulse propagation in an infinitely long nerve cell can be written in the form u,=u~+f(u v~u), v,=g(u,v4u),
(1)
where f and g are smooth functions of the variables u(x, t) and v(x, t), and ~t is a real parameter [1—3]. We assume thatf(0, 0; 4u) =g(0, 0; ~u)= 0, so the spatially homogeneous stationary state (u,v) = (0,0) is always a solution. A travelling solution (u(x,t), v(x,t)) = (U(z), V(z)), for z=x+ct (c is the propagation speed), can be found by solving the ordinary differential equations U” = cU’ —ft U, V~u), cV’ =g(
~,
V~)
,
(2)
where the primes denote differentiation with respect to z. Periodic solutions of (2) correspondto periodic travelling waves in (1), and homoclinic orbits in (2) (trajectories biasymptotic to the same stationary point) correspànd to solitary waves in (1). Roughly speaking, an n-pulse solitary wave has n large humps, which propagate at the appropriate velocity, c, given ~u,for which the correspondinghomoclinic orbit exists for (2). Some work has been done to reinterpret the theorems of Shil’nikov [4,5] in terms of the FitzHugh—Nagumo equations [1—3,61,particularly those results for systems which depend upon parameters, since we have two natural parameters, ~t and c, in (2). In this paper we shall restrict attention to results for ordinary differential equations in ~, 0375-960 1/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
although they can all be extended to l~’~ provided certam genericity conditions are satisfied. The results of
refs. [5,7,8] can be reinterpreted to give scaling laws for the infinite sequence of double-pulse solitary waves associated with certain types of single-pulse solitary waves (cf. refs. [1—3]).In the statement of these theorems we have assumedthat the velocity, C, of a travelling wave is related to a natural parameter of the system in the simplest possible way [61. If this is not the case, the scaling laws derived below must be modified appropriately. .
Theorem 1. Suppose that for (p0,c0) eq. (2) pos-
sesses a homoclinic orbit biasymptotic to the stationary point at the origin, and that the jacobian at the origin has eigenvalues (Ad, —Po ±i&0) with co~>0 and ,%~>Po>O. Then (i) eq. (1) has countably many double-pulse solitary waves at (4u0,c1) where c—’c0 as i—~,,oand lim C~~1 C~= exp( xp0/co0); —
i-~
C
—
—
Ci_i
(ii) there exists K such that for all L> K and c sufficiently close to c0 the system (1) has a travelling wave with velocity c and wavelength L where L
c—c0 Be P° cos(w0L+ 0), for B and 0 constant to lowest order.
The conditions of this theorem may be statisfied 411
Volume 121, number 8,9
PHYSICS LETTERS A
for the FitzHugh—Nagumo equations [1—3],but it is also possible to find regions ofparameter spacewhere the eigenvalues of the jacobian matrix closest to the imaginary axis on each side of the imaginary axis do not satisfy the conditions of theorem 1. In this case we have the following result [4]. Theorem 2. Suppose that for (jz~,c0)eq. (2) possesses a homoclinic orbit, 1, biasymptotic to the stationary point at the origin, 0, and that the jacobian matrix at 0 has eigenvalues (A0,r1,r2). Then if A0eR, 0> Re r1 ~ Re r2 and A~< Re r1, there is at most one periodic orbit in a neighbourhood off’ in phase space and a neighbourhood Nof (p0,c0) in parameterspace. —
This theorem also has the obvious interpretation for (1) as described in ref. [6]. Note that both these theorems refer to codimension-one possibilities: there is only one condition which needs to be satisfied, so, in principle, the parameter ~t is redundant. The aim ofthis letter is to examine how the behaviour of trayelling solutions to (1) varies in a two-parameter space with a codimension-two point of the sort discussed in ref. [9]. Yanagida considers the case where all the eigenvalues of the jacobian matrix are real, and asks when it is possible to find a double-pulse homoclinic orbit near a single-pulse homoclinic orbit for (2). Although he is able to give a complete answer to this question he does not describe the local bifurcation structure completely and it is this, together with the implications for travelling wave solutions, that we will investigate here. Theorem 3[9]. Suppose that for (~,c0)= (~t,c1(‘u)) eq. (2) possesses a homoclinic orbit biasymptotic to the stationary point at the origin, and that the jacobian matrix at 0 has real eigenvalues (2~Cu),A2Cu), A~(p))with 21(~u)>0,and 0<,~.2(4u)<23(~u).Then there is a constant A, such that if ,~(jt~)+)~2(pa) = 0 for some ~, and (d/diz)(21 +~2.2)I ~ then ifA<0, no double-pulse solitary wave solution branches at (~,c1(pa)) whilst if A>0 a double-pulse solution branches tangentially from the curve c= c1 (a) along c= c2 (a) in the direction for which A1+22<0
>0 412
if A>l, if 0
.
18 May 1987
2W
Is
lu
l~ —
214 1 u,
2s_ ~P1
PD~~“ lu ________________
0
—
1H
Is
_______________________
0
~
°
Fig. 1. Bifurcation diagrams in (~,v)space. The curve PD is the locus of period-doubling bifurcations, 2H is the locusof doublepulse homoclinic orbits (c=c2Cu) in the text), and the axis u=O is the locus ofsingle-pulse homocinic orbits, 1H (c=c1 (jt) in the text). The periodic orbits are indicated by nu or ns where the u (s) denotes unstable (stable) orbits and the n gives the number of pulses per period. 0 denote no periodic orbits (a) 0 !.
This situation is depicted in fig. 1, where the curve c = c1 (it) is labelled 1 H, and the curve c= c2 (it) is labelled 2H. Yanagida proves considerably more than this, but I suspect that this is the result that will be of most relevance to examples. As suggested earlier, this cannot be the complete bifurcation picture near (~t0,c1(4u0)). Consider a circular path in parameter space, centred on (~,c1(pa)). Each time the curve c= c2(au) is crossed, a doublepulse periodic solution is added to the dynamics, but it is never taken away! So there must be at least one curve of bifurcations which intersects the circle and removes the periodic orbit created at c = c2(u). The analysis of the single pulse homoclinic orbit can be reduced (via standard techniques [4,10]) to the map x’ = v(~,c)-A (j~,c)x~,x> 0, x small, where a= —)~2Cu)/~~.1Cu) and x represents the eigenvector corresponding to the eigenvalue 2~(p).The plane x= 0 is the local stable manifold of the origin, and so the single pulse homoclinic orbit occurs when v(~u,c)= 0, with solution ~i = c1 (a). Yanagida’s theorem is for ,~(a) near )~2Cu), so setting a = 1 + we obtain the map —
x’=f(x)=v—Ax’~~, x>0,
(3)
for values of (u,c) near (~,c0)and x small but positive. From now on we shall work in the new param-
Volume 121, number 8,9
PHYSICS LETTERS A
18 May 1987
eter space (~,v). Double-pulse honioclinic orbits occur when
point of (3)) elsewhere. In the second case, A> 1, there is a similar bifurcation structure in e>0 as
0=v—Av’~, v>0.
shown in fig. lb. In both cases it is easy to show that these are the only bifurcations in a sufficiently small neighbourhood of the codimension-two point (~,v) = (0,0). However, outside this small neighbourhood it is possible to find regions of parameter values for which there are saddle-node bifurcations of two-pulse
Clearly this has no solutions ifA <0, so we only need to consider the case A>0. For A >0 we obtain the solution VV
(4)
2HA.
Since v is small and positive, this gives a curve in e>.O if A> 1, and c <0 if 0
(5)
The derivative of the map is (1 + ~)Ax~,so this stationary point period-doubles when —
l=(l+e)Ax~.
(6)
Substituting for x~in (5) we get x=v(l+~)/(2+a)
(7)
as the condition that x be a non-hyperbolic station-
ary point of (3). But t~ x=[l/A(l+e)] (from (6)) so v=vPD=A~(2+a)/(l+a)11~~~ (8) Thus the curve defined by (8) (for small a and v) gives the locus of period-doubling bifurcations for the periodic orbit of the flow corresponding to the stationary point of (3). Now note that
periodic orbits, but we shall not describe these here. For the FitzHugh—Nagamo equations this implies that whenever the situation described above arises there is a very small region in (a,c) space near (p~,c1(u~))in which there are travelling waves with velocity c near c,(a,~) which have two pulses per period. However, the bifurcation schemes described above are considerably more general, and apply to any set of ordinary differential equations or reaction—diffusion equations on a one-dimensional, infinite domain where this codimension-two situation arises.
A similar analysis can be done for pairs of homoclinic orbits which are mappedonto each other under a symmetry (as, for example, in the Lorenz equations). The description of this bifurcation, in which chaos must be avoided locally, can be found in ref. [11]. I am grateful to the SERC for support during this work.
.
log vpD
—
log ~2H=log 2— 1 + f2/24 + 0( a3) ,
which is less than zero for small a, 50 near (,v) =(0,0), VPDO given by (8) on which the periodic orbit~has a Floquet exponent of 1. Thus there are two periodic orbits between these two curves, as shown in fig. 1, and only the original periodic orbit (the stationary —
References [1] J. Evans, N. Fenichel and J.A. Feroe, SIAM J. AppI. Math. 42(1982) 219. [2) J.A. Feroe, SIAM J. App!. Math. 42 (1982) 235. [3) S.P. Hastings, SIAM J. AppI. Math. 42 (1982) 247. [4] L.P. Shil’nikov Math. USSR Sb. 6 (1968) 427. [5] L.P. Shil’nikov Math. USSR Sb. 10(1970)91. [6]P. Glendinning, Homoclinic bifurcations in ordinary differential equations, in: NATO ASI Life Sciences Series, Chaos in biological systems, eds. H. Degn, A.V. Holden and L Olsen (Plenum, New York, 1987), to be published. [7] P. Gaspard, R. Kapral and 0. Nicolis, J. Stat. Phys. 35 (1984) 697. [8] P. Glendinning and C. Sparrow, J. Stat. Phys. 35 (1984) 645. [9) E. Yanagida, J. Duff. Equ. 66(1986) 243. [10] C. Tresser, Ann. Inst. H. Poincaré 40 (1984) 441. [11] P. Glendinning, Ph.D. thesis, University of Cambridge (1985).
413
PHYSICS LETTERS A
18 May 1987
TRAVELLING WAVE SOLUTIONS NEAR ISOLATED DOUBLE-PULSE SOLITARY WAVES OF NERVE AXON EQUATIONS
Paul GLENDINNING MathematicsInstitute, University of Warwick, Coventry CV4 7AL, UK
Received 16 February 1987; accepted for publication 25 March 1987
It is known that a countable number of double-pulse solitary waves can be associated with a single-pulse solitary wave in reaction diffusion equations such as the FitzHugh—Nagumo equations of nerve conduction. Even when the conditions for this occur do not hold, a double-pulse solution can branch from a single-pulse wave. We describe how this happens, and show that double-pulse travelling waves exist over very small regions of the bifurcation parameters.
Simple models of the Hodgkin—Huxley equations of impulse propagation in an infinitely long nerve cell can be written in the form u,=u~+f(u v~u), v,=g(u,v4u),
(1)
where f and g are smooth functions of the variables u(x, t) and v(x, t), and ~t is a real parameter [1—3]. We assume thatf(0, 0; 4u) =g(0, 0; ~u)= 0, so the spatially homogeneous stationary state (u,v) = (0,0) is always a solution. A travelling solution (u(x,t), v(x,t)) = (U(z), V(z)), for z=x+ct (c is the propagation speed), can be found by solving the ordinary differential equations U” = cU’ —ft U, V~u), cV’ =g(
~,
V~)
,
(2)
where the primes denote differentiation with respect to z. Periodic solutions of (2) correspondto periodic travelling waves in (1), and homoclinic orbits in (2) (trajectories biasymptotic to the same stationary point) correspànd to solitary waves in (1). Roughly speaking, an n-pulse solitary wave has n large humps, which propagate at the appropriate velocity, c, given ~u,for which the correspondinghomoclinic orbit exists for (2). Some work has been done to reinterpret the theorems of Shil’nikov [4,5] in terms of the FitzHugh—Nagumo equations [1—3,61,particularly those results for systems which depend upon parameters, since we have two natural parameters, ~t and c, in (2). In this paper we shall restrict attention to results for ordinary differential equations in ~, 0375-960 1/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
although they can all be extended to l~’~ provided certam genericity conditions are satisfied. The results of
refs. [5,7,8] can be reinterpreted to give scaling laws for the infinite sequence of double-pulse solitary waves associated with certain types of single-pulse solitary waves (cf. refs. [1—3]).In the statement of these theorems we have assumedthat the velocity, C, of a travelling wave is related to a natural parameter of the system in the simplest possible way [61. If this is not the case, the scaling laws derived below must be modified appropriately. .
Theorem 1. Suppose that for (p0,c0) eq. (2) pos-
sesses a homoclinic orbit biasymptotic to the stationary point at the origin, and that the jacobian at the origin has eigenvalues (Ad, —Po ±i&0) with co~>0 and ,%~>Po>O. Then (i) eq. (1) has countably many double-pulse solitary waves at (4u0,c1) where c—’c0 as i—~,,oand lim C~~1 C~= exp( xp0/co0); —
i-~
C
—
—
Ci_i
(ii) there exists K such that for all L> K and c sufficiently close to c0 the system (1) has a travelling wave with velocity c and wavelength L where L
c—c0 Be P° cos(w0L+ 0), for B and 0 constant to lowest order.
The conditions of this theorem may be statisfied 411
Volume 121, number 8,9
PHYSICS LETTERS A
for the FitzHugh—Nagumo equations [1—3],but it is also possible to find regions ofparameter spacewhere the eigenvalues of the jacobian matrix closest to the imaginary axis on each side of the imaginary axis do not satisfy the conditions of theorem 1. In this case we have the following result [4]. Theorem 2. Suppose that for (jz~,c0)eq. (2) possesses a homoclinic orbit, 1, biasymptotic to the stationary point at the origin, 0, and that the jacobian matrix at 0 has eigenvalues (A0,r1,r2). Then if A0eR, 0> Re r1 ~ Re r2 and A~< Re r1, there is at most one periodic orbit in a neighbourhood off’ in phase space and a neighbourhood Nof (p0,c0) in parameterspace. —
This theorem also has the obvious interpretation for (1) as described in ref. [6]. Note that both these theorems refer to codimension-one possibilities: there is only one condition which needs to be satisfied, so, in principle, the parameter ~t is redundant. The aim ofthis letter is to examine how the behaviour of trayelling solutions to (1) varies in a two-parameter space with a codimension-two point of the sort discussed in ref. [9]. Yanagida considers the case where all the eigenvalues of the jacobian matrix are real, and asks when it is possible to find a double-pulse homoclinic orbit near a single-pulse homoclinic orbit for (2). Although he is able to give a complete answer to this question he does not describe the local bifurcation structure completely and it is this, together with the implications for travelling wave solutions, that we will investigate here. Theorem 3[9]. Suppose that for (~,c0)= (~t,c1(‘u)) eq. (2) possesses a homoclinic orbit biasymptotic to the stationary point at the origin, and that the jacobian matrix at 0 has real eigenvalues (2~Cu),A2Cu), A~(p))with 21(~u)>0,and 0<,~.2(4u)<23(~u).Then there is a constant A, such that if ,~(jt~)+)~2(pa) = 0 for some ~, and (d/diz)(21 +~2.2)I ~ then ifA<0, no double-pulse solitary wave solution branches at (~,c1(pa)) whilst if A>0 a double-pulse solution branches tangentially from the curve c= c1 (a) along c= c2 (a) in the direction for which A1+22<0
>0 412
if A>l, if 0
.
18 May 1987
2W
Is
lu
l~ —
214 1 u,
2s_ ~P1
PD~~“ lu ________________
0
—
1H
Is
_______________________
0
~
°
Fig. 1. Bifurcation diagrams in (~,v)space. The curve PD is the locus of period-doubling bifurcations, 2H is the locusof doublepulse homoclinic orbits (c=c2Cu) in the text), and the axis u=O is the locus ofsingle-pulse homocinic orbits, 1H (c=c1 (jt) in the text). The periodic orbits are indicated by nu or ns where the u (s) denotes unstable (stable) orbits and the n gives the number of pulses per period. 0 denote no periodic orbits (a) 0 !.
This situation is depicted in fig. 1, where the curve c = c1 (it) is labelled 1 H, and the curve c= c2 (it) is labelled 2H. Yanagida proves considerably more than this, but I suspect that this is the result that will be of most relevance to examples. As suggested earlier, this cannot be the complete bifurcation picture near (~t0,c1(4u0)). Consider a circular path in parameter space, centred on (~,c1(pa)). Each time the curve c= c2(au) is crossed, a doublepulse periodic solution is added to the dynamics, but it is never taken away! So there must be at least one curve of bifurcations which intersects the circle and removes the periodic orbit created at c = c2(u). The analysis of the single pulse homoclinic orbit can be reduced (via standard techniques [4,10]) to the map x’ = v(~,c)-A (j~,c)x~,x> 0, x small, where a= —)~2Cu)/~~.1Cu) and x represents the eigenvector corresponding to the eigenvalue 2~(p).The plane x= 0 is the local stable manifold of the origin, and so the single pulse homoclinic orbit occurs when v(~u,c)= 0, with solution ~i = c1 (a). Yanagida’s theorem is for ,~(a) near )~2Cu), so setting a = 1 + we obtain the map —
x’=f(x)=v—Ax’~~, x>0,
(3)
for values of (u,c) near (~,c0)and x small but positive. From now on we shall work in the new param-
Volume 121, number 8,9
PHYSICS LETTERS A
18 May 1987
eter space (~,v). Double-pulse honioclinic orbits occur when
point of (3)) elsewhere. In the second case, A> 1, there is a similar bifurcation structure in e>0 as
0=v—Av’~, v>0.
shown in fig. lb. In both cases it is easy to show that these are the only bifurcations in a sufficiently small neighbourhood of the codimension-two point (~,v) = (0,0). However, outside this small neighbourhood it is possible to find regions of parameter values for which there are saddle-node bifurcations of two-pulse
Clearly this has no solutions ifA <0, so we only need to consider the case A>0. For A >0 we obtain the solution VV
(4)
2HA.
Since v is small and positive, this gives a curve in e>.O if A> 1, and c <0 if 0
(5)
The derivative of the map is (1 + ~)Ax~,so this stationary point period-doubles when —
l=(l+e)Ax~.
(6)
Substituting for x~in (5) we get x=v(l+~)/(2+a)
(7)
as the condition that x be a non-hyperbolic station-
ary point of (3). But t~ x=[l/A(l+e)] (from (6)) so v=vPD=A~(2+a)/(l+a)11~~~ (8) Thus the curve defined by (8) (for small a and v) gives the locus of period-doubling bifurcations for the periodic orbit of the flow corresponding to the stationary point of (3). Now note that
periodic orbits, but we shall not describe these here. For the FitzHugh—Nagamo equations this implies that whenever the situation described above arises there is a very small region in (a,c) space near (p~,c1(u~))in which there are travelling waves with velocity c near c,(a,~) which have two pulses per period. However, the bifurcation schemes described above are considerably more general, and apply to any set of ordinary differential equations or reaction—diffusion equations on a one-dimensional, infinite domain where this codimension-two situation arises.
A similar analysis can be done for pairs of homoclinic orbits which are mappedonto each other under a symmetry (as, for example, in the Lorenz equations). The description of this bifurcation, in which chaos must be avoided locally, can be found in ref. [11]. I am grateful to the SERC for support during this work.
.
log vpD
—
log ~2H=log 2— 1 + f2/24 + 0( a3) ,
which is less than zero for small a, 50 near (,v) =(0,0), VPD
References [1] J. Evans, N. Fenichel and J.A. Feroe, SIAM J. AppI. Math. 42(1982) 219. [2) J.A. Feroe, SIAM J. App!. Math. 42 (1982) 235. [3) S.P. Hastings, SIAM J. AppI. Math. 42 (1982) 247. [4] L.P. Shil’nikov Math. USSR Sb. 6 (1968) 427. [5] L.P. Shil’nikov Math. USSR Sb. 10(1970)91. [6]P. Glendinning, Homoclinic bifurcations in ordinary differential equations, in: NATO ASI Life Sciences Series, Chaos in biological systems, eds. H. Degn, A.V. Holden and L Olsen (Plenum, New York, 1987), to be published. [7] P. Gaspard, R. Kapral and 0. Nicolis, J. Stat. Phys. 35 (1984) 697. [8] P. Glendinning and C. Sparrow, J. Stat. Phys. 35 (1984) 645. [9) E. Yanagida, J. Duff. Equ. 66(1986) 243. [10] C. Tresser, Ann. Inst. H. Poincaré 40 (1984) 441. [11] P. Glendinning, Ph.D. thesis, University of Cambridge (1985).
413