homoclinic orbits in a parametrized saddle-focus system

Physica D 62 (1993) 254-262 North-Holland

Homoclinic orbits in a parametrized saddle-focus system

J o h n A . F e r o e Department o f Mathematics, Vassar College, Poughkeepsie, N Y 12601, USA

We consider a parametrized dynamical system with a saddle-focus equilibrium point and which, for one value of the parameter, has a homoclinic orbit. Conditions on the eigenvalues for the equilibrium point, together with transversality conditions, imply the existence of an infinite discrete set of parameter values for which the system has a homoclinic orbit. Such systems arise in the study of nerve axon equations where a homoclinic orbit corresponds to a finite train of nerve impulses traveling at a velocity identified by the associated parameter value.

1. Introduction

We are interested in a smooth parameter ized system of differential equations

X'=Fo(X) , 3 Fc(O)=O (1)

which for one value of the parameter , c o , has the

following conditions: l a . The eigenvalues of DFc0(0 ) are

A, -o r -+/3i, h > a > 0 a n d / 3 > 0 .

l b . One branch of the (1-dimensional) unstable manifold 0g at 0 is a homoclinic orbit.

l c . As c pass through c o the unstable manifold transversally crosses the stable manifold ~. This

situation is illustrated in fig. 1. As was originally observed by Sil 'nikov

[11,12], the presence of the homoclinic orbit for c = c o implies the existence of a large family of periodic and nonperiodic bounded orbits in a ne ighborhood of the homoclinic orbit. The central issue here is the characterization of parameter values near c o for which the system also has a homoclinic orbit. As illustrated in fig. 1 for c > c 0, there are values of c such that the branch of the unstable manifold labeled ~ ÷ leaves 0,

loops through phase space, returns near 0, and then follows a trajectory close to its original loop. The homoclinic orbits identified here cor- respond to such trajectories that return to 0 after two or more such loops.

Orbits consisting of two loops were shown to exist by several investigators, including Evans et al. [2], Gaspard [6,7], Glendinning and Sparrow [8], and Tresser [13]. Such analyses immediately imply orbits of 2 n loops. This paper extends the author ' s more general approach in [4], which shows the existence of an arbitrary number of loops. A similar approach is taken by Wang [14].

Because of the sensitive dependence of this system on the pa ramete r c, one has difficulty observing the associated homoclinic orbits either numerically or in many of the physical problems governed by such a system. It is worth noting therefore , that such orbits do arise in a stable, and therefore observable, way in the study of nerve impulse equations. Section 2 offers a brief presentat ion of that setting. The section also expands the range of applications covered in these proceedings. The subsequent t reatment of the general problem is not dependent on the particulars of nerve impulse equations.

Section 3 states the main result. In section 4 we sketch the argument that this problem can be reduced to the question of locating the zeros of a

0167-2789/93/$06.00 (~) 1993- Elsevier Science Publishers B.V. All rights reserved

J.A. Feroe / Homoclinic orbits in a parametrized saddle-focus system 255

3 iu -

Fig. 1. The stable and unstable manifolds for c < 0, c = 0, and c > 0.

recursively defined family of real valued functions of a single real variable. Section 5 discusses the reduced problem.

A nerve impulse is a voltage disturbance that arises from and returns to the rest state and travels along the axon with constant velocity c and unchanging shape. This corresponds to a traveling wave solution of (2) with

2. Nerve axon equations

Nerve axon equations are reaction-diffusion equations of the form

020 OD Ox z = ~ - + f ( v , f ) ,

Off at = ( , (v , ff ) , (2)

where v ( x , t) and ff(x, t) take values in R and R" respectively, n-> 1. These equations model the t ransmembrane voltage v along the length x of a nerve axon at time t. The components of f are referred to as recovery variables and their number and physiological interpretation vary with the particular model. In the Hodgkin-Huxley model , n = 3 and the componen t sa r e related to the conductance of the membrane to sodium and potassium ions. The simpler F i tzHugh-Nagumo equations have n = 1. See [5] for a survey of nerve axon models. Physiological considerations require the existence of a constant stable rest solution which we can take to be (v, ~5) = (0, 0).

We will limit ourselves here to the case n = 1 to avoid technical complications needed to ex- tend the results. See [2,4] for a t reatment of the more general case.

( v ( x , t ) , w ( x , t)) = ( v ( x + ct, 0), w ( x + ct, 0))

def = w o ( z ) )

for z = x + ct and such that limz__,+_=(v¢, w~)-~ (0, 0). Thus a nerve impulse is a homoclinic solution of the traveling wave equations

v: = cv" + f (vc , w¢) ,

cw: = g(vc , w¢) .

For X = (v ' , v~, w~) t (the superscript t indicates transpose) the traveling wave equations can be written as the 3-dimensional system

x ' = Fc(X).

Evans [1] has shown that the exponential stability of the rest state as a solution of (2) implies that DFc(0 ) has one positive eigenvalue and the rest have negative real parts. Thus the existence of an homoclinic solution is equivalent to the existence of a velocity value c such that one of the two branches of the 1-dimensional unstable manifold 0// for 0 lies in the stable manifold b °. In nerve models this is a possibility for only one of the two branches, the branch we designate as q/+.

256 J.A. Feroe / Homoclinic orbits in a parametrized saddle-focus system

Thus the neurophysiologically reasonable re- quirements of a stable rest state and the existence of a nerve impulse give us eq. (1) with condition (la). There is no physiological basis to expect conditions (lb) and (lc), although they are met by versions of the Fi tzHugh-Nagumo equations [3,10].

If we think of the homoclinic orbit for c = c o as a single nerve impulse, then the associated homoclinic orbits for values of c near c o corre- spond to solutions of (2) that identify a finite train of two or more approximate copies of the single impulse. Some of these wave forms are stable solutions of (2) [3,15,16], and so appropri- ate initial conditions to (2) evolve to multiple impulse solutions. This is the sense in which the associated homoclinic orbits arise in the stable observable way referred to above.

3. Statement of the main result

We need to define a family of vectors with integer coordinates to use as an indexing set for the parameter values.

D e f i n i t i o n . For p > 1 and k ~ Z+ define

Fk(p ) = { ( n , , . . . , rig): n i ~ i e +,

n i _ l / p < n, < P n l }

and

1"(o) =UG(p). k

T h e o r e m . For eqs. (1) with conditions ( l a ) - ( l c ) and p with 1 < p < A/a there exists a family of distinct parameter values c indexed by F ( p ) for which (1) has a homoclinic orbit.

A proof of this result appears in [4]. The following analysis gives a simplified approach for F(p) , and indicates the basis for conjecturing the extension of the result to a larger indexing set r * ( p ) .

The indices carry information about the associated orbits. In rough terms, the orbit indexed by (n~ . . . . , n k ) ~ F k ( P ) consists of k + 1 similar loops through phase space, with each loop re- turning near the equilibrium point. The value of the component n i corresponds to the time the orbit spends near the equilibrium point between the ( i - 1)st and ith loops. An increase of one unit in n i corresponds to an increase of that time by approximately ~r//3, that is, half the period of the rotation in the linearization of the stable manifold. Thus in the nerve model the components ni identify the spacing between succes- sive pulses. A more complete analysis of the information carried by the indices appears in [4].

The indexing set F ( p ) does not provide a complete itemization of the homoclinic orbits, although one can uniquely associate any homoclinic orbit with c sufficiently near c o with some n-tuple of positive integers. On the other hand, not every n-tuple indexes a homoclinic orbit. The central problem then is to characterize com- pletely the set of indices for homoclinic orbits. The following discussion conjectures the extension of F(p) to an indexing set to a significantly larger, but still incomplete, set F * ( p ) .

De f in i t i on . Define the binary operation*:Zk x 7/--. Z ~+t+kt by

ff * fit = (1~, m~, ~, m 2, ~, . . . , m I, 1~)

= ( n l , . . . , n k , m l , n l , . . . , n k , . . . , m l ,

n I , . . . , n k ) •

Def in i t i on . Define ,~, 5~: zk-->Z by

~ ( r t ) = n I and ~( f f ) = n k .

De f in i t i on . Define

F*(p) = { r ~ * f f E * . - - , f f / j ~ Z + ,

~, ~ r(p), ~(~,)p > ~ q , ~ i _ , ) } .

The indexing set F * ( p ) is well defined since the binary operation * is associative.


The basis for conjecturing that F*(p) indexes homoclinic orbits arises from the analysis for F(p). Each homoclinic orbit indexed by F(p) can be viewed as multiple loops through phase space, with each loop approximating the trajectory of a homoclinic orbit that is thought of as the base orbit. Since we can take any homoclinic orbit to be a base orbit, each homoclinic orbit gives rise to its own associated family of homoclinic orbits. An index of the form if* rh ~ F*(p) corresponds to a homoclinic orbit for which the base orbit is indexed by ri and which relative to that base would be indexed by nq. Recursive application leads to indices of the form

J~l ~ /~2 ~ . • • ~ /~j.

4. Reduction of the problem

Fig. 2 shows the dynamical system with homoclinic orbit for c = c 0. A cylindrical region P isolates the equilibrium point. The unstable manifold 07/+ leaves P through a planar section labeled T O P and re-enters through the section labeled SIDE. By rescaling and rotating we can assume that in cylindrical coordinates R × C,

P = {(x, r e~°): [xl-< 1, I r l~ 1}

with 0 - / / + A T O P = ( 1 , 0 ) and a / / ÷ N S I D E =

(0, e °') = (0, 1).

The strategy for analyzing such a system is to view the dynamics inside P as the linear constant coefficient system given by the linearization about the equilibrium point. This simplification of the analysis can be made rigorous through appeal to standard linearization theorems [9]. In the linear system the trajectory for initial condition (x, e i ° ) ~ S I D E is (xe~t,e i° e (-~+~)') and since x e At = 1 at t = In x -I/A the flow maps

SIDE--> T O P by

(X, e i ° ) ~ (1, X (a-~i ) /A e i°) .

Now choose the local complex coordinates for SIDE that associates the point (x, e i°) ~ S I D E with the coordinates x + i0. Similarly, take the local complex coordinates for TO P that identifies (1, e i°) E T O P with e i°. The mapping from TOP to SIDE induced by the flow is a diffeomorph- ism. As part of our simplification of the problem, we take this mapping to be the identity (in the local coordinates). Thus for c = c o the Poincar6 mapping SIDE--~SIDE in local coordinates is x + i0 ~--~x (~-t3i)/x e i°.

The next simplification assumes that for c near co the only changes in the system are that, in local coordinates, o-//÷ n SIDE = c - c 0, and the mapping T O P - - S I D E is the translation r e i° ~ (c - Co) + r e i°. This satisfies condition (lc) . The mapping SIDE---~ SIDE is thus

x + i0 ~-~ (c - Co) + x (~-'i)/~ el0 .

Fig. 2. The t ra jectory of the unstable manifold for c = c o, with the equi l ibr ium point enclosed in the region P.

The final simplification is to eliminate the ef- fect of 0 in this mapping by making

x + i0 ~ (c - Co) + x <~-~i)/~ .

This is reasonable for a bounded set of values of 0 since the feature of interest in this mapping is the logarithmic spiral image of the line with constant 0, not the rotation of the spiral that results from changes in 0.

For c - c o small and positive, the trajectory of 0//÷ leaves P through TOP, re-enters at SIDE


near S¢, spirals up through TOP a second time near its first intersection with TOP, and so is carried near its previous path outside of P to a point in SIDE near its first intersection with SIDE. Because of the spiraling inside P, that second intersection with SIDE may be on 5e, in which case the orbit is homoclinic. Thus the system has a homoclinic orbit for any value of c for which the kth intersection of the trajectory with SIDE is a point whose local coordinates has zero real part. Let x = c - Co, and define the complex valued function fk(X) to be the local coordinates of the kth intersection of 9/+ and SIDE. By the above,

f l ( x ) = x ,

f2(x) = X + X ('~-t3i)/A ,

f3(x) = x + [Re f2(x)] ("-ai)/x if 0 < Re f l (x) < 1 ,

f k (x ) = x + [Re fk_,(x)] ("-t3i)/x

i f0 < Re fj(x) < 1 f o r j = l , . . . , k - 1 ,

and the problem has been reduced to studying the zeros of Re fk(X).

5. Investigation of the reduced problem

To simplify the following calculations, we assume that h = 1, or equivalently, substitute a, fl for a / A , / 3 / h . Select p so that l < p < l / a .

First, partition (0, 1] by the intervals

I(m ) ----" (e -m~/:, e -(m-1)~/~] for rn E Z + .

A ( k - 1)-tuple of integers (n 1, n 2 , . . . , nk_ l ) that indexes a zero of Re fk will be such that

Re f / (x (~ : : ...... k-l)) E I(,,,)

f o r i = 1 . . . . , k - 1 .

We approach this problem by considering the functions f l , f2, f3, and f4 in turn. At that point some generalizations and speculations are possible.

Zeros o f Re fl : Trivially, Re fl (x) = x = 0 only at x = 0 .

Zeros o f Re f2 : The function f2 is given by

f2(x) = x + [Re fl(x)] "-~i = x + x "-~i

and the image f2([0, 1]) in the complex plane is shown in fig. 3 (all figures are actual calculations for the case a = 0.5 and/3 = 5).

-0.~

0.5i-

-0.5i

x (nl) . . . . I

1(~11

fl((0, 1])

fl(/(n~))

1 -0.5

0.5i~ " /

Fig. 3. The images of (0, 1] and a representative interval 1(,1)under f~ and f2.

f2((O, 1])


Each half rotation in the upper or lower half- plane is the image of I(. 0 for some n I ~ Z +. Since a zero of Re f2 corresponds to a crossing of the imaginary axis, it is clear from the figure that for each n~ E Z +, there is a unique zero X(.l) I(. O. The analytical verification of this is that the values of f2 at the endpoints of I(.1) are

f z ( e - " : n3) = e-- : / t~ + ( - 1 ) "I e - -n : "o

f2(e-(nl -D~/O) = e-(.~-l)~/a

+ ( - - 1 ) n1-1 e-a(nl-1)~r//3

Because of the image f2((0, 1]) spirals, the subin- tervals ~-i) alternate between having x(.x) as its right or left endpoint. Fig. 4 shows the images of one such ~-1) under f2 and f3-

If it exists, let I(-1,n2) be the subinterval of I(. 0 such that Re f2(I(.~,.2))= I(.2). As seen in the figure, each half-rotation in the spiral image f3(~.1)) is the image f~(I(.1,.2)) for some n z. Moreover , the spiral f3(I(.o) is centered at x(. 0 since

f3(x) = x + [Re fz(x)]"-t3i

and since for any n > 0 the condition that 0 < a < 1 gives us that e - ~ n " / ~ > e - ' ' /~ , the two values of f2 differ in sign. The intermediate value theorem gives the existence of the zero x(.1) 1(.1). The zero is unique since Re fz is monotonic o n I(nl).

Thus there are countably many zeros of f2 indexed by Z + = Fl(p) . We limit our attention to values of n~ with n a > N where N is an integer chosen large enough to satisfy conditions that are developed below.

Zeros o f Re f3 : The function f3 is defined on the subinterval of I(. 0 given by

I(.1) = {x ~ I(-o: Re fz(x) --> 0}.

and

lira Re f2(x) = O . x-"X(nl)

The values of n 2 for which 1(. 1,.z) exists include those satisfying

n 2 > nil p .

This follows since

e -(n2-1)~'# < e -' ': 'It3 < max R e f2 (~n l ) ) .

The restriction that n~ > N is needed here for N sufficiently large (N > p/(1 -po t ) will do).

Because the spiral image f3(l( .0) crosses the

-0.~

0.5i

-0.5i

\

/(-~) -,~ ,',~ -/t~

e

I:(i(.,)) .~ff( ..... ))

/•05i• ~ f3(i("l))

I ! ~ ~ fa(l( ..... ))

, \ , ~ x(,,~)

Fig. 4. The images of representative intervals I(,1) and l(nl.n2 ) under f2 and f3.


imaginary axis only finitely many times, there are only finitely many zeros of Re f3 in 1(.1). There will be a zero of Re f~ in the interval I(.~,.:)C I~.,) if one of its endpoints is sent to a negative value by f3. We claim that this is the case for those nz satisfying

n 2 < pnx

since for such an n 2 we have that e-~"z=/~> e -~=;~, which in turn implies that f3 will be negative at one of the two endpoints of I~.1,.~ ). The requirement that n I > N for sufficiently large N is needed here as well ( N > p / ( p - 1 )

will do). Thus there is a zero x(. , , .~) of Re f3 for n~ > N

and

n~/p < n 2 < pn~ . (3)

These inequalities give that (n~, n2)~ Fz(p).

Z e r o s o f Re f4: The picture becomes somewhat more complicated with f4. Zeros of Re f4 are indexed by triples (n~, n 2, rt3). For a given value of n~, the possible values of n 2 continue to be bounded below by the inequality n e > n l / p as given in condition (3). But since the upper bound on n2 in condition (3) is there to assure a zero of Re f3, we look for zeros x(n~,.~,.~) of Re f4 in two separate cases: those for which n~ < p n l and those for which n 2 >- pn~. ~

Consistent with the definition of I(.n, let I'(~,~) be the values in I(.~,.~) in the domain off4. Also, we will now limit our attention to n2 with n z > N .

C A S E i : n 2 ~ p n ~ . In this case, the image fa(~;,~z)) shown in fig, 5 looks very much like the image f 3 ( ~ ) ) in fig. 4.

The analysis in this case closely follows the analysis of f3- The image f4(l(~,~)) crosses the imaginary axis only finitely many times as its spirals in towards its center, which is x(.~,~). Each half rotation in that spiral is the image of the subinterval of I(~,n~) that Re f3 sends to some I(~). Denote each such subinterval by I(n~,ne,n3).

¢ ¢ . J r - .

tO "N

I : .y."" ~ S

.5

v

.... X

I

y "

ol)

F

/ /.

i /

/

t \ d

}?

I

d,


The possible values of n 3 include all those satisfying n 3 > n2/p since x + IXC~-'flil > e -""2~/t3 for all

X ~. 1(. 1,.2). There will be a zero of Re f4 in the interval

I(.1..2,.3) if one of its endpoints is sent to a negative value by f4- We claim that this is the case for those n 3 satisfying

n21p < n 3 < pn I . (4)

The lower bound was established in the previous paragraph. The upper bound implies that e-~"3~/o> e -("1-1)~/t3 and so f4(x) will be negative when x is one of the two endpoints of

I(n ,n2,n3)" We can generalize conditions (3) and (4) to

claim for any integer k > 2 that Re fk has a zero indexed by any (nt, n 2 . . . . . n k _ l ) E F ~ _ l ( p ) w h e r e n ~ > N f o r i = l . . . . . k - 1 .

C A S E 2: n e->pn~. In this case, the image f3(I(.,,.2)) shown in fig. 6 does not cross the imaginary axis, but it is still a half rotation of the spiral image f3(I(.0) centered at x(. 0. As is the case in the figure, for n2 large enough, the set Re f3(l(nt)) is a subset of 1(,1). We claim that even though the image f4(I(,1,,2)) is not a half rotation, it still will cross the imaginary axis, and thus there will be a zero of Re f4 indexed by (nl , n2, n ,) .

For the analytic justification for this claim, we approximate the mapping Re f2 on 1(,1) by the linear mapping L 2 for which

L2: ~"1) o.to) R e f 2 ( ~ n l ) ) •

We can write this mapping as

L 2 ( x ) = c . , ( x - x(.1) )

where the only observation we need about c,~ is , that c,1 > 1.

We can use L 2 to estimate Re f3(I(,1.,2)). First, estimate the endpoints of 1(,1,,2 ) as

L z ' ( e - " 2 È/tj) = x(. 0 + e-"2"/t~/c.~,

L21(e-(n2-1)~/~) = X(,l) + e-("2-~)=/~/c,,.

The images under f3 of those endpoints are

f3(x(.,) + e-"2=/~/c.)

m X(nl ) -~ e-n2~r/g/Cnl --i- e -an2 ~r/13 ,

.f3(X(nl) + e-(nz-1)~r/O/Cnl )

= X(nl) q- e-(n2-1)~r/#/Cnl -~ e-a(n2-1)~r/l 3 ,

which, since, c,1 > 1, are on opposite sides of x(,1). Thus there is a zero of Re f4 indexed by (n 1 , n 2, n ,) .

/ 0 . 5 i

/

-0°

f~(i(.1))

x(,~l)

0.5i

/

-0.5i

Fig. 6. The images of representative intervals 1(.1) and 1(.1,.2 ) for n 2 1> pn~ under f3 and f4.


The index (n~, n2, r t l ) is not in F(p) , but it is in F * ( p ) since we can write it as (nl)* (nz). The basis for the extension of the main result to all of F*(p) is to treat the homoclinic orbit indexed by (na) as a new base homoclinic orbit which has an associated homoclinic orbit indexed by (n2). In the language of nerve impulse solutions, (n~, n 2, nl) is a quadruple pulse train composed of two almost identical double pulse trains. More generally, any homoclinic orbit indexed by ri F(p) could serve as the base orbit, so that associated orbits indexed by rh E F(p) relative to the new base would be indexed by t~ * n] relative to the original base. The condition that ~(rfi) × p > LP(ri) puts the lower bound on the first (and therefore subsequent) coordinates of th that is inherited from the final position of tL

Finally, although the treatment of the reduced problem required that the components of an index ( n ~ , . . . , nk) be bounded below by N, we can remove this restriction by associating any (n~ . . . . . nk) E F*(p) with the zero found above indexed by (n I + N . . . . . n k + N) , which is also in F*(p) .

References

[1] J.W. Evans, Indiana Univ. Math. J. 22 (1972) 75. [2] J.W. Evans, N. Fenichel and J. Feroe, SIAM J. Appl.

Math. 42 (1982) 219. [3] J.A. Feroe, SIAM J. Appl. Math. 42 (1982) 235. [4] J.A. Feroe, SIAM J. Appl. Math. 46 (1986) 1079. [5] R. FitzHugh, Mathematical models of excitation and

propagation in nerve, in: Biological Engineering, H.P. Schwan, Ed., (McGraw-Hill, New York, 1969), Ch. 1, pp. 1-85.

[6] P. Gaspard, M6canismes d'appartition de comporte- ments chaotiques dans les syst6mes dynamiques dis- sipatifs, M6moire de Licence, Universit6 de BruxeUes (1982).

[7] P. Gaspard, Phys. Lett. A 97 (1983) 1. [8] P. Glendinning P. and C. Sparrow, J. Stat. Phys. 35

(1984) 645. [9] P. Hartman, Ordinary Differential Equations (Wiley,

New York, 1969). [10] S.P. Hastings, SIAM J. Appl. Math. 42 (1982) 247. [11] L.P. Sil'nikov, Soviet Math. Dokl. 6 (1965) 1 63. [12] L.P. Sil'nikov, Math. USSR-Sp. 10 (1970) 91. [13] C. Tresser, Ann. Inst. H. Poincar6 40 (1984) 440. [14] W.-P. Wang, Commun. Pure Appl. Math. 41 (1988) 71. [15] W.-P. Wang, Commun. Pure Appi. Math. 41 (1988) 997. [16] E. Yanagida and K. Maginu, SIAM J. Appl. Math. 49

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homoclinic orbits in a parametrized saddle-focus system

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