EXISTENCE AND STABILITY OF ASYMPTOTICALLY ...

EXISTENCE AND STABILITY OF ASYMPTOTICALLY OSCILLATORY DOUBLE PULSES

J. C. Alexander† and C. K. R. T. Jones‡ Department of Mathematics University of Maryland College Park, Maryland 20742

June, 1991 Revised: October, 1992 To be published in Crelles Journal f¨ ur die reine und angewandte Mathematik

†

Research partially supported by the National Science Foundation under grant DMS-

90-01788. ‡ Research partially supported by the National Science Foundation under grant DMS88-01627

EXISTENCE AND STABILITY OF ASYMPTOTICALLY OSCILLATORY DOUBLE PULSES J. C. Alexander† and C. K. R. T. Jones‡ Department of Mathematics University of Maryland College Park, Maryland 20742

1. Introduction and statement of results. We are concerned with the existence and stability of travelling-wave solutions of semilinear parabolic systems on a line: ∂2U ∂U = B 2 + F (U ), ∂t ∂x

(1.1)

where U ∈ IRn and x, t ∈ IR are space and time respectively, B is a positive diagonal n × n matrix of diffusion coefficients, and F is C 2 with derivatives through order two bounded on IRn . Systems (1.1) appear in a variety of models of transport and diffusion phenomena, for examples, systems modelled by cable equations (including physiological models), and in population biology of species distributed over the line. Perhaps the most important nontrivial solutions of (1.1) are the travelling waves U (ξ) where ξ = x − ct for some wave speed c. Many observed solutions are travelling waves or asymptotically tend to travelling waves. In a travelling-wave solution, the disturbance propagates uniformly along the line with no change in shape. Travelling waves can effect the change of the state from one stationary point at ξ = −∞ to another one at ξ = ∞. For the present paper, we consider travelling wave solutions which decay exponentially to the same constant state: U (ξ) → U as ξ → ±∞,

(1.2)

where F (U ) = 0. We call such solutions pulse solutions. Some authors reserve the terminology pulse solution for a travelling-wave solution with a sharp peak; however we use the term more generally. If (1.1) models, say, a predator-prey dynamics on the line, a pulse solution has the following behavior at any point x. It remains near a stationary solution for a long †

Research partially supported by the National Science Foundation under grant DMS-

90-01788. ‡ Research partially supported by the National Science Foundation under grant DMS88-01627.

Alexander & Jones: Asymptotically oscillatory double pulses

2

time. Then there is a large excursion in the populations—the pulse (presumably an increase in the prey followed by an increase in the predator)—and then a decay to the same stationary value. Under certain conditions on a pulse solution (asymptotically oscillatory), a system (1.1) can admit solutions which look like two copies of the pulse solution widely separated in time (or space). These are the double pulse solutions of the title. Indeed, under appropriate conditions, there are a countably infinite number of them, differing mostly in the amount of separation between the pulses. In general, only stable travelling-wave solutions of (1.1)–(1.2) model realizable pheˆ (x−ct) nomena. Let k·k denote the L∞ (sup) norm on vectors in IRn . A pulse solution U of (1.1)–(1.2) is stable if there exists ζ > 0 so that if U (x, t) is any solution of (1.1) with ˆ (x)k < ζ, then there exists a constant k such that kU (x, 0) − U ˆ (x − ct + k)k = 0 lim kU (x, t) − U

t→∞

(1.3)

It happens that if the original pulse solution is stable, the double-pulse solutions are alternately stable and unstable. In earlier work, Evans, Fenichel and Feroe [1982] considered nerve-axon equations, which are systems (1.1) with all but one entry in B equal to 0 (that is, one parabolic partial differential equation with some subsidiary ordinary differential equations). They showed the existence of double-pulse solutions under certain conditions. Yanagida and Maginu [1989] solved the question of stability. Double-pulse solutions correspond to homoclinic solutions of a certain autonomous ordinary differential system, and the technique is to control the stable and unstable manifolds of the stationary point. In the case of nerve-axon equations, the unstable manifold is one-dimensional. In this paper, we develop the machinery to establish this for the system (1.1) with U two-dimensional, which is the minimal dimension for which the phenomenon occurs. In this case the unstable and stable manifolds are two-dimensional, and it is necessary to utilize the machinery of Alexander, Gardner and Jones [1990]. From one point of view, this paper can be considered a demonstration of that paper’s techniques. Suppose U (ξ) is a pulse solution of (1.1)–(1.2). A double pulse solution of (1.1)–(1.2) at distance η from U with delay δ is a pulse solution U2 (x − cˆt) which satisfies sup kU (x − ct) + U (x − ct − δ) − U2 (x − cˆt) − U k = η

(1.4)

x,t

To within η, a double pulse solution looks like two copies of U separated by distance δ travelling at almost the same speed as U . Insert Figure 1 near here.


3

The main object of study in this paper is double pulse solutions of a particular type of two-variable system. See Figure 1. Consider two coupled C 2 reaction-diffusion equations ut = b1 uxx + f (u, w),

(1.5)

wt = b2 wxx + g(u, w).

Here f and g are C 2 and the diffusion coefficients b1 , b2 are both strictly positive. We ¡ ¢ consider pulse solutions u(ξ), w(ξ) with lim

¡

ξ→±∞

¢ u(ξ), w(ξ) = (u, w).

(1.6)

A pulse solution of (1.5)–(1.6) is positively asymptotically oscillatory if there exists a decay constant r > 0, frequency θ and phase ν such that ° µ ¶ µ ¶° ° rξ u(ξ) − u cos(θξ − ν) ° °e °→0 −A ° w(ξ) − w sin(θξ − ν) °

(1.7)

as ξ → ∞ for some nonsingular 2 × 2 matrix A. The decay constant and frequency are uniquely defined. Negatively asymptotically oscillatory is defined similarly, and a solution is asymptotically oscillatory if it is both positively and negatively asymptotically oscillatory (with possibly different decay constants, frequencies and phases at ±∞). ¡ ¢ Suppose u(ξ), w(ξ) is an oscillatory pulse with decay constants and frequencies r1 and θ1 at +∞ and r2 and θ2 at −∞. Suppose r1 6= r2 ; let p be the half-period corresponding to the slower decay constant. That is ½ p=

π/θ1 , π/θ2 , ½

Also let q = p max{r1 , r2 } =

if r1 < r2 , if r2 < r1 .

(1.8)

πr2 /θ1 , πr1 /θ2 ,

(1.9)

if r1 < r2 , if r2 < r1 .

We collect our major results in the following omnibus theorem. In section 2, the concept of a transversally constructed pulse is defined. Basically it means that the wave can be constructed by a general-position argument. Theorem 1.1. Suppose (u, w) is an exponentially stable solution of (1.5). Suppose ¢ ¡ u(x − c0 t), w(x − c0 t) is a stable transversally constructed asymptotically oscillatory pulse solution of (1.5)–(1.6), with decay constants and frequencies r1 and θ1 at +∞ and r2 and θ2 at −∞. Suppose r1 6= r2 . Then there exists a sequence of wave speeds cn , n = 1, 2, . . . , |cn − c0 | ≤ Ce−nq ,

(1.10)


4 (n)

for some constant C with q as in (1.9), for which there exist double pulse solutions U2 . (n)

These double pulses, are asymptotically oscillatory; if ri decay constants and frequencies of

(n) U2 ,

(n)

|θi −

are the asymptotic

then

|ri − ri | < Cr e−nq , (n) θi |

(n)

and θi

−nq

< Cθ e

,

i = 1, 2,

(1.11)

i = 1, 2.

(1.12) (n)

for some constants Cr and Cθ . Also if ηn is the distance (1.4) from U2

to U , then

ηn < Cη e−nq

(1.13)

for some constant Cη . Asymptotically, the delays δn of the solutions corresponding to (n)

(cn , U2 ) obey δn+1 − δn = p, (n)

where p is the half-period (1.8). The U2

(1.14)

are the only double-pulse solutions close to U

in the sense of (1.4). Finally, the solutions are alternately (in n) stable and unstable. In section 2, an index is defined which detects stability and the final statement of the theorem is sharpened in section 4. In addition to being asymptotically oscillatory, the two pulses of such a doublepulse solution are separated by oscillations. The index n essentially counts the number of half-oscillations between the two pulses. If r1 = r2 , double pulses may also exist, but details depend on higher-order information. In the case of nerve-axon equations, the cn of the double pulses approach c0 monotonically from one side. In the present case, monotonicity in c does not occur except for certain discrete values of θ1 /θ2 . However, there is a phase monotonicity, which it is possible to use to differentiate between the stable and unstable double pulses in terms of their behavior. For large negative ξ (thus large negative x or large positive t), the original pulse behaves like a growing sinusoid in its variables, as do the double pulses. The oscillations of the stable double pulses have slightly larger (ratio 1 + Ce−nq ) humps than the original pulse, whereas the unstable double pulses have slightly smaller humps. The situation is reversed at large positive ξ. If the original pulse is oscillatory at only one end, generically there may or may not exist nearby double pulses depending on the sizes of r1 and r2 . The existence reduces to the same case of Evans, Fenichel and Feroe [1982], and the stability can be determined by the methods of this paper. If the original pulse is not oscillatory at either end, nearby double pulses generically do not exist, by Evans, Fenichel and Feroe [1982].


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It is possible to effectively determine the asymptotic oscillatory behavior of a pulse from the behavior of f and g at (u, w). In case b1 = b2 , the criterion is simple to state. µ

Let ∆= Let T =

1 2

fu gu

¶¯ fw ¯¯ . gw ¯ u=u w=w

tr ∆ be half the trace of the Jacobian matrix ∆ (which must be negative is a

pulse is stable), and D = det ∆ the determinant. Suppose there is a stable pulse solution of (1.5)–(1.6). Proposition 1.2. Suppose b1 = b2 in (1.5), and T 2 < D.

(1.15)

Then any pulse solution of (1.5)–(1.6) is asymptotically oscillatory. Note that (1.15) implies that fv and gu have opposite signs; in particular the conditions are satisfied for predator-prey systems for which the interactive terms are large compared to the diagonal terms at (u, w). Existence is established by studying the dynamics near a certain homoclinic solution of a four-variable ordinary differential system. The dynamics near such a homoclinic ˇ in IR4 was also studied by Silnikov [1967] (see also Wiggens [1988]); he showed the existence of horseshoes in the dynamics. The authors also recently became aware of Glendinning [1989], which discusses such dynamics and overlaps with our section 3. Stability is established by linearizing and computing the spectrum of the linearization. Sattinger [1976] first proved that linear stability implies stability in the sense of (1.3) for travelling-wave solutions of some parabolic systems. This work was extended by Henry [1981]. Evans [1972], Evans [1975] proved that linear stability implies stability for travelling-wave solutions of nerve-axon systems. In these papers, techniques to compute the spectrum of the linearized system of nerve-axon equations were also developed. They were refined by Jones [1984] to establish stability of the FitzHugh-Nagumo pulse (see also Yanagida [1985]), and then developed into a systematic procedure for studying stability of (1.1) in Alexander, Gardner and Jones [1990]. Using certain limiting processes, it is possible to permit some of the bi in (1.1) to be zero; see Alexander, Gardner and Jones [1990], Gardner and Jones [1991]. We remark that with our methods, showing existence and determining stability are intrinsically related. Although sophisticated machinery is needed, the knowledge required for existence is the knowledge required for stability. Of course, in some philosophical sense, the stability of a solution, as one of the properties of the solution, is accessible once existence is proved. However, we mean a more concrete relation. The


6

existence argument is geometric, and the geometric invariants arising in the construction are the ones used to characterize stability. An analogy might be drawn with degree theory. Existence (say of a fixed-point) might be shown by determining that a degree is nonzero; at the same time the sign of that degree contains information about the stability of the solution. For example, sometimes this kind of argument shows that stable and unstable solutions pair off in saddle-node pairs. In fact, this analogy is quite close to our situation, especially if one thinks of degree as the orientation of some transversal intersection. Indeed, we show existence via a transversality argument (transversal intersection of stable and unstable manifolds) and the orientation of that intersection (which we call the orientation index), combined with a certain winding number (another kind of degree), characterizes the stability of the solution. In section 2, we develop some general concepts about pulse solutions. Here transversal intersections are studied and the orientation index is defined. In section 3 of doublepulse solutions under the conditions of the theorem is established. It is proved that such transversal intersections obtain. In section 4, stability is established by showing the orientation index characterizes stability and the remarks about monotonicity after the theorem are proved. The proposition is proved in section 5. 2. Transversal constructions, orientation indices and Evans functions. In this section, we work in more generality than in later sections. Some of the notation is unique to this section. We consider a more general parabolic system in variables ui (x, t), i = 1, . . . , h:

³n ∂ l u o ´ ∂ui ∂ ri u i i = bi ri + fi . i ∂t ∂x ∂xl 0≤l 0. This proves Lemma 4.2. We turn to the zeroes of Dn (λ), and count them by a winding-number argument. Lemma 4.3. For large n each Dn (λ) has precisely two zeroes, counting multiplicity, in the region { − 12 β} Proof. Let K ◦ ⊂ C be the set {λ : |λ| ≤ 2M, −ξ 2 }, B± = K × {±∞} × [−1, 1],

outer seams,

B∩ = K × {(ξ, −ξ 2 ) : −1 ≤ ξ ≤ 1},

inseams.

The “bottom” of B (y = −1) consists of two disjoint copies of S 2 ; over the bottom we put two copies of E0 . The “top” of B (y = 1) consists of one copy of S 2 ; over the top we put a copy of En . Since the bundles are close, we can fill in a bundle over the rest of B. This gives us a cobordism of bundles (Conner and Floyd [1964]) between En and 2E0 . Chern numbers are additive over the disjoint union on the bottom and are cobordism invariants; hence (4.7) is proved. In more detail: Consider the values ξs and ξu where γn first intersects the stable torus Ts (3.8) and last intersects the unstable torus Tu , respectively. We can suppose ξs < 0 < ξu . Consider two copies of [−∞, ∞]; denote them [−∞, ∞]1 and [−∞, ∞]2 , with coordinates ξ1 and ξ2 . These are the coordinates for the two copies of the single pulse. We need to recoordinatize so they fit on [−∞, 0] and [0, ∞], respectively. Define π : [−∞, ∞]1 ∪ [−∞, ∞]2 → [−∞, ∞] as follows ( ξ1 if −∞ ≤ ξ1 ≤ ξs ξs π(ξ1 ) = if ξ1 > ξs , 1 + ξ12 ( ξu if ξ2 < ξu , π(ξ2 ) = 1 + ξ22 if ξu ≤ ξ2 ≤ ∞ ξ2 The function 1/(1+ξ 2 ) is a smooth monotonically decreasing function which takes [0, ∞] to [1, 0]. By construction, the unstable bundles are bundles induced by Gauss maps (“classifying maps”) of S 2 to G2,4 (C), the Grassmannian of 2-dimensional complex subspaces of C 4 . Denote the classifying maps for E0 and En by e0 : S 2 → G2,4 (C) and en : S 2 → G2,4 (C), respectively. The first pulse of the orbit γn is uniformly close to γ; suppose γn and a copy of γ, denoted γ (1) , are parametrized so that γn (ξ) is uniformly (1)

close to γ (1) (ξ) for ξ ≤ ξs . Thus so are the associated operators L0 and Ln . Hence the (1)

classifying maps e0 (λ, ξ) and en (λ, ξ) are also uniformly close for ξ ≤ ξs , λ ∈ K. Since (1)

there are no eigenvalues of L0

(1)

on K, the fibers Φ− (λ, ξs ) of E0

approach Φ− (λ, −∞) (1)

uniformly for λ ∈ K, and we can thus assume the fibers Φ− (λ, ξs ) of E0 close to Φ− (λ, −∞). Since

(1) e0 (λ, ξ)

are uniformly

and en (λ, ξ) are uniformly close for ξ = ξs , λ ∈ K,

we may assume the fibers Φ− (λ, ξs ) of En are uniformly close to Φ− (λ, −∞). Since the


27

tangent space to the unstable manifold is an attractor of the induced flow for λ ∈ K, the fibers Φ− (λ, ξ) of En are uniformly close to Φ− (λ, −∞) for ξs ≤ ξ ≤ ξu . Finally we can take another copy of γ, denoted γ (2) and parametrize it so that γn (ξ) and γ (2) (ξ) are uniformly close for ξ ≥ ξu ; and hence the maps en (λ, ξ) and e(2) (λ, ξ) are uniformly close for ξ ≥ ξu , λ ∈ K. Let us define a map e : B → G2,4 (C). On the complement of the set {0 < y < 1} ⊂ B, it is defined by  ξ),  en (λ, ¢ √ (1) ¡ e(λ, ξ, y) = e0 λ, π −1 (ξ − −y) , ¢  (2) ¡ √ e0 λ, π −1 (ξ + −y) ,

if y = 1, if ξ ≤ 0, y ≤ 0, if ξ ≥ 0, y ≤ 0.

By the previous paragraph e(λ, ξ, 0) is uniformly close to e(λ, ξ, 1). Hence by using a metric on the manifold G2,4 (C), the map e may be filled in on the set {0 < y < 1} so that the resulting bundle over B is smooth. This bundle on B forms a cobordism between 2E0 and En , as required to complete the proof of Lemma 4.3. Theorem 4.1 can now be proved. The Evans function Dn (λ) has a zero at the origin. By (4.7) and (4.4), it has one other zero in the region { − 12 β}. Since D(λ) is complex analytic, this zero has to lie on the real axis. By (4.5), this zero is in the positive half-plane if and only if D0 (0) < 0. By (2.16), the first part of Theorem 4.1 is proved. For the final part, we use (3.40). The o-index can be determined by the orientation of the intersection of F (Σs ) (with tangent vectors Vcu and Vu ) and Γ−1 c (Cs ) (with tangent (1)

(1)

vector Vs ) in Tu (with normal vectors Vcs and V ). Suppose we orient F (Σs ) with a (u) (u) vector Vt transverse to {ψ˜2 ≡ 0} = Γ−1 (Cs ) and a vector Vp tangent to {ψ˜2 ≡ 0}. The second spirals around with the logarithmic spiral of Lemma 3.5. On the other hand (1) (u) Vs , by (3.26), is horizontal in {ψ˜ ≡ 0}. The successive γn correspond to intersections 2

of F (Σs ) and Γ−1 c (Cs ) at polar angles approximately π apart. Thus Vp and Vs

(1)

intersect

in opposite orientations at successive γn . This completes the proof of Theorem 4.1 and of Theorem 1.1. Our indexing of the γn is rather arbitrary, even up to parity. Changing normal coordinates so the sign of αs changes, for example, changes the parity. However, it is possible to control the geometry sufficiently well to characterize the stable and unstable double pulses in terms of their behavior, as asserted after the statement of the main theorem. For this purpose, it is useful to consider a three-dimensional cross section at some point, say γs , of the original homoclinic γ in IR4 . We can suppose that this cross (1)

section is a small cube, and that the tangent space at γs is spanned by Vcu , Vu , and

Alexander & Jones: Asymptotically oscillatory double pulses (1)

Vs

28

(see lemma 3.2). Let Lc0 denote the oriented image of the unstable core T (Cu ) in (1)

this cross section; it is a line segment with Vu

tangent in its positive direction. For

nearby values of c, the image of the unstable core is an oriented line segment Lc very close to Lc0 . If we visualize the oriented Lc as lines with arrows pointing in the positive direction, we can speak of the beginning and end segments of Lc —the parts near the − tail and head of the arrows, respectively. Denote these L+ c and Lc , respectively. We − can think of L+ c as “coming in” towards the stable core Cs (decreasing ρ1 ) and Lc as

“going out” (increasing ρ1 ). The image of Lc under F is something like a hairpin, circling around Tu in ψ1 (as well as spiralling around Cu in ψ2 ). The two legs of the hairpin are − + the images of L+ c and Lc . By (3.15), Lc is mapped by F in the sense of increasing ψ1

and L− c in the sense of decreasing ψ1 . The image ΓF (Lc ) intersected with the cubic cross section is a number of line segments (the number increases without bound as c → c0 ); − those segments from ΓF (L+ c ) have the same orientation as Lc and those from ΓF (Lc )

the opposite orientation. By hypothesis γ has positive crossing direction ((2.14)). Hence those double pulses corresponding to the intersection of ΓF (L+ c ) with Cs has positive crossing direction, and hence are stable by Theorem 4.1, and the reverse for the others. Ã+ A point on Cu is taken to L c if and only if its ψ1 coordinate is (slightly) smaller than the ψ1 coordinate of γu . We consider the two homoclinics γ(ξ) and γn (ξ), and coordinates as functions of ξ. Thus ¢ ¡ ¡ ¢ ψ1 γn (ξ) = ψ1 γ(ξ) − ²n , for some ²n > 0. The state variables u and w are some linear combinations of x1 , y1 , x2 ˆ u is u and y2 . Suppose, for example, the projection of u onto W ˆ = αρ1 cos(ψ1 − χ) for some phase χ and nonzero amplitude α. Up to small errors, ¡ ¢ u ˆ γ(ξ) = αer1 ξ cos θ1 (ξ − χ), ¡ ¢ u ˆ γn (ξ) = αer1 ξ cos θ1 (ξ − χ − ²n ). ¡ ¢ Thus the maximum of u ˆ γn (ξ) at any oscillation occurs for a slightly later value of ξ ¡ ¢ than does the maximum of u ˆ γ(ξ) , and hence is larger. Also x2 and y2 are essentially zero. Thus the maximum of an oscillation is larger for u(γn ) than for u(γ), and similarly for w. This proves the characterization of stable and unstable double pulses of the introduction. 5. An example: A predator-prey system. In this section, we prove the proposition of the introduction. We consider the pair of equations on the line: ut = buxx + f (u, w), wt = bwxx + g(u, w),

(5.1)


29

as in the proposition in the introduction. Here f and g are smooth. The diffusion µ

coefficient b. Recall ∆= and that T =

1 2

fu gu

fw gw

¶ ,

tr ∆ < 0 and D = det ∆. Suppose there is a travelling-wave solution of

(5.1) with wave speed c0 . We may assume u = 0, w = 0. If we let ξ = x − ct and assume a travelling-wave solution u = u(ξ), w = w(ξ), the functions u and w satisfy the ordinary differential equations

−cu0 = bu00 + f (u, w), −cw0 = bw00 + g(u, w),

Let

u0 = v,

bv 0 = −cv − f (u, w),

w0 = y,

by 0 = −cy − g(u, w),

(5.2)

(5.3)

be the 4-dimensional first order system derived from (5.2). The linearization of (5.3) at (u, w, v, y) = (0, 0, 0, 0) is 

0  0  −fu /b −gu /b

0 0 −fw /b −gw /b

 0 1  . 0 −c0 /b

1 0 −c0 /b 0

(5.4)

To prove the proposition, we need to show that the eigenvalues of (5.4) are two pairs of complex conjugates with real parts of opposite signs. The lower-left 2 × 2 block of (5.4) is −∆/b. Denote the lower-right block by −C/b. µ

Let P =

u w

¶ Q=

µ ¶ v . y

Then σ is an eigenvalue of (5.4) if and only if Q = σP, −∆P − CQ = bσQ. This holds if and only if

³ ¢ ∆ + c0 σ + bσ 2 I √ is singular. Let T = 12 tr ∆, S = det ∆ − T 2 , so that the eigenvalues of ∆ are T ± iS (using (1.15)). Thus σ satisfies the equation σ 2 + c0 σ + b(T ± iS) = 0,

Alexander & Jones: Asymptotically oscillatory double pulses so c0 σ=− ± 2

r³ ´ c 2 0

2

30

− b(T ± iS).

Since S 6= 0, σ is complex. The four choices of signs give the four values of σ, two pairs of complex conjugates. Since c0 /2 6= 0, the real parts of the two different conjugate pairs are unequal. It remains to show that the real parts of the two different conjugate pairs have opposite signs. This occurs if |µ| > c0 /2, where µ + iν =

r³ ´ c 2 0

2

− b(T ± iS).

Squaring, we find µ2 − ν 2 =

³ c ´2 0

2 2µν = bS.

− bT,

(5.5) (5.6)

Suppose that |µ| < c0 /2. By (5.5), ν 2 < bT . But then, by squaring (5.6) and substituting, b2 S 2 = 4µ2 ν 2 < bc20 T , which contradicts (1.15). This completes the proof of the proposition. References J. C. Alexander, R. Gardner and C. Jones [1990], A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math. 410, 167–212. E. A. Coddington and N. Levinson [1955], Theory of Ordinary Differential Equations, McGraw-Hill, New York, NY. P. E. Conner and E. E. Floyd [1964], Differentiable Periodic Maps, Springer-Verlag, New York–Heidelberg–Berlin. J. W. Evans [1972], Nerve axon equations, III: Stability of the nerve impulse, Indiana Univ. Math. J. 22, 577–594. [1975], Nerve axon equations, IV: The stable and unstable impulse, Indiana Univ. Math. J. 24, 1169–1190. J. W. Evans, N. Fenichel and J. A. Feroe [1982], Double impulse solutions in nerve axon equations, SIAM J. Appl. Math. 42, 219–234. R. Gardner and C. Jones [1991], Stability of travelling wave solutions of diffusive predator-prey systems, Trans. Amer. Math. Soc. 292, in press. P. Glendinning [1989], Subsidiary bifurcations near bifocal homoclinic orbits, Math. Proc. Cambridge Philos. Soc. 105, 597–605.


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P. Hartman [1964], Ordinary Differential Equations, John Wiley & Sons, New York, NY. D. Henry [1981], The geometric theory of semilinear parabolic equations, Lect. Notes in Math., no. 840, Springer-Verlag, New York–Heidelberg–Berlin. C. K. R. T. Jones [1984], Stability of the travelling wave solution of the FitzHughNagumo system, Trans. Amer. Math. Soc. 286, 431–469. D. Sattinger [1976], On the stability of waves of nonlinear parabolic systems, Adv. in Math. 22, 312–355. ˇ L. P. Silnikov [1967], The existence of a denumerable set of periodic motions in fourdimensional space in an extended neighborhood of a saddle-focus, Soviet Math. Dokl. 8, 54–58. S. Wiggens [1988], Global Bifurcations and Chaos, Applied Mathematical Sciences, no. 73, Springer-Verlag, New York–Heidelberg–Berlin. E. Yanagida [1985], Stability of fast travelling pulse solutions of the FitzHugh-Nagumo equations, J. Math. Biol. 22, 81–104. E. Yanagida and K. Maginu [1989], Stability of double-pulse solutions in nerve axon equations, SIAM J. Appl. Math. 49, 1158–1173.


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6. Figure Captions. Figure 1. An asymptotically oscillatory double pulse. The horizontal axis is x; the vertical axis is one of the variables of U . This is a snapshot at some time t. As indicated, the wave is moving, unchanged in shape, to the right at speed c. The distance between peaks is δ. The wave is asymptotically oscillatory, in that its takeoff and decay are oscillatory. The theory of the paper implies that generically, if there is a single asymptotically oscillatory wave, there is an infinite sequence of such double waves, alternately stable and unstable, indexed by the number of semi-oscillations between the two peaks. Figure 2. The stable and unstable tori Ts and Tu , and their cores Cs and Cu . The figure is very schematic, in that the tori are three dimensional solid tori and their union is the complete boundary of a neighborhood of the the origin in IR4 . The two-dimensional surfaces of Tu and Ts actually coincide. The figure illustrates that the original homoclinic orbit γ leave the neighborhood from a point γu ∈ Cu and reenters at a point γs ∈ Cs . We let Mu ⊂ Tu be a small neighborhood of γu . The method of establishing existence and stability is to study the section maps Γc : Mu → Ts and F : Ts r Cs → Tu r Cu . Figure 3. The images Γc of Cu ∩ Mu under Γ (c projected out). As c varies near c0 , the image Γ(Cu ∩ Mu ) is a disk in Ts transversal to Cs . The disk is foliated by the individual images Γc (Cu ∩ Mu ) which are lines in the disk. In normal coordinates, the disk is the set {ψ˜s = 0}. 1

Figure 4. The image under F of the disk of Figure 3. The point of intersection of the disk and the core Cs has no image (since all points on the core remain in the neighborhood N² for all positive time under the flow). The image of the remainder of the disk is a two-dimensional spiral around the core Cu in Tu . The image F Γc (Cu ∩Mu ) is a one-dimensional spiral which (because of the aysmptotic oscillatory conditions) spirals around Cu in a meridinal direction, and also in a longitudinal direction. The image Γc F Γc (Cu ∩ Mu ) ⊂ Ts is thus a spiral which, for a sequence cn of c, intersects Cs . Each such intersection corresponds to a double pulse solution.