Bifurcation to instability through the lens of the Maslov index
BBIFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOVINDEX
PAUL CORNWELL, CHRISTOPHER K. R. T. JONES, AND CLAIRE KIERS
Abstract.
The Maslov index is a powerful tool for assessing the stability of solitary waves. Al-though it is difficult to calculate in general, a framework for doing so was recently established forsingularly perturbed systems [14]. In this paper, we apply this framework to standing wave solu-tions of a three-component activator-inhibitor model. These standing waves are known to becomeunstable as parameters vary. Our goal is to see how this established stability criterion manifestsitself in the Maslov index calculation. In so doing, we obtain new insight into the mechanism forinstability. We further suggest how this mechanism might be used to reveal new instabilities insingularly perturbed models.
Contents
1. Introduction 12. A 3-component activator-inhibitor model 32.1. Standing single pulses 43. The stability problem and the Maslov index 73.1. The Maslov index 94. Calculation of the Maslov index 104.1. The reference plane 114.2. The fast front 124.3. Passage near the right slow manifold 134.4. The fast back 144.5. Corner 2.5: Arrival at the right slow manifold 145. Conclusion 175.1. Discussion 18References 191.
Introduction
Many physical systems conserve energy. In such systems, the notion that states which minimizeenergy are stable is a fairly reliable heuristic. Solitary waves, for example, often arise as criticalpoints of an energy functional. One checks whether a wave is a minimizer by analyzing the secondderivative of the energy, which happens to be the linearization about the wave. The existence ofunstable spectrum therefore indicates that there are directions in which the energy decreases. For a r X i v : . [ m a t h . D S ] F e b P. CORNWELL, C. K. R. T. JONES, AND C. KIERS variational problems in general, the
Morse index is defined to be the dimension of the maximalsubspace on which the Hessian of the energy evaluated at a critical point is negative definite.Computing the Morse index amounts to an infinite-dimensional eigenvalue problem, which istypically difficult to solve. The celebrated Morse Index Theorem [27, §
15] is a valuable tool inthe special case where the functional is the energy of a path, and critical points are geodesics ona manifold M . It states that the Morse index, evaluated at a critical path γ ( t ), is equal to thenumber of conjugate points along γ . Without diving into definitions, the crux of this result is thatthe Morse index is determined by how γ is situated in the tangent bundle T M ; there is no need toanalyze the “spectrum” of the Hessian explicitly.The subject of this paper is an adaptation–in fact, a generalization–of the Morse Index Theoremto the context of stability of solitary waves. The number of conjugate points for a critical path isreplaced by the Maslov index of the wave, which is an intersection number assigned to curves ofLagrangian planes. One can show that the Maslov index counts (or at least gives a lower boundfor) real, unstable eigenvalues. Moreover, it is computed by fixing the spectral parameter λ = 0,which yields the equation of variations for the wave. This important fact is the key to extractingstability information from phase space geometry, which we discuss later.The equality of the Morse and Maslov indices has been worked out in a number of settings inrecent years, e.g. [11, 12, 13, 20, 22, 24]. The bigger challenge is arguably calculating the Maslovindex, which has proven difficult to do. In the spirit of the Morse Index Theorem, two schools haveemerged with techniques for doing so. The first is based on the calculus of variations, owing to Chenand Hu [11, 12]. They use the Maslov index to show that the Morse index is 0 for energy minimizers.The “dirty work” of the calculation is then to construct an admissible class of functions and find aminimizer. This strategy was employed to prove the existence and (in)stability of standing wavesin a doubly-diffusive FitzHugh-Nagumo system [10, 12].The second school strives to calculate or estimate the Maslov index directly by locating inter-sections. This approach has produced instability results for generic standing waves of gradientreaction-diffusion systems [3], as well as (in)stability results for various standing and travelingwaves [4, 8, 9, 14, 23]. The main technical tool for this approach is the crossing form of Robbinand Salamon [28]. A conjugate point corresponds to the intersection of a curve of Lagrangianplanes with a codimension one set (the “singular cycle”), and the crossing form determines thecontribution to the Maslov index at a conjugate point.We now briefly describe the challenge of calculating the Maslov index directly. Let L be theoperator obtained by linearizing about a solitary wave. The eigenvalue equation Lp = λp canbe cast as a non-autonomous dynamical system on R n (for λ ∈ R ), where n is the number ofcomponents. The curve of interest in the Maslov index calculation is the n -dimensional subspace ofsolutions to Lp = 0 which decay at −∞ , called the unstable bundle. Due to translation invariance,the derivative of the wave is everywhere part of this space. In the scalar case, this is the onlysolution, and conjugate points can be related to zeros of the velocity (from which Sturm-Liouvilletheory follows almost immediately). It is the presence of other solutions in the higher-dimensionalcase that makes the calculation difficult. Finding these solutions is tantamount to solving a linear,non-autonomous equation on the real line.In [14], two of the authors of this paper established a framework for calculating the Maslovindex in singularly perturbed equations using geometric singular perturbation theory (GPST). Thestrategy is to use the fact that the unstable bundle is everywhere tangent to the unstable manifold containing the wave in phase space. Using Fenichel theory [17] and subsequent developments suchas the Exchange Lemma [25], one can accurately determine the orientation of the unstable manifold IFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOV INDEX 3 as it evolves along the wave. Thus it is possible to calculate the Maslov index without having tosolve the linear system explicitly. As a proof of concept, this framework was applied to show thatfast traveling waves for a FitzHugh–Nagumo equation (with equal diffusion rates) are stable.The aim of this paper is to apply the framework of [14] to a system of three reaction-diffusionequations (2.1) introduced by Schenk et al [30]. Using GSPT, Kaper et al showed that (2.1)supports a multitude of interesting standing and traveling wave solutions [15]. They subsequentlyobtained stability results using a fast-slow Evans function decomposition [32]. These results werethen reproved and expanded upon by van Heijster et al using a variational approach [31]. Thelatter stability proof uses the Maslov index in the manner of the first school described above. Wewill use a conjugate point-based Maslov index calculation to re-derive the known stability result inthe case of standing single pulses.We have several objectives in this work. The first is to demonstrate the robustness of thecalculation method developed in [14]. In particular, we prove its utility in systems of dimensiongreater than two–the simplest non-trivial case. Second, the pulses that we study can be stableor unstable depending on the model parameters. By tuning the parameters, we can therefore seeexactly how the instability manifests itself in the Maslov index calculation. This stands in contrastto the calculation in [14], where the waves are stable regardless of the parameters (at least locally).Using this new example, we will argue that the Maslov index can be used to identify and manipulatemechanisms for (in)stability in singularly perturbed systems.Finally, we believe that the calculation carried out herein will be of interest to GSPT itself.Since the original work of Fenichel [17], the two most important advances to the geometric theoryof singularly perturbed systems are the Exchange Lemma and geometric desingularization–the“blowup method” [16]. The former is useful when considering passage near a normally hyperboliccritical manifold, whereas the latter is used to study dynamics at a point where normal hyperbolicityis lost. We shall see later that the Maslov index calculation requires a hybrid approach. Althoughthe system we analyze has no fold points, the fast-slow transitions in the tangent bundle above thewave are critical. As such, it is necessary to zoom in on the exact point in phase space where thistransition occurs. Instead of studying the dynamics near these points on a sphere–as one woulddo in the blowup method–we aim to understand the dynamics of the equation of variations on theLagrangian Grassmannian. An interesting extension of this work would be to calculate the Maslovindex of a wave which passes through a fold point, e.g. [7].The rest of this paper is organized as follows. In section 2, we state the model equations andemphasize the fast-slow structure. We then describe the standing pulse solutions of interest bybreaking them into fast and slow components. In section 3, we lay out the eigenvalue problem anddefine the Maslov index of the pulses. In section 4, we carefully compute the Maslov index andshow how an instability can appear when the model parameters are changed. Finally, we concludein section 5 by explaining the insight gained from this calculation as well as how it might apply toother singularly perturbed systems.2.
A 3-component activator-inhibitor model
In this work, we consider the following three-component system of reaction-diffusion equations: U t = (cid:15) U xx + U − U − (cid:15) ( αV + βW + γ ) τ V t = V xx + U − VθW t = D W xx + U − W. (2.1) The reader must decide whether tracking the unstable manifold should be done with [25] or without [5] the useof differential forms.
P. CORNWELL, C. K. R. T. JONES, AND C. KIERS
We assume that τ, θ > D >
1, 0 < (cid:15) (cid:28)
1, and α, β, γ ∈ R . We further assume that all parametersare O (1) in (cid:15) . This model has its roots in the high-level study of gas-discharge dynamics. For morebackground on the physical aspects of the problem, we refer the reader to [31]. In [15, 32], geometricsingular perturbation theory was used to prove the existence and stability of various standing andtraveling waves for (2.1). These results were then revisited and expanded in [31] by combining theGSPT analysis with an action functional approach.We will consider standing pulse solutions of (2.1), which are time-independent, localized struc-tures. By setting U t = V t = W t = 0 and introducing the variables P = (cid:15)U x , Q = V x , and R = DW x ,such solutions are seen to be homoclinic orbits for the standing-wave ODE (cid:15)U (cid:48) = P(cid:15)P (cid:48) = − U + U + (cid:15) ( αV + βW + γ ) V (cid:48) = QQ (cid:48) = V − UW (cid:48) = 1 D RR (cid:48) = 1 D ( W − U ) , (2.2)where (cid:48) = ddx . (2.2) is formulated on the “slow” timescale. We will also use the “fast” version of(2.2), obtained by setting (cid:15)ξ = x. Denoting ˙ = ddξ , we rewrite (2.2) as˙ U = P ˙ P = − U + U + (cid:15) ( αV + βW + γ )˙ V = (cid:15)Q ˙ Q = (cid:15) ( V − U )˙ W = (cid:15)D R ˙ R = (cid:15)D ( W − u ) . (2.3)When (cid:15) >
0, (2.2) and (2.3) define the same dynamics. However, the limiting systems obtained bysending (cid:15) → V, Q, W, and R actingas parameters. On the other hand, (2.2) is a differential-algebraic equation where the dynamics arefour-dimensional and restricted to the set M = { ( U, P, V, Q, W, R ) ∈ R : P = 0 , U ∈ {− , , }} . (2.4) M is called the critical manifold . Observe that M is precisely the set of critical points for (2.3)with (cid:15) = 0.The goal of GSPT is to construct solutions to (2.3) for small (cid:15) > (cid:15) >
0. Since this theory is well understood (at least as applied to the system at hand), wewill simply describe the (cid:15) = 0 object, which is all that is needed to compute the Maslov index. Theinterested reader can find more detail on the existence proofs in [15, § Standing single pulses.
As mentioned earlier, there are many permanent structures hidingin this model. Since the objective of this paper is to observe the bifurcation to instability through
IFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOV INDEX 5 the lens of the Maslov index, we will focus on the simplest structure that exhibits this behavior–thestanding single pulse.In phase space, a solitary pulse is a homoclinic orbit to a fixed point. For (cid:15) >
0, once sees that(2.3) has three fixed points, which are distinguished by the U component. The wave we consider ishomoclinic to the smallest value of U , which we see from (2.3) and (2.4) is U − (cid:15) := − O ( (cid:15) ) . (2.5)The other components of the fixed point follow directly from (2.5). We define the fixed point X − (cid:15) := ( U − (cid:15) , , U − (cid:15) , , U − (cid:15) ,
0) (2.6)of (2.3) as the rest state of the wave. The homoclinic orbit itself will be O ( (cid:15) ) close to a singularversion consisting of five segments: three slow and two fast. We denote the singular orbit ϕ ( x ) ∈ R , where the subscript refers to (cid:15) = 0. To describe ϕ in more detail, we will need the two-dimensional fast system ˙ U = P ˙ P = − U + U , (2.7)and the four-dimensional slow reduced system V (cid:48) = QQ (cid:48) = V − UW (cid:48) = 1 D RR (cid:48) = 1 D ( W − U ) . (2.8)The two fast orbits are heteroclinic connections from ( U, P ) = ( − ,
0) to (
U, P ) = (+1 ,
0) andback again. Observe that (2.7) is Hamiltonian with H ( U, P ) = P / U / − U / , so we can solve H ( U, P ) = H ( − ,
0) to see that the heteroclinic connections are given by P = ± (cid:114) − U + U ± √ − U ) . (2.9)(The positive branch goes from ( − ,
0) to (1 , U is now a parameter. Setting U = − V, Q, W, R ) = ( − , , − ,
0) := X − . (2.10)Moreover, the ( V, Q ) and (
W, R ) equations decouple, so it is easy to compute the (2D) stable andunstable manifolds of X − : W s ( X − ) = { ( V, Q, W, R ) : Q = − ( V + 1) , R = − ( W + 1) } W u ( X − ) = { ( V, Q, W, R ) : Q = V + 1 , R = W + 1 } . (2.11)Likewise, for U = +1, (2.8) is linear with the origin shifted to( V, Q, W, R ) = (1 , , ,
0) := X + . (2.12)The stable and unstable manifolds of X + are W s ( X + ) = { ( V, Q, W, R ) : Q = − ( V − , R = − ( W − } W u ( X + ) = { ( V, Q, W, R ) : Q = V − , R = W − } . (2.13) P. CORNWELL, C. K. R. T. JONES, AND C. KIERS
Figure 1.
Projection of slow flows on M +0 and M − onto V Q − space (or W R − space). Segment 3 (on M +0 ) serves to carry the slow coordinates from W u ( X − )to W s ( X − ).We can now describe how the segments fit together to form ϕ . First, since these manifolds willplay a prominent role later, we officially define the relevant subsets of the critical manifold: M − = { ( U, P, V, Q, W, R ) : U = − , P = 0 } M +0 = { ( U, P, V, Q, W, R ) : U = 1 , P = 0 } . (2.14)To simplify notation, we also append the appropriate ( U, P ) coordinates to W u/s ( X ± ) so that W u/s ( X − ) ⊂ M − , and W u/s ( X + ) ⊂ M +0 . The two fast segments, as mentioned above, are hetero-clinic connections between M − and M +0 . Two of the slow segments will turn out to be trajectoriesinside of W u/s ( X − ); the first segment leaves X − along W u ( X − ), and the fifth segment returns to X − along W s ( X − ).The third segment is the part of the slow flow on M +0 . Since the slow coordinates do not changealong the fast jumps, the role of this segment is to flow the slow coordinates from W u ( X − ) to W s ( X − ). The only way for this to happen is for the V and W coordinates of the jump-off point tobe negative. This is equivalent to the jump-off occurring prior to the intersection of W u ( X − ) and W s ( X + ) projected onto the slow coordinates. (See Figure 1.) Since (2.8) is linear, segment 3 lieson the product of two hyperbolas–one each in V Q − and W R − space.At this point, we have completely described the singular orbit qualitatively. However, one detailremains: the values of V and W at the jump-off points. We write z i , i ∈ { , , , } , for the pointin R where segment i meets segment i + 1. From Figure 1, it is clear that specifying V and W at z determines all four variables at each z i . It turns out that fixing these values is an (cid:15) > Theorem 1 (Theorem 2.1 of [15], Theorem 1 of [31]) . Let ( α, β, γ, D ) be such that αe − x + βe − x/D = γ (2.15) IFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOV INDEX 7 has a positive solution x = x ∗ . Then, for (cid:15) > sufficiently small, (2.3) possess a homoclinic orbit ϕ (cid:15) ( x ) , whose U − coordinate approaches U − (cid:15) defined by (2.5) as x → ±∞ . Moreover, the image of ϕ (cid:15) ( x ) is O ( (cid:15) ) -close to ϕ with jump-off point z = ( − , , − e − x ∗ , − e − x ∗ , − e − x ∗ /D , − e − x ∗ /D ) . (2.16) Remark 2.1.
Depending on the parameters, equation (2.15) can have zero, one, or two solutions.If there are zero solutions, no single pulse waves exist. If there are two solutions, then two wavesexist with different jump-off values.
Remark 2.2.
Comparing (2.15) and (2.16), we see that the jump-off condition can be rewrittenin terms of the slow coordinates as αV z + βW z + γ = 0 . (2.17) As expected, Q z = V z + 1 , and R z = W z + 1 due to the constraint that z ∈ W u ( X − ) . We close this section by recapitulating the singular orbit. ϕ consists of the five segments below.(S1) ϕ is the segment of W u ( X − ) between X − and z . ( X − is reached as x → −∞ .)(F2) ϕ is the heteroclinic orbit for (2.7) connecting ( U, P ) = ( − ,
0) to (
U, P ) = (+1 , ϕ is the unique trajectory for (2.8) on M +0 connecting z (= z ) and z (= z ). Note thatthe V and W coordinates at z and z are the same, and ( W z , R z ) = ( − W z , − R z ). Thisensures both that the jump-off condition (2.15) is met and that ϕ lands on W s ( X − ) afterthe fast jump.(F4) ϕ is the heteroclinic orbit for (2.7) connecting ( U, P ) = (+1 ,
0) to (
U, P ) = ( − , ϕ is the segment of W s ( X − ) which begins at q and approaches X − as x → ∞ .3. The stability problem and the Maslov index
Now assume that ( α, β, γ, D, (cid:15), x ∗ ) satisfy the conditions of Theorem 1. Let ϕ (cid:15) ( x ) be the corre-sponding standing pulse. We will write ϕ (cid:15) ( x ) for both the homoclinic orbit of (2.2) and the standingpulse solution ( U ( x ) , V ( x ) , W ( x )) of (2.1). The goal of this section is to analyze the stability of ϕ (cid:15) by using the Maslov index. We begin by briefly reviewing the stability problem for ϕ (cid:15) . Sincethe following theory is well understood, we omit many of the details. For more background on thestability analysis of solitary waves, we refer the reader to [1, 29]. Definition 1.
The standing wave ϕ (cid:15) ( x ) is asymptotically stable relative to (2.1) if there is aneighborhood V ⊂ BU ( R , R ) of ϕ (cid:15) ( x ) such that if ψ ( x, t ) solves (2.1) with ψ ( x, ∈ V , then || ϕ (cid:15) ( x + k ) − ψ ( x, t ) || ∞ → as t → ∞ for some k ∈ R . Although Definition 1 refers to nonlinear stability, it is sufficient in this case to prove linearstability [19]. Moreover, the essential spectrum is bounded away the imaginary axis in C = { z ∈ C : Re z < } in our parameter regime [32, Lemma 3.1]. P. CORNWELL, C. K. R. T. JONES, AND C. KIERS
It follows that the stability of ϕ (cid:15) is entirely determined by eigenvalues, which are values λ ∈ C such that λu = (cid:15) u xx + u − (3( U ( x )) ) u − (cid:15) ( αv + βw ) λτ v = v xx + u − vλθw = D w xx + u − w (3.1)has a solution P λ ( x ) := ( u ( x ) , v ( x ) , w ( x )) ∈ BU ( R , C ). (Notice that the right hand side of (3.1)is the linearization of (2.1) about ϕ (cid:15) ( x ).)Due to translation invariance, λ = 0 is necessarily an eigenvalue. However, it is shown in [32]that this eigenvalue is simple. We therefore have the following theorem. Theorem 2.
The standing pulse ϕ (cid:15) ( x ) of (2.1) is asymptotically stable if and only if all of thenon-zero eigenvalues of (3.1) have negative real part. To employ the Maslov index, we rewrite (3.1) as a first order system on the fast timescale ( ξ ):˙ Y ( ξ ) = uvwpqr ξ = (cid:15)
00 0 0 0 0 (cid:15)D λ − U α(cid:15) β(cid:15) − (cid:15) (cid:15) ( λτ + 1) 0 0 0 0 − (cid:15)D (cid:15)D ( λθ + 1) 0 0 0 uvwpqr = A ( λ, ξ ) Y ( ξ ) . (3.2)Notice that we switched the order of the variables in order to make the Hamiltonian structure ofthe equations clearer. Conversely, for the nonlinear problem it was more convenient to have thefast and slow variables separated.We follow the standard approach [1] to analyzing (3.2), which is to define the stable and unstablebundles of solutions decaying at −∞ and ∞ respectively: E s ( λ, ξ ) = { Y ( ξ ) ∈ C : Y ( ξ ) solves (3.2) and Y ( ξ ) → ξ → ∞} E u ( λ, ξ ) = { Y ( ξ ) ∈ C : Y ( ξ ) solves (3.2) and Y ( ξ ) → ξ → −∞} . (3.3)Since the only dependence on ξ of (3.2) is through U ( ξ )–which decays exponentially to − ±∞ –one would expect the dynamics of (3.2) to be influenced by those of the constant coefficient system˙ Y ( ξ ) = A ∞ ( λ ) Y ( ξ ), where A ∞ ( λ ) := lim ξ →±∞ A ( λ, ξ ) . (3.4)This is indeed the case. Defining S ( λ ) and U ( λ ) to be the stable and unstable subspaces for A ∞ ( λ )respectively, we have lim ξ →∞ E s ( λ, ξ ) = S ( λ )lim ξ →−∞ E u ( λ, ξ ) = U ( λ ) . (3.5)In particular, the dimensions of S ( λ ) and U ( λ ) are equal to the dimensions of the respective bundlesfor any λ and fixed ξ . It is easy to check in this case [32, §
3] that S ( λ ) and U ( λ ) are each three-dimensional for all λ with non-negative real part. It follows that E u/s ( λ, ξ ) define two-parametercurves in Gr ( C ), or Gr ( R ), if λ ∈ R . This fact is the basis for analyzing (3.2) with the Maslovindex in the next section. IFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOV INDEX 9
The Maslov index.
We are now ready to discuss the Maslov index. While we do presentall of the necessary definitions and theorems in this paper, the reader may find the more detailedaccount in [13, §
3] to be helpful. We shall henceforth restrict to λ ∈ R . Suppose that Y ( ξ ) and Y ( ξ ) are two solutions of (3.2) for fixed λ . Consider the two-form ω := du ∧ dp − αdv ∧ dq − βDdw ∧ dr. (3.6)It is straightforward (eg., [13, Theorem 2.1]) to calculate that ddξ ω ( Y , Y ) = ω ( Y , ˙ Y ) + ω ( ˙ Y , Y ) = 0 . (3.7)In other words, the form ω is constant in ξ when evaluated on two solutions of (3.2). Furthermore,if α and β are nonzero, ω is also non-degenerate. It thus defines a symplectic form on R . Define thesymplectic complement of a subspace V as V ⊥ = { w ∈ R : ω ( v, w ) = 0 , ∀ v ∈ V } . A (necessarilythree-dimensional) subspace V ⊂ R such that V = V ⊥ is called Lagrangian . We can thereforerephrase (3.7) as follows: The set of Lagrangian planes, denoted Λ(3), is an invariant set of thedynamical system induced on Gr ( R ) by (3.2).The set Λ(3) happens to be a submanifold of Gr ( R ) of dimension 6. The interesting fact forour purposes is that π (Λ(3)) = Z , which means that an integer winding number can be associatedwith curves in this space. (By contrast, π (Gr ( R )) = Z / Z , so no such integer index exists in thefull Grassmannian.) This winding number is the Maslov index.Since we will be dealing with non-closed curves in general, we will define the Maslov index asan intersection number instead of a winding number (`a la Poincar´e duality [18]). More precisely,we will fix a Lagrangian plane V –called the reference plane–and count the number of times thata curve of Lagrangian planes intersects V . Arnol’d established this definition of the Maslov indexin the case of one-dimensional intersections [2], and then Robbin and Salamon generalized it forintersections of any dimension [28].To handle the accounting for multidimensional intersections, Robbin and Salamon developedthe “crossing form.” This is a quadratic form whose signature determines the contribution to theMaslov index at each point. We remind the reader that the signature of a quadratic form Q is thedifference of the positive and negative indices of inertia:sign( Q ) = n + ( Q ) − n − ( Q ) . (3.8)For a reference plane V ∈ Λ(3) and a curve of Lagrangian planes γ ( t ) ∈ Λ(3), a conjugate point is a time t ∗ such that γ ( t ∗ ) ∩ V (cid:54) = { } . The crossing form is defined on the intersection, so thecontribution to the Maslov index at a k − dimensional crossing can be anywhere between − k and k .A crossing is called regular if Q is non-degenerate.Below we will formally define the Maslov index of the solitary wave ϕ (cid:15) . The curve of interestis the unstable bundle E u (0 , λ ), which we can think of as containing all potential 0-eigenfunctions(i.e., those which satisfy the left boundary conditions). In the spirit of a shooting argument, thereference plane should be the right boundary data for a potential 0-eigenvector, namely S (0); see(3.5). Actually, for technical reasons, it is untenable to consider E u (0 , ξ ) on all of R due to thepresence of a conjugate point at + ∞ . (See Remark 3.1.) The rigorous definition of the Maslovindex for a standing wave–due to Chen and Hu [11]–fixes this issue by truncating the curve andusing the stable bundle at the new endpoint as the reference plane. Definition 2.
Let ξ ∞ be large enough so that U (0) ∩ E s (0 , ξ ) = { } for all ξ ≥ ξ ∞ . (3.9) An ostensibly different symplectic form is used in [13]. However, this difference is artificial; it is a byproduct ofthe way that we convert (3.1) to a first-order system. (2.1) is indeed of the skew-gradient variety studied in [11, 13].
We define the
Maslov index of ϕ (cid:15) to be Maslov( ϕ (cid:15) ) := (cid:88) ξ ∗ ∈ ( −∞ ,ξ ∞ ) sign Γ( E u , E s (0 , ξ ∞ ) , ξ ∗ ) + n + (Γ( E u , E s (0 , ξ ∞ ) , ξ ∞ )) , (3.10) where the sum is taken over all interior crossings of ξ (cid:55)→ E u (0 , ξ ) with Σ , the train of E s (0 , ξ ∞ ) .If ξ ∗ is a conjugate point, and ψ ∈ E u (0 , ξ ∗ ) ∩ E s (0 , ξ ∞ ) , then the crossing form is given by Γ( E u , E s (0 , ξ ∞ ) , ξ ∗ ) = ω ( ψ, A (0 , ξ ∗ ) ψ ) . (3.11) Remark 3.1.
We only count negative crossings at −∞ (of which there are none) and positivecrossings at ξ ∞ by convention. Notice that there is a guaranteed to be a crossing at ξ = ξ ∞ , since ϕ (cid:48) (cid:15) ( ξ ∞ ) ∈ E s (0 , ξ ∞ ) ∩ E u (0 , ξ ∞ ) . This crossing is one-dimensional–equivalently, the translationeigenvalue is simple–by virtue of the transverse construction of the wave. A key feature of Definition 2 is that λ = 0 is fixed. One can typically show that Maslov( ϕ (cid:15) )either counts real, unstable eigenvalues or gives a lower bound on the count. Moreover, (3.2) is theequation of variations for (2.3) when λ = 0, which means that one can glean spectral informationfrom how the wave itself is constructed. Indeed, this is the premise for the calculation in the nextsection.The next step would be to prove that the Maslov index actually counts all unstable eigenvalues.Since our focus here is on the calculation of the index more than stability per se, we will not godown that path. However, the authors in [31] verified that the Maslov index does indeed give thedesired count. For a blueprint on how to prove this equality in general, we refer the reader to[12, 13]. 4. Calculation of the Maslov index
We are now prepared to calculate the Maslov index using the framework developed in [14]. Inorder to motivate the calculation, we first state the known stability result for ϕ (cid:15) . Theorem 3 (Theorem 4.1 of [32], Theorem 1 of [31]) . Let ϕ (cid:15) ( ξ ) be a standing single pulse solutionof (2.1), as described in Theorem 1. Then ϕ (cid:15) is stable in the sense of Definition 1 if and only if αV + βD W < , (4.1) where V and W are the values (to leading order in (cid:15) ) of V, W at the jump-off points z i . Remark 4.1.
Recall that V , W ∈ ( − , , so at least one of α, β must be negative in order for ϕ (cid:15) to be unstable. Remark 4.2.
Values of ( α, β, D ) yielding equality in (4.1) correspond to a saddle-node bifurcationof homoclinic orbits [32, Theorem 2.1]. Our perspective in this section will be to consider α and β as parameters. In light of Theorem3, we expect to see a conjugate point appear or disappear as α and β are tuned to cause a signchange in (4.1).The ensuing calculation rests on two ideas. The first is that the stable and unstable bundles(3.3) are everywhere tangent to the stable and unstable manifolds for X − (cid:15) as a fixed point of (2.3).The second is that the nonlinear objects W u/s ( X − (cid:15) ) can be tracked using Fenichel theory [17, 21].We know that E u (0 , ξ ) is three-dimensional, with one direction given by the wave velocity ϕ (cid:48) (cid:15) .Using GSPT, we are able to discern the other two dimensions using the geometry of phase spaceas opposed to having to solve the non-autonomous linear system (3.2). IFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOV INDEX 11
The crux of GSPT is that the critical manifolds M ± perturb to locally invariant manifolds M ± (cid:15) when 0 < (cid:15) (cid:28)
1. Moreover, the flow on M ± (cid:15) , to leading order, is is given by (2.8). Actually,something even stronger is true. Recall that M ± is the union of fixed points of the fast system(2.7). Each of these fixed points has one-dimensional stable and unstable manifolds, obtained bylinearizing (2.7) at U = ±
1. It is therefore possible to define W u/s ( M ± ) as the union of thecorresponding invariant manifolds for the points comprising M ± . These, too, perturb to locallyinvariant manifolds W u/s ( M ± (cid:15) ). The fact that (2.7) is 2D Hamiltonian and (2.8) is essentially linearwill make it quite easy to describe these objects.The smooth convergence of the slow manifolds and their attendant invariant manifolds allowsus to work primarily with (cid:15) = 0, provided that all conjugate points are regular. However, thetransitions from fast-to-slow dynamics and vice-versa require some care. Indeed, the Maslov indexcalculation is properly happening in the tangent bundle along the wave. Although the wave itselfis continuous in the limit, its derivative (and more generally the tangent space to W u ( X − )) has ajump discontinuity at these transitions. To figure out the (cid:15) > z i , it is shown in [14] that one must treat (2.3) as a constant coefficient system at z i .As we shall see, these ‘corners’ are the most interesting part of the calculation since the bifurcationmanifests itself there.In light of the preceding paragraph, there should be nine segments (= five components of thewave and four corners) that we must analyze to calculate the index. However, our flexibility inchoosing the right endpoint of the unstable bundle will shorten the calculation considerably. Sincea similar calculation is done in full detail in [14], we will omit many of the details and focus onthe pieces that influence stability. In referring to the corners, we will use the notation “ i.
5” for thetransition from segment i to i + 1.4.1. The reference plane.
To utilize Definition 2, we need to select a reference plane. Selectinga reference plane is tantamount to choosing a value ξ ∞ so that U (0) ∩ E s (0 , ξ ) = { } for all ξ ≥ ξ ∞ .To that end, we begin by linearizing (2.3) about X − . We can obtain the leading-order terms bysetting (cid:15) = 0 separately in (2.7) and (2.8). This way the ( U, P ) , ( V, Q ), and (
W, R ) decouple nicely,and we we compute the following eigenvectors in ( u, v, w, p, q, r ) coordinates: S (0) = span −√ , − , − , U (0) = span , , √ . (4.2)Without loss of generality, we assume that D >
1. With this assumption, the eigenvectors(4.2) appear from left to right in order of increasing eigenvalue. We label these η , . . . , η , withcorresponding eigenvalues µ i . To leading order in (cid:15) , the eigenvalues are µ = −√ µ = − (cid:15)µ = − (cid:15)/Dµ = (cid:15)/Dµ = (cid:15)µ = √ To choose the reference plane, we flow E s (0 , ξ ) backwards in time from + ∞ until we reach aconvenient point. Because (2.8) itself is affine linear, there is no change to the slow directions η and η along segment ( S η , η ∈ E s (0 , ξ ) for any point along the back. The fast stable direction also does not change along( S η , which is the direction in which ϕ approaches z along the back. Thisproves that E s (0 , ξ ) is tangent to W s ( M − ) at the landing point z . Along the back itself, thestable direction is tangent to the heteroclinic orbit. This can be computed from (2.9) as (1 , √ U )in ( u, p ) space.Let ξ be some time for which ϕ (cid:15) ( ξ ) is on the back. To verify condition (3.9), we compute thedeterminant det[ U (0) , E s (0 , ξ )] = det √ √ U − − = 4 √ − U ) . (4.4)Evidently we are free to select any point along the back besides ±
1, so we will pick the jumpmidpoint where U = 0. We will therefore count intersections of E u (0 , ξ ) with the reference plane V = span , − , − = span { η + η , η , η } . (4.5)4.2. The fast front.
This sections begins the calculation of the Maslov index. As mentionedearlier, E u (0 , ξ ) is everywhere tangent to W u ( X − (cid:15) ) along ϕ (cid:15) ( ξ ). By the same reasoning as § W u ( X − (cid:15) ) is spanned by { η , η , η } throughout segment ( S
1) and corner 1 .
5. It isclear from (4.5) that there are no conjugate points for these pieces.Along the fast front, the slow directions remain the same, and the fast unstable direction is givenby the tangent vector ϕ (cid:15) . Again using (2.9), we find the ( u, p ) components to be (1 , −√ U ). Todetect conjugate points (i.e., intersections with V ), we computedet[ E u (0 , ξ ) , V ] = − √ U, (4.6)which is zero if and only if U = 0. This is the midpoint of the jump, just like we chose for thereference plane. If we selected a different ξ ∞ for Definition 2, then this conjugate point would havemoved accordingly. To compute the dimension of this crossing, we first verify that the intersectionis one-dimensional, spanned by ψ = . (4.7)We next apply (3.11) with (cid:15) = 0 and ξ such that U = 0 to see that ω ( ψ, A (0 , ξ ) ψ ) = du ∧ dp ( ψ, A (0 , ξ ) ψ ) = − . (4.8) IFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOV INDEX 13
Thus the crossing form is one-dimensional and negative definite, so the contribution to the Maslovindex along the fast front ( F
2) is − . Passage near the right slow manifold.
It turns out that the stability result will hinge oncorner 2 .
5, so we save that section until the end. We now consider the passage of E u (0 , ξ ) near M + (cid:15) . The slow flow takes over for this section, so the fast direction will be constant to leadingorder. As one would expect, it is the unstable fast direction η that persists for this segment; see[14, § M + (cid:15) . Byflowing this set forward in time, we obtain a two-dimensional manifold with boundary S foliatedby the slow trajectories. The tangent space to this manifold along ( S
3) gives the slow directions of T ϕ (cid:15) ( ξ ) W u ( X − (cid:15) ) near this segment.To get an explicit expression, we first solve (2.8) generically on M +0 (i.e., with U ≡ V ( x ) = c e x + c e − x + 1 Q ( x ) = c e x − c e − x W ( x ) = c e x/D + c e − x/D + 1 R ( x ) = c e x/D − c e − x/D (4.9)Because the steady state equation for (2.1) has reversibility symmetry, we know that V and W willachieve their maxima at the same point. We may assume that this maximum is x = 0 for (4.9),from which we conclude that c = c and c = c . The solutions of interest for (4.9) satisfy theequations ( V − − Q = 4 c ( W − − R = 4 c , (4.10)which intersect the lines Q = V + 1 and R = W + 1 from (2.16) when V = − c W = − c . (4.11)We combine (4.11) with (2.17) to see that αc + βc = γ, (4.12)which we can solve for c , knowing that both c , <
0. We can therefore describe S as the graph ofa function of x and c : ( V, Q, W, R ) = h ( x, c ) = c cosh( x ) + 1 c sinh( x ) c ( c ) cosh (cid:0) xD (cid:1) c ( c ) sinh (cid:0) xD (cid:1) (4.13)We differentiate to determine a basis for the tangent space to S : T ϕ ( x ) S = span (cid:26) ∂h∂x , ∂h∂c (cid:27) = span c sinh( x ) cosh( x ) c cosh( x ) sinh( x ) c D sinh (cid:0) xD (cid:1) c (cid:48) cosh (cid:0) xD (cid:1) c D cosh (cid:0) xD (cid:1) c (cid:48) sinh (cid:0) xD (cid:1) (4.14)where (cid:48) = ddc . Since the fast direction is unchanged, it is clear that we can check for conjugatepoints by seeing if (4.14) intersects the plane spanned by η and η . To that end, a somewhat messy computation shows thatdet (cid:20) η , η , ∂h∂x , ∂h∂c (cid:21) = ( c /D − c c (cid:48) ) e x (1+1 /D ) . (4.15)Clearly, the sign of (4.15) is independent of x . Differentiating (4.12) implicitly, we find that c D − c c (cid:48) = c D + αc βc = ( β/D ) c + αc βc . (4.16)We must now evaluate this expression along ϕ ( x ). Using (2.17) and (4.11), the numerator of (4.16)simplifies to − (cid:18) αV z + βD W z (cid:19) . (4.17)Notice that this is exactly the expression appearing in the stability condition (4.1). We are assumingthat we are not in the regime where the bifurcation occurs, so (4.15) does not vanish anywherealong ( S The fast back.
Corner 3.5 turns out to be trivial after the pain of the previous section. Thefast direction is already in the correct position for the fast back–tangent to W u ( X + )–from thepassage near M + (cid:15) . An application of the ( k + σ ) Exchange Lemma [21, § σ = 2 shows thatthe slow directions also remain unchanged. We therefore have no contribution to the Maslov index.For the back, we can focus entirely on the two-dimensional fast system. Indeed, the calculations(4.15) and (4.16) show that the slow directions at launch (which don’t change over the back) aretransverse to the slow directions in V .We again use (2.9) to get a U -dependent expression for the fast direction of W u ( X + ). This time,we use the branch of (2.9) with negative P , so the tangent vector is given by ( u, p ) = (1 , √ P ). Asin § P = 0. We could easily compute the crossing formagain, but we’ll instead argue geometrically that the crossing is negative. The two fast jumps forma separatrix for (2.7). For a planar system, the Maslov index measures the winding of the anglethat the tangent vector to the wave makes. It is clear that the tangent vector is rotating clockwiseat both of (0 , / ± √ ξ ∞ so that this is an endpoint crossing. By Definition 2, onlypositive crossings are counted at the right endpoint.Excluding corner 2.5, it follows that Maslov( ϕ (cid:15) ) = −
1, with the only contributing conjugatepoint being the midpoint of the fast front. In the final section, we analyze corner 2.5. We nowknow that in order for the wave to be stable, there must be a conjugate point in the positivedirection to offset the one from the front.4.5.
Corner 2.5: Arrival at the right slow manifold.
As the wave ϕ ( x ) approaches z , weknow from § W u ( X − ) will be tangent to Y = span { η , η , η } . (4.18)The stable direction η is present because it is the limit as U → , −√ U ). In other words,the fast front much approach the fixed point (1 ,
0) tangent to the stable manifold of the fixed point.On the other hand, the jump-off condition (2.17) must be satisfied by incoming points if theyare to spend a long time near M + (cid:15) . This forces an abrupt reorientation of W u ( X − ) right at z . The IFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOV INDEX 15 fast direction will be unstable, and the slow directions are given by (4.14) evaluated at the landingpoint. It is not difficult to see that we can write these in terms of the η i as Z = span (cid:26) η , η + 1 D η + W z D η + V z η , − αβ η + η (cid:27) . (4.19)Our task is to figure out the (cid:15) > Y and Z in Λ(3). To do this,we first observe that the tangent space to W u ( X (cid:15) − ) evolves on the fast timescale because it alwayscontains at least one fast direction. It follows that this reorientation must happen arbitrarily closeto z as (cid:15) →
0. The relevant dynamical system to study is therefore Y (cid:48) ( x ) = BY ( x ) , (4.20)where B = A (0 , ξ ∗ ), and U ( ξ ∗ ) = 1. System (4.20) induces an equation on Gr ( R ), which we knowfrom (3.7) has Λ(3) as an invariant submanifold. The phase portrait of (4.20) is fairly simple anddescribed in detail in [14, Appendix B]. The highlights are that the fixed points are direct sums ofeigenspaces, each of which is hyperbolic. This includes Y . We will find the orbit connecting Y and Z by treating (4.20) as a boundary value problem. In this sense, our analysis is similar in spirit tothe Brunovsky approach to inclination lemmas [5, 6].Since Y is a fixed point, any candidate link between Y and Z must lie in W u ( Y ) ⊂ Λ(3). Usingrow reduction, we can compute that all such planes must be expressible in the form span { v , v , v } ,with v = η + a η + a η + a η v = η + a η v = η + a η . (4.21)Actually, (4.21) gives W u ( Y ) in the full Grassmannian. We can reduce this set to three dimensionsby applying the form ω pairwise on the v i and imposing the condition that it vanishes. As a result,we can express W u ( Y ) ⊂ Λ(3) as W u ( Y ) = sp { η + δ η + δ η + δ η , η + βD √ δ η , η + α √ δ η } , (4.22)for δ , , ∈ R .Comparing (4.22) and (4.19), it appears that we are in trouble since Z / ∈ W u ( Y ). However, theseexpressions all hold to leading order only, and we shall see that there are 3-planes arbitrarily closeto Z which do lie in W u ( Y ). To prove this, we express each of Z and a generic plane in W u ( Y ) inPl¨ucker coordinates [14, Appendix A] using the basis { η i } :Basis Vectors W u ( Y ) Z { , , } { , , } { , , } { , , } { , , } { , , } Basis Vectors W u ( Y ) Z { , , } { , , } { , , } α √ δ { , , } − βD √ δ { , , } { , , } { , , } { , , } δ { , , } α √ δ − αβ { , , } − βD √ δ δ { , , } δ { , , } α √ δ δ − αβD { , , } − βD √ δ D { , , } δ β ( W z /D ) + αV z β Pl¨ucker coordinates are projective, so only the ratio of terms has meaning. We use the notation p ijk for the Pl¨ucker coordinates of the plane spanned by η i , η j , and η k . The goal is to choose δ , , so that the resulting Lagrangian 3-plane is arbitrarily close to Z . We first compute the ratio p p = δ δ = − α/β − α/ ( βD ) = D. (4.23)One can verify that this choice is consistent with any other p ijk that are degree 2 in δ and δ . Tofix δ , we compute the ratio p p = αδ √ δ = − αβ ( W z /D ) + αV z , (4.24)from which we conclude that δ = − β ( W z /D ) + αV z √ δ . (4.25)Choosing δ and δ to satisfy (4.23) and (4.25) ensures that the Pl¨ucker coordinates have the properratios whenever the corresponding coordinates for Z are nonzero. Notice that the remaining non-zero coordinates for the generic 3-plane in W u ( Y ) are all first order in δ or δ . It follows that we IFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOV INDEX 17 can make the projective coordinates as close to 0 as desired by sending δ towards infinity. Thiscompletes the proof.Now that we have a plane X ∈ W u ( Y ) that is arbitrarily close to Z , all that remains is to computethe trajectory (in backwards time) through X . This is easily done since the η i are eigenvectors of B . Denoting the trajectory through (4.22) as Φ( x ), we see thatΦ( x ) = span e µ x η + δ e µ x η + δ e µ x η + δ e µ x η ,e µ x η + βD √ δ e µ x η ,e µ x η + α √ δ e µ x η , (4.26)for x ∈ ( −∞ , V, Φ( x )] , takingcare to write V in the basis { η i } :det e µ x δ e µ x δ e µ x e µ x e µ x
01 0 0 βD √ δ e µ x α √ δ e µ x δ e µ x = e ( µ + µ ) x ( δ e µ x − e µ x ) . (4.27)Using the fact that µ = − µ (for any (cid:15) ), we see that (4.27) vanishes if δ = e µ x . (4.28)The right hand side of (4.28) takes all values in [1 , ∞ ). Since | δ | (cid:29) Z , it followsthat there is a conjugate point if and only if δ >
0, or equivalently, β ( W z /D ) + αV z < . (4.29)Comparing Theorem 3 and (4.29), the conclusion is that there is a conjugate point at this corner ifand only if the wave is stable. We know from the previous sections that Maslov( ϕ (cid:15) ) = − ϕ (cid:15) ) = 0 for the whole wave.To verify this, we observe that the intersection of V and Φ( x ) is spanned by ψ = e µ x η + δ e µ x η + δ e µ x η + e µ x η . (4.30)At this corner, A (0 , ξ ) = B , and Bη i = µ i η i . It is straightforward then to calculate that ω ( ψ, A (0 , ξ ) ψ ) = ω ( e µ x η + e µ x η , µ e µ x η + µ e µ x η )= e µ x ( µ − µ ) ω ( η , η )= 2 √ e µ x ( µ − µ ) > . (4.31)5. Conclusion
The calculation of the previous section sheds new light on how the stability criterion (4.1)manifests itself in the construction of the wave. Considering (4.16) and (4.22), the sign of δ (i.e.,the stability criterion) is telling us about the exchange of the fast directions near M + (cid:15) . To betterunderstand this, we consider the projection of E u (0 , ξ ) onto the fast subsystem (2.7).The fast component of E u (0 , ξ ) along the jumps is tangent to ϕ . Along both F F ξ increases. As a result, the crossing form is negativedefinite at all crossings. The challenge is what happens at ( U, P ) = (1 , F F Figure 2.
The Maslov index is a winding number for ϕ (cid:48) ( ξ ) in the U P − plane. Theinequality (4.1) reflects the two ways in which ϕ (cid:48) can reorient itself for the back at( U, P ) = (1 , F F in the figure), ϕ (cid:48) ( ξ ) is tangent to η . In order to get back to η forsegment F
4, it can either continue rotating clockwise or reverse course and go counter-clockwise.In the latter case, the net angular rotation along the front ends up being 0, since the fixed points U = ± η + η ), and it must cross the horizontal subspace in the positive direction.This is exactly what we observed in § E u (0 , ξ ) with V had slowcomponents, the crossing form calculation (4.31) reduced to a calculation in the fast components.As expected, the crossing form was positive definite, so the winding along the front was “undone”at the corner. Conversely, in the unstable case we observed no conjugate points at corner 2.5. Thusthe fast components of E u (0 , ξ ) continued to rotate in the clockwise direction.The conclusion matches the intuition that we have from Sturm-Liouville theory; the stablestructures are the ones that oscillate less. The difference in this case is that the profiles of thestable and unstable waves look exactly the same. Instead, it is the the unstable bundles for thesewaves that do or do not oscillate. To connect this to phase space, the stable waves are the oneswhose unstable manifolds don’t contain any full twists as they go from −∞ to + ∞ .5.1. Discussion.
We will close by comparing the Maslov index calculation of this paper with thatof [14]. The latter paper analyzed traveling waves for the following FitzHugh-Nagumo system: u t = u zz + f ( u ) − vv t = v zz + (cid:15) ( u − γv ) . (5.1)The traveling wave and eigenvalue equations are each four-dimensional for (5.1), hence the unstablebundle is a curve in Λ(2). Nonetheless, beyond the difference in dimension, the two Maslov indexcalculations are quite similar. Most notably, the waves in both cases are homoclinic orbits consistingof two fast jumps between slow dynamics. Moreover, the Maslov index in each case comes down toa corner trajectory in the Lagrangian Grassmannian, as in § IFURCATION TO INSTABILITY THROUGH THE LENS OF THE MASLOV INDEX 19
The key difference between the two models is the mechanics of the corner calculation. In § § We showedthat the wave is stable by keeping track of the orientation of the unstable manifold during thefast-slow transitions.The technical reason for this difference is that the transversality condition needed to proveexistence of the waves (via the Exchange Lemma) requires (cid:15) > M + (cid:15) , and hencethe orientation of W u ( X − ) at z . The tangent space here is given by (4.19), which is not a fixedpoint for (4.20). In the traveling wave case, varying the speed c allows one to obtain transversalitywith (cid:15) = 0, so the reorientation of the unstable bundle at the corners is simply an exchange ofeigenvectors.Ignoring the technical reason, the two calculations give the impression that (in)stability is “in-evitable” in the standing wave case but “fortuitous” in the traveling wave case. To be more precise,the fate of the standing wave (with respect to stability) is sealed along the first fast jump. Forthe traveling wave, on the other hand, the potential conjugate point lives in the second corner.More importantly, there is no obvious reason to select one heteroclinic orbit over the other for thereorientation leading into the fast back. One is tempted to see if the FitzHugh-Nagumo fast wavescan be destabilized by somehow altering the equations to select the other orbit instead.The preceding discussion speaks to the motivation for using the Maslov index to study stabilityin the first place. By using detailed information about the wave and its phase space, we can obtainintrinsic reasons for stability or instability. Furthermore, this information can be used to distinguishdifferent mechanisms for instability, and perhaps show us how to generate such instabilities. Acknowledgments:
C.J. acknowledges support from the US Office of Naval Research under grantnumber N00014-18-1-2204.
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