Travelling Pulse Solutions for the Discrete FitzHugh-Nagumo System

Transcription

1 Travelling Pulse Solutions or the Discrete FitzHugh-Nagumo System H. J. Hupkes a,, B. Sandstede a a Brown University - Division o Applied Mathematics Astract The existence o ast travelling pulses o the discrete FitzHugh Nagumo equation is otained in the weakrecovery regime. This result extends to the spatially discrete setting the well-known theorem that states that the FitzHugh Nagumo PDE exhiits a ranch o ast waves that iurcates rom a singular pulse solution. The key technical result that allows or the extension to the discrete case is the Exchange Lemma that we estalish here or unctional dierential equations o mixed type. Key words: lattice dierential equations, travelling waves, singular perturation theory, exchange lemma, Lin s method, discrete FitzHugh-Nagumo. 1 Introduction In this paper we consider the discrete FitzHugh Nagumo equation u i (t) = α[u i+1 (t) + u i 1 (t) 2u i (t)] + g ( u i (t) ) w i (t), ẇ i (t) = ɛ ( u i (t) γw i (t) ), (1.1) where u i, w i R or each i Z. The nonlinearity g is taken rom a class o istale nonlinearities that includes the cuic polynomial g(u; a) = u(1 u)(u a) or some 0 < a < 1 2. We consider aritrary positive coupling coeicients α > 0, take 0 < ɛ 1 to e small, and assume that γ > 0 is not too large so that {(u i, w i )} i Z = (0, 0) is the only i-independent rest state o (1.1); this requires that g(γw) w or all w 0. Our primary reason or looking at the spatially discrete FitzHugh Nagumo equation is its relevance in modelling. For example, when studying the propagation o electrical signals through nerve iers, it turns out to e more natural to study the discrete system (1.1) instead o its continuous counterpart that is traditionally used or this purpose. This is related to the act that a nerve axon is almost entirely surrounded y an insulating myeline coating that admits small gaps at regular Corresponding author: H.J. Hupkes Division o Applied Mathematics Brown University 182 George Street Providence, RI tel address: hjhupkes@dam.rown.edu (H. J. Hupkes). Preprint sumitted to... Novemer 4, 2009

2 intervals. These nodes were irst oserved in 1878 y Ranvier [42] and now carry his name. The insulation induced y the myeline causes excitations o the nerve at these nodes o Ranvier to eectively jump rom one node to the next, through a process called saltatory conduction. This mechanism was irst suggested in 1925 y Lillie [37] and demonstrated convincingly in 1949 y Huxley and Stämpli [29]. Other discrete lattice models have appeared in a wide range o scientiic disciplines, including chemical reaction theory [19, 36], material science [2, 5] and image processing and pattern recognition [13]. Our goal is to show that (1.1) admits travelling pulse solutions (u i, w i )(t) = (u, w )(i + ct) or some positive wave speed c > 0, where the proiles (u, w ) are localized so that (u, w )(ξ) 0 as ξ ±. Sustituting our ansatz into (1.1), we see that these proiles must satisy the system cu (ξ) = α[u (ξ + 1) + u (ξ 1) 2u (ξ)] + g ( u (ξ) ) w (ξ), cw (ξ) = ɛ ( u (ξ) γw (ξ) ). (1.2) Such equations are reerred to as unctional dierential equations o mixed type (MFDEs), since they contain oth advanced and retarded terms. This class o equations is notoriously diicult to analyse. Previous work on the discrete FitzHugh Nagumo equation and variants thereo can e split into two main directions. On the one hand, rigorous results have een otained or specially tailored nonlinearities g. Tonnelier [45] and Elmer and Van Vleck [18], or example, considered the McKean sawtooth caricature o the cuic, while Chen and Hastings [10] studied a discrete Morris Lecar type system with a nonlinearity that vanishes identically on certain critical regions o u and w. On the other hand, using asymptotic techniques, ormal results have een otained or (1.1) y Carpio and coworkers [7 9]. We are, however, not aware o any rigorous results or (1.1) that hold or the cuic polynomial or more general istale nonlinearities, and it is this issue that we shall address in this paper. Beore outlining our result in more detail, we riely discuss the spatially continuous case, or which a large ody o literature exists. 1.1 Travelling waves or the FitzHugh Nagumo PDE Let us thereore consider the spatially continuous FitzHugh Nagumo system u t = u xx + g(u) w, w t = ɛ(u γw), (1.3) where x R. This partial dierential equation (PDE) plays an important role as a tractale simpliication o the Hodgkin Huxley equations that are widely used to model the propagation o signals through myelinated nerve iers [25]. As a consequence, (1.3) has een analysed extensively in the literature. A large portion o the results that have een otained concern travelling-wave solutions to (1.3), that is, solutions o the orm (u, w)(x, t) = (u, w)(x + ct) that depend on the single argument ξ = x + ct. Such solutions must satisy the ordinary dierential equation (ODE) u = v, v = cv g(u) + w, (1.4) w = ɛ c (u γw). For simplicity, we take g to e the cuic nonlinearity g(u; a) = u(1 u)(u a). Note that the origin p 0 = (0, 0, 0) is an equilirium or (1.4) regardless o the precise values o a, c and ɛ. Finding travelling pulses o (1.3) then amounts to constructing homoclinic orits or (1.4) that are i-asymptotic to p 0. 2

3 Fig. 1: Phase diagram or the travelling-wave equation (1.4) o the FitzHugh Nagumo PDE. In the regime 0 < ɛ 1, this question can e answered using geometric singular perturation theory. We will now outline this approach and reer to Figure 1 or an illustration. First, we set ɛ = 0 in (1.4) to get the system u = v, v = cv g(u) + w, w = 0, (1.5) which admits a maniold M o equiliria that consists o all points (u, 0, w) that have w = g(u). Oviously, this maniold contains the points p 0 and p 1 = (1, 0, 0). One can now choose a neighorhood M L M around p 0 together with a neighorhood M R M around p 1. I these neighorhoods do not contain the knees o the cuic polynomial, they are normally hyperolic invariant maniolds that hence persist as locally invariant sets or small ɛ > 0 as a consequence o Fenichel s irst theorem [20]. It is well-known that, or each ixed 0 < a < 1 2, there exists a heteroclinic solution Q = (q, q, 0) to (1.5) that connects p 0 to p 1 and has wave speed c = c or some c > 0: indeed, these solutions correspond to travelling ronts o the Nagumo equation u t = u xx + g(u). (1.6) In addition, or any such a there exists a w > 0 such that (1.4) with ɛ = 0 and c = c admits a heteroclinic solution Q = (q, q, w ) that connects M R to M L. We can now write Γ s 0 or the singular orit that arises y comining these orits with the segments o M R and M L that connect w = 0 to w = w. The superscript s is used in view o the act that we are considering ast waves with speed c > 0. The ollowing well-known result is the ODE analogue o the result we set out to otain or the unctional dierential equation (1.2). Proposition 1.1 Consider (1.4) with the cuic nonlinearity g = g( ; a) or any ixed 0 < a < 1 2, then there exists a unique curve in the (ɛ, c)-plane emanating rom the point (0, c ) that consists o homoclinic solutions to (1.4) that are i-asymptotic to 0, while eing O(ɛ)-close to Γ s 0 and winding around Γ s 0 exactly once. The irst proos estalishing the existence o the ranch o homoclinics descried in the result aove are due to Carpenter [6] and Hastings [24], who otained their results independently using 3

4 Fig. 2: Illustration o the geometric setting o the Exchange Lemma. classical singular perturation theory [6] and the Conley index [24]. A more streamlined proo o Proposition 1.1 that also gives transversality and local uniqueness is ased on a geometrical construction developed y Jones and coworkers [33]. The idea is to construct the desired homoclinic orits as an intersection o the unstale maniold W u (0) and the stale maniold W s (M L ). The main diiculty is to track the unstale maniold W u (0) as it passes close to M R, since it spends time o order O(ɛ 1 ) here. The tool developed in [33] to deal with this tracking is reerred to as the Exchange Lemma. We reer to Figure 2 or an illustration o the geometric setting o this result, which we descrie here riely. The statement o the Exchange Lemma can e explained most easily in terms o the Fenichel normal orm [20, 32]: in a neighorhood o M R, the ODE (1.4) can e put into the orm x = A s (x, y, z)x, y = A u (x, y, z)y, z = ɛ[1 + B(x, y, z)xy], (1.7) where the new coordinates x, y and z are real-valued, the unctions A s, A u and B are smooth, and A s and A u are ounded rom elow y some constant η > 0. The Exchange Lemma then states that (1.7) has, or each z 0 R, each suiciently large T and each suiciently small ɛ > 0 and > 0, a solution that satisies the oundary conditions x(0) =, z(0) = z 0 and y(t ) =. Furthermore, the norms y(0), x(t ) and z(t ) z 0 ɛt and their derivatives with respect to T, ɛ, z 0 and any other parameters that may appear in the prolem are o order e ηt as T. Instead o attempting to analyse the intersection o W u (0) with W s (M R ) directly, one can now decouple the prolem or large T and study separately how W u (0) and W s (M L ) ehave near x = and y =, respectively, which are ar easier to analyse and lead to a two-dimensional nonlinear system that involves the three variales ɛ, c and T [35]. This system can e solved to yield the ranch o homoclinic orits descried in Proposition 1.1. A great deal more is known aout (1.4). For example, the PDE staility o the resulting ast travelling pulses was proved independently y Jones [30] and Yanagida [46]. It is also known that there is a second slow travelling wave that exists, or ixed 0 < a < 1 2, in the limit c 0 and ɛ/c 0. The resulting singular homoclinic orit Γ sl 0 or ɛ = ɛ/c = 0 is actually a regular homoclinic orit to the origin that lives in the plane w = 0. Since Γ sl 0 does not contain any segments o M L and M R, perturations are easier to analyse and one may show that a ranch o slow homoclinic solutions can e constructed near Γ sl 0 or small c > 0 and ɛ > 0 [44]. A conjecture due to Yanagida [46] states that these ranches o ast and slow waves connect to each other. At the moment, this has only een conirmed or a near the critical point a = 1 2, where the two singular orits Γ s 0 and Γ sl 0 coalesce [35]. We remark that [35] also contains a proo that, 4

5 somewhere along this connecting curve, the homoclinic orits undergo an inclination-lip iurcation. The presence o such a iurcation makes it very likely to ind homoclinic solutions that wind around the singular orit an aritrary numer o times. To estalish this rigorously, a speciic non-degeneracy condition needs to e veriied. At the moment, this is only easile when considering orits that have winding numer two, in which case a result due to Nii [41] can e invoked. 1.2 The discrete Nagumo equation The construction o ast pulses o the continuous FitzHugh Nagumo equation relied on gluing travelling ronts and acks o appropriate Nagumo equations (1.6) together. Thus, it is natural to start our discussion o the discrete FitzHugh Nagumo system y summarizing a ew key eatures o the discrete Nagumo equation u i (t) = α[u i+1 (t) + u i 1 (t) 2u i (t)] + g(u i (t); a) (1.8) with g(u; a) = u(1 u)(u a). Travelling waves u i (t) = u (i+ct) o (1.8) satisy the scalar unctional dierential equation cu (ξ) = α[u (ξ + 1) + u (ξ 1) 2u (ξ)] + g(u (ξ); a) (1.9) o mixed type. The irst numerical study o travelling ronts o (1.1) was conducted y Chi, Bell and Hassard [11]. Since that early paper, the discrete Nagumo equation and the associated travelling-wave MFDE (1.9) have served as prototype systems or investigating the properties o lattice dierential equations. In contrast to the continuous case where travelling ronts with positive wave speeds exist or each 0 < a < 1 2, the discrete Nagumo equation may not support travelling ronts or each such a. The reason is that the limit c 0 in (1.9) is highly singular. Indeed, the limiting system is a map which may admit transverse heteroclinic orits that preclude the existence o travelling ronts. More precisely, the comined results o Keener [34] and Mallet-Paret [39, Theorem 2.6] give the ollowing: or each suiciently small α > 0, there exists an 0 < a 0 < 1 2 such that, or each a [a 0, 1 2 ], heteroclinic solutions to (1.9) that connect the two equiliria u = 0 and u = 1 exist i and only i c = 0. This eature is called propagation ailure and distinguishes (1.8) rom its continuous counterpart (1.6). By now there is an aundance o numerical evidence showing that this phenomenon may occur in an extremely roust ashion throughout a wide range o discrete systems [1, 15 17]. One implication o this eature or the present work is that we need to assume that a does not lie inside the region o propagation ailure or the discrete Nagumo equation. We remark that propagation ailure in the underlying discrete Nagumo equation is the reason why slow waves do not exist or the discrete FitzHugh Nagumo equation in the same way as they do or the continuous case. Hence, we ocus on ast waves in this paper. 1.3 Travelling waves or the discrete FitzHugh Nagumo system We now turn to the travelling-wave equation (1.2), cu (ξ) = α[u(ξ + 1) + u(ξ 1) 2u(ξ)] + g ( u(ξ) ) w (ξ), cw (ξ) = ɛ ( u(ξ) γw(ξ) ), (1.10) associated with the discrete FitzHugh Nagumo equation (1.1). Our goal is to ind an appropriate value o c > 0 and construct solutions (u, w)(ξ) o this MFDE or 0 < ɛ 1 that converge to zero as ξ. Similar to the case o delay equations, the state space associated with (1.10) will necessarily e ininite-dimensional, and we will consequently work with (u, w) Y = C([ 1, 1], R) R in this paper. In contrast to the case o delay equations, however, the initial-value prolem associated with 5

6 (1.10) on the space Y is ill-posed 1 due to the presence o oth advanced and retarded terms. This issue prevents us rom using the semigroup techniques developed or retarded dierential equations [14]. An alternative strategy is to utilize Fredholm properties and exponential dichotomies, which were developed or MFDEs y Mallet-Paret [38], Verduyn Lunel [40] and Härterich, Sandstede and Scheel [23]. This approach was recently used successully y Hupkes and Verduyn Lunel to extend Lin s method to MFDEs [27]. The key complication that needs to e overcome eore homoclinic solutions to (1.10) can e constructed in the singular limit ɛ 0, is that geometric singular perturation theory is not readily availale or MFDEs. Indeed, this theory relies heavily on the existence o semilows which, as we outlined aove, do not exist in our MFDE setting. For instance, almost all proos o Fenichel s irst theorem [20] aout the persistence o normally hyperolic slow maniolds are ased on Hadamard s graph transorm technique [21]. The approach that we use in this paper to resolve these issues is ased on a comination o the ideas contained in [26, 27, 35, 43]. First, the work o Sakamoto [43] uses analytic techniques to estalish Fenichel s irst theorem or ODEs through a systematic use o the concept o slowly varying coeicients. Comining this approach with our recent results [26] concerning linear MFDEs that have slowly varying coeicients we construct one-dimensional slow maniolds M L and M R or (1.10) that persist or small ɛ > 0. To prove an appropriate version o the Exchange Lemma, we exploit the ideas in [35] in which an analytic proo was given that is ased on Lin s method. The key eature o this approach is that, unlike earlier proos using dierential orms [31] or oundary-value techniques [4], the construction o the stale and unstale iers o M R can e done gloally, therey allowing us to immediately reduce the existence prolem to a inite set o nonlinear equations, similar to those that we need to solve. Borrowing the techniques used in [27] to generalize Lin s method to MFDEs and again applying the slowly-varying coeicient ramework developed in [26], we can imitate this construction in the current setting. While we concentrate in this paper on the concrete discrete FitzHugh Nagumo equation, we elieve that our techniques can e used in a much wider context than in just the construction o pulses or the speciic system (1.1). For example, we expect that ater some minor adaptations it should e possile to study travelling multi-pulse solutions or long-period wave train solutions to general MFDEs in which a slow time-scale can e identiied. In addition, we currently use our singular perturation ramework to assess the staility o the ast waves constructed here with respect to the dynamics o the underlying lattice equation. The rest o this paper is organized as ollows. We state our main result in 2 and give a detailed outline o the main steps that are need to prove this result in 3, while hiding most o the technical details ehind a sequence o propositions. The invariant slow maniolds M L (c, ɛ) and M R (c, ɛ) are constructed throughout 4. We then study the stale and unstale oliations o these slow maniolds in 5 and develop a suitale version o the Exchange Lemma in 6. Section 7 contains a rie discussion. 2 Main result Recall that travelling waves o the discrete FitzHugh Nagumo equation (1.1) can e ound as solutions o the system cu (ξ) = α[u(ξ + 1) + u(ξ 1) 2u(ξ)] + g ( u(ξ) ) w(ξ), cw (ξ) = ɛ ( u(ξ) γw(ξ) ). (2.1) Throughout this paper we will assume that α > 0 and γ > 0. The prototype nonlinearity that we have in mind is given y the cuic polynomial g = g(u; a) = u(1 u)(u a) or some 0 < a < 1 2. However, we will ocus on a roader class o istale nonlinearities in order to illustrate the generality 1 In general, given an initial condition in Y, we cannot solve (1.10) orward or ackward in ξ. 6

7 Fig. 3: Illustration o the assumptions on the nonlinearity g(u) and the constant γ. o our approach. We thereore impose the ollowing generic assumptions on the nonlinearity g, which are also illustrated in Figure 3. Hypothesis (H1) The nonlinearity g is C r+3 -smooth or some integer r 2. Hypothesis (H2) We have g(0) = g(1) = 0, g (0) < 0 and g (1) < 0. On account o condition (H2), we may choose closed intervals I L and I R, with 0 I L and 1 I R, that have non-empty interior and in addition have g (u) < 0 or all u I L I R. We pick constants w min < 0 and w max > 0 in such a way that oth w min, w max g(i L ) g(i R ). The implicit unction theorem can now e used to deine two C r+3 -smooth unctions s L : [w min, w max ] I L and s R : [w min, w max ] I R in such a way that g ( s L (w) ) = g ( s R (w) ) = w or all w [w min, w max ]. Notice that s L (0) = 0 and s R (0) = 1. Our next assumption roughly states that g is N-shaped, admitting precisely one extra solution to g(u) = w. Hypothesis (H3) For any w [w min, w max ], there exists a ρ ( s L (w), s R (w) ) such that g(ρ) = w. In addition, we have g(u) > w, u (, s L (w) ) ( ρ, s R (w) ), g(u) < w, u ( s L (w), ρ ) ( s R (w), ). In order to ensure that (2.1) admits a suitale singular orit, we will need to assume that this equation with ɛ = 0 admits a ront and a ack solution that propagate at the same wave speed. Hypothesis (H4) There exist two constants w (0, w max ) and c > 0 such that (2.1) with ɛ = 0 and c = c admits two solutions (q, 0) and (q, w ) that satisy the limits lim (ξ) ξ = 0, lim (ξ) ξ = 1, lim (ξ) ξ = s R (w ), lim (ξ) ξ = s L (w ). (2.2) We remark here that [38, Proposition 5.3] in comination with the act that g (0) 0, g (1) 0, g ( s L (w ) ) 0 and g ( s R (w ) ) 0 allows us to conclude that q and q approach their limits at ± at an exponential rate. Such an argument is made explicit in the proo o [39, Theorem 2.2]. In contrast to the setting o Proposition 1.1, the possiility o propagation ailure prevents us rom otaining results that hold or the cuic polynomial g( ; a) with aritrary 0 < a <

8 Fig. 4: Panel (i) shows the graph o a given unction u, while panel (ii) illustrates the associated unction ev ξ u : [ 1, 1] R or a ixed ξ. Lemma 2.1 Fix any positive coupling coeicient α > 0, then the conditions (H1)-(H4) are satisied or the cuic nonlinearity g(u) = u(1 u)(u a) provided a > 0 is suiciently small. Proo. The conditions (H1) through (H3) are oviously satisied. Using [39, Theorem 2.6], one can conclude the existence o a wave speed c > 0 and a wave proile q such that the pair (q, 0) satisies (2.1) with ɛ = 0 and c = c, while satisying the limits given in the irst line o (2.2). The requirement that a is suiciently small is needed here to ensure that the wave speed c does not vanish. To otain the existence o the pair (q, w ) that solves (2.1) at the same speed c, one may exploit the mirror symmetry o cuic polynomials. Our inal assumption concerns the parameter γ and ensures that, or any ɛ > 0 and c 0, the only equilirium solution to (2.1) is given y (0, 0). Hypothesis (H5) The parameter γ > 0 is so small that g(γw) w or all w 0. Let us now write Γ s 0 or the singular homoclinic orit that arises y ollowing the heteroclinic connection q rom (0, 0) to (1, 0), moving along the maniold M R := {( s R (w), w)} rom (1, 0) to ( s R (w ), w ), ollowing q rom ( s R (w ), w ) to ( s L (w ), w ) and inally moving ack to (0, 0) along the maniold M L := {( s L (w), w)}. Our main result is concerned with homoclinic solutions to (2.1) that iurcate o Γ s 0 as ɛ moves away rom zero and wind around this singular homoclinic exactly once. In order to deine this winding numer properly, we need to have a notion o transversality that will allow us to construct Poincaré sections. The winding numer can then e related to the numer o times a homoclinic orit passes through these sections. Let us thereore write X = C([ 1, 1], R) or the state space associated with the irst component o (2.1). The state o a unction u C(R, R) at ξ R will e denoted y ev ξ u X = C([ 1, 1], R) and is deined y [ev ξ u](θ) := u(ξ + θ), θ [ 1, 1]; see Figure 4. We can now pick two suspaces X and X o X such that X = span{ev 0 q } X, X = span{ev 0 q } X. (2.3) We are now ready to descrie the type o solutions to (2.1) that we are interested in and reer to Figure 5 or an illustration. Deinition 2.2 (Homoclinic solution) For each 0 < δ 1 and ξ 1, we say that a pair (u, w) C(R, R 2 ) is a (δ, ξ )-homoclinic solution i (u, w) satisies (2.1) or all ξ R and meets the ollowing conditions: (i) There exists exactly one ξ ev 0 q + X. R or which evξ u ev 0 q < δ, w(ξ ) < δ, and ev ξ u 8

9 Fig. 5: Shown are the singular homoclinic orit Γ s 0 and the location o the two Poincaré sections along the ront (ev ξ q, 0) and the ack (ev ξ q, w ) in the underlying phase space C([ 1, 1], R) R. (ii) There exists exactly one ξ R or which ev ξ u ev 0 q < δ, w(ξ ) w < δ, and ev ξ u ev 0 q + X. (iii) We have lim ξ ± ( u(ξ), w(ξ) ) = 0. (iv) The solution (u, w) is close to Γ s 0 in the sense that u(ξ) q (ξ( ξ ) ) < δ and w(ξ) < δ or ξ ξ + ξ, u(ξ) s R w(ξ) < δ or ξ + ξ ξ ξ ξ, u(ξ) q (ξ ( ξ ) ) < δ and w(ξ) w < δ or ξ ξ ξ ξ + ξ, u(ξ) s L w(ξ) < δ or ξ + ξ ξ. Our main result shows that y varying the wave speed c, one may otain a one-parameter ranch o such solutions that iurcates away rom Γ s 0. Theorem 1 Consider the nonlinear system (2.1) and suppose that Hypotheses (H1)-(H5) hold, then there are constants 0 < δ 1 and ξ 1 with the ollowing property: or each c < c that is suiciently close to c, there exists a unique ɛ = ɛ(c) > 0 or which (2.1) admits a (δ, ξ )-homoclinic solution (u, w). This pair (u, w) is O(c c )-close to Γ s 0 and unique up to translations. 3 Proo o Theorem 1 Our proo o Theorem 1 is split into our main parts. In this section we will descrie each o these steps, hiding the technical details ehind our propositions that will e proved throughout the remainder o this paper. At the end o this section, our main claim will have een reduced to a statement concerning the roots o a two-dimensional nonlinear system involving three variales. The desired one-parameter ranch o (δ, ξ )-homoclinic solutions to (2.1) can susequently e read o rom these equations. The our main parts o our argument can e outlined roughly as ollows. First, we consider the equilirium maniolds M L = {( s L (w), w)} and M R = {( s R (w), w)}. We show that these curves can e pertured to yield slow maniolds M L (c, ɛ) and M R (c, ɛ) that remain invariant when considering (2.1) with small ɛ > 0 and c c. In the next step, we show that, or each ɛ 0 and c c, there are two unique solutions near the ront (q, 0) so that the irst orit lies in the ininite-dimensional unstale maniold o (0, 0), the second solution lies in the ininite-dimensional stale maniold o M R (c, ɛ), and their dierence at ξ = 0 is contained in a certain one-dimensional suspace. Thus, up to this one-dimensional jump, these maniolds have a unique intersection near the ront. We reer to such connections as quasi-ront 9

10 solutions and reer to Figure 7 elow or an illustration. Similarly, or each such ɛ and c, and or each choice o w 0, there are unique quasi-ack solutions in the unstale maniold o M R (c, ɛ) and the strong stale ier o M L (c, ɛ) elonging to w = w 0, respectively, so that their dierence at ξ = 0 again lies in an appropriate ixed one-dimensional suspace. Using the Hale inner product, which is tailored speciically or unctional dierential equations, the derivatives o the aorementioned jumps with respect to the three ree parameters can e related to Melnikov-type integrals whose signs we can evaluate. In the third step, we prove an Exchange Lemma or MFDEs that allows us to match quasi-ronts and quasi-acks as they pass near the maniold M R (c, ɛ). This can e done up to two extra jumps that lie in the same one-dimensional spaces that we discussed aove. These extra jumps turn out to e C 1 -exponentially small with respect to the time spent near the slow maniold M R (c, ɛ). This allows us to set up and analyse the resulting two-dimensional nonlinear system that descries the size o the remaining gaps in the inal step. 3.1 Step 1 - The slow maniolds We now descrie the slow maniolds M L (c, ɛ) and M R (c, ɛ) in more detail. In order to avoid complications that arise when w leaves the region [w min, w max ], we will need to modiy (2.1). To this end, we choose a C -smooth cut-o unction χ sl : R R as shown in Figure 6 and consider the system cu (ξ) = α[u(ξ + 1) + u(ξ 1) 2u(ξ)] + g ( u(ξ) ) w(ξ), cw (ξ) = ɛ ( u(ξ) γw(ξ) ) χ sl ( w(ξ) ), (3.1) instead o working directly with (2.1). The ollowing result will e estalished in 4. Proposition 3.1 Consider the nonlinear system (3.1) and suppose that (H1)-(H3) are satisied, then there exist constants δ ɛ > 0 and δ c > 0, together with two C r+2 -smooth unctions such that the ollowing is true: s R, s L : [w min, w max ] [c δ c, c + δ c ] [0, δ ɛ ] R, (i) For each ϑ [w min, w max ], c [c δ c, c +δ c ] and ɛ [0, δ ɛ ], we have the identities s R (ϑ, c, 0) = s R (ϑ) and s L (ϑ, c, 0) = s L (ϑ). (ii) For each ϑ [w min, w max ], c [c δ c, c + δ c ] and ɛ [0, δ ɛ ], consider the unique solution o the ODE cθ (ξ) = ɛ ( s R (θ(ξ), c, ɛ) γθ(ξ) ) χ sl ( θ(ξ) ), θ(0) = ϑ, (3.2) then the pair (u, w) deined y u(ξ) = s R (θ(ξ), c, ɛ) and w(ξ) = θ(ξ) satisies (3.1). The same statement holds upon replacing the suscript R y L. (iii) There exists a constant δ > 0 such that any solution (u, w) to (3.1) with c c < δ c and 0 ɛ δ ɛ that has oth w min w(ξ) w max and u(ξ) sr ( w(ξ) ) < δ or all ξ R must in act satisy u(ξ) = s R (w(ξ), c, ɛ) or all ξ R. The same statement holds or the suscript L. Fig. 6: The deinition o the cut-o unction χ sl (w). 10

11 The unctions s L and s R can e used to deine the invariant maniolds M L (c, ɛ) and M R (c, ɛ) mentioned at the start o this section. In particular, we take M L (c, ɛ) = {(s L (w, c, ɛ), w)} and M R (c, ɛ) = {(s R (w, c, ɛ), w)}, letting w run through the interval [w min, w max ]. In the sequel we will oten need to reer to the low on these maniolds M L and M R, so we will introduce some notation here or convenience. Recall the constants δ c > 0 and δ ɛ > 0 that appear in Proposition 3.1 and introduce the unctions that are given y T R, T L : [w min, w max ] [c δ c, c + δ c ] [0, δ ɛ ] R T R (ϑ, c, ɛ) = [s R (ϑ, c, ɛ) γϑ]χ sl (ϑ), T L (ϑ, c, ɛ) = [s L (ϑ, c, ɛ) γϑ]χ sl (ϑ). For each ϑ [w min, w max ], c [c δ c, c + δ c ] and ɛ [0, δ ɛ ], we write Θ s R (ϑ, c, ɛ) C(R, R) to denote the unique solution o the ODE cθ (ξ) = ɛt R (θ(ξ), c, ɛ), θ(0) = ϑ. (3.3) Similarly, we introduce the notation Θ s L (ϑ, c, ɛ) C(R, R) to denote the unique solution o the ODE cθ (ξ) = ɛt L (θ(ξ), c, ɛ), θ(0) = ϑ. (3.4) The superscript s reers to the act that (3.3) and (3.4) are written in terms o the ast time scale. In contrast, we will write Θ sl R (ϑ, c, ɛ) or the unique solution o the ODE cθ (ζ) = T R (θ(ζ), c, ɛ), θ(0) = ϑ, where the superscript now indicates that we solve with respect to the slow time scale. Note that (ϑ, c, ɛ)(ξ) = Θsl R (ϑ, c, ɛ)(ɛξ). Θ s R 3.2 Step 2 - Quasi-ronts and quasi-acks: stale and unstale oliations We now construct the quasi-ront connections etween (0, 0) and M R (c, ɛ) that are illustrated in Figure 7. As shown there, the construction depends on the choice o the one-dimensional suspace Γ that contains the dierence o two solutions o the underlying MFDE. Thus, we irst ocus on outlining our choice o Γ and o the space Γ that we shall use to construct quasi-ack solutions which connect M R (c, ɛ) ack to (0, 0). We will use the decomposition (2.3), X = span{ev 0 q } X = span{ev 0 q } X, o the phase space X = C([ 1, 1], R) and the associated Poincaré sections that we introduced in Deinition 2.2(i)-(ii) or this purpose. Our goal is to ind suitale suspaces Γ and Γ o X and X that contain the jumps. As a preparation, we sustitute the ansatz u(ξ) = q (ξ) + v(ξ) into the irst equation o (2.1) and set w = 0 and ɛ = 0. We otain the variational MFDE cv (ξ) = α[v(ξ + 1) + v(ξ 1) 2v(ξ)] + g ( q (ξ) + v(ξ) ) g ( q (ξ) ) whose linearization aout v = 0 gives the operator Λ : W 1,1 loc (R, R) L1 loc (R, R) with [Λ v](ξ) = cv (ξ) α[v(ξ + 1) + v(ξ 1) 2v(ξ)] g ( q (ξ) ) v(ξ). We also deine the ormal adjoint Λ : W 1,1 loc (R, R) L1 loc (R, R) o Λ via [Λ v](ξ) = cv (ξ) + α[v(ξ + 1) + v(ξ 1) 2v(ξ)] + g ( q (ξ) ) v(ξ). 11

12 Fig. 7: Shown is a quasi-ront solution which consists o two solutions that lie respectively in the unstale maniold o the equilirium (u, w) = 0 and the stale oliation o the slow maniold M R. These solutions will, in general, not coincide ut we will show that they can e chosen so that their dierence at ξ = 0 lies in the one-dimensional suspace Γ o the phase space C([ 1, 1], R). The new equiliria inside M R are created y the cut-o unction in (3.1). The dual product etween Λ and Λ is provided through the Hale inner product [22], which is given y [ 1 1 ] ψ, φ = cψ(0)φ(0) α ψ(σ 1)φ(σ) dσ + ψ(σ + 1)φ(σ) dσ 0 0 or any pair φ, ψ X. It was estalished in [40] that the Hale inner product is non-degenerate in the sense that i ψ, φ = 0 or all ψ X then necessarily φ = 0. A key eature o the Hale inner product is the identity d dξ ev ξψ, ev ξ φ = ψ(ξ)[λ φ](ξ) + [Λ ψ](ξ)φ(ξ), (3.5) which holds or any pair ψ, φ C 1 (R, R) and ξ R. Indeed, i we pick ψ in such a way that Λ ψ = 0, one readily sees that integration o (3.5) will yield Melnikov-type identities. In view o these considerations, it is important to understand the kernels K = {φ C 1 (R, R) Λ φ = 0 and φ < }, K = {ψ C 1 (R, R) Λ ψ = 0 and ψ < }. (3.6) In addition, we will also need to consider the kernels K and K that arise in the exact same ashion ater sustituting the ansatz u(ξ) = q (ξ) + v(ξ) into (2.1), while keeping w = w and ɛ = 0 ixed. The ollowing result ollows directly rom [39, Theorem 4.1]. Lemma 3.2 Consider the nonlinear system (2.1) and suppose that (H1)-(H4) are satisied, then we have q (ξ) > 0 and q (ξ) < 0 or all ξ R, together with K = span{q }, K = span{q }. In addition, there exist two ounded unctions d and d that decay exponentially at oth ± and have d (ξ) > 0 and d (ξ) > 0 or all ξ R, such that K = span{d }, K = span{d }. Let us consider any non-zero d K, write X = {φ X ev 0 d, φ = 0} 12

13 and deine X in the analogous ashion. Note that X X is closed and o codimension one and that the same holds or the inclusion X spaces Γ X and Γ X and write X. This allows us to choose appropriate one-dimensional X = span{ev 0 q } X Γ = span{ev 0 q } X Γ. (3.7) By construction, any φ Γ satisies φ = 0 i and only i ev 0 d, φ = 0, which in comination with (3.5) ensures that Γ and Γ are ideally suited to capture the jumps that the quasi-ronts and quasi-acks make when they pass through the hyperplanes ev 0 q X and ev 0 q X. As a inal preparation, let us consider the homogeneous MFDEs cv (ξ) = α[v(ξ + 1) + v(ξ 1) 2v(ξ)] g ( s j (ϑ) ) v(ξ), or j = L, R, where the quantity ϑ is taken rom [w min, w max ]. Looking or solutions o the orm v(ξ) = e zξ we otain the characteristic equations j,ϑ (z) = 0, with j,ϑ (z) = cz α[e z + e z 2] g ( s j (ϑ) ). (3.8) Notice that Im j,ϑ (iκ) = cκ or any κ R, while Re j,ϑ (0) = g ( s j (ϑ) ). Our choice o the constants w min and w max hence ensures that we can pick η > 0 in such a way that the characteristic equations j,ϑ (z) = 0 have no roots with Re z η or any c 0, any ϑ [w min, w max ] and j = L, R. This constant η will e used uiquitously throughout this paper. We are now ready to deine the concept o a quasi-ront solution; see again Figure 7. Recall the quantities δ c > 0 and δ ɛ > 0 that appear in Proposition 3.1 and ix c [c δ c, c +δ c ] and ɛ [0, δ ɛ ]. Deinition 3.3 (Quasi-ront solution) For each 0 < δ 1 and ξ 1, we say that the quadruplet (u, u +, w, ϑ) C((, 1], R) C([ 1, ), R) C(R, R) [ δ, δ] is a (δ, ξ )-quasi-ront solution i the ollowing is true: (i) The pair (u ±, w) satisies (3.1) on the interval R ±. (ii) We have lim ξ (u (ξ), w(ξ)) = 0 and u(ξ) q (ξ) < δ and w(ξ) < δ or ξ ξ u + (ξ) s R ( w(ξ) ) < δ or ξ ξ, where u(ξ) should e read as u (ξ) or ξ 1, as u + (ξ) or ξ 1, and as oth u ± (ξ) in the region 1 ξ 1. (iii) We have lim ξ e η ξ [w(ξ) Θ s R (ϑ, c, ɛ)(ξ)] = 0. (iv) We have ev 0 u ev 0 q X, ev 0 u + ev 0 q X, and ev 0 [u u + ] Γ. Roughly speaking, these properties imply that u ± and w can e comined to uild a solution to (3.1) that remains δ-close to the portion o the singular orit Γ s 0 that consists o q and M R and that is continuous everywhere except on the interval [ 1, 1]. On this interval the solution is doule-valued, with a dierence that is contained in Γ. We need one more deinition eore we can state our result concerning the existence o quasiront-solutions. To this end, consider any interval I R. We introduce the ollowing amily o Banach spaces, parametrized y η R, { } BC η (I, R) = x C(I, R) x η := sup ξ I e η ξ x(ξ) <. In the sequel, we will also use the spaces BC 1 η(i, R) = {y BC η (I, R) y BC η (I, R)}. 13

14 Fig. 8: Shown is a quasi-ack solution that connects M R to (0, 0) and has a discontinuity at ξ = 0 with a jump that lies in the one-dimensional Γ. Proposition 3.4 Consider the nonlinear equation (3.1) and assume that (H1)-(H4) are satisied, then there are constants ξ 1 and 0 < δ, δ c, δ ɛ 1 and a set o maps that satisies the ollowing properties. u : [c δ c, c + δ c ] [0, δ ɛ ] C((, 1], R), u + : [c δ c, c + δ c ] [0, δ ɛ ] C([ 1, ), R), w : [c δ c, c + δ c ] [0, δ ɛ ] C(R, [w min, w max ]), ϑ : [c δ c, c + δ c ] [0, δ ɛ ] [ δ, δ] (i) For any c [c δ c, c + δ c ] and ɛ [0, δ ɛ ], the quadruplet ( u (c, ɛ), u+ (c, ɛ), w (c, ɛ), ϑ (c, ɛ) ) is the unique (δ, ξ )-quasi-ront solution to (3.1). (ii) Write ξ (c, ɛ) := ev 0[u (c, ɛ) u+ (c, ɛ)] Γ and pick a non-zero d K with d (0) > 0, then the ollowing Melnikov inequalities hold, D c [ ev 0 d, ξ (c, ɛ) ] c=c,ɛ=0 < 0, D ɛ [ ev 0 d, ξ (c, ɛ) ] c=c,ɛ=0 < 0. (3.9) (iii) The maps (c, ɛ) ϑ (c, ɛ) R and u (c, ɛ) q BC η ((, 1], R) w (c, ɛ) (c, ɛ) R BC η ((, 0], R) w (c, ɛ) R+ Θ s R (ϑ (c, ɛ), c, ɛ) BC η ([0, ), R) u + (c, ɛ) s R(w (c, ɛ)( ), c, ɛ) BC η ([ 1, ), R) are C r -smooth with values in the spaces indicated aove, where r appeared in (H1). Moving on to study the connections etween M R (c, ɛ) and M L (c, ɛ), we now deine quasi-ack solutions, which are illustrated in Figure 8. Deinition 3.5 (Quasi-ack solution) For each 0 < δ 1 and ξ 1, we say that the quintuplet (u, u +, w, ϑ, ϑ + ) C((, 1], R) C([ 1, ), R) C(R, R) [w δ, w + δ] 2 is a (δ, ξ )-quasi-ack solution i the ollowing is true: (i) The pair (u ±, w) satisies (3.1) on the interval R ±. 14

15 (ii) We have lim ξ (u + (ξ), w(ξ)) = 0 and ( ) u (ξ) s R w(ξ) < δ or ξ ξ, u(ξ) ( q (ξ) ) < δ and w(ξ) w < δ or ξ ξ ξ, u + (ξ) s L w(ξ) < δ or ξ ξ, where u(ξ) should e read as u (ξ) or ξ 1, as u + (ξ) or ξ 1, and as oth u ± (ξ) in the region 1 ξ 1. (iii) We have lim ξ e η ξ [w(ξ) Θ s R (ϑ, c, ɛ)(ξ)] = 0, lim ξ e η ξ [w(ξ) Θ s L (ϑ+, c, ɛ)(ξ)] = 0. (iv) We have ev 0 u ev 0 q X, ev 0 u + ev 0 q X, and ev 0 [u u + ] Γ. Compared to the existence result or the quasi-ronts, an additional degree o reedom arises when constructing quasi-ack solutions to (3.1). This reedom is used in the ollowing result to ix w(0). Proposition 3.6 Consider the nonlinear equation (3.1) and suppose that (H1)-(H5) are satisied. Then there exist constants ξ 1 and 0 < δ, δ ϑ, δ c, δ ɛ 1, together with a set o maps u : [w δ ϑ, w + δ ϑ ] [c δ c, c + δ c ] [0, δ ɛ ] C((, 1], R), u + : [w δ ϑ, w + δ ϑ ] [c δ c, c + δ c ] [0, δ ɛ ] C([ 1, ), R), w : [w δ ϑ, w + δ ϑ ] [c δ c, c + δ c ] [0, δ ɛ ] C(R, [w min, w max ]), ϑ : [w δ ϑ, w + δ ϑ ] [c δ c, c + δ c ] [0, δ ɛ ] [w min, w max ], ϑ + : [w δ ϑ, w + δ ϑ ] [c δ c, c + δ c ] [0, δ ɛ ] [w min, w max ] that satisies the ollowing properties. (i) For any ϑ 0 [w δ ϑ, w + δ ϑ ], c [c δ c, c + δ c ] and ɛ [0, δ ɛ ], the quintuplet ( u (ϑ0, c, ɛ), u + (ϑ0, c, ɛ), w (ϑ 0, c, ɛ), ϑ (ϑ0, c, ɛ), ϑ + (ϑ0, c, ɛ) ) is the unique (δ, ξ )-quasi-ack solution to (3.1) that has w(0) = ϑ 0. (ii) Write ξ (ϑ0, c, ɛ) := ev 0 [u (ϑ0, c, ɛ) u + (ϑ0, c, ɛ)] Γ and pick a nonzero d K d (0) > 0, then the ollowing Melnikov inequalities hold, that has In addition, we have D ϑ 0ϑ ± (w, c, 0) 0. D c [ ev 0 d, ξ (ϑ0, c, ɛ) 0 ] ϑ 0 =w,c=c,ɛ=0 > 0, D ϑ 0[ ev 0 d, ξ (ϑ0, c, ɛ) 0 ] ϑ 0 =w,c=c,ɛ=0 < 0. (iii) The maps (ϑ 0, c, ɛ) ϑ ± (ϑ0, c, ɛ) R and w (ϑ 0, c, ɛ) R Θ s R (ϑ (ϑ0, c, ɛ), c, ɛ) BC η ((, 0], R) (ϑ 0 u, c, ɛ) (ϑ0, c, ɛ) s R (w (ϑ 0, c, ɛ)( ), c, ɛ) BC η ((, 1], R) w (ϑ 0, c, ɛ) R+ Θ s L (ϑ+ (ϑ0, c, ɛ), c, ɛ) BC η ([0, ), R) u + (ϑ0, c, ɛ) s L (w (ϑ 0, c, ɛ)( ), c, ɛ) BC η ([ 1, ), R) are C r -smooth with values in the spaces indicated aove, where r appeared in (H1). 15

16 Fig. 9: An illustration o quasi-solutions and their passage near the slow maniold M R. 3.3 Step 3 - The passage near M R : the Exchange Lemma We now proceed to connect the quasi-ront solutions to the quasi-ack solutions somewhere near the maniold M R (c, ɛ): Figure 9 illustrates the solutions we shall construct in this section. We will use the time T that solutions spend near M R as our primary parameter. We note that (H5) allows us to deine the slow time T sl > 0 as the unique time or which Θ sl R(0, c, 0)(T sl ) = w. (3.10) Since we need solutions to ollow the ack q with w w, we require that ɛt T sl. In particular, this means that ɛ and the ast time T cannot e treated as independent parameters. To accommodate this requirement, we introduce the slow time variale T sl = ɛt and treat c, T sl and T as the independent parameters. We thereore introduce the parameter space Ω = Ω(δ c, δ sl, T ) = [c δ c, c + δ c ] [T sl δ sl, T sl + δ sl ] [T, ). (3.11) Recall the unctions ϑ and ϑ + that appear in Propositions 3.4 and 3.6: these unctions select the speciic ier o M R (c, ɛ) that quasi-ronts and quasi-acks approach as ξ or ξ. We will use the additional parameter ϑ 0 that is availale when selecting a quasi-ack to ensure that these iers match up properly ater the time T spent near M R (c, ɛ). Speciically, we introduce the unction ϑ 0 : Ω [w min, w max ] that is uniquely deined y the requirement that ϑ (ϑ0 (ω), c, T sl /T ) = Θ sl R(ϑ + (c, T sl /T ), c, T sl /T )(T sl ) or all ω = (c, T sl, T ) Ω. On account o Proposition 3.6(ii), the unction ϑ 0 is well-deined provided that T is chosen to e suiciently large and δ c > 0 and δ sl > 0 are chosen to e suiciently small. In addition, we have the expansion ϑ 0 (ω) w = κ 1 (ω)[t sl T sl ] + κ 2 (ω)/t + κ 3 (ω)(c c ), (3.12) in which κ 1, κ 2 and κ 3 are o class C r on Ω, with κ 1(c, T sl, ) 0. Deinition 3.7 (Quasi-solution) Pick T 1 and 0 < δ sl, δ c 1, choose ω = (c, T sl, T ) Ω(δ c, δ sl, T ), and consider (3.1) with ɛ := T sl /T. For each 0 < δ 1 and ξ 1, we say that the quadruplet (u, u, u xc, w) C((, 1], R) C([T 1, ), R) C([ 1, T + 1], R) C(R, R) 16

17 is a (δ, ξ )-quasi-solution i the ollowing holds: (i) The pairs (u, w), (u xc, w), and (u, w) satisy (3.1) on the intervals (, 0], [0, T ], and [T, ), respectively. (ii) We have lim ξ (u (ξ), w(ξ)) = 0 and lim ξ (u (ξ), w(ξ)) = 0. (iii) We have u(ξ) ( q (ξ) ) < δ and w(ξ) < δ or ξ ξ, u(ξ) sr w(ξ) < δ or ξ ξ T ξ, u(ξ) q (ξ ( T ) ) < δ and w(ξ) w < δ or T ξ ξ T + ξ, u(ξ) sl w(ξ) < δ or T + ξ ξ, where u(ξ) should e read as u (ξ) or ξ 1, u xc (ξ) or 1 ξ T 1, u (ξ) or ξ T + 1, oth u (ξ) and u xc (ξ) in the region 1 ξ 1, and oth u (ξ) and u xc (ξ) in the region T 1 ξ T + 1. (iv) We have ev 0 u, ev 0 u xc ev 0 q X, ev T u, ev T u xc ev 0 q X, (v) We have ev 0 [u u xc ] Γ and ev T [u u xc ] Γ. Our next result, which can e interpreted as an extension o the Exchange Lemma to MFDEs, is concerned with the existence o quasi-solutions. Proposition 3.8 Consider the nonlinear equation (3.1) and suppose that (H1)-(H5) are satisied, then there are constants ξ 1, T 1 and 0 < δ, δ c, δ sl 1 with the ollowing property. For each ω = (c, T sl, T ) Ω = Ω(δ c, δ sl, T ), there exists a quadruplet ( u (ω), u (ω), u xc (ω), w(ω) ) with that satisies the ollowing properties. u (ω) C((, 1], R), u (ω) C([T 1, ), R), u xc (ω) C([ 1, T + 1], R), w(ω) C(R, [w min, w max ]), (i) For any ω Ω, the quadruplet ( u (ω), u (ω), u xc (ω), w(ω) ) is the unique (δ, ξ )-quasi-solution to (3.1) with ɛ = T sl /T. (ii) The maps ω ξ (ω) and ω ξ (ω) deined y ξ (ω) := ev 0 [u (ω) u xc (ω)] Γ, ξ (ω) := ev T [u (ω) u xc (ω)] Γ are C r -smooth, where the integer r appeared in (H1). (iii) Consider the maps ξ : ω ξ (ω) ξ (c, T sl /T ), ξ : ω ξ (ω) ξ (ϑ0 (ω), c, T sl /T ), then there exists a constant C > 0 such that, or any integer 0 l r and any ω Ω, we have the estimate Dω l ξ (ω) + Dω l ξ (ω) Ce η T. (3.13) With this result in hand we have gathered all the ingredients we need to estalish our main theorem. 17

18 3.4 Step 4 - Proo o Theorem 1 The remaining arguments are almost identical to those used in the proo o [35, Theorem 1]. Let Ω e as in Proposition 3.8. Finding (δ, ξ )-homoclinic solutions to (3.1) has now een reduced to inding ω Ω that have ξ (ω) = ξ (ω) = 0. This leads to the system 1 (ω)e η T = c 1 (ω)(c c ) c 2 (ω)/t, 2 (ω)e η T = c 3 (ω)(c c ) c 4 (ω)[ϑ 0 (ω) w ] + c 5 (ω)/t, (3.14) in which the unctions 1, 2 and c 1 through c 5 and their derivatives are ounded on Ω. In addition, setting ω 0 = (c, T sl, ), we have c 1 (ω 0 ) 0, c 2 (ω 0 ) 0, c 4 (ω 0 ) 0 and sign ( c 1 (ω 0 ) ) = sign ( c 2 (ω 0 ) ). Using (3.12) and solving the second equation in (3.14), we ind T sl T = T sl T ( 1 + O T 2 + (c c ) 1 ). T Sustituting this expression into the irst line o (3.14) and solving, we otain 1 T = c 1(ω 0 ) c 2 (ω 0 )T sl (c c ) + O ( (c c ) 2), which, using ɛ = T sl /T, yields the desired expansion ɛ = c 1(ω 0 ) c 2 (ω 0 ) (c c ) + O ( (c c ) 2). This completes the proo o our main result suject to proving the propositions that we stated in the preceding sections. Their proos will occupy the remainder o this paper. 4 Persistence o slow maniolds In this section we set out to prove Proposition 3.1. The approach in this section is ased heavily on the construction developed in [43 2] to estalish the persistence o slow maniolds in the context o singularly pertured ODEs. At the appropriate points in the analysis, the machinery that was developed in [26 6] or MFDEs with slowly modulating coeicients will e put to work. We will ocus on the construction o the unction s R, noting that s L can e constructed in a similar ashion. Our approach will e to ix w 0, c and ɛ > 0 and look or a ounded solution (u, w) to (3.1) that remains close to M R and has w(0) = w 0. We will then write s R (w 0, c, ɛ) = u(0) and show that this unction has the desired properties. In essence, we are constructing a center maniold around M R. Let us start y introducing the new variale v via u(ξ) = s R ( w(ξ) ) + v(ξ). (4.1) Sustituting this ack into (3.1) and recalling the identity g ( s R (w) ) = w, we ind that the pair (v, w) must satisy cv (ξ) = L ( s R (w(ξ)) ) ( ( ) ] ( ) ev ξ v ɛd s R w(ξ) )[ sr w(ξ) + v(ξ) γw(ξ) χ1 w(ξ) +G ( v(ξ), w(ξ) ) ( ) + H(ev ξ w), ( ) (4.2) cw (ξ) = ɛ[ s R w(ξ) + v(ξ) γw(ξ)]χsl w(ξ), in which the operator L : R L(X, R) is given y L(u)ev ξ v = α[v(ξ + 1) + v(ξ 1) 2v(ξ)] + g (u)v(ξ), (4.3) 18

19 while the nonlinear operators G : R R R and H : C([ 1, 1], [w min, w max ]) R are given y G(v, w) = g ( s R (w) + v ) g ( s R (w) ) g ( s R (w) ) v, H(ev ξ w) = α[ s R ( w(ξ + 1) ) + sr ( w(ξ 1) ) 2 sr ( w(ξ) ) ]. (4.4) In order to stay as close as possile to the setting in [43] and in particular to reproduce the estimate [43, Equation (2.3)], we will split the operator H into a part H lin that is linear in w and a nonlinear part H nl. Ater using the dierential equation or w to transorm H lin, we write H lin (ev ξ v, ev ξ w) = ɛα c D s ( ) [ ξ+1 ( R w(ξ) [ sr ξ w(ξ ) ) + v(ξ ) γw(ξ ) ] ( χ sl w(ξ ) ) dξ + ξ 1 ( [ sr ξ w(ξ ) ) + v(ξ ) γw(ξ ) ] ( χ sl w(ξ ) ) ] dξ, ( ) ( ) ( ) H nl (ev ξ w) = α[ s R w(ξ ( + 1) ) + sr w(ξ 1) 2 sr w(ξ) ] αd s R w(ξ) [w(ξ + 1) + w(ξ 1) 2w(ξ)]. (4.5) Since we are only interested in solutions or which v is small, we will add a cut-o to v. In addition, to ound the Lipschitz constant associated with H nl, we will need to apply a special cut-o to w. To this end, let us introduce or any w C(R, R), the notation cev ξ w = ( w(ξ + 1) w(ξ), w(ξ 1) w(ξ) ) R 2. (4.6) We pick an aritrary C -smooth unction χ : [0, ) R that has χ(ζ) = 1 or 0 ζ 1 and χ(ζ) = 0 or ζ 2. For any δ > 0, we write χ δ or the unction χ δ (ζ) = χ(ζ/δ). We are now ready to deine, or small quantities δ v > 0 and δ w > 0, the cut-o nonlinearities G c (v, w) = χ δv ( v )G(v, w), Hlin c (ev ξv, ev ξ w) = ɛα c D s ( ) [ ξ+1 ( R w(ξ) [ sr ξ w(ξ ) ) + v(ξ ) γw(ξ ) ] ( χ 1 ( v(ξ ) )χ sl w(ξ ) ) dξ + ξ 1 ( [ sr ξ w(ξ ) ) + v(ξ ) γθ(ξ ) ] ( χ 1 ( v(ξ ) )χ sl w(ξ ) ) ] dξ, Hnl c (ev ξw) = χ δw ( cev ξ w )H nl (ev ξ w). (4.7) Putting this together, we pick ζ 0, introduce the nonlinearity R c cm : BC ζ (R, R) C(R, [w min, w max ]) R R BC ζ (R, R) (4.8) that is given y R c cm(v, w, ɛ, c)(ξ) = ɛd s R ( w(ξ) )[ sr ( w(ξ) ) + v(ξ) γw(ξ) ] χ1 ( v(ξ) )χ sl ( w(ξ) ) +G c( v(ξ), θ(ξ) ) + H c lin (ev ξv, ev ξ w) + H c nl (ev ξw) (4.9) and study the equation cv (ξ) = ( ( ) ) L s R w(ξ) ev ξ v + R c cm(v, w, ɛ, c)(ξ), (4.10a) cw (ξ) = ɛ[ s R ( w(ξ) ) + v(ξ) γw(ξ)]χsl ( w(ξ) ) χ1 ( v(ξ) ). (4.10) Let us pick small constants δ c > 0 and δ ɛ > 0. For quantities a and that depend on the various cut-os δ v, δ w, δ c and δ ɛ that we have introduced, we will use the notation a (4.11) to express the act that there exists a C > 0 that does not depend on these cut-os, such that a C holds or all δ v 1, δ w 1, δ c 1 and δ ɛ 1. 19

20 Notice that or any w, w 1, w 2 C(R, [w min, w max ]), v, v 1, v 2 BC ζ (R, R), ɛ [0, δ ɛ ] and c [c δ c, c + δ c ], we have the inequalities R c cm(v, w, c, ɛ)(ξ) δ ɛ + δv 2 + δw, 2 R c cm(v 1, w 1, c, ɛ) R c cm(v 2, w 2, c, ɛ) ζ (δ ɛ + δ v + δ w )[ v 1 v 2 ζ + w 1 w 2 ζ ]. (4.12) We proceed with our analysis y considering the linear part o (4.10a). We will write this as Λ(w, c)v = h, (4.13) in which the linear operator Λ(w, c) : C 1 (R, R) C(R, R) acts as ( [Λ(w, c)v](ξ) = cv ( ) ) (ξ) L s R w(ξ) ev ξ v, (4.14) or any w C(R, [w min, w max ]) and c 0. The next result shows that or any η [ η, η ], an inverse can e deined or the operator Λ(w, c) on the space BC η (R, R) provided that w is suiciently small. Lemma 4.1 Consider the linear system (4.13) and suppose that (H1)-(H3) are satisied. Then there exists a constant δ c > 0, together with a amily o maps K η : C(R, [w min, w max ]) [c δ c, c + δ c ] L ( BC η (R, R), BC η (R, R) ) (4.15) deined or all η [ η, η ], such that the ollowing properties are satisied. (i) There exists κ > 0, such that i w C 1 (R, [w min, w max ]) and w (ξ) < κ or all ξ R, then v = K η (w, c)h satisies Λ(w, c)v = h or any η [ η, η ], c [c δ c, c + δ c ] and h BC η (R, R). (ii) The norm K η (w, c) can e ounded independently o η [ η, η ], c [c δ c, c + δ c ] and w C(R, [w min, w max ]). (iii) There exists a constant C > 0 such that or any η 1 > 0, any η 2, η 3 [ η, η ] that have η 1 + η 2 η 3, any two unctions w 1, w 2 C(R, [w min, w max ]), any two c 1, c 2 [c δ c, c + δ c ] and any h BC η2 (R, R), we have the estimate K η2 (w 1, c 1 )h K η2 (w 2, c 2 )h η3 C[ w 1 w 2 η1 + c 1 c 2 ] h η2. (4.16) (iv) Consider a pair η 1, η 2 [ η, η ] together with a unction h BC η1 (R, R) BC η2 (R, R). (4.17) Then or any w C(R, [w min, w max ]) and c [c δ c, c + δ c ], we have K η1 (w, c)h = K η2 (w, c)h. (4.18) (v) Recall the integer r that appears in (H1). Consider any integer 0 l r + 2 and pick η 1 > 0 and η 2, η 3 [ η, η ] in such a way that η 3 > lη 1 + η 2. Then the map (θ, c) K(θ, c) is C l -smooth when considered as a map rom BC η1 (R, [w min, w max ]) [c δ c, c + δ c ] into L ( BC η2 (R, R), BC η3 (R, R) ). In addition, or any pair o integers p 1, p 2 0 with p 1 + p 2 = l, the derivative D p1 K is well-deined even when interpreted as a map D p1 1 Dp2 with η = lη 1 + η 2. 1 Dp2 2 2 K : BC η 1 (R, [w min (, w max ]) [c δ c, c + δ c ] L (l) BC η1 (R, R) p1 R p2, L ( BC η2 (R, R), BC η (R, R) )) (4.19) 20

21 (vi) For any ξ 0 R, η [ η, η ], w C(R, [w min, w max ]), c [c δ c, c +δ c ] and h BC η (R, R), we have in which T ξ0 denotes the shit [T ξ0 h](ξ ) = h(ξ 0 + ξ ). K η (T ξ0 w, c)t ξ0 h = T ξ0 K η (w, c)h, (4.20) Proo. We irst consider the linear system Λ(w, c)v = h in the special case that w is a constant, i.e., w(ξ) = w 0 or all ξ R. In this case (4.13) reduces to a linear constant-coeicient inhomogeneous MFDE that has een studied extensively [28, 38]. The characteristic unction associated with this MFDE can e otained y seeking a solution o the orm v(ξ) = exp(zξ) to the homogeneous system Λ(w 0 1, c)v = 0. Recalling (3.8), we ind that (z) = R,w0 (z), rom which it ollows that the characteristic equation (z) = 0 admits no roots with Re z η. Ater picking δ c to e suiciently small, the constructions in [28 5] can e used to deine, or any w 0 [w min, w max ], any c [c δ c, c + δ c ] and any η [ η, η ], the operators K cs η (w 0, c) : BC η (R, R) BC 1 η(r, R) (4.21) that solve the constant coeicient system Λ(w 0 1, c)v = h. More precisely, or any v BCη(R, 1 R) we have Kη cs (w 0, c)λ(w 0 1, c)v = v and or any h BC η (R, R) we have Λ(w 0 1, c)kη cs (w 0, c)h = h. One can now employ simpliied versions o the arguments in [26 6] and use these operators Kη cs (w 0, c) to construct a amily K η that satisies the properties (i) through (vi). Lemma 4.2 Consider the linear system (4.13) and suppose that (H1)-(H3) are satisied. Then there exist constants κ > 0 and δ c > 0, such that or any c [c δ c, c +δ c ] and any w C(R, [w min, w max ]) that has w (ξ) < κ or all ξ R, the homogeneous equation Λ(w, c)v = 0 admits no non-zero solutions v BC 1 η (R, R). Proo. In the special case that w is a constant unction, the claim ollows rom [28, Proposition 5.2], in view o the oservation contained in the proo o Lemma 4.1 that the characteristic unction (z) = 0 admits no roots with Re z η. A simpliied version o the proo o [26, Lemma 6.4] can now e used to generalize the claim to unctions w that have a suiciently small derivative. We now turn our attention to the equation or w given y (4.10). For any ixed v C(R, R) and c 0, this equation is an ODE with a smooth right-hand side. This allows us to introduce, or δ c > 0 suiciently small and any δ ɛ > 0, the operator W : [w min, w max ] C(R, R) [c δ c, c + δ c ] [0, δ ɛ ] C(R, [w min, w max ]) (4.22) that is uniquely deined y the property that w = W (w 0, v, c, ɛ) solves (4.10) with w(0) = w 0. Our next result is the equivalent o [43, Lemma 2.4] and can e proved using Gronwall s inequality and variational equations. Lemma 4.3 There exist constants L 0 > 0 and L 1 > 0 such that the ollowing hold true. (i) For any w 0 [w min, w max ], ɛ [0, δ ɛ ], c [c δ c, c + δ c ] and v C(R, R), we have W (w 0, v, c, ɛ)(ξ) ξ + ɛl 0 ξ. (4.23) (ii) Consider any ɛ [0, δ ɛ ], c [c δ c, c + δ c ] and any η > ɛl 1. Then or any two pairs (w 1 0, v 1 ), (w 2 0, v 2 ) [w min, w max ] BC η (R, R), we have the estimate W (w 1 0, v 1, c, ɛ)(ξ) W (w0, 2 v 2, c, ɛ)(ξ) w 1 0 w0 2 e ɛl 1ξ + ɛl 1 v 1 v 2 η ɛl η e ηξ. (4.24) 1 21