Wave speeds for the FKPP equation with enhancements of the reaction function

Transcription

1 Z. Angew. Math. Phys. 66 (015), c 014 Springer Basel /15/ published online May 3, 014 DOI /s Zeitschrift für angewandte Mathematik und Physik ZAMP Wave speeds for the FKPP equation with enhancements of the reaction function Freddy Dumortier and Tasso J. Kaper Abstract. In classes of N-particle systems and lattice models, the speed of front propagation is approximated by that of the corresponding continuum model, and for many such systems, the rate of convergence to the continuum speed is known to be slow as N. This slow convergence has been captured by including a cutoff function on the reaction terms in the continuum models. For example, the Fisher Kolmogorov Petrowskii Piscounov (FKPP) equation with a cutoff has fronts that travel at the speed c c FKPP π (ln(n)), which agrees well with data from numerical simulations of the corresponding N-particle systems, where c FKPP is the linear spreading speed. In Panja and van Saarloos (Phys Rev E 66:01506, 00), an example is presented in which a small enhancement of the reaction function causes the propagation speeds of fronts to be larger than c FKPP. Such front speeds are also observed in stochastic lattice models where the growth rates in the regime of few particles are modified. In this article, we analyze the dynamics of traveling fronts in the FKPP equation with the constant enhancement function employed by Panja and van Saarloos. We present formulas for the wave speeds, develop the criteria on the parameters for which the front speeds are larger than the linear spreading speed even in the limit in which the size of the cutoff domain vanishes, study the rate of approach as N, and identify the mechanisms in phase space by which the constant enhancement of the reaction function makes possible the larger than linear wave speeds. In addition, we extend these results to the FKPP equation with two other enhancement functions, which are also of interest for continuum level modeling of lattice models and many-particle systems in the regimes of small numbers of particles, namely a linear enhancement function and an enhancement that is uniform above the linearized reaction function. We also derive explicit formulas for the parameters in these problems. The mathematical techniques used herein are geometric singular perturbation theory, geometric desingularization, invariant manifold theory, and normal form theory, all from dynamical systems. Mathematics Subject Classification (1991). 35K57 34E15 34E05. Keywords. Reaction diffusion equations Enhancement: Cutoff Traveling waves Blow-up technique. 1. Introduction The Fisher Kolmogorov Petrowskii Piscounov (FKPP) equation u t = u x + u(1 um ), (1) with m = 1, [6, 31] arises ubiquitously as a model in biology, combustion, ecology, interacting particle systems, optics, phase transitions, physiology, plasma physics, and other fields, see e.g., [ 4, 11, 8]. In these problems, general data often evolve toward traveling fronts, which are solutions that are stationary in a frame moving at a constant speed c>0 and that connect the spatially homogeneous states u =0 and u = 1 ahead of and behind the front, respectively. For example, the FKPP equation may be derived in the large-scale limit of interacting particle systems or in the mean-field limit of microscopic or lattice models, and in these systems, the fronts travel with speeds approximated by the classical FKPP speed c FKPP =, see e.g., [8 10,14,15,9,30]. Hence, considerable research has been devoted to finding fronts and traveling waves in reaction diffusion models, as well as to determining their propagation speeds. For broad classes of interacting particle systems, the characteristic propagation speed is known to converge only slowly to the value predicted by the continuum limit as the number of particles, or average

2 608 F. Dumortier and T. J. Kaper ZAMP occupancy number, N. Even for large values of N such as N = 10 6, the propagation speeds are substantially smaller than expected, as has been observed in systematic numerical simulations, see [9,10,14,9,30,34]. This slow convergence as N grows has been captured successfully in the context of continuum models by using cutoff functions for the reaction terms. Indeed, the pioneering analysis in [1] introduced a Heaviside cutoff function in the FKPP equation to set the reaction function to zero when and wherever the concentration u decreases below a threshold. The motivation for introducing the Heaviside cutoff was that few or no particles are present at points in the domain at which the concentration is less than ε = 1 N, and, hence, at these points, few or no reactions take place. For the FKPP (1) with f set to zero on the interval 0 u ε =1/N, they derived asymptotics showing logarithmic (i.e., slow) convergence to the continuum limit: c π (ln ε) as ε 0. () Moreover, itwasshownin[1] that this logarithmic convergence agrees well with the data from sequences of numerical simulations with successively larger values of N. Since publication of [1], this slow convergence in the large-n limit has been analyzed for several classes of problems with pulled and fluctuating fronts, see, for example, Section 7.1 of [4]. In addition, mathematical analysis of the corresponding continuum models with cutoff functions has been presented in [5,6,19,1,,35], where it was shown that the first-order correction in () is universal for pulled fronts within a large class of cutoff functions and where the impact of cutoffs was also studied on bistable fronts and pushed fronts in other prototypical reaction diffusion equations, such as the Nagumo equation. More recently, a particular case of the FKPP equation with cutoff has been studied in which the propagation speed is substantially larger than c FKPP, and in which the speed remains larger than c FKPP even in the limit as ε 0. In [36], Panja and van Saarloos introduced an enhancement function to the reaction function f on a narrow interval of concentration values adjacent to the cutoff threshold, and they showed that with this enhancement the propagation speed c(ε, α) =c 0 (α)+ c(ε, α)hasc 0 (α) >c FKPP, with c = o(ε), where α is a parameter that controls the width of the enhancement interval, see (4) below. These problems are of interest, because these larger wave speeds have been observed in stochastic lattice models which have modified growth rates in the regimes of low particle concentrations, see [36]. In this article, we study the FKPP equation (1) with the enhancement of [36]: u t = u + f(u, ε, α), (3) x where the reaction function f is defined in the following piecewise manner: 0 if 0 u ε, f(u, ε, α) = εα if ε<u αε, (4) u(1 u m ) if αε < u, with α 1andm =1,,... We label the three regions as Regions 1 3, respectively. Region 1, u [0,ε], corresponds to the by-now standard cutoff region, and we have taken the Heaviside cutoff for convenience in the calculations. Several of the key quantities that enter into the formulas for the wave speeds and into the determination of the critical width of the enhancement interval depend on the choice of the cutoff function that is used in Region 1. Extensions to other cutoffs of the type used in [1,1] are possible following the same type of explicit calculation presented here. Region, u (ε, αε], corresponds to the new interval of enhancement introduced in [36], and Region 3, αε < u 1, corresponds to the bulk of the domain in which the FKPP reaction function f is unmodified. We note that there are discontinuities in f at u = ε and u = αε; see also the proposition in Sect We establish the following theorem, which rigorously justifies some of the calculations for (3) (4) presented in [36] and which presents additional results about the convergence of the speeds to the limiting speeds. The analysis also identifies the geometry responsible for the existence of the critical value α ;see

3 Vol. 66 (015) Wave speeds for the FKPP equation 609 also Remark 4. Moreover, the proof of the theorem also identifies the mechanisms in the phase space that are responsible for making the larger speeds possible. When Region is sufficiently wide, the enhancement causes the fronts to become pushed. Theorem 1. For each α 1 and for each m =1,,..., there exists an ε 0 = ε 0 (α, m) > 0 such that for ε (0,ε 0 ), the reaction diffusion equation (3) with enhancement (4) has a unique traveling wave solution. There also exists a critical value of α given by α = 4e 1+e such that the speeds c(ε, α) of the traveling waves are as follows: For α<α, For α = α, c(ε, α) = π (ln ε) + O( (ln ε) 3). (5) c(ε, α) = π 4(ln ε) + O( (ln ε) 3). (6) For α>α, there exists a monotonically increasing, analytic diffeomorphism c 0 :(α, ) (, ) such that the propagation speed is given by c(ε, α) =c 0 (α)+ c(ε, α). (7) The function c 0 (α) is the unique solution of the equation α = α 0 (c), where α 0 (c) = 1 ( 1 c + c c 4 ) e, (8) c (c+ c 4) and dα0 dc (c) > 0 for c>, so that c = c 0(α) is a simple root of α = α 0 (c). Also, there exists a monotone, analytic diffeomorphism β 0 :(, ) (0, ) with c β 0 (c) = 1 ( c 4+c c 4 ), (9) such that for any c (, ) and for any m =1,,..., the following properties hold for the correction function c(ε, α): O(ε β0(c) ) if c< m+ m+1, c(ε, α 0 (c)) = O(ε m ln(1/ε)) if c = m+ m+1, (10) O(ε m ) if c> m+ m+1, where the symbol O is used in the strict sense. For α<α, the result of this theorem follows from the calculations performed in [1,1]. The enhancement is insufficiently large to cause any change in the propagation speed. Moreover, we observe that the first-order correction term is universal, as was shown in [1] in the absence of enhancement functions, though the value of α depends on the choice of the cutoff function. Then, for α = α, the enhancement is only large enough to alter the first-order asymptotic correction. In the most interesting case, α>α, the enhancement is sufficiently large to increase the leading-order wave speeds, c 0 (α), above the linear spreading speed c FKPP =. This is the primary impact and interest of this example. In the course of finding the formula for c 0 (α) and demonstrating that c 0 (α) >c FKPP for α>α, we also show that the function α 0 (c) extends continuously down to c =, with α 0 () = α and that 1 α 0 (c) <c for all c [, ). Finally, we observe that the resonances in the normal form of the governing equations give rise to the fractional powers and logarithmic dependence of c(ε, α) on ε. Remark 1. The value of α is the same critical value as obtained in [36], translated to their parameter r =1/α. Forα>α, the formula for c 0 (α) differs in one term from that in equation (7) in [36]. We note that there may be a small error in the calculation in [36]. Also, the expressions for c(ε, α) differ, as there are subtleties about the function c(ε, α) which the geometric approach here captures naturally.

4 610 F. Dumortier and T. J. Kaper ZAMP Traveling waves of (3) with enhancement (4) satisfy the system U = V, V (11) = cv f(u, ε, α), where U(ξ) =u(x, t), ξ = x ct, and the prime denotes differentiation with respect to ξ. They correspond to heteroclinic orbits in (11) which connect the homogeneous states U =1andU = 0 behind and ahead of the front, lim (U, V )(ξ) =(1, 0) ξ Q and lim (U, V )(ξ) =(0, 0) ξ Q+. In the limit ε 0, this equation is degenerate at Q + due to the cutoff in Region 1 and the enhancement in the reaction function f in Region. To remove the degeneracy at Q + in the traveling wave ODE system (11), we apply the method of geometric desingularization, also known as the blow-up method. It expands the degenerate fixed point into a topological sphere, removing the degeneracy and making all of the fixed points on this sphere hyperbolic or semi-hyperbolic. The orbits of the desingularized vector field will be tracked in Regions 1 3 on and near this sphere in the appropriate coordinate charts. We remark that this same coordinate change was also used in [19,1,,38,39] in the analysis of the impact of cutoffs on pulled fronts, bistable fronts, and pushed fronts. Hence, this study may also be viewed as a natural extension of those earlier works. For the FKPP equation (3) with the constant enhancement (4), each traveling front will then be obtained as a heteroclinic orbit of the desingularized vector field. In particular, for each (α, m) with α 1andm =1,,..., we construct a locally unique singular heteroclinic solution that connects Q and Q +. This singular heteroclinic corresponds to the ε = 0 limit of the desired heteroclinic orbit of the full system, and it is independent of m. This limiting analysis directly yields the value of α,and it establishes the properties of the singular solutions, as well as formula (8) forα 0 (c). Then, for each (α, m) with α 1andm =1,,..., we establish the persistence of the singular heteroclinic for ε>0 and sufficiently small. The persistence proof also establishes the uniqueness of the traveling wave for each (α, m) and the asymptotic formulas for c(ε, α) for each (α, m), completing the proof of Theorem 1 for the constant enhancement function in Region introduced in [36]. After completing the proof of Theorem 1, we extend the above results to the FKPP equation (3) with two different types of enhancement functions in Region. We consider a linear enhancement function, in which the reaction function f is given in Region by f LE (u, ε, α, γ) =(1+γ)u with γ>1, (1) and a uniform enhancement function, in which the reaction function f is given in Region by f UE (u, ε, α, δ) =u + εδ with δ>1, (13) which represents a uniform enhancement above the linearized reaction function. These different enhancements are useful for other types of modifications of the growth rates used in the few-particle regimes of stochastic lattice models, microscopic models, and interacting particle models. For the FKPP equation (3) with the linear enhancement function f LE (1), we prove. Theorem. For each (α, γ, m) with α 1, γ>1, andm =1,,..., there exists an ε 0 = ε 0 (α, γ, m) > 0 such that for ε (0,ε 0 ), the reaction diffusion equation (3) with enhancement (1) has a unique traveling wave solution. There exists a critical value of α given by 1+γ α γ = α (γ) = exp γ [ 1 γ arctan ( 1 γ )] such that for each γ>1 there exists an analytic, monotonically increasing diffeomorphism c γ 0 :(αγ, ) (, γ +1)such that the speeds c(ε, α, γ) of the traveling waves are as follows: For α<α γ, c(ε, α, γ) is given by (5). Forα = α γ, c(ε, α, γ) is given by (6). Forα>α γ, (14)

5 Vol. 66 (015) Wave speeds for the FKPP equation 611 c(ε, α, γ) =c γ 0 (α)+ c(ε, α, γ). (15) Here, c γ 0 (α) is the unique solution of ( ( )) ( ( )) ω(c, γ) q(c, γ) c 4 ω(c, γ) q(c, γ) γ cos ln c α γ sin ln ω c α γ =, (16) q(c, γ) where ω(c, γ) = 1 4(1 + γ) c, q(c, γ) =ω 4 + ω (c ) + c 16 (c 4), and4(γ +1) >c.also, c satisfies the same properties as the function c in Theorem 1. Finally, for the FKPP equation (3) with the uniform enhancement function f UE (13), we prove Theorem 3. For each (α, δ, m) with α 1, δ>1, andm =1,,..., there exists an ε 0 = ε 0 (α, δ, m) > 0 such that for ε (0,ε 0 ), the reaction diffusion equation (3) with enhancement (13) has a unique traveling wave solution. There also exists a critical value of α given by α δ = α (δ) = δ δ 1 ( 1+ δ 1 ln ( 1 1 )), (17) δ ( such that for ) each δ>1there exists an analytic, monotonically increasing diffeomorphism c δ 0 :(α, δ ), δ + δ 1 which is the unique solution of [ ] δ α(c) = d (1 + δ cd)d c 4 (dδ) d c 4 (d(1 + δ) c) 1, (18) d ( with d = 1 c c 4 ), such that the speeds c(ε, α, δ) of the traveling waves are as follows: For α<α, δ c(ε, α, δ) is given by (5). Forα = α, δ c(ε, α, δ) is given by (6). Forα>α, δ c(ε, α, δ) =c δ 0(α)+ c(ε, α, δ), (19) where here c also satisfies the same properties as the function c in Theorem 1. Remark. The ranges of the functions c γ 0 (α)andcδ 0(α) obtained with the linear and uniform enhancement functions f LE and f UE in Theorems and 3 are finite, in contrast to the infinite range of the function c 0 (α) obtained with the constant enhancement function in Theorem 1. The distinction that the speeds remain finite for linear and uniform enhancement functions is useful for modeling N particle systems and for lattice models, [36]. The blow-up technique was first used in studying limit cycles in planar vector fields near a cuspidal loop in [5]. In addition, it has since been applied successfully to a broad array of different systems of ODEs, including especially in [3] where an extension of the more classical geometric singular perturbation theory is presented to problems in which normal hyperbolicity is lost; see also [16,18,0,4,3,33,40] and the references therein. Moreover, the blow-up method has also been used for analyzing propagation speeds of pulled fronts, bistable fronts, and pushed fronts in continuum reaction diffusion models with cutoffs, see [19,1,,38,39]. This article is organized as follows. In Sect., we employ the geometric desingularization method (blow-up) to the degenerate equilibrium at Q +, and we construct the singular heteroclinic orbit. In Sect. 3, we establish the persistence of this singular heteroclinic for ε>0 and sufficiently small, thus completing the proof of Theorem 1. In Sect. 4, we prove Theorems and 3, for the other two primary enhancement functions studied in this article.. Geometric desingularization of (11) and the singular heteroclinics In this section, we desingularize the origin in system (11) via a blow-up transformation; and, for each α 1, we construct a locally unique singular heteroclinic orbit Γ between Q and Q +, which corresponds

6 61 F. Dumortier and T. J. Kaper ZAMP to the ε = 0 limit of the traveling front solution of (3) (4). For each fixed value of α 1, these singular heteroclinics are independent of m. The blow-up coordinate change for (11) is U = rū, V = r v, and ε = r ε, (0) where (ū, v, ε) S = { (ū, v, ε) ū + v + ε =1 }, and r [0,r 0 ]forr 0 > 0 sufficiently small. With this coordinate change, the degenerate equilibrium at the origin is transformed into the two-sphere S. Moreover, since we are interested in ε 0, we only need to consider the half-sphere S + defined by restricting S to ε 0. As is the case when working with spheres in differential geometry, it is natural to use coordinate charts here. We analyze the induced vector field on S + in the following two charts: the rescaling chart K ( ε = 1), which is used to study the dynamics of (11) in Regions 1 and, and the phase-directional chart K 1 (ū = 1), which we employ to analyze Region 3, see Sects..1 and., respectively. Remark 3. For any object in the original (U, V, ε)-variables, we will denote the corresponding object in the phase space of the desingularized vector field by. Moreover, in charts K i (i =1, ), the object will be denoted by i..1. Dynamics in the rescaling chart K In this section, we use the rescaling chart K, defined by ε =1in(0), to study the dynamics of system (11) in Regions 1 and. It provides a top view of the sphere. For each α 1, we determine the portion of the singular heteroclinic orbit Γ that lies in K. With ε = 1, the blow-up transformation (0) in this chart is given by U = r u, V = r v, and ε = r. (1) One readily observes that U = ε corresponds to u =1andU = αε corresponds to u = α. Hence, Region 1 corresponds to 0 u 1, and in this region, system (11) is equivalent to u = v, (a) v = cv, (b) r =0. (c) The phase space of this linear system is illustrated in Fig. 1. The entire strip 0 u 1 is foliated with diagonal lines of slope c, since the system may be written as dv /du = c. Moreover, for each r, there is a line segment of fixed points (u, 0,r ) parametrized by u [0, 1], and each fixed point on it attracts the initial conditions on the diagonal line through it. We focus on the fixed point (0, 0, 0) and denote it by Q +. This fixed point is semi-hyperbolic for (), with eigenvalues λ 1 = c, λ =0,, andλ 3 = 0. The corresponding eigenspaces are spanned by (1, c, 0) T, (1, 0, 0) T,and(0, 0, 1) T, respectively. The solution of this equation through Q + is v (u )= cu. (3) This solution hits the right boundary of Region 1 at the point P =(1, c, 0); see Figs. 1 and. The diagonal line segment between P and Q + is (the lower branch of) the stable manifold Ws (Q + ), and for each α 1, it is precisely the portion Γ of the singular solution Γ in Region 1. It will also be useful to introduce the cross-section Σ in = {(1,v,r ) (v,r ) [ v 0, 0] [0,r 0 ]}, where r 0 > 0 is sufficiently small and where v 0 is appropriately chosen. For the parameter values α α, where we recall that α =4e /(1 + e ), we will chose c =, while for α>α, we will chose c = c 0 (α) where c 0 (α) is the solution of (8). As will become clear below, these

7 Vol. 66 (015) Wave speeds for the FKPP equation 613 Fig. 1. The phase space of system (11) in Regions 1 and, with 1 α<c Fig.. The geometry in chart K

8 614 F. Dumortier and T. J. Kaper ZAMP are the limiting values as ε 0, and for the remainder of this section, we will construct the singular heteroclinics in terms of c. Next, we analyze the dynamics of system (11) in Region (1 <u α). In this region, the reaction term is given by the constant enhancement. The governing equations are u = v, (4a) v = cv α, (4b) r =0. (4c) The phase space of this linear system is the strip 1 <u α, and we may continuously extend solutions to u = 1. The horizontal line v = α c is invariant. Initial conditions in the strip approach it at an exponential rate in the v direction. See Fig. 1. We focus on orbits that enter the strip at {u = α} below the invariant line, i.e., that have initial conditions (α, v in, 0) with v in < α c, since the orbits with vin α c remain above the line v = α c and hence cannot connect up to W s (Q + ) and cannot form the desired heteroclinic. For each orbit with v in < α c, there is an implicit function relating the entry value to the v coordinate (v out )onexitat {u =1}; and, this implicit relation is found by integrating the scalar equation dv du from u =1tou = α: v out v in α ( v out c ln + α ) c v in + α = c(α 1). (5) c Of particular importance is the orbit that exits Region through the point P =(1, c, 0), which we recall is the entry point of W s (Q + ) into Region 1, introduced above. This orbit has vout = c. Hence, by the implicit function (5), it has v coordinate on entry into Region given by v in, where v in + α ( c + α ) c ln c = cα. (6) v in + α c = (cv+α) v We label the entry point (α, v in, 0) by P in, and for each α 1, this orbit connecting P in to P is precisely that portion of the singular heteroclinic orbit Γ in Region. In summary, for each α 1, we have identified the portion of the singular heteroclinic orbit Γinchart K to be that segment of W s (Q + in ) that lies between P and Q + and that goes through the point P. We label it Γ. Also, for reference below, we observe that v in decreases monotonically with non-zero speed as c increases... Dynamics in the phase-directional chart K 1 In this section, we use the directional chart K 1, which is defined by ū = 1, to study the dynamics of system (11) in Region 3. For each α 1, we identify the portion of the singular orbit ΓlyinginK 1. With ū = 1, the blow-up transformation in this chart is U = r 1, V = r 1 v 1, and ε = r 1 ε 1, (7) and system (11) is equivalent to r 1 = r 1 v 1, (8a) v 1 = (1 r1 m ) cv 1 v1, (8b) ε 1 = ε 1 v 1. (8c) For all c, there is a line of fixed points with v 1 =0,l 1 = { (1, 0,ε) ε [0,ε0 ] }, which corresponds to the original point Q before blowup. In particular, for ε = 0, we will denote the point (1, 0, 0) on l 1 by Q 1. The unstable manifold, W1 u (Q 1 ), is part of the desired singular solution. We will track it in Region 3.

9 Vol. 66 (015) Wave speeds for the FKPP equation 615 We will carry out the tracking separately in the cases c>andc =, beginning with the former. For c>, system (8) has two additional equilibria, P ± 1 =(0,v± 1, 0) where v± 1 = c ± 1 c 4. (9) We will focus on P 1 + here, and we introduce the notation d = v 1 +, (30) which will be useful throughout. These fixed points merge at v 1 = 1 forc = and then disappear for c<. For c>, the eigenvalues of (8) linearized at P 1 + are given by d, c 4, and d, with eigenvectors (1, 0, 0) T,(0, 1, 0) T,and(0, 0, 1) T, respectively. Hence, P 1 + has a two-dimensional stable manifold given by {ε 1 =0}, and a one-dimensional unstable manifold corresponding to the invariant line v 1 = d, r 1 =0. The key observation is that, in the plane ε 1 = 0, solutions along the unstable manifold W1 u (Q 1 ) approach P 1 + with v 1 v 1 + from above, and we remark that the details of the approach depend on the size of c, as there is a 1 : 1 resonance among the two stable eigenvalues at c = 3 Lemma.1. Orbits in the unstable manifold W1 u (Q 1 ) approach the fixed point P 1 + with v 1 + <v 1, i.e., 3 from above, and this approach is tangent to the v 1 -axis for c 0 (, ) and perpendicular to the axis for c 0 > 3. Proof. The proof of this lemma follows from the same type of phase plane argument used in the proof of Lemma.5 in [1]. Consider system (11) in the limit ε = 0, in which it is the traveling wave ODE system for the original FKPP equation. We construct a trapping region in the (U, V ) phase plane and show that orbits on W u (Q ) must enter it and approach Q + inside it. The fixed point Q + is a stable node for c>and a stable improper node for c =. The upper boundary of the trapping region is given by the U-axis, i.e., {V =0}. On this axis, the vector field simplifies to U =0andV = U(1 U m ). Hence, the vector field points downward there for U (0, 1), into the trapping region. The lower boundary of the trapping region is the line V =( c + 1 c 4)U. This line corresponds to the weak stable eigendirection at Q +, corresponding to the weak stable eigenvalue λ + = c + 1 c 4 d. Projecting the vector field (11) onto this eigendirection, we find U(1 U m )+cv +dv. Now, on V = du, the sign of this expression is such that, for all U [0, 1] and for both c>and c =, the vector field points into the trapping region at all points on the lower boundary. Putting these two results together, we see that orbits on W u (Q ) must enter the trapping region and hence approach Q +. Finally, from the governing equations in chart K 1, one sees that the two stable eigenvalues, d and d c, are equal at c = 3. Therefore, since the approach to P 1 + is along the weak stable direction, the 3 approach is either tangent to the v 1 -axis or perpendicular to it, depending on which side of the speed c lies on. This completes the proof of the lemma.. For all α 1, the first portion of the singular solution in chart K 1 is then precisely W1 u (Q 1 ), and we label it by Γ 1. The second portion, denoted Γ + 1, of the singular orbit Γ 1 depends on the magnitude of α relative to α.forα>α, the second portion of the singular orbit Γ 1 is given by the segment of the one-dimensional manifold W1 u (P 1 + ), which coincides with {v 1 = d, r 1 =0}, uptothepointp1 out =(0, d, 1 α ). Then, for α = α, it is given by the segment of the one-dimensional unstable manifold of (0, 1, 0) up to the point P1 out =(0, 1, 1 α ). Finally, for α<α, it corresponds to the trajectory that leaves (0, 1, 0) tangent to the v 1 -axis with v 1 < 1 and that hits the plane ε 1 = 1 out α at the point P1 =(0,v1 out, 1 α ), with vout 1 < 1. The illustration in Fig. 3 is for the case α>α. where the cross-section Σ in 1 = {(r 0,v 1,ε 1 ) (v 1,ε 1 ) [ v 0, 0] [0, 1]}, where v 0 is the same as in the previous section. Also, for reference below, we observe that, for each α 1, v1 out is strictly monotonically increasing as c increases.

10 616 F. Dumortier and T. J. Kaper ZAMP Fig. 3. The geometry in chart K 1 for c>andα>α The illustration is for the case <c< 3.3. Completing the construction of the singular heteroclinic Γ The final step in the construction of the singular orbit for each α 1 is to identify the exit point P1 out, where it leaves Region 3 and chart K 1, with the point P in, where it enters Region and chart K. With this identification, we will have hooked up the pieces, Γ 1,Γ+ 1, and Γ, of the singular orbit and connected Q 1 to Q+ in {r =0}. Here, we present the constructions for α>α and α = α. In the third (and remaining) case α<α, we chose the unique orbit inside {r 1 =0} for chart K 1 that cuts the plane {ε 1 = 1 in α } at the point P. We refer the reader to Section.3 of [1] for the details of the construction for α<α. To relate the analyses of the previous sections, we use the following relationship between the variables in (1) and (7) on the domain of overlap between charts K 1 and K : Lemma.. [1] The change of coordinates κ 1 : K 1 K is given by u = 1, v = v 1, and r = r 1 ε 1. ε 1 ε 1 For the inverse change κ 1 = κ 1 1 : K K 1, there holds r 1 = r u, v 1 = v, and ε 1 = 1. u u Both κ 1 and κ 1 are well defined as long as ε 1 and u, respectively, are finite and bounded away from zero. Correspondingly, the overlap domain between K 1 and K includes {U = αε}, where ε 1 = 1 α and u = α. Recall P1 out = (0, d, 1 in α ) and P = (α, v in, 0). Hence, using the lemma, we find that v in = v1 out /ε out 1 = αd. Therefore, after substitution of this into (6) and some algebra, we find the desired relation between c and α:

11 Vol. 66 (015) Wave speeds for the FKPP equation 617 Fig. 4. The global geometry of the blown-up vector field (c α)e c(c d) + α αcd =0, (31) where c>ford<1andα>α. Solving this for α as a function of c, we find (8). In addition, in the limit c +, we see from (31) that α α, where α = 4e 1+e, as stated in Theorem 1. Moreover, these results may be extended continuously to v 1 = 1 in the case in which c =andα = α. Finally, one may verify that dα0 dc (c 0) > 0. A direct calculation yields dα 0 dc = (α + ep (c) (c α))(c c c 4) + e P (c) c c 4(c α +1) e P (c) 1+ c (c, c 4) where P (c) = c (c + c 4) and 1 <α α<c <e P (c), so that the numerator and denominator are strictly positive. Alternatively, one may see this from the observations that the values v 1 + (= d) andvin are monotonically increasing and decreasing, respectively, as c increases, so that the manifolds W u (Q ) and W s (Q + ) pass through each other transversely at c = c 0 (α). Therefore, we have completed the construction of the desired singular heteroclinic orbit for each parameter α 1. This construction is summarized in the following proposition, and global geometry of the singular heteroclinic solution is illustrated in Fig. 4. Proposition.1. For each (α, m) with α 1 and m =1,,..., there exists a singular heteroclinic orbit Γ that connects Q 1 to Q+ and that consists of the union of the segments Γ 1, Γ+ 1,andΓ of equations (), (4), and(8). There exists α = 4e 1+e such that for each α α, c 0 (α) =, whereas for each α>α, c is given by the unique solution c 0 (α) of the equation α = α 0 (c), whereα 0 is given by (8). Moreover, c 0 (α) > for all α>α, dα0 dc (c) > 0 for c>, andatc = c 0(α), the singular orbit lies in the transverse intersection of W u (Q ) and W s (Q + ) for ε =0. Finally, in chart K 1 for c>, the singular

12 618 F. Dumortier and T. J. Kaper ZAMP heteroclinic passes through P + 1 =(0, d, 0), whered = c 1 c 4, and for c =, it passes through (0, 1, 0). Remark 4. The geometry in Region gives rise naturally to the existence of the critical value α stated in the proposition. For each value α>α, we see that v in = αd < α, and there is a unique value of c>, for which a singular heteroclinic connection can be formed involving the unstable manifold of the saddle at v 1 = v 1 + = d, as stated in the proposition. At α = α, v in = α, and the singular heteroclinic connection can only be formed for c =. Finally, for values α<α, the situation is similar to that in which there is no enhancement. 3. Persistence of the singular heteroclinic orbit Γ and completion of the Proof of Theorem 1 In this section, we establish the persistence of the singular solution for each (α, m) with α 1and m =1,,... and, hence, complete the proof of Theorem 1. We start in Sect. 3.1 by showing that, for each such (α, m) and for each ε>0 sufficiently small, there exists a unique value c(ε, α) ofc in (11), for which there is a heteroclinic orbit connecting Q to Q + that lies in the intersection of W u (Q )andw s (Q + ), and that is close to the singular orbit Γ constructed in the previous section. Then, in Sect. 3., we derive the corresponding necessary conditions involving the dependence of c(ε, α) on ε in order for Γ to persist. Combining these two aspects of the analysis, we obtain the existence of the unique traveling waves with wave speeds c(ε, α), as stated in Theorem 1. We carry out the construction explicitly first for the case in which c 0 (α) > andα>α and then for the case in which c 0 (α) =andα = α. The construction for c 0 (α) =andα [1,α ) is similar to that given in Section 3 of [1] for the FKPP equation without an enhancement. For convenience of notation, we suppress the dependence of the invariant manifolds W u (Q )andw s (Q + ) on the system parameters in (11), including c, as is customary in dynamical systems theory Existence and uniqueness of persistent heteroclinics In this section, we establish the existence and uniqueness of the persistent heteroclinics with c(ε, α) c 0 (α) for each (α, m) with α 1. We prove that, for each (α, m) with α 1andm =1,... and for ε>0 sufficiently small in (11), the unstable manifold of Q intersects the stable manifold of Q + for a unique value of c, labeled c(ε, α). Proposition 3.1. For each (α, m) with α 1 and m =1,,..., there exists an ε 0 sufficiently small such that for 0 <ε<ε 0 and c(α, ε) c 0 (α), there exists a unique heteroclinic connecting Q to Q +. Proof. The proof of this proposition is similar to that of Proposition 3.1 in [1]. In the system with ε =0 (i.e., r 1,r =0)andc = c 0 (α), the intersection of W u (Q )andw s (Q + ) is transverse for each α 1, as shown in the previous section. There, it was shown that d, thev 1 coordinate of the point P1 out at which W1 u (Q 1 ) intersects Σout 1, is strictly monotonically increasing as the parameter c increases, whereas v in, the v coordinate of the point P in at which W s (Q + ) intersects Σin = κ 1 (Σ out 1 ), is strictly monotonically decreasing as the parameter c increases. Hence, the manifolds pass through each other with non-zero speed at c = c 0 (α). Then, by standard persistence theory for invariant manifolds under smooth perturbations, the full system with ε positive and sufficiently small has a smooth stable manifold W s (Q + )on{0 U<ε},which may be extended smoothly to U = ε. Similarly, by standard persistence theory for invariant manifolds under smooth perturbations, the full system with ε positive and sufficiently small has a smooth unstable manifold W u (Q ) on the domain αε < U 1, and it may be extended smoothly to U = αε. In addition, solutions on the unstable manifold W u (Q ) may be extended continuously across the discontinuity at

13 Vol. 66 (015) Wave speeds for the FKPP equation 619 U = αε along smooth solutions of the full problem on the domain ε U<αε. These manifolds are O(ε) close to their ε = 0 counterparts, and hence, due to the transverse intersection of their ε = 0 counterparts, these perturbed manifolds intersect transversally for a unique value of c(ε, α), where c c 0 (α) as ε Transition through chart K 1 and the asymptotics of c(ε, α) In this section, we study the passage of trajectories through chart K 1 under the flow of the full equation (8) with r 1 > 0, and we derive the asymptotics of c. It is convenient to work with appropriate sections for the flow. We will employ the sections Σ out 1 = κ 1 (Σ in ) and Σ in 1 defined above. The transition of orbits through chart K 1 from Σ in 1 to Σ out 1 is governed by the transition map Π 1 :Σ in 1 Σ out 1. We will derive asymptotically accurate representations of this map in both cases, α>α and α = α, beginning with the former. The analysis of the transition through K 1 for r 1 > 0 in the third case, α<α, follows closely that of Section 3. in [1] for the FKPP equation without enhancement. We begin with the case α>α. Taking into account that c(ε, α) c 0 (α) foreachα>α in the singular limit as ε 0, we define c(ε, α) =c(ε, α) c 0 (α), where we observe that c(ε, α) =O(1) as ε 0 by Proposition 3.1. We shift the point P 1 =(0,v 1 + = d, 0) to the origin by introducing the new variable w = v 1 + d = v 1 v 1 +. (3) As before, let P1 in denote the point of intersection of Γ 1 describing Π + 1 on Σin 1 {v 1 >v 1 + } in the following. With the above transformations, system (8) is equivalent to r 1 = r 1 (d w), w = (1 r1 m )+c(d w) (d w), ε 1 = ε 1 (d w). with Σin 1. Since c 0, we restrict ourselves to (33a) (33b) Then, recalling that 1 cd+d = 0 and rescaling time by the positive factor d w, we find that system (33) is equivalent to ṙ 1 = r 1, (33c) (34a) ẇ = ( c d )w + 1 d (rm 1 w ) 1 w, d (34b) ε 1 = ε 1. (34c) (Here, the overdot denotes differentiation with respect to the new, rescaled time ξ 1.) Plainly, the ε 1 - equation decouples, and ε 1 (ξ 1 )=ε 1,0 e ξ1 = ε r 0 e ξ1. Also, r 1 (ξ 1 )=r 0 e ξ1, but it will be useful to keep the equations for r 1 and w together as a system. To study the (r 1,w)-system in (34), we binomially expand the second component for w d < 1 and put the system into normal form. Lemma 3.1. There exists a smooth near-identity coordinate change of the form (r 1,w) (r 1,W) with W = wk(r 1,w), withk(0, 0) = 1, which transforms (34) into ṙ 1 = r 1, ( Ẇ = c ) W + rm 1 d d. (35a) (35b)

14 60 F. Dumortier and T. J. Kaper ZAMP Proof. Let ν = c d. We have ν>0. At the level of formal power series, a calculation with Lie brackets, following the general presentation [13, 7], yields [ r 1 νw r 1 w,wi r j ] 1 =( j (i 1)ν)w i r j 1 w w. Hence, it clearly follows that all terms of the form w i r j 1 w with i 1 can be removed by a transformation w W 1 = w h(r 1,w) for some formal power series h. Now, by taking any smooth function h(r 1,w) with infinite jet j h(0, 0) = h, one has that the transformation (r 1,w) (r 1,W 1 ) with W 1 = wh(r 1,w)brings(34) intotheform ṙ 1 = r 1, (36a) ( Ẇ 1 = c ) W 1 + rm 1 d d + W 1l(r 1,W 1 ), (36b) where l is a smooth function whose formal power series expanded about the point (r 1,W 1 )=(0, 0) is identically zero. Finally, from the normal form theory presented in [41], the flat function l in the second component of the system may also be removed by a near-identity coordinate change that preserves {W 1 = 0}. Hence, (36) has been transformed into system (35) via a sequence of smooth, near-identity coordinate changes that preserve {w = 0}. Next, we solve the normal form (35) for arbitrary α>α. For convenience, we set c(α, ε) = η in the decomposition c(α, ε) =c 0 (α) + c(α, ε), suppressing the α-dependence, where η is a new variable. The initial conditions are r 0 (0) and W (0) = w 0 + b(η). Using explicit integration, we find r 1 (ξ 1 )=r 1 (0)e ξ1 (37a) W (ξ 1 )= (w 0 + b(η)+ (r 1(0)) m )e ( c d )ξ1 (r 1(0)) m (m +)d c (m +)d c e mξ1. (37b) Then, from the solution ε 1 (ξ 1 ), we know that e ξ1 = εα r, where ξ 1(0) 1 denotes the time at which the solutions hit the section Σ out 1 at {u = α}. Therefore, the solutions are given by W (ε, η) = (w 0 + b(η)+ (r 1(0)) m )( εα ) c d α m ε m (m +)d c r 1 (0) (m +)d c. With the expansion c = c 0 η, we also have d = d 0 + δ(η), where d 0 = c0 1 c 4andδ (0) < 0, as well as c c0 d = d 0 +τ(η), with τ(0) = 0. Therefore, on the cross-section Σ out 1 at the boundary between charts K 1 and K, the final form of the solution for each (α, m) is ( (r 1 (0)) m )( εα ) c 0 d +τ(η) 0 α m ε m W (ε, η) = w 0 + b(η)+ (m +)d 0 c 0 + δ(η) r 1 (0) (m +)d 0 c 0 + δ(η) (38) For each (α, m), this solution, which lies on W u (Q ) and which we tracked through Region 3 in chart K 1, must also lie on W s (Q + ) to represent the persistent heteroclinic. This is the geometric condition that uniquely determines η (i.e., c) as a function of ε and α. To impose this geometric condition, we observe that for small positive values of ε = r > 0, the manifold W s (Q + ) may also be tracked in chart K, just as we did in Sect..1 in the limiting case ε = r = 0. The equations in chart K depend on ε only through the value of c = c 0 (α) η. Hence, by following the calculations presented in Sect..1, we see that orbits on W s (Q + ) hit the line u =1at the boundary of the cutoff domain with v = (c 0 (α) η). Moreover, we can also follow the calculations presented there for Region. In particular, using formula (5) with v out = (c 0 (α) η), one can calculate the corresponding value of v, denoted v in (η), at which the orbits on W s (Q + ) cross the line u = α, i.e., hit the section Σ in = κ 1 (Σ out 1 ). From this calculation, one also sees that v in (η) approaches v in

15 Vol. 66 (015) Wave speeds for the FKPP equation 61 from below and at a linear rate in the limit η 0, where v in is given by (6). Namely, we may write v in (η) =v in ηs(η) for some bounded function s satisfying s(0) < 0. Hence, imposing the geometric condition and using only the leading-order terms in (38), we find that to leading order ( w 0 + (r 1 (0)) m )( εα ) c 0 d 0 (m +)d 0 c 0 r 1 (0) α m ε m = η s(0). (39) (m +)d 0 c 0 α For each α>α, this formula determines how η and, hence, c depend on ε, and we recall that w 0 depends on r 1 (0). We demonstrate this as follows. The term on the left-hand side that dominates, asymptotically as ε 0, depends on the value of c. In particular, we see that the exponent c0 d 0 onε in the first term equals the exponent m of the second term when c = m+ m+1,foreachm =1,,... These are precisely the values at which there are resonances in (34). Hence, for c< m+ m+1, we see that c0 d 0 <m, which means that the first term dominates, and )( ) ( α c 0 η d 0 1 s(0) lim r 1(0) 0 w 0 (r 1 (0)) c 0 d 0 ε c 0 d 0. In contrast, for c> m+ m+1, we see that c0 d 0 >m, which means that the second term dominates, and ( η α m+1 s(0)((m +)d 0 c 0 ) ) ε m Summarizing, formula (39) reveals that η = O(ε c 0 d 0 ) if c< m+ m+1, O(ε m ln(1/ε)) if c = m+ m+1, O(ε m ) if c> m+ m+1. (40) In the boundary case, the gauge function ln(ε) arises due to the resonance. The coefficient may be computed directly from (38). In all cases, we have that η>0 (where we recall s(0) < 0), and hence, c(ε, ( α) = η <0. Also, the coefficients will depend on α. Finally, we recall that β 0 (c) was defined to be 1 c + c ) c 4 4 in (9), and this is precisely c0 d 0. Therefore, asymptotically as ε 0, formula (40) exactly yields the function for c(ε, α) = η stated in (10). This completes the proof of Theorem 1 for each (α, m) with α>α and m =1,,... Now, we prove Theorem 1 in the case α = α = 4e 1+e. For this case, c 0 (α) =, and the analysis proceeds closely along the lines used in Section 3. of [1] in the case of the FKPP equation without enhancement, although it differs in one crucial aspect. Following the analysis of [1], we write c(α, ε) = η, where we make the Ansatz that the correction to the wave speed is negative. This Ansatz is made without loss of generality, since the wave speed is unique and we find a solution with c <0. Since v 1 + = 1, system (33) is equivalent in this case to r 1 = r 1 (1 w), w = (1 r1 m )+( η )(1 w) (1 w), ε 1 = ε 1 (1 w). (41a) (41b) (41c)

16 6 F. Dumortier and T. J. Kaper ZAMP Hence, after simplifying the middle component and rescaling the time in the full system by the positive factor 1 w, we arrive at the main system to be studied in this case, ṙ 1 = r 1, (4a) ẇ = η + rm 1 w 1 w, (4b) η =0. (4c) This is a system of two first-order equations which depend on the parameter η. To analyze the transition through chart K 1 as the heteroclinics pass near P 1 for ε (0,ε 0 ) small, we compute the map Π 1 in more detail. The functional relation between η and ε will follow from requiring that the unstable manifold W u (Q ) connects to the stable manifold W s (Q + )inσ in = κ 1 (Σ out 1 )after the transition past P 1. Proposition 3.. Let α = α and c 0 (α) =. For a heteroclinic connection between Q and Q + to be possible when ε>0 in (11), η must be given by η(ε) = π lnε + O( (ln ε) ). (43) Proof. To simplify the analysis of (4), we make a normal form transformation which decouples the dynamics of r 1 and w in (4). For each r 1, there exists, by Theorem 1 of [7], a C r coordinate change (r 1,w) ( R 1 (r 1,w,η),W(r 1,w,η) ) (44) with R 1 (0,w,η) = 0 which transforms the r 1 w system (4) into Ṙ 1 = R 1, Ẇ = η W,. (45b) 1 W This normal form with η as a parameter respects the invariance of {r 1 =0}. For η sufficiently small, we calculate the transition time Ξ 1 of solutions of system (45) between the two sections corresponding to Σ in 1 and Σ out 1 after transformation by (44). Let W in > 0andW out < 0 denote the corresponding values of W. We will see that, to leading order, Ξ 1 = Ξ 1 (W in,w out,η)is independent of the exact values of W in and W out. Since the equations in (45) are decoupled, we can solve (45b) by separation of variables. Introducing the new variable Z = W η in (45b), we find ) (1 Z η dz d ξ 1 = ( ). Z + η 1 η 4 Integrating, we find Ξ 1 = 1 η Z arctan Z out 1 ( ) η 1 η 4 η 1 η Z ln Z + η 1 η Z out. (46) 4 in Z in 4 Here, Z in and Z out are the values of Z obtained from W in and W out, respectively. Reverting to W in (46) and dividing out a factor of η 1, we find Ξ 1 = 1 1 η arctan W out η arctan W in η η 1 η 4 η 1 η 4 η 1 η 4 η [ ln (W out ) W out η + η ln (W in ) W in η + η ]]. (47) (45a)

17 Vol. 66 (015) Wave speeds for the FKPP equation 63 Since we are only interested in deriving a leading-order expression for η, weexpand(1 η )(1 η 4 ) 1 = 1+O(η ). Also, since W out < 0andW in > 0 and since (44) is near identity, we conclude that W out < 0 and W in > 0areO(1) as ε 0 and independent of η to leading order. To derive expansions for the arctangent terms in (47), we make use of the identity ( 1 arctan(x) + arctan = ± x) π, where the sign equals the sign of x. In particular, for x large, we have [1] arctan(x) =± π ( 1 x 1 ) 3x In our case, x = W η = W ( 1+O(η ) ) η η 1 η 4 and, hence, arctan W out η = η W η 1 out + O(η3 ) and arctan W in η = π η W η 4 η 1 in + O(η3 ). η 4 The crucial difference lies with this first arctan, which is zero to leading order here due to the fact that Γ + 1 lies on the line defined by w =0andr 1 = 0 perpendicular to the w-axis, whereas that arctan term is asymptotic to π for α<α due to the fact that Γ + 1 is then tangent to the w-axis. For the logarithmic terms in (47), one has ln W Wη + η =ln W +ln 1 η W (1 W ) = ln W + O(η ). Hence, Ξ 1 = 1 [ [ π 1 η + η W out 1 +ln W out ] ] W in W in + O(η 3 ). (48) On the other hand, we know that R 1 evolves according to R 1 = R in 1 e ξ 1, where R in 1 > 0 denotes the initial value R 1 (0). Hence, the time Ξ is also given by Ξ 1 = ln R 1 R1 in = ln ( r 1 β(r 1,w,η) ), (49) where β(r 1,w,η) is a strictly positive, C r -smooth function that depends on the choice of normalizing coordinates in (44). (Note that the ε-dependence of β is implicitly encoded in its arguments r 1, w, and η and that β = O(1) as ε 0.) Now, during that same time Ξ 1 introduced above, the R 1 - variable has to evolve from R1 in to a value R1 out permitting a connection between W1 u (l 1 )andws (l + )in Σ in = κ 1 (Σ out 1 ). This will impose a relation between η and ε that we calculate to leading order in the normal form coordinates (R 1,W,η). To that end, recall that r 1 in Σ out 1 is fixed at r1 out = ε. Hence, (49) yields Ξ 1 = ln ( ε β(ε, η) ) (50) for some function β(ε, η) which is strictly positive and C r -smooth, with β = O(1) for ε 0. Combining (48) and (50) and recalling that W in and W out are O(1) as ε 0, as noted below Eq. (47), we find ln ε = 1 [ π η + ηθ(ε)+o ( η )] (51)

18 64 F. Dumortier and T. J. Kaper ZAMP for some bounded function θ. Solving (51) forη, we obtain η = π + η, (5) lnε where η defines a relative correction in (5), i.e., there holds η = O((ln ε) 1 ). In fact, substituting (5) into (48), one can check that η = O((ln ε) ): Given (51), it follows that ( π ) η(ln ε) = η ln ε θ(ε)+o ( (ln ε) 1, η, η ln ε ). Since η ln ε = O(1) by assumption, we have η = O((ln ε) ). This concludes the proof. Now, we show that the assertions of Theorem 1 for α = α follow immediately from Propositions 3.1 and 3.. By Proposition 3.1, given ε>0 sufficiently small, for α = α and for each m =1,,..., there exists a heteroclinic orbit in (11) connecting Q and Q + and each heteroclinic lies close to the corresponding singular heteroclinic orbit Γ with c π 4(ln(ε)). Then, by Proposition 3., the heteroclinics lie in the intersection of the two manifolds W u (Q )andw s (Q + ) and correspond to the traveling wave solutions of (3) with enhancement (4). This establishes (6) and completes the proof of Theorem 1 also for the case α = α. 4. Proofs of Theorems and 3 In this section, we present the proofs of Theorems and 3, beginning with the former. Proof. For the PDE (3) with reaction function f given by (1), the traveling wave ODE is again (11), and the blow-up coordinate change is again given by (0). Moreover, the function f LE is the same as the constant enhanced reaction function (4) in both Region 1 (0 U ε) and Region 3 (αε < U 1). Hence, in Region 1, the desingularized vector field in chart K is equivalent to (), and the desired solution is again (3). It enters Region 1 at the point P =(1, c, 0). It is a segment of W s (Q + ) and constitutes the portion of the singular heteroclinic orbit Γ in Region 1, exactly as in Sect..1. Similarly, the governing system of equations in Region 3 is equivalent to (8), because f LE is also the unperturbed reaction function here. Moreover, all of the analysis in Sect.. applies directly. The portions of the singular heteroclinic in Region 3 are given by Γ 1 Γ+ 1, where Γ 1 is the connection from Q 1 =(1, 0, 0) to P 1 + =(0,v+ 1, 0) with v+ 1 = d = c + 1 c 4, and Γ + 1 is that part of W u (P 1 + )on the line {r 1 =0,v 1 = d} up to the point P1 out =(0, d, 1/α). The governing system of equations in Region (1 <u α) differs from that generated by the constant enhancement function (4). In Region, in terms of the variables of chart K, the governing equations are u = v, (53a) v = cv (1 + γ)u, (53b) r =0. (53c) This is the equation for a damped harmonic oscillator. To find the portion of the singular heteroclinic orbit Γ in Region, we look for a solution of (53) that connects the point P1 out =(r 1 =0,v 1 = d, ε 1 =1/α), at which the singular heteroclinic exits Region 3 and enters Region, to the point P =(u =1,v = c, r = 0), at which the singular heteroclinic exits Region and enters Region 1. Also, we recall that under the coordinate change between charts K 1 and K, P1 out is the same as P in =(u = α, v = dα, r = 0). For 4(γ +1)>c, the general solution of (53) is u (ξ) =Ae δξ cos(ωξ)+be δξ sin(ωξ). (54)