Bichromatic travelling waves for lattice Nagumo equations
BBichromatic travelling waves for lattice Nagumoequations
Hermen Jan Hupkes ∗ , Leonardo Morelli † , and Petr Stehl´ık ‡ Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands Department of Mathematics and NTIS, Faculty of Applied Sciences, University of West Bohemia,Univerzitn´ı 8, 306 14 Plzeˇn, Czech Republic
September 8, 2018
Abstract
We discuss bichromatic (two-color) front solutions to the bistable Nagumo lattice differential equa-tion. Such fronts connect the stable spatially homogeneous equilibria with spatially heterogeneous2-periodic equilibria and hence are not monotonic like the standard monochromatic fronts. We pro-vide explicit criteria that can determine whether or not these fronts are stationary and show that thebichromatic fronts can travel in parameter regimes where the monochromatic fronts are pinned. Thepresence of these bichromatic waves allows the two stable homogeneous equlibria to both spread outthrough the spatial domain towards each other, buffered by a shrinking intermediate zone in whichthe periodic pattern is visible.
Keywords: reaction-diffusion equation; lattice differential equation; travelling waves; nonlinear al-gebraic equations.
MSC 2010:
In this paper we consider the Nagumo lattice differential equation (LDE)˙ u j ( t ) = d (cid:2) u j − ( t ) − u j ( t ) + u j +1 ( t ) (cid:3) + g (cid:0) u j ( t ); a (cid:1) , (1.1)posed on the spatial lattice j ∈ Z , with t ∈ R . We assume d > g ( u ; a ) = u (1 − u )( u − a ) with a ∈ (0 , u = 0 and u = 1 in a discrete spatial environment.A crucial role is reserved for so-called travelling front solutions, which have the form u j ( t ) = Φ( j − ct ) , Φ( −∞ ) = 0 , Φ(+ ∞ ) = 1 . (1.2)Such solutions are often referred to as invasion waves, as they provide a mechanism by which the ener-getically preferred state can invade the spatial domain. ∗ [email protected] † corresponding author, [email protected] ‡ [email protected] a r X i v : . [ m a t h . A P ] M a y ur work focuses on the case where c = 0 holds for these primary invasion waves, indicating a delicatebalance between the two competing states. In this case (1.1) can admit stable spatially periodic rest-states. Numerical results indicate that these states can act as a buffer between regions of space where u = 0 and u = 1 dominate the dynamics. This buffer shrinks as these two stable states appear to movetowards each other. This latter process is governed by secondary two-component invasion waves that weanalyze in detail in this paper. Nagumo PDE
The LDE (1.1) can be seen as the nearest-neighbour discretization of the Nagumoreaction-diffusion PDE [35] u t = u xx + g ( u ; a ) , x ∈ R (1.3)on a spatial grid with size h = d − / . This PDE has been used as a highly simplified model for thespread of genetic traits [1] and the propagation of electrical signals through nerve fibres [3]. In higherspace dimensions it also serves as a desingularization of the standard mean-curvature flow that is oftenused to describe the evolution of interfaces [18].Fife and McLeod [21] used phase plane analysis to show that (1.3) admits a front solution for each a ∈ [0 , u ( x, t ) = Φ( x − ct ) , Φ( −∞ ) = 0 , Φ(+ ∞ ) = 1 , (1.4)for some smooth waveprofile Φ and wavespeed c that has the same sign as a − . These fronts henceconnect the two stable spatially homogeneous equilibria u ( x, t ) ≡ u ( x, t ) ≡ u ( x,
0) = u ( x ) that has u ( x ) ≈ x (cid:28) − u ( x ) ≈ x (cid:29) +1 will converge to a shiftedversion of this front as t → ∞ .These front solutions can be used as building blocks to capture the behaviour of a more general classof solutions to (1.3). Consider for example the two-parameter family of functions u plt; α ,α ( x, t ) = Φ( x − ct + α ) + Φ( − x − ct + α ) − , (1.5)with α ≥ α . Each of these functions can be interpreted as a shifted version of the front solution (1.4)that is reflected in a vertical line to form a plateau.If c <
0, then any initial configuration that has u ( x ) ≈ | x | (cid:29) L and u ( x ) ≈ | x | ≤ L will converge to a member of the family (1.5) as t → ∞ . This provides a mechanism by which compactregions where u ∼ c > t → −∞ and tend to zero as t → + ∞ . These solutions are stable under small perturbations [44]. Inparticular, they can be viewed as a robust elimination process whereby compact regions that have u ∼ t → ∞ . Nagumo LDE
For many physical phenomena such as crystal growth in materials [6], the formation offractures in elastic bodies [39] and the motion of dislocations [10] and domain walls [14] through crystals,the discreteness and topology of the underlying spatial domain have a major impact on the dynamicalbehaviour. It is hence important to develop mathematical modelling tools that can incorporate suchstructures effectively. Indeed, by now it is well known that discrete models can capture dynamicalbehaviour that their continuous counterparts can not.The LDE (1.1) has served as a prototype system in which such effects can be explored. It arisesas a highly simplified model for the propagation of action potentials through nerve fibers that haveregularly spaced gaps in their myeline coating [3]. Two-dimensional versions have been used to describephase transitions in Ising models [2], to analyze predator-prey interactions [38] and to develop pattern2 .0 0.2 0.4 0.6 0.8 1.0 a0.010.020.030.040.05d monochromatic travelling waves, c mc ≠ c bc ≠ c bc = Figure 1: Existence regions for monochromatic and bichromatic wave solutions to (1.1).recognition algorithms in image processing [12, 13]. Recently, an interest has also arisen in Nagumoequations posed on graphs [40], motivated by the network structure present in many biological systems[36].Many authors have studied the LDE (1.1), focusing primarily on the richness of the set of equilibria[31] and the existence of travelling and standing front solutions [33, 45]. Such solutions have the form(1.2), which leads naturally to the waveprofile equation − c Φ (cid:48) ( ξ ) = d (cid:2) Φ( ξ − − ξ ) + Φ( ξ + 1) (cid:3) + g (cid:0) Φ( ξ ); a (cid:1) . (1.6)Since the behaviour of every lattice point is governed by the same profile Φ, we refer to these frontsolutions as monochromatic waves in this paper (in order to distinguish them from the bichromatic waveswe discuss in the sequel). The seminal results by Mallet-Paret [33] show that for each a ∈ [0 ,
1] and d > c = c mc ( a, d ) for which such monochromatic (mc) solutions exist. Pinning
Upon fixing a ∈ (0 , \ { } , Zinner [45] established that c mc ( a, d ) (cid:54) = 0 for d (cid:29)
1, while Keener[30] showed that c mc ( a, d ) = 0 for 0 < d (cid:28)
1. Upon fixing d >
0, Mallet-Paret established [33] that c mc ( a, d ) (cid:54) = 0 for a ≈ a ≈
0. In addition, again for fixed d >
0, the results in [24, 34] stronglysuggest that there exists δ > c mc ( a, d ) = 0 whenever (cid:12)(cid:12) a − (cid:12)(cid:12) ≤ δ ; see Figure 1.This last phenomenon is called pinning and distinguishes the LDE (1.1) from the PDE (1.3). Itis a direct consequence of the fact that we have broken the translational invariance of space. Indeed,(1.6) becomes singular in the limit c → g it is possible to design systems for which this pinning isabsent [15, 26]. Understanding the pinning phenomenon is an important and challenging mathematicalproblem that also has practical ramifications. Periodicity
In this paper we study waves that connect spatially homogeneous stationary solutions of(1.1) with spatially heterogeneous 2-periodic stationary solutions. It is well known that many physicalsystems exhibit spatially periodic features [22, 23, 37]. Examples that also feature spatial discretenessinclude the presence of twinning microstructures in shape memory alloys [4] and the formation of domain-wall microstructures in dielectric crystals [41].In many cases the underlying periodicity comes from the spatial system itself. For example, in [19, 20,25] the authors consider chains of alternating masses connected by identical springs (and vice versa). The3igure 2: A monochromatic travelling wave of (1.1) (left panel) connects two spatially homogeneous sta-tionary solutions. A bichromatic travelling wave of (1.1) (right panel) connects a spatially homogeneousstationary solution with a spatially heterogeneous one.dynamical behaviour of such systems can be easily modelled by LDEs with periodic coefficients. In certainlimiting cases the authors were able to construct so-called nanopterons, which are multi-component wavesolutions that have low-amplitude oscillations in their tails.However, periodic patterns also arise naturally as solutions to spatially homogeneous discrete systems.Indeed, we shall see in § d > d <
0. Thiscan be seen by introducing new variables v j = ( − j u j , which restores the applicability of the comparisonprinciple. This choice essentially decomposes the lattice sites Z into two groups Z odd and Z even that eachhave their own characteristic behaviour.Such anti-diffusion models have been used to describe phase transitions for grids of particles thathave visco-elastic interactions [8, 9, 42]. In [5] this problem has been analyzed in considerable detail.The authors show that the resulting two component system admits co-existing patterns that can be bothmonostable and bistable in nature. Similar results with piecewise linear nonlinearities but more generalcouplings between neighbours can be found in [43]. Bichromatic waves
In this paper we are interested in the parameter region where c mc ( a, d ) = 0. Ina subset of this region it is possible to show that (1.1) has spatially heterogenous stable equilibria. Wefocus on the simplest case and consider so-called bichromatic (two-color) equilibria, which are spatiallyperiodic with period two. As such, they are closely connected to solutions of the Nagumo equation posedon a graph with two vertices. We set out to construct bichromatic front-solutions to (1.1), which can beseen as waves that connect the spatially homogeneous equilibrium u ≡ §
3, where we give a detailed description of the set ofparameters ( a, d ) where such 2-periodic equilibria exist and where they are stable. In contrast to thesetting encountered in [5], the relevant bifurcation curves cannot all be described explicitly. Besides aglobal result stating that the number of such equilibria decreases as d is increased, we also obtain preciseasymptotics that describe the boundaries near the three corners ( a, d ) ∈ { (0 , , (1 / , / , (1 , } inFigure 1.As in [5], these preparations allow the existence of bichromatic fronts to be established in a straight-forward fashion. Indeed, one can apply the general theory developed by Chen, Guo and Wu in [11]for discrete periodic systems that admit a comparison-principle. These results imply that there exists4 unique wavespeed c bc ( a, d ) for which such bichromatic (bc) fronts exist. If c bc ( a, d ) (cid:54) = 0, the Fred-holm theory developed in [28] together with the techniques from [27, §
3] can be used to show that thesetravelling fronts depend smoothly on ( a, d ) and are nonlinearly stable.However, these general results cannot distinguish between the cases c bc ( a, d ) = 0 and c bc ( a, d ) (cid:54) = 0where we have standing respectively travelling fronts. This should be contrasted to the situation for thePDE (1.3), where the sign of the wavespeed is given by the sign of a simple integral [21]. Indeed, thereis a large set of parameters ( a, d ) for which the discrete bichromatic fronts fail to travel, even though theanalogous integral does not vanish.Our second main contribution is that we provide explicit criteria in § c bc = 0or c bc >
0. Together these results cover most of the parameter region where bichromatic fronts exist.In any case, they provide a two-component generalization of the coercivity conditions introduced in [33],which ensure c mc ( a, d ) (cid:54) = 0 for the boundary regions a ≈ a ≈ c bc = 0 are closely related to the setup used by Keener [30] to establishthat monochromatic waves are pinned for 0 < d (cid:28)
1. In particular, for small values of d one can neglectthe diffusion term in § § ,
0) and (1 , Colliding fronts
One of the main reasons for our interest in these bichromatic fronts is that theypresent mechanisms via which the stable homogeneous states u = 0 and u = 1 can spread throughoutthe domain, even though the primary invasion waves are blocked from propagation. By using similartechniques as in [44], we believe it should be possible to construct entire solutions consisting of a right-travelling bichromatic front connection between the homogeneous equilibrium u ≡ u ≡
1; see Figure 3. The resulting state after the collisionis then a pinned monochromatic front that connects 0 with 1. We have been able to numerically verifythe existence of these solutions in the parameter regions predicted by the theory developed in this paper.
Acknowledgements
HJH acknowledges support from the Netherlands Organization for Scientific Re-search (NWO) (grant 639.032.612). LM acknowledges support from the Netherlands Organization forScientific Research (NWO) (grant 613.001.304).
Our interest here is in the lattice differential equation˙ x j ( t ) = d (cid:2) x j − ( t ) − x j ( t ) + x j +1 ( t ) (cid:3) + g (cid:0) x j ( t ); a (cid:1) (2.1)posed on the one-dimensional lattice, i.e., j ∈ Z . The bistable nonlinearity is explicitly given by g ( u ; a ) = u (1 − u )( u − a ) , (2.2)5igure 3: Colliding front of (1.1) consisting of a right-travelling bichromatic front connection betweenthe homogeneous equilibrium u ≡ u ≡ a ∈ (0 , x j ( t ) = Φ u ( j − ct ) if j is even , Φ v ( j − ct ) if j is odd , (2.3)for some wavespeed c ∈ R and R -valued waveprofileΦ = (Φ u , Φ v ) : R → R . (2.4)Substituting this Ansatz into (2.1) we obtain the travelling wave system − c Φ (cid:48) u ( ξ ) = d (cid:2) Φ v ( ξ − − u ( ξ ) + Φ v ( ξ + 1) (cid:3) + g (cid:0) Φ u ( ξ ); a (cid:1) , − c Φ (cid:48) v ( ξ ) = d (cid:2) Φ u ( ξ − − v ( ξ ) + Φ u ( ξ + 1) (cid:3) + g (cid:0) Φ v ( ξ ); a (cid:1) . (2.5)Upon introducing the functions G ( u, v ; a, d ) = G ( u, v ; a, d ) G ( u, v ; a, d ) = d ( v − u ) + g ( u ; a )2 d ( u − v ) + g ( v ; a ) , (2.6)we see that any stationary solution (Φ u , Φ v )( ξ ) = (cid:0) u, v (cid:1) (2.7)to (2.5) must satisfy the nonlinear algebraic equation G ( u, v ; a, d ) = 0 . (2.8)The full bifurcation diagram for this equation is described in §
3. For our purposes here however itsuffices to summarize a subset of the conclusions from this analysis, which we do in our first result below.In particular, there exists a region Ω bc in the ( a, d )-plane for which the spatially homogeneous system( ˙ u, ˙ v ) = G ( u, v ; a, d ) has a stable equilibrium (cid:0) u bc ( a, d ) , v bc ( a, d ) (cid:1) that can be interpreted as a bichromaticequilibrium state for the LDE (2.1). 6 roposition 2.1 (see § . There exists a continuous curve d bc : [0 , → [0 , ] with d bc ( ) = and d bc (1 − a ) = d bc ( a ) so that for every 0 ≤ d < d bc and 0 < a < u, v ) ∈ [0 , . Upon writingΩ bc = { < d < d bc ( a ) and 0 < a < } , (2.9)there exist C ∞ -smooth maps ( u bc , v bc ) : Ω bc → (0 , (2.10)with u bc < a < v bc so that for every ( a, d ) ∈ Ω bc we have G (cid:0) u bc ( a, d ) , v bc ( a, d ); a, d (cid:1) = 0 (2.11)together withdet D , G (cid:0) u bc ( a, d ) , v bc ( a, d ); a, d (cid:1) > , Tr D , G (cid:0) u bc ( a, d ) , v bc ( a, d ); a, d (cid:1) < . (2.12)We note that the statements (2.11)-(2.12) are also valid upon replacing the bichromatic rest-state( u bc , v bc ) by the monochromatic equilibria (0 ,
0) and (1 , ξ →−∞ Φ( ξ ) = (0 , , lim ξ → + ∞ Φ( ξ ) = ( u bc , v bc ) , (2.13)or the ’upper’ boundary conditionslim ξ →−∞ Φ( ξ ) = ( u bc , v bc ) lim ξ → + ∞ Φ( ξ ) = (1 , . (2.14)The result below summarizes several key facts concerning the existence and uniqueness of such waves. Itintroduces subregions of Ω bc denoted by T low and T up where the bichromatic travelling waves (2.3) existwith nonzero speeds c low > c up <
0; see Figure 4. With the exception of the inequalities c low ≥ c up ≤
0, these properties follow directly from the theory developed in [11, 28].
Theorem 2.2 (see § . There exist continuous maps c low : Ω bc → [0 , ∞ ) , c up : Ω bc → ( −∞ ,
0] (2.15)that satisfy the following properties.(i) Upon introducing the open sets T low = { ( a, d ) ∈ Ω bc : c low > } , T up = { ( a, d ) ∈ Ω bc : c up < } , (2.16)the functions c low and c up are C ∞ -smooth on T low respectively T up .(ii) There exist C ∞ -smooth functionsΦ low : T low → W ∞ ( R ; R ) , Φ up : T up → W ∞ ( R ; R ) , (2.17)such that for any ∈ { low , up } and any ( a, d ) ∈ T , the pair( c, Φ) = (cid:0) c ( a, d ) , Φ ( a, d ) (cid:1) (2.18)satisfies (2.5) together with the boundary condition (2.13) if (cid:48) > (0 , up low low ⋂ up Ω bc ∖ ( low ⋃ up ) Figure 4: Numerical bounds for the parameter sets Ω bc , T low and T up introduced in Theorem 2.2, in theneighbourhood of the cusp ( , ).(iii) For any ∈ { low , up } and any ( a, d ) ∈ Ω bc \ T , there exists a non-decreasing function Φ : R → R that satisfies (2.5) with c = 0 together with the boundary condition (2.13) if ∈ { low , up } and ( a, d ) ∈ Ω bc and consider any c (cid:54) = 0 together with a function Φ ∈ W , ∞ ( R ; R ) that satisfies (2.5) together with the boundary condition (2.13) if c = c ( a, d ) and Φ = Φ ( a, d )( · − ϑ ) for some ϑ >
0. In particular,we have ( a, d ) ∈ T .(v) Pick ∈ { low , up } and ( a, d ) ∈ Ω bc and consider any non-decreasing function Φ : R → R thatsatisfies (2.5) with c = 0 together with the boundary condition (2.13) if a, d ) ∈ Ω bc \ T .We numerically determined the locations of the sets T low and T up in Figure 4. In particular, wesimulated (2.1) with an initial condition that consists of the stable periodic pattern multiplied by ahyperbolic tangent. By checking if this solution converges to a travelling or stationary pattern one candecide whether ( a, d ) ∈ T low .We now introduce the notation γ ± ( a ) = 13 (cid:104) a + 1 ± (cid:112) a − a + 1 − d bc ( a ) (cid:105) . (2.19)Writing α a for the inverse of the strictly increasing function[0 , γ − ( a )] (cid:51) v (cid:55)→ v − g ( v ; a )2 d bc ( a ) , (2.20)we formally introduce the quantityΓ( a ) = 2 γ + ( a ) − g (cid:48) ( γ + ( a ); a ) d bc ( a ) − u bc (cid:0) a, d − bc ( a ) (cid:1) − max (cid:8) u ∈ (cid:2) , u bc (cid:0) a, d − bc ( a ) (cid:1)(cid:3) : 2 u − g ( u ; a ) d bc ( a ) − v bc (cid:0) a, d − bc ( a ) (cid:1) = α a ( u ) (cid:9) (2.21)for any 0 < a <
1. Here the notation d − bc ( a ) refers to the limit d ↑ d bc ( a ). The geometric interpretationof this definition will be clarified in § d bc , u bc , v bc ) associated to the two-dimensional algebraic problem G ( u, v ; a, d ) = 0 in order to compute Γ( a ). In particular, there is an essential difference betweencomputing Γ( a ) and using the numerical procedure above to check whether c (cid:54) = 0.The main contribution of the present paper is contained in our final result, which provides analyticalbounds for the parameter regions T low and T up where the bichromatic waves actually travel (i.e., where c low > c up < a, d ) = ( , ). Inaddition, the corners (0 ,
0) and (1 ,
0) are accumulation points for the sets T up respectively T low . Theorem 2.3 (see § . The sets T low and T up satisfy the following properties.(i) For each ( a, d ) ∈ T up we have d > a , while for each ( a, d ) ∈ T low we have d > (1 − a ) .(ii) If ( a, d ) ∈ T low then also ( a (cid:48) , d ) ∈ T low for all ( a (cid:48) , d ) ∈ Ω bc that have a (cid:48) ≥ a . On the other hand, if( a, d ) ∈ T up then also ( a (cid:48) , d ) ∈ T up for all ( a (cid:48) , d ) ∈ Ω bc that have a (cid:48) ≤ a .(iii) There exists (cid:15) > a, d ) ∈ T low ∩ T up (2.22)for all ( a, d ) ∈ Ω bc that have 0 < (cid:12)(cid:12)(cid:12)(cid:12) a − (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) d − (cid:12)(cid:12)(cid:12)(cid:12) < (cid:15). (2.23)(iv) The expression (2.21) is well-defined for all 0 < a <
1. If Γ( a ∗ ) > < a ∗ <
1, then thereexists (cid:15) > a, d ) ∈ T low for all ( a, d ) ∈ Ω bc that have0 < | a − a ∗ | + | d − d bc ( a ∗ ) | < (cid:15). (2.24)(v) The inequality Γ( a ) > − a > , ∈ T up and (1 , ∈ T low .Using numerics we have verified that Γ( a ) > a ∈ (cid:2) . , . § T low is a connected set that extends towards the right boundaryof Ω bc . By symmetry, this is equivalent to the statement that T up is a connected set that extends towardsthe left boundary of Ω bc . In this section we uncover the structure of the solution set to G ( u, v ; a, d ) = 0 as a function of theparameters ( a, d ). Our first result shows that for d (cid:29) , a, a ) and (1 , d + ( a ) between this region and the region with five distinct roots canbe explicitly computed. However, we only have qualitative and asymptotic results for the boundary d − ( a )where the root-count increases to the maximal value of nine. In Figure 5 we compare these asymptoticsto numerically computed values for d − ( a ). We remark here that the monotonicity of the root count withrespect to d does not hold for general bistable nonlinearities g . Proposition 3.1 (see § . There exist two continuous functions d ± : [0 , → [0 , ∞ ) (3.1)that satisfy the following properties. The first problem is three dimensional, while the second problem is infinite dimensional and hence involves truncations. - a ) + ( - a ) a + a ( d - / ) =( a - / ) Ω - Figure 5: Comparison of the asymptotics for d − described in Proposition 3.1 (v) with the numericallycomputed border of the set Ω − .(i) For any 0 < a < d + ( a ) = g (cid:48) ( a ; a )4 , (3.2)together with the identities d − ( a ) = d − (1 − a ) , d + ( a ) = d + (1 − a ) (3.3)and the inequality d − ( a ) < d + ( a ). In addition, we have d − (0) = d + (0) = d − (1) = d + (1) = 0 (3.4)together with d − ( ) = .(ii) We have d − ∈ C ∞ (cid:0) [0 , ) (cid:1) ∩ C ∞ (cid:0) ( , (cid:1) . In addition, d − is strictly increasing on [0 , ] and strictlydecreasing on [ , a ∈ (0 , G ( u, v ; a, d ) = 0 has nine distinct roots for 0 ≤ d < d − ( a ), fivedistinct roots for d − < d < d + ( a ) and three distinct real roots for d ≥ d + ( a ).(iv) Pick any a ∈ (0 , G ( u, v ; a, d ) = 0 has seven distinct roots for d = d − ( a ) if a (cid:54) = and five if a = .(v) We have the expansion d − ( a ) = a + a + O ( a ) for a ↓
0. In addition, writing a − : [0 , ] → [0 , ]for the inverse function of d − on [0 , ], we have the expansion a − ( d ) = 12 − (cid:114) − d −
124 ) + O (cid:0) ( d −
124 ) (cid:1) (3.5)as d ↑ .In order to break the symmetry caused by the swap u ↔ v , we set out to describe the roots of G ( u, v ; a, d ) = 0 that have v > u . To this end, we introduce two regionsΩ − = { ( a, d ) : 0 < a < < d < d − ( a ) } , Ω + = { ( a, d ) : 0 < a < d − ( a ) < d < d + ( a ) } (3.6)10 B CD d + ( ) d - ( ) d - ( ) d - ( ) d - ( ) v AB C D d + ( ) d - ( ) d - ( ) v Figure 6: Illustration of the functions ( u A , v A ), ( u B , v B ), ( u C , v C ) from Proposition 3.2 and the function( u D , v D ) from Proposition 3.3 for a = .
45 (left panel) and a = . d + ( a ) and d − ( a ) are indicated by squares and circles; see Proposition 3.1.that are studied separately in the two results below. In Ω − there are three such bichromatic equilibriawith v > u . These equilibria can be ordered and the middle one is the only stable one. Two (or three) ofthese equilibria collide at d = d − ( a ) in a saddle node (or pitchfork) bifurcation, leaving a single unstablebichromatic equilibrium in Ω + . This equilibrium in turns collides with its swapped counterpart and themonochromatic equilibrium ( a, a ) on the boundary d + ( a ). These processes are illustrated in Figure 6. Inparticular, we see that Ω − coincides with the set Ω bc introduced in §
2; cf. Figures 4 and 5.
Proposition 3.2 (see § . There exist continuous functions( u A , v A ) : Ω − → [0 , ( u B , v B ) : Ω − → [0 , , ( u C , v C ) : Ω − → [0 , (3.7)that satisfy the following properties.(i) Pick any ( a, d ) ∈ Ω − . Then we have G ( u ( a, d ) , v ( a, d ); a, d ) = 0 (3.8)for all ∈ { A, B, C } . If also ( a, d ) ∈ Ω − , then the matrix D , G ( u ( a, d ) , v ( a, d ); a, d ) (3.9)has two strictly negative eigenvalues if B or one strictly positive and one strictly negativeeigenvalue if ∈ { A, C } .(ii) For any 0 ≤ a ≤ u A , v A )( a,
0) = (0 , a ) , ( u B , v B )( a,
0) = (0 , , ( u C , v C )( a,
0) = ( a, . (3.10)(iii) For any ( a, d ) ∈ Ω − we have the ordering0 < u A ( a, d ) < u B ( a, d ) < u C ( a, d ) < a < v A ( a, d ) < v B ( a, d ) < v C ( a, d ) . (3.11)11iv) For any a ∈ [0 , ] we have ( u B , v B ) (cid:0) a, d − ( a ) (cid:1) = ( u C , v C ) (cid:0) a, d − ( a ) (cid:1) , (3.12)while for any a ∈ [ ,
1] we have( u A , v A ) (cid:0) a, d − ( a ) (cid:1) = ( u B , v B ) (cid:0) a, d − ( a ) (cid:1) . (3.13) Proposition 3.3 (see § . There exist continuous functions( u D , v D ) : Ω + → [0 , (3.14)that satisfy the following properties.(i) Pick any ( a, d ) ∈ Ω + . Then we have G ( u D ( a, d ) , v D ( a, d ); a, d ) = 0 . (3.15)If also ( a, d ) ∈ Ω + , then the matrix D , G ( u D ( a, d ) , v D ( a, d ); a, d ) (3.16)has one strictly positive and one strictly negative eigenvalue.(ii) For any 0 ≤ a ≤ u D , v D ) (cid:0) a, d + ( a ) (cid:1) = ( a, a ) . (3.17)(iii) For any ( a, d ) ∈ Ω + we have the ordering0 < u D ( a, d ) < a < v D ( a, d ) < . (3.18)(iv) For any a ∈ [0 , ] we have the identity( u D , v D ) (cid:0) a, d − ( a ) (cid:1) = ( u A , v A ) (cid:0) a, d − ( a ) (cid:1) = ( u B , v B ) (cid:0) a, d − ( a ) (cid:1) , (3.19)while for any a ∈ [ ,
1] we have( u D , v D ) (cid:0) a, d − ( a ) (cid:1) = ( u B , v B ) (cid:0) a, d − ( a ) (cid:1) = ( u C , v C ) (cid:0) a, d − ( a ) (cid:1) . (3.20) Corollary 3.4.
For any ( a, d ) ∈ Ω − , we have the identities (cid:0) u A , v A (cid:1) ( a, d ) = (cid:0) − v C , − u C (cid:1) (1 − a, d ) , (cid:0) u B , v B (cid:1) ( a, d ) = (cid:0) − v B , − u B (cid:1) (1 − a, d ) , (cid:0) u C , v C (cid:1) ( a, d ) = (cid:0) − v A , − u A (cid:1) (1 − a, d ) . (3.21)In addition, for any ( a, d ) ∈ Ω + we have the identity (cid:0) u D , v D (cid:1) ( a, d ) = (cid:0) − v D , − u D (cid:1) (1 − a, d ) . (3.22)12 roof. The symmetry g (1 − u, − a ) = − g ( u, a ) implies that G (1 − u, − v ; 1 − a, d ) = − G ( u, v ; a, d ) . (3.23)In addition, we have G ( u, v ; a, d ) = 0 if and only if G ( v, u ; a, d ) = 0. The statements hence follow fromthe ordering (3.11).Our final result concerns the special case a = , in which case it is possible to be more explicit. Inparticular, the bichromatic roots ( u B , v B ) and ( u D , v D ) lie on the line u + v = 1 and collide preciselywhen g (cid:48) ( u ; ) = g (cid:48) ( v ; ) = 0. Corollary 3.5.
For any 0 ≤ d ≤ we have u B ( 12 , d ) = 1 − v B ( 12 , d ) , (3.24)while for any ≤ d ≤ = d + (1 /
2) we have u D ( 12 , d ) = 1 − v D ( 12 , d ) . (3.25)In addition, we have the identities u A ( , ) = u B ( , ) = u C ( , ) = u D ( , ) = − √ ,v A ( , ) = v B ( , ) = v C ( , ) = u D ( , ) = + √ . (3.26) a = 0 In this section we construct the branches ( u B , v B ) and ( u C , v C ) of solutions to G ( u, v ; a, d ) = 0 in theregime where ( a, d ) ≈ (0 , H ( u, v ; a, d ) = G ( u, v ; a, d ) (3.27)and determine the zeroes of H for which ( u, v, a, d ) are small. Proposition 3.6.
There exist constants δ a > δ d > (cid:15) > d c : (0 , δ a ) → (0 , δ d ) and a constant K ≥ < a < δ a and 0 < d < d c ( a ) the equation H ( u, v ; a, d ) = 0 has precisely two solutionson the set {| u | + | v | < (cid:15) } .(ii) For every 0 < a < δ a and d c ( a ) < d < δ d the equation H ( u, v ; a, d ) = 0 has no solutions on the set {| u | + | v | < (cid:15) } .(iii) For every 0 < a < δ a we have the estimate (cid:12)(cid:12)(cid:12)(cid:12) d c ( a ) − a − a (cid:12)(cid:12)(cid:12)(cid:12) ≤ Ka . (3.28)(iv) For every 0 < a < δ a and the equation H ( u, v ; d c ( a ) , a ) = 0 has precisely one solution (cid:0) u c ( a ) , v c ( a ) (cid:1) on the set {| u | + | v | < (cid:15) } . We have the estimates (cid:12)(cid:12) u c ( a ) − a (cid:12)(cid:12) ≤ Ka , (cid:12)(cid:12) v c ( a ) + a + a (cid:12)(cid:12) ≤ Ka . (3.29)13riting H ( u, v ; a, d ) = (cid:0) H ( u, v ; a, d ) , H ( u, v ; a, d ) (cid:1) T we can compute H ( u, v ; a, d ) = u − ua − u + u a + 2 d + 2 dv − du,H ( u, v ; a, d ) = − v − v + va − v + v a + 2 du − d − dv. (3.30)Our strategy is to use the identity H = 0 to eliminate v and then recast H = 0 into the normal formof a saddle-node bifurcation. Lemma 3.7.
Pick δ > (cid:15) > K ≥ α : ( − δ, δ ) → R , α : ( − δ, δ ) → R , R α ;2 : ( − δ, δ ) → R (3.31)that satisfy the following properties.(i) For every ( u, a, d ) ∈ ( − δ, δ ) the equation H ( u, v ; a, d ) = 0 has a unique solution v = v ∗ in the set {| v | < (cid:15) } . This solution is given by v ∗ ( u ; a, d ) = α ( a, d ) + α ( a, d ) u + u R α ;2 ( u ; a, d ) . (3.32)(ii) Upon writing α ( a, d ) = − d − ad + S α ( a, d ) ,α ( a, d ) = 2 d + S α ( a, d ) , (3.33)the bounds | S α ( a, d ) | ≤ K ( d + | d | a ) , | S α ( a, d ) | ≤ K | d | ( | a | + | d | ) , (3.34)together with | ∂ d S α ( a, d ) | ≤ K ( | d | + a ) , | ∂ d S α ( a, d ) | ≤ K ( | a | + | d | ) (3.35)hold for all ( a, d ) ∈ ( − δ, δ ) .(iii) For every ( u, a, d ) ∈ ( − δ, δ ) we have the bounds | R α ;2 ( u ; a, d ) | + (cid:12)(cid:12) R (cid:48) α ;2 ( u ; a, d ) (cid:12)(cid:12) + (cid:12)(cid:12) R (cid:48)(cid:48) α ;2 ( u ; a, d ) (cid:12)(cid:12) + (cid:12)(cid:12) R (cid:48)(cid:48)(cid:48) α ;2 ( u ; a, d ) (cid:12)(cid:12) ≤ Kd , (3.36)together with | ∂ d R α ;2 ( u ; a, d ) | + (cid:12)(cid:12) ∂ d R (cid:48) α ;2 ( u ; a, d ) (cid:12)(cid:12) + (cid:12)(cid:12) ∂ d R (cid:48)(cid:48) α ;2 ( u ; a, d ) (cid:12)(cid:12) + (cid:12)(cid:12) ∂ d R (cid:48)(cid:48)(cid:48) α ;2 ( u ; a, d ) (cid:12)(cid:12) ≤ K | d | . (3.37) Proof.
Substituting the Ansatz (3.32) into H , we obtain the fixed point problems α = − d − dα + α a + α a − α − α ,α = 2 d − α α + 2 α α a − dα + α a − α α , (3.38)together with R = 2 α R a + R a + α a − α α − dR − α − α R − α R +(2 α R a − α α R − α R − α ) u +( R a − α R − R − α R ) u − α R u − R u . (3.39)14hese fixed point problems can be successively solved for small a , d and u , along with their differentiatedcounterparts. The desired estimates can subsequently be obtained in a standard fashion by computingTaylor expansions.In order to eliminate v from H , we define the function J ( u ; a, d ) = H ( u, v ∗ ( u ; a, d ); a, d )= β ( a, d ) + β ( a, d ) u + u (cid:0) a − u + R β ;2 ( u ; a, d ) (cid:1) , (3.40)which forces us to write β ( a, d ) = 2 d + 2 dα ( a, d ) ,β ( a, d ) = − a − d + 2 dα ( a, d ) ,R β, ( u ; a, d ) = 2 dR α ;2 ( u ; a, d ) . (3.41)By applying a shift to u the linear term in (3.40) can be removed, transforming (3.40) into a normal formfor saddle node bifurcations. Lemma 3.8.
Pick δ > (cid:15) > K ≥ u ∗ : ( − δ, δ ) → R , ζ : ( − δ, δ ) → R , R ζ ;2 : ( − δ, δ ) → R (3.42)that satisfy the following properties.(i) For every (˜ u, a, d ) ∈ ( − δ, δ ) , we have the identity J ( u ∗ ( a, d ) + ˜ u ; a, d ) = ζ ( a, d ) + ˜ u (cid:2) R ζ ;2 (˜ u ; a, d ) (cid:3) . (3.43)(ii) Upon writing u ∗ ( a, d ) = a + d − a + ad − a + S u ∗ ( a, d ) ,ζ ( a, d ) = 2 d − a − da − d + a + da − a + S ζ ( a, d ) , (3.44)the bounds | S u ∗ ( a, d ) | ≤ K (cid:2) d + a | d | + a (cid:3) , | S ζ ( a, d ) | ≤ K ( a + | d | + d | a | + | a | | d | ) , (3.45)together with | ∂ d S u ∗ ( a, d ) | ≤ K (cid:2) | d | + a (cid:3) , | ∂ d S ζ ( a, d ) | ≤ K ( d + | d | | a | + | a | ) (3.46)hold for all ( a, d ) ∈ ( − δ, δ ) .(iii) For every (˜ u, a, d ) ∈ ( − δ, δ ) we have the bounds | R ζ ;2 (˜ u ; a, d ) | ≤ K ( | a | + | d | ) , (cid:12)(cid:12)(cid:12) R (cid:48) ζ ;2 (˜ u ; a, d ) (cid:12)(cid:12)(cid:12) ≤ K | d | . (3.47)15 roof. We first introduce the notation N β ;2; u ∗ (˜ u ; a, d ) = ˜ u − (cid:104) R β ;2 ( u ∗ + ˜ u ) − R β ;2 ( u ∗ ; a, d ) − R (cid:48) β ;2 ( u ∗ ; a, d )˜ u (cid:105) (3.48)for ˜ u (cid:54) = 0, together with N β ;2; u ∗ (0; a, d ) = R (cid:48)(cid:48) β ;2 ( u ∗ ; a, d ). This allows us to compute J ( u ∗ + ˜ u, a, d ) = γ ( a, d, u ∗ ) + γ ( a, d, u ∗ )˜ u + (cid:0) a − u ∗ + R γ ;2 (˜ u ; a, d, u ∗ ) (cid:1) ˜ u , (3.49)in which γ ( a, d, u ∗ ) = β + β u ∗ + u ∗ [1 + a − u ∗ + R β ;2 ( u ∗ )] ,γ ( a, d, u ∗ ) = β + u ∗ [ − R (cid:48) β ;2 ( u ∗ ; a, d )] + 2 u ∗ [1 + a − u ∗ + R β ;2 ( u ∗ )] ,R γ ;2 (˜ u ; a, d, u ∗ ) = ( u ∗ + ˜ u ) N β ;2; u ∗ (˜ u ; a, d ) + R (cid:48) β ;2 ( u ∗ ; a, d )(2 u ∗ ˜ u + ˜ u ) + R β ;2 ( u ∗ ; a, d ) . (3.50)On account of (3.36) we have |N β ;2; u ∗ (˜ u ; a, d ) | + (cid:12)(cid:12) N (cid:48) β ;2; u ∗ (˜ u ; a, d ) (cid:12)(cid:12) ≤ C (cid:48) (3.51)and hence also |R γ ;2 (˜ u ; a, d, u ∗ ) | + (cid:12)(cid:12) R (cid:48) γ ;2 (˜ u ; a, d, u ∗ ) (cid:12)(cid:12) ≤ C (cid:48) . (3.52)Setting γ = 0 leads to the fixed point problem u ∗ = − β − au ∗ + 32 u ∗ − u ∗ R (cid:48) β ;2 ( u ∗ ; a, d ) − u ∗ R β ;2 ( u ∗ ; a, d ) , (3.53)which has a unique small solution that we write for the moment as u ∗ ( a, d, β ) = − β + 12 aβ + 38 β − a β − aβ − β + ˜ S u ∗ ( a, d, β ) . (3.54)In a standard fashion one obtains the bound (cid:12)(cid:12)(cid:12) ˜ S u ∗ ( a, d, β ) (cid:12)(cid:12)(cid:12) ≤ C (cid:48) (cid:2) β + | a | | β | + ( a + d ) β + ( | a | + | d | ) | β | (cid:3) . (3.55)Differentiating (3.53) we obtain D d u ∗ = − ∂ d β − u ∗ ∂ d R (cid:48) β ;2 ( u ∗ ; a, d ) − u ∗ ∂ d R β ;2 ( u ∗ ; a, d ) − aD d u ∗ + 3 u ∗ D d u ∗ − u ∗ R (cid:48) β ;2 ( u ∗ ; a, d ) D d u ∗ − R β ;2 ( u ∗ ; a, d ) D d u ∗ , (3.56)which yields the estimate (cid:12)(cid:12)(cid:12)(cid:12) D d u ∗ − (cid:2) −
12 + 12 a + 34 β (cid:3) ∂ d β (cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:48) (cid:2) d | β | + a + | a | | β | + β (cid:3) | ∂ d β | . (3.57)Using the bounds (cid:12)(cid:12) β ( a, d ) − [2 d − d ] (cid:12)(cid:12) ≤ C (cid:48) d ( | a | + | d | ) , | β ( a, d ) + ( a + 2 d ) | ≤ C (cid:48) d , (3.58) All primed constants in this paper are strictly positive and do not depend on the variables appearing on the left handside of the inequalities where they appear. | ∂ d β ( a, d ) − [2 − d ] | ≤ C (cid:48) | d | ( | a | + | d | ) , | ∂ d β ( a, d ) + 2 | ≤ C (cid:48) | d | (3.59)and finally | R β ;2 ( u ∗ ; a, d ) | + (cid:12)(cid:12) R (cid:48) β ;2 ( u ∗ ; a, d ) (cid:12)(cid:12) + (cid:12)(cid:12) R (cid:48)(cid:48) β ;2 ( u ∗ ; a, d ) (cid:12)(cid:12) + (cid:12)(cid:12) R (cid:48)(cid:48)(cid:48) β ;2 ( u ∗ ; a, d ) (cid:12)(cid:12) ≤ C (cid:48) | d | , (3.60)the desired estimates follow by writing ζ ( a, d ) = γ (cid:0) a, d, u ∗ ( a, d ) (cid:1) ,R ζ ;2 (˜ u ; a, d ) = a − u ∗ ( a, d ) + R γ ;2 (cid:0) ˜ u, a, d, u ∗ ( a, d ) (cid:1) (3.61)and computing ∂ d ζ directly. Proof of Proposition 3.6.
We note first that the map˜ u (cid:55)→ ˜ u (cid:113) R ζ ;2 (˜ u ; a, d ) (3.62)is invertible for (˜ u, a, d ) ∈ ( − δ, δ ) . In order to find d c it hence suffices to solve ζ ( a, d c ) = 0, which givesthe fixed-point problem d c = 18 a − a + 1128 a + 12 ad c − d c a + 52 d c − S ζ ( a, d c ) . (3.63)Our estimate on ∂ d S ζ guarantees the existence of a unique small solution for small a . The estimate(iii) now follows in a standard fashion. Writing u c = u ∗ (cid:0) a, d c ( a ) (cid:1) and v c = v ∗ (cid:0) u ∗ ( a, d c ( a )); a, d c ( a ) (cid:1) theexpansions in (iv) follow directly by substitution. a = 1 / Our goal here is to unfold the structure of the solution-set to G ( u, v ; a, d ) near the critical point ( a, d ) =(1 / , /
24) where the branches ( u , u ) with ∈ { A, B, C, D } all collide in a cusp bifurcation; seeFigures 1 and 6. In particular, we introduce the cusp location( u cp , v cp , a cp , d cp ) = (cid:0) u min ( ) , u max ( ) , , (cid:1) = (cid:0) − √ , + √ , , (cid:1) (3.64)together with the function H cp ( u, v ; a, d ) = G (cid:0) u cp + u, v cp + v ; a cp + a, d cp + d (cid:1) (3.65)and set out to determine the zeroes of H cp for which ( u, v, a, d ) are small. Proposition 3.9.
There exist constants δ a > δ d > (cid:15) > a c : ( − δ d , → (0 , δ a ) and a constant K ≥ ≤ d < δ d and any a ∈ ( − δ a , δ a ), the equation H cp ( u, v ; a, d ) = 0 has precisely onesolution on the set {| u | + | v | < (cid:15) } .(ii) For every − δ d < d < a ∈ ( − δ a , δ a ) with | a | > a c ( d ), the equation H cp ( u, v ; a, d ) = 0 hasprecisely one solution on the set {| u | + | v | < (cid:15) } .17iii) For every − δ d < d < a ∈ (cid:0) − a c ( d ) , a c ( d ) (cid:1) , the equation H cp ( u, v ; a, d ) = 0 has precisely threesolutions on the set {| u | + | v | < (cid:15) } .(iv) For any − δ d < d <
0, the equation H cp ( u, v ; a c ( d ) , d ) = 0 has precisely two solutions on the set {| u | + | v | < (cid:15) } .(v) For any d ∈ ( − δ d ,
0) we have the estimate (cid:12)(cid:12)(cid:12) a c ( d ) − (cid:112) − d (cid:12)(cid:12)(cid:12) ≤ Kd . (3.66)In order to recast our equation into an efficient form, we introduce two functions ( h , h ) by writing h ( p, q ; a, d ) h ( p, q ; a, d ) = − H cp (cid:0) p + q, p − q ; a, d (cid:1) , (3.67)which can be evaluated as h ( p, q ; a, d ) = − a + 2 ap − pq + 2 aq − √ aq − p + 2 √ pq,h ( p, q ; a, d ) = − √ ap − q + √ d + √ p + √ q − p q − dq + 4 apq − q . (3.68)Since h contains a term that is linear in q , we set out to eliminate this variable by demanding h = 0. Lemma 3.10.
Pick δ > (cid:15) > K ≥
1, together withfunctions α : ( − δ, δ ) → R , α : ( − δ, δ ) → R , α : ( − δ, δ ) → R , R α ;3 : ( − δ, δ ) → R (3.69)that satisfy the following properties.(i) For every ( p, a, d ) ∈ ( − δ, δ ) the equation h ( p, q ; a, d ) = 0 has a unique solution q = q ∗ in the set {| q | < (cid:15) } . This solution is given by q ∗ ( p ; a, d ) = α ( a, d ) + α ( a, d ) p + α ( a, d ) p + p R α ;3 ( p ; a, d ) . (3.70)(ii) Upon writing α ( a, d ) = 4 √ d + S α ( a, d ) ,α ( a, d ) = − √ a + S α ( a, d ) ,α ( a, d ) = 3 √ S α ( a, d ) , (3.71)the bounds | S α ( a, d ) | ≤ Kd , | DS α ( a, d ) | ≤ K | d | , | S α ( a, d ) | ≤ K | a | | d | , | DS α ( a, d ) | ≤ K ( | a | + | d | ) , | S α ( a, d ) | ≤ K ( | d | + a ) , | DS α ( a, d ) | ≤ K (3.72)hold for all ( a, d ) ∈ ( − δ, δ ) .(iii) For every ( p, a, d ) ∈ ( − δ, δ ) we have the bounds | R α ;3 ( p ; a, d ) | ≤ K ( | a | + | p | ) , | D a,d R α ;3 ( p ; a, d ) | ≤ K, (3.73)together with (cid:12)(cid:12)(cid:12) R ( i ) α ;3 ( p ; a, d ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) D a,d R ( i ) α ;3 ( p ; a, d ) (cid:12)(cid:12)(cid:12) ≤ K (3.74)for all 1 ≤ i ≤
6. 18 roof.
Substituting the Ansatz (3.70) into h , we obtain the fixed point problems α = √ d − dα + √ α − α , α = − √ a + 4 α a + 2 √ α α − α α − α d, α = √ − α + √ α + 4 aα − α α + 2 √ α α − α α − dα , (3.75)together with R α ;3 = − α − α + 2 √ α α − α α α + 4 aα + 2 √ α R α ;3 − α R α ;3 − dR α ;3 + (cid:16) √ α − α − α α − α α + 2 √ α R α ;3 − α α R α ;3 + 4 aR α ;3 (cid:17) p + (cid:16) √ α R α ;3 − α α − α R α ;3 − R − α α R α ;3 (cid:17) p + (cid:16) √ R α ;3 − α − α α R α ;3 − α R α ;3 (cid:17) p + (cid:16) − α R α ;3 − α R α ;3 (cid:17) p − α R α ;3 p − R α ;3 p . (3.76)These fixed point problems can be successively solved for small a , d and p , which yields the desiredestimates.In order to eliminate q from h , we define the function J ( p ; a, d ) = ˜ h ( p, q ∗ ( p ; a, d ) , a, d ) (3.77)and obtain the following representation. Corollary 3.11.
Pick δ > K ≥
1, together withfunctions β : ( − δ, δ ) → R , β : ( − δ, δ ) → R , β : ( − δ, δ ) → R , R β ;3 : ( − δ, δ ) → R (3.78)that satisfy the following properties.(i) For every ( p, a, d ) ∈ ( − δ, δ ) we have J ( p ; a, d ) = β ( a, d ) + β ( a, d ) p + β ( a, d ) p + p [16 + R β ;3 ( p ; a, d )] . (3.79)(ii) Upon writing β ( a, d ) = − a + S β ( a, d ) ,β ( a, d ) = 24 d + S β ( a, d ) ,β ( a, d ) = − a + S β ( a, d ) , (3.80)the bounds | S β ( a, d ) | ≤ K | a | | d | , | DS α ( a, d ) | ≤ K ( | a | + | d | ) , | S β ( a, d ) | ≤ K ( a + d ) , | DS α ( a, d ) | ≤ K ( | a | + | d | ) , | S β ( a, d ) | ≤ K | a | ( | d | + a ) , | DS α ( a, d ) | ≤ K ( | a | + | d | ) (3.81)hold for all ( a, d ) ∈ ( − δ, δ ) . 19iii) For every ( p, a, d ) ∈ ( − δ, δ ) we have the bounds | R β ;3 ( p ; a, d ) | ≤ K ( | d | + a + | a | | p | + p ) , (cid:12)(cid:12)(cid:12) R (cid:48) β ;3 ( p ; a, d ) (cid:12)(cid:12)(cid:12) ≤ K ( | a | + | p | ) , | D a,d R β ;3 ( p ; a, d ) | + (cid:12)(cid:12)(cid:12) D a,d R (cid:48) β ;3 ( p ; a, d ) (cid:12)(cid:12)(cid:12) ≤ K, (3.82)together with (cid:12)(cid:12)(cid:12) R ( i ) β ;3 ( p ; a, d ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) D a,d R ( i ) β ;3 ( p ; a, d ) (cid:12)(cid:12)(cid:12) ≤ K (3.83)for all 2 ≤ i ≤ Proof.
Substitution yields β = − a − √ aα + 2 aα ,β = 4 aα α − √ aα − α + 2 √ α ,β = 4 aα α − √ aα + 2 a − α α + 2 √ α + 2 aα , (3.84)together with R β ;3 = 4 aα α − α − α α + 2 √ α − √ − √ aR α ;3 + 4 aα R α ;3 + (cid:16) aα − α α + 2 √ R α ;3 + 4 aα R α ;3 − α R α ;3 (cid:17) p + (cid:16) aα R α ;3 − α − α R α ;3 (cid:17) p + (cid:16) aR α ;3 − α R α ;3 (cid:17) p − R α ;3 p . (3.85)The desired estimates can be determined directly from these expressions.By applying a (small) shift to p the undesired quadratic term in (3.79) can be eliminated. Thebifurcation curve in ( a, d ) space can subsequently be found by determing when the remaining equationhas roots of order two or higher. Lemma 3.12.
Pick δ > K ≥ ζ : ( − δ, δ ) → R , ζ : ( − δ, δ ) → R , ζ : ( − δ, δ ) → R (3.86)and p ∗ : ( − δ, δ ) → R , R ζ ;4 : ( − δ, δ ) → R (3.87)that satisfy the following properties.(i) For every (˜ p, a, d ) ∈ ( − δ, δ ) , we have the identity J ( p ∗ ( a, d ) + ˜ p ; a, d ) = ζ ( a, d ) + ζ ( a, d )˜ p + ζ ( a, d )˜ p + ˜ p R ζ ;4 (˜ p ; a, d ) . (3.88)(ii) Upon writing ζ ( a, d ) = − a + S ζ ( a, d ) ,ζ ( a, d ) = 24 d + S ζ ( a, d ) ,ζ ( a, d ) = 16 + S ζ ( a, d ) , (3.89)20he bounds | S ζ ( a, d ) | ≤ K ( | a | | d | + | a | ) , | DS ζ ( a, d ) | ≤ K ( | a | + | d | ) , | S ζ ( a, d ) | ≤ K ( a + d ) , | DS ζ ( a, d ) | ≤ K ( | a | + | d | ) , | S ζ ( a, d ) | ≤ K ( | a | + | d | ) , | DS ζ ( a, d ) | ≤ K (3.90)hold for all ( a, d ) ∈ ( − δ, δ ) .(iii) For every (˜ p, a, d ) ∈ ( − δ, δ ) we have the bounds | R ζ ;4 (˜ p ; a, d ) | + (cid:12)(cid:12)(cid:12) R (cid:48) ζ ;4 (˜ p ; a, d ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) R (cid:48)(cid:48) ζ ;4 (˜ p ; a, d ) (cid:12)(cid:12)(cid:12) ≤ K, | D a,d R ζ ;4 (˜ p ; a, d ) | + (cid:12)(cid:12)(cid:12) D a,d R (cid:48) ζ ;4 (˜ p ; a, d ) (cid:12)(cid:12)(cid:12) ≤ K. (3.91) Proof.
We first introduce the notation N β ;3; p ∗ (˜ p ; a, d ) = ˜ p − (cid:104) R β ;3 ( p ∗ + ˜ p ; a, d ) − R β ;3 ( p ∗ ; a, d ) − R (cid:48) β ;3 ( p ∗ ; a, d )˜ p − R (cid:48)(cid:48) β ;3 ( p ∗ ; a, d )˜ p (cid:105) (3.92)for ˜ p (cid:54) = 0, with N β ;3; p ∗ (0; a, d ) = R (cid:48)(cid:48)(cid:48) β ;3 ( p ∗ ; a, d ). This allows us to compute J ( p ∗ + ˜ p ; a, d ) = γ ( a, d, p ∗ ) + γ ( a, d, p ∗ )˜ p + γ ( a, d, p ∗ )˜ p + (cid:2)
16 + R γ ;3 (˜ p ; a, d, p ∗ ) (cid:3) ˜ p , (3.93)in which γ ( a, d, p ∗ ) = β ( a, d ) + β ( a, d ) p ∗ ( a, d ) + β ( a, d ) p ∗ ( a, d ) + p ∗ (cid:2)
16 + R β ;3 ( p ∗ ; a, d ) (cid:3) ,γ ( a, d, p ∗ ) = β ( a, d ) + 2 β ( a, d ) p ∗ + p ∗ R (cid:48) β ;3 ( p ∗ ; a, d ) + 3 p ∗ (cid:2)
16 + R β ;3 ( p ∗ ; a, d ) (cid:3) ,γ ( a, d, p ∗ ) = β ( a, d ) + p ∗ R (cid:48)(cid:48) β ;3 ( p ∗ ; a, d ) + 3 p ∗ R (cid:48) β ;3 ( p ∗ ; a, d ) + 3 p ∗ (cid:2)
16 + R β ;3 ( p ∗ ; a, d ) (cid:3) , (3.94)together with R γ ;3 (˜ p ; a, d, p ∗ ) = ( p ∗ + ˜ p ) N β ;3; p ∗ (˜ p ; a, d ) + (3 p ∗ + 3 p ∗ ˜ p + ˜ p ) R (cid:48)(cid:48) β ;3 ( p ∗ ; a, d )+(3 p ∗ + ˜ p ) R (cid:48) β ;3 ( p ∗ ; a, d ) + R β ;3 ( p ∗ ; a, d ) . (3.95)On account of (3.83) we have |N β ;3; p ∗ (˜ p ; a, d ) | + (cid:12)(cid:12) N (cid:48) β ;3; p ∗ (˜ p ; a, d ) (cid:12)(cid:12) + (cid:12)(cid:12) N (cid:48)(cid:48) β ;3; p ∗ (˜ p ; a, d ) (cid:12)(cid:12) + (cid:12)(cid:12) N (cid:48)(cid:48)(cid:48) β ;3; p ∗ (˜ p ; a, d ) (cid:12)(cid:12) ≤ C (cid:48) (3.96)and hence also |R γ ;3 (˜ p ; a, d, p ∗ ) | + (cid:12)(cid:12) R (cid:48) γ ;3 (˜ p ; a, d, p ∗ ) (cid:12)(cid:12) + (cid:12)(cid:12) R (cid:48)(cid:48) γ ;3 (˜ p ; a, d, p ∗ ) (cid:12)(cid:12) + (cid:12)(cid:12) R (cid:48)(cid:48)(cid:48) γ ;3 (˜ p ; a, d, p ∗ ) (cid:12)(cid:12) ≤ C (cid:48) (3.97)for small | ˜ p | .Setting γ ( a, d, p ∗ ) = 0 leads to a fixed-point equation for p ∗ , which has a unique small solution p ∗ ( a, d ) that admits the bounds | p ∗ ( a, d ) | ≤ C (cid:48) | β ( a, d ) | ≤ C (cid:48) a, | D a,d p ∗ ( a, d ) | ≤ C (cid:48) . (3.98)21he results hence follow by writing ζ ( a, d ) = γ (cid:0) a, d, p ∗ ( a, d ) (cid:1) ,ζ ( a, d ) = γ (cid:0) a, d, p ∗ ( a, d ) (cid:1) ,ζ ( a, d ) = 16 + R γ ;3 (cid:0) a, d, p ∗ ( a, d ) (cid:1) , (3.99)together with R ζ ;4 (˜ p ; a, d ) = ˜ p − (cid:2) R γ ;3 (cid:0) ˜ p ; a, d, p ∗ ( a, d ) (cid:1) − R γ ;3 (cid:0) a, d, p ∗ ( a, d ) (cid:1)(cid:3) (3.100)for ˜ p (cid:54) = 0 and R ζ ;4 (0; a, d ) = R (cid:48) γ ;3 (cid:0) a, d, p ∗ ( a, d ) (cid:1) . (3.101)Here we use (3.97) to get bounds on R ζ ;4 and its derivatives. Proof of Proposition 3.9.
In order to find roots of order two or higher we need to solve the system ζ ( a, d ) + ζ ( a, d )˜ p + ζ ( a, d )˜ p + ˜ p R ζ ;4 (˜ p, a, d ) = 0 ,ζ ( a, d ) + 3 ζ ( a, d )˜ p + 4˜ p R ζ ;4 (˜ p, a, d ) + ˜ p R (cid:48) ζ ;4 (˜ p, a, d ) = 0 . (3.102)Solving the second equation we find two branches˜ p ± ( a, d ) = ± (cid:112) − d/ S ˜ p ± ( a, d ) , (3.103)in which we have S ˜ p ± ( a, d ) ≤ C (cid:48) ( | a | + | d | ) ,D a,d S ˜ p ± ( a, d ) ≤ C (cid:48) . (3.104)Plugging this into the first line of (3.102) we obtain13 a = ± d (cid:112) − d/ O ( d + | ad | + | a | + a (cid:112) | d | ) , (3.105)from which the desired expression for a c follows. Our strategy to establish Propositions 3.1-3.3 hinges upon geometric properties of the cubic g ( u ; a ). Asa preparation, we introduce the notation u infl ( a ) = 13 ( a + 1) (3.106)together with u min ( a ) = u infl ( a ) − (cid:112) − a (1 − a ) , u max ( a ) = u infl ( a ) + 13 (cid:112) − a (1 − a ) (3.107)and note that g (cid:48) (cid:0) u min ( a ); a (cid:1) = 0 , g (cid:48)(cid:48) (cid:0) u infl ( a ); a (cid:1) = 0 , g (cid:48) (cid:0) u max ( a ); a (cid:1) = 0 . (3.108)In addition, we note that g (cid:48)(cid:48) ( u ; a ) > u < u infl ( a ) and g (cid:48)(cid:48) ( u ; a ) < u > u infl ( a ). When 0 < a < we have the ordering 0 < u min ( a ) < a < u infl ( a ) < u max ( a ) < . (3.109)22 + v - u min ( a ) v + ;min v - ;max v + v - v + v - u min ( ) u max ( ) Figure 7: The functions v ± defined in Lemmata 3.13-3.14 for a = 0 .
45 (left) and a = 0 . u min ( ).Recalling the definition (2.6), one readily sees that G ( u, v ; a, d ) + G ( u, v ; a, d ) = g ( u ; a ) + g ( v ; a ) . (3.110)In order to exploit the fact that this identity does not depend on d , we first use basic properties of thecubic to parametrize solutions to g ( u ; a ) + g ( v ; a ) = 0. Restricting ourselves to a ∈ [0 , ], it is possible toconstruct two solution curves v ± ( u ) that are defined for u ∈ [0 , a ]; see Figure 7. Lemma 3.13.
Fix 0 < a < . Then there are two constants a < v − ;max < v +;min < C ∞ -smooth functions v − : [0 , a ] → [ a, v − ;max ] , v + : [0 , a ] → [ v +;min ,
1] (3.112)that satisfy the following properties.(i) We have g ( u ; a ) = − g ( v − ( u ); a ) = − g ( v + ( u ); a ) for all 0 ≤ u ≤ a .(ii) If g ( u ; a ) = − g ( v ; a ) for some pair u ∈ (0 , a ) and v ∈ [0 , v = v − ( u ) or v = v + ( u ).(iii) We have v − (0) = v − ( a ) = a and v + (0) = v + ( a ) = 1.(iv) We have the identities v (cid:48)± ( u ) = − [ g (cid:48) ( v ± ( u ); a )] − g (cid:48) ( u ; a ) (3.113)for all 0 ≤ u ≤ a .(v) We have v (cid:48)− ( u ) > − ≤ u < a , together with v (cid:48)− ( a ) = − Proof.
Since 0 < a < we have g ( u max ( a ); a ) > − g ( u min ( a ); a ), which implies that v ± ( u ) (cid:54) = u max ( a ).Properties (i)-(iv) hence follow immediately from the implicit function theorem.To obtain (v), we take u min ( a ) < u < a and recall a < v − ( u ) < u max . If v − ( u ) ≤ u infl ( a ) thenclearly g (cid:48) ( v − ( u ); a ) > g (cid:48) ( u ; a ) >
0, as desired. In order to hande the remaining case v − ( u ) > u infl ( a ), weintroduce the reflection u refl = 2 u infl ( a ) − u . Exploiting the point symmetry of the graph of g ( · ; a ) aroundits inflection point (cid:0) u infl ( a ) , g ( u infl ( a ); a ) (cid:1) , the inequality g ( u infl ( a ); a ) > u infl ( a ) < v − ( u ) < u refl . (3.114)23ince g (cid:48)(cid:48) (˜ u ; a ) < u > u infl ( a ), we obtain g (cid:48) (cid:0) v − ( u ); a (cid:1) > g (cid:48) ( u refl ; a ) = g (cid:48) ( u ; a ) > , (3.115)which implies v (cid:48)− ( u ) > − Lemma 3.14.
Fix a = . Then there are two functions v − : [0 , a ] → [ a, u max ( a )] , v + : [0 , a ] → [ u max ( a ) ,
1] (3.116)that satisfy items (i) - (iii) from Lemma 3.13 together with the following additional properties.(i) We have v − (cid:0) u min ( a ) (cid:1) = v + (cid:0) u min ( a ) (cid:1) = u max ( a ) . (3.117)(ii) For any u ∈ [0 , a ] \ { u min ( a ) } we have the identities v (cid:48)± ( u ) = − [ g (cid:48) ( v ± ( u ); a )] − g (cid:48) ( u ; a ) . (3.118)(iii) We have v − ( u ) = 1 − u for all u min ( a ) ≤ u ≤
1, while v + ( u ) = 1 − u for all 0 ≤ u ≤ u min ( a ).(iv) We have the limits lim u ↑ u min ( a ) v (cid:48)− ( u ) = lim u ↓ u min ( a ) v (cid:48) + ( u ) = +1 , (3.119)together with lim u ↑ u min ( a ) v (cid:48)(cid:48)− ( u ) = lim u ↓ u min ( a ) v (cid:48)(cid:48) + ( u ) = − √ . (3.120) Proof.
Items (i) and (iii) follow directly from the symmetry of g ( · ; a ), while (ii) follows from the implicitfunction theorem. To obtain (iv), we first compute g ( u min + u ; a ) = g ( u min ; a ) + 12 √ u − u , g ( u max + v ; a ) = − g ( u min ; a ) − √ v − v . (3.121)In particular, the identity g ( u min + u ; a ) = − g ( u max + v ; a ) (3.122)can be rewritten as v = u − √ (cid:2) u + v (cid:3) , (3.123)which can be interpreted as a fixed point problem for v upon assuming that v and u have the same sign.For small | u | this problem has a solution that can be expanded as v = u − √ u + O ( u ) , (3.124)which yields v = u − √ u + O ( u ) . (3.125)The desired limits follow directly from this expansion.24 .4 Tangencies Let us now fix a ∈ [0 , ]. In order to find solutions to G ( u, v ; a, d ) = 0 with d >
0, we introduce thefunction v d ( u ) = u − g ( u ; a )2 d (3.126)and note that the results above show that it suffices to find u ∈ [0 , a ] for which one of the equations v ± ( u ) = v d ( u ) (3.127)holds.Our goal here is to show that non-transverse intersections of this type can only occur at local minimaof v ± − v d . Together with the strict monotonicity ∂ d v d ( u ) < u ∈ (0 , a ), this will allow us to obtain global results in § v − ( a ) = v d ( a ) = a (3.129)for every d >
0. In addition, we may compute v (cid:48) d ( a ) = 1 − g (cid:48) ( a ; a )2 d , (3.130)together with v (cid:48)− ( a ) = − [ g (cid:48) ( a ; a )] − g (cid:48) ( a ; a ) = − . (3.131)In particular, when g (cid:48) ( a ) = 4 d the intersection (3.129) is tangential. In the sequel we show that in fact v − > v d on [0 , a ) for this critical value of d .For intersections with u ∈ (0 , a ) such explicit computations are significantly harder to carry out, whichis why we pursue a more indirect approach here. As a preparation, we compute v (cid:48)(cid:48)± ( u ) = − g (cid:48)(cid:48) ( u ) g (cid:48) (cid:0) v ± ( u ) (cid:1) − g (cid:48) ( u ) g (cid:48) (cid:0) v ± ( u ) (cid:1) g (cid:48)(cid:48) (cid:0) v ± ( u ) (cid:1) (3.132)together with v (cid:48)(cid:48) d ( u ) = − g (cid:48)(cid:48) ( u )2 d . (3.133)In addition, for any κ ≤ g (cid:48) ( u infl ( a ); a ) = ( a − a + 1), we introduce the expressions u l ( κ ) = u infl ( a ) − (cid:113) ( g (cid:48) ( u infl ( a ); a ) − κ ) ,u r ( κ ) = u infl ( a ) + (cid:113) ( g (cid:48) ( u infl ( a ); a ) − κ ) . (3.134)It is easy to verify that u l ( κ ) and u r ( κ ) are the two solutions to the quadratic equation g (cid:48) ( u ; a ) = κ . Lemma 3.15.
Fix 0 < a ≤ and d > v (cid:48) ( u ) = v (cid:48) d ( u ) = β (3.135)for some ∈ {− , + } and 0 ≤ u ≤ a , with u (cid:54) = u min ( a ) in case a = . Then the following statementshold. 25i) We have 2 d [ v (cid:48)(cid:48) ( u ) − v (cid:48)(cid:48) d ( u )] = 11 − β g (cid:48)(cid:48) ( u ; a ) + β − β g (cid:48)(cid:48) (cid:0) v ( u ); a (cid:1) . (3.136)(ii) If − then we have β ∈ [ − , ∪ (1 , ∞ ). On the other hand, the inclusion β ∈ (0 ,
1) holds if u = u l (cid:0) d (1 − β ) (cid:1) .(iv) Suppose that − . Then the identity v ( u ) = u l (cid:0) d (1 − β − ) (cid:1) (3.137)holds if v − ( u ) ≤ u infl ( a ). On the other hand, we have v ( u ) = u r (cid:0) d (1 − β − ) (cid:1) (3.138)if v − ( u ) > u infl ( a ).(v) The identity v ( u ) = u r (cid:0) d (1 − β − ) (cid:1) (3.139)holds if Proof.
We first consider the case 0 < a < . For any 0 ≤ ˜ u ≤ a , one sees that v (cid:48) d (˜ u ) = 1 holds if andonly if g (cid:48) (˜ u ; a ) = v (cid:48)± (˜ u ) = 0, which shows that β / ∈ { , } .For any 0 ≤ ˜ u ≤ a we have g (cid:48) ( v + (˜ u ); a ) < (cid:2) v (cid:48) + (˜ u ) (cid:3) = sign (cid:2) g (cid:48) (˜ u ; a ) (cid:3) . (3.140)If g (cid:48) ( u ; a ) > v (cid:48) d ( u ) >
0, hence β ∈ (0 , g (cid:48) ( v − (˜ u ); a ) > ≤ ˜ u ≤ a . If g (cid:48) ( u ; a ) < v (cid:48) d ( u ) = β >
1, while if g (cid:48) ( u ; a ) > − we may use item (v) from Lemma 3.13 to conclude − ≤ v (cid:48)− ( u ) = β < , (3.141)which establishes (ii).In order to obtain (iii), it suffices to recall the bound u ≤ a ≤ u infl ( a ) and note that the identity v (cid:48)∗ ( u ) = β implies that g (cid:48) ( u ) = 2 d (1 − β ) . (3.142)On the other hand, the identity v (cid:48)± ( u ) = β implies g (cid:48) (cid:0) v ± ( u ) (cid:1) = − dβ − (1 − β ) . (3.143)Items (iv) and (v) now follow directly, remembering that v + ( u ) ≥ u max ( a ) ≥ u infl ( a ).Exploiting (3.132) we may now compute v (cid:48)(cid:48)± ( u ) = β d (1 − β ) g (cid:48)(cid:48) ( u ; a ) + (2 d ) (1 − β ) β (2 d ) (1 − β ) g (cid:48)(cid:48) ( v ± ( u ); a )= β d (1 − β ) g (cid:48)(cid:48) ( u ; a ) + β d (1 − β ) g (cid:48)(cid:48) ( v ± ( u ); a ) . (3.144)The desired identity in (i) hence follows directly from (3.133). In order to conclude the proof, it sufficesto note that the arguments above remain valid when a = . Indeed, the critical cases g (cid:48) ( u ; a ) = 0 and g (cid:48) ( v ± ; a ) = 0 are excluded by the requirement that u (cid:54) = u min ( a ).26 emma 3.16. Fix 0 < a ≤ and suppose that v (cid:48)− ( u ) = v (cid:48) d ( u ) (3.145)for some u min ( a ) ≤ u < a and d >
0, with u (cid:54) = u min ( a ) if a = . Then we have v (cid:48)(cid:48)− ( u ) > v (cid:48)(cid:48) d ( u ) . (3.146) Proof.
Exploiting (v) of Lemma 3.13 and (ii) of Lemma 3.15 we have v (cid:48)− ( u ) ∈ ( − , v − ( u ) > u and g (cid:48)(cid:48)(cid:48) < g (cid:48)(cid:48) ( v − ( u ); a ) ≤ g (cid:48)(cid:48) ( u ; a ). Writing β = v (cid:48)− ( u ) we may henceestimate 2 d [ v (cid:48)(cid:48)− ( u ) − v (cid:48)(cid:48) d ( u )] ≥ − β g (cid:48)(cid:48) ( u ; a ) + β − β g (cid:48)(cid:48) (cid:0) u ; a (cid:1) = β +11 − β g (cid:48)(cid:48) ( u ; a ) > , (3.147)in which we used g (cid:48)(cid:48) ( u ; a ) > v (cid:48)± ( u ) > h l ( β ) = d (1 − β ) (cid:104) g (cid:48)(cid:48) (cid:16) u l (cid:0) d (1 − β ) (cid:1) ; a (cid:17) + β g (cid:48)(cid:48) (cid:16) u l (cid:0) d (1 − β − ) (cid:1) ; a (cid:17)(cid:105) ,h r ( β ) = d (1 − β ) (cid:104) g (cid:48)(cid:48) (cid:16) u l (cid:0) d (1 − β ) (cid:1) ; a (cid:17) + β g (cid:48)(cid:48) (cid:16) u r (cid:0) d (1 − β − ) (cid:1) ; a (cid:17)(cid:105) . (3.148) Lemma 3.17.
Pick 0 < a ≤ and 0 < d ≤ g (cid:48) ( a ; a )4 . Then for any β > h l ( β ) < β ∈ (0 , ∞ ) \ { } we have h r ( β ) > . (3.150) Proof.
Observe first that for β > { d (1 − β ) , d (1 − β − ) } ≤ d ≤ g (cid:48) ( a ; a )2 ≤ g (cid:48) ( u infl ( a ); a )2 , (3.151)which implies that h l ( β ) and h r ( β ) are well-defined.A little algebra yields h l ( β ) = √ d (1 − β ) (cid:104)(cid:112) g (cid:48) ( u infl ( a ); a ) + 2 d ( β −
1) + β (cid:112) g (cid:48) ( u infl ( a ); a ) + 2 dβ − (1 − β ) (cid:105) ,h r ( β ) = √ d (1 − β ) (cid:104)(cid:112) g (cid:48) ( u infl ( a ); a ) + 2 d ( β − − β (cid:112) g (cid:48) ( u infl ( a ); a ) + 2 dβ − (1 − β ) (cid:105) . (3.152)It is clear that h l ( β ) < β >
1. Upon writing∆( β ) = g (cid:48) ( u infl ( a ); a ) + 2 d ( β − − (cid:0) β g (cid:48) ( u infl ( a ); a ) + 2 dβ (1 − β ) (cid:1) = (1 − β ) (cid:0) g (cid:48) ( u infl ( a ); a ) − d (cid:1) + 2 dβ (1 − β ) , (3.153)it is easy to verify that ∆( β ) < β > β ) > < β <
1. This yields the final inequality(3.150). 27 emma 3.18.
Pick 0 < a ≤ and 0 < d ≤ g (cid:48) ( a ; a )4 . Then we have v (cid:48)− (0) < v (cid:48) d (0) . (3.154)In addition, suppose that v (cid:48)− ( u ) = v (cid:48)∗ ( u ) (3.155)for some 0 ≤ u ≤ a , with u (cid:54) = u min ( a ) if a = . Then one of the following two statements must hold.(a) We have the inequality v (cid:48)(cid:48)− ( u ) > v (cid:48)(cid:48) d ( u ) . (3.156)(b) We have the identities u = a, d = g (cid:48) ( a )4 , v (cid:48)(cid:48)− ( u ) = v (cid:48)(cid:48) d ( u ) . (3.157) Proof.
An easy computation yields v (cid:48) d (0) = 1 − g (cid:48) (0; a )2 d ≥ − g (cid:48) (0; a ) g (cid:48) ( a ; a ) > − g (cid:48) (0; a ) g (cid:48) ( a ; a ) = v (cid:48)− (0) . (3.158)We introduce the critical value u c = sup { ≤ u ≤ a : v − (˜ u ) ≤ u infl ( a ) for all 0 ≤ ˜ u ≤ u } (3.159)and remark that u c = 0 when a = . This allows us to define the value u I = sup (cid:8) ≤ u ≤ min { u c , u min ( a ) } : v (cid:48) d ( u ) > v (cid:48)− ( u ) (cid:9) , (3.160)which again satisfies u I = 0 when a = .We claim that also v (cid:48) d ( u I ) > v (cid:48)− ( u I ). Indeed, assuming this is false we can define β = v (cid:48) d ( u I ) = v (cid:48)− ( u I ) ≥
0. Item (ii) of Lemma 3.15 the implies β >
1. Since h l ( β ) < v (cid:48)(cid:48)− ( u I ) < v (cid:48)(cid:48) d ( u I ) , (3.161)which yields a contradiction.In particular, if (3.155) holds then we must have u ≥ min { u min ( a ) , u c } . If u min ( a ) ≤ u < a , Lemma3.16 shows that (a) must hold. On the other hand, if u c ≤ u < u min ( a ), then we can define β = v (cid:48) d ( u ) = v (cid:48)− ( u ) ≥ β >
1. In addition, we have v − ( u ) ≥ u infl ( a ), which allows us touse h r ( β ) > u = a , theremarks at the start of this section together with a direct computation of v (cid:48)(cid:48) d ( a ) and v (cid:48)(cid:48)− ( a ) imply theidentities in (b).In the remainder of this section we collect several consequences of these computations. In each case,we either rule out non-transverse intersections of v ± with v d or show that they must occur at local minimaof v ± − v d . Corollary 3.19.
Fix 0 < a ≤ together with 0 < d ≤ g (cid:48) ( a ; a )4 and suppose that v (cid:48) + ( u ) = v (cid:48) d ( u ) (3.162)for some 0 ≤ u ≤ a , with u (cid:54) = u min ( a ) if a = . Then we have v (cid:48)(cid:48) + ( u ) > v (cid:48)(cid:48) d ( u ) . (3.163)28 roof. Using the fact that h r ( β ) > β ∈ (0 , Corollary 3.20.
Fix 0 < a ≤ together with 0 < d < g (cid:48) ( a ; a )4 and suppose that v (cid:48)− ( u ) = v (cid:48)∗ ( u ) (3.164)for some 0 ≤ u ≤ a , with u (cid:54) = u min ( a ) if a = . Then we have v (cid:48)(cid:48)− ( u ) > v (cid:48)(cid:48) d ( u ) . (3.165)In addition, we have v (cid:48)− ( a ) > v (cid:48) d ( a ) . (3.166) Proof.
The first inequality follows directly from the fact that option (b) in Lemma 3.18 cannot holdbecause of the restriction on d . The final inequality can be verified directly by noting that v (cid:48) d ( a ) = 1 − g (cid:48) ( a ; a )2 d < − g (cid:48) ( a ; a ) g (cid:48) ( a ; a ) = − v (cid:48)− ( a ) . (3.167) Corollary 3.21.
Fix 0 < a ≤ together with d = g (cid:48) ( a ; a )4 . Then we have v (cid:48)− ( u ) < v (cid:48) d ( u ) (3.168)for all 0 ≤ u < a , with the exception of u = u min ( a ) in the special case a = . Proof.
It is easy to verify that v (cid:48)− ( a ) = v (cid:48) d ( a ) and v (cid:48)(cid:48)− ( a ) = v (cid:48)(cid:48) d ( a ). We also compute v (cid:48)(cid:48)(cid:48) d ( a ) = − g (cid:48)(cid:48)(cid:48) ( a ; a ) g (cid:48) ( a ; a ) = 12 a (1 − a ) > v (cid:48)(cid:48)(cid:48)− ( u ) = − g (cid:48)(cid:48)(cid:48) ( u ; a ) g (cid:48) ( v − ( u ); a ) − g (cid:48)(cid:48) ( u ; a ) g (cid:48) ( u ; a ) g (cid:48) ( v − ( u ); a ) g (cid:48)(cid:48) ( v − ( u ); a ) − g (cid:48) ( u ; a ) g (cid:48) ( v − ( u ); a ) g (cid:48)(cid:48) ( v − ( u ); a ) + g (cid:48) ( u ; a ) g (cid:48) ( v − ( u ); a ) g (cid:48)(cid:48)(cid:48) ( v − ( u ); a ) , (3.170)which gives v (cid:48)(cid:48)(cid:48)− ( a ) = − g (cid:48)(cid:48) ( a ; a ) g (cid:48) ( a ; a ) ≤ . (3.171)In particular, we see that v (cid:48)− ( a − (cid:15) ) < v (cid:48) d ( a − (cid:15) ) (3.172)for all sufficiently small (cid:15) >
0. If a (cid:54) = , the conclusion now follows from (3.154) together with (a) fromLemma 3.18.For a = , one also needs to use the identities v (cid:48) d (cid:0) u min ( a ) (cid:1) = 1 , v (cid:48)(cid:48) d (cid:0) u min ( a ) (cid:1) = − √ v (cid:48)− ( u min ( a ) − (cid:15) ) < v (cid:48) d ( u min ( a ) − (cid:15) ) (3.174)for all sufficiently small (cid:15) >
0. The arguments above allow us to extend this to (cid:15) ∈ (0 , u min ( a )]. Inaddition, we have v (cid:48)− (˜ u ) = − < v (cid:48) d (˜ u ) for u min ( a ) < ˜ u < a .29 orollary 3.22. Pick 0 < a ≤ and 0 < d ≤ g (cid:48) ( a ; a )4 . Then we have v (cid:48)− ( u ) < v (cid:48) d ( u ) (3.175)for all 0 ≤ u < u min ( a ). Proof.
Writing d c = g (cid:48) ( a ; a )4 , we may use Corollary 3.21 to compute v (cid:48) d ( u ) = 1 − d g (cid:48) ( u ; a ) ≥ − g (cid:48) ( a ; a ) g (cid:48) ( u ; a ) = v (cid:48) d c ( u ) > v (cid:48)− ( u ) (3.176)for u ∈ [0 , u min ( a )). Corollary 3.23.
Fix a = and 0 < d ≤ g (cid:48) ( a ; a )4 . Then we have v (cid:48) + ( u ) > v (cid:48) d ( u ) (3.177)for all u min ( a ) < u ≤ a . Proof.
Item (iv) of Lemma 3.14 allow us to compute v (cid:48) d (cid:0) u min ( a ) (cid:1) = 1 = lim u ↓ u min ( a ) v (cid:48) + ( u ) (3.178)together with v (cid:48)(cid:48) d (cid:0) u min ( a ) (cid:1) < lim u ↓ u min ( a ) v (cid:48)(cid:48) + ( u ) , (3.179)which allows us to conclude that v (cid:48) + ( u min ( a ) + (cid:15) ) > v (cid:48) d ( u min ( a ) + (cid:15) ) (3.180)for all sufficiently small (cid:15) >
0. Corollary 3.19 allows us to extend this conclusion to the desired interval (cid:15) ∈ (cid:0) , a − u min ( a ) (cid:1) . We are now ready to analyze the global structure of the solution set to G ( u, v ; a, d ) = 0. Our first tworesults fix a ∈ (0 , ] and track the intersections of the curves v ± that were introduced in § v d introduced in § d is increased; see Figure 8. Lemma 3.24.
Fix 0 < a ≤ . Then there exists a continuous strictly increasing function u AD : [0 , g (cid:48) ( a ; a )4 ] → [0 , a ] (3.181)that satisfies the following properties.(i) We have u AD (0) = 0 and u AD ( g (cid:48) ( a ; a )4 ) = a .(ii) The identity v − (cid:0) u AD ( d ) (cid:1) = v d (cid:0) u AD ( d ) (cid:1) holds for any 0 < d ≤ g (cid:48) ( a ; a )4 .(iii) Suppose that v − ( u ) = v d ( u ) for some 0 < d ≤ g (cid:48) ( a ; a )4 and 0 ≤ u ≤ a . Then in fact u ∈ { u AD ( d ) , a } .(iv) Consider any 0 < d < g (cid:48) ( a ; a )4 for which ( a, d ) (cid:54) = ( , ). Then we have the inequality v (cid:48)− (cid:0) u AD ( d ) (cid:1) < v (cid:48) d (cid:0) u AD ( d ) (cid:1) . (3.182)30 + v - v d with d < d - ( a ) v d - ( a ) ( u AD ,v AD ) ( u B ,v B ) ( u C ,v C ) v + v - v d with d - ( a )< d < d + ( a ) v d + ( a ) ( u AD ,v AD ) Figure 8: Fix 0 < a < . The branches u B and u C described in Lemma 3.25 arise as the two intersectionsof v + and v d on [0 , a ], which collide as d ↑ d − ( a ) (left). On the other hand, the branch u AD described inLemma 3.24 arises as the unique intersection of the curves v − and v d on [0 , a ), which converges to a as d ↑ d + ( a ) (right).(v) For any d > g (cid:48) ( a ; a )4 and 0 ≤ u < a we have v − ( u ) > v d ( u ). Proof.
For convenience, we introduce the function h d ( u ) = v − ( u ) − v d ( u ) and set out to count the zeroesof h d on the interval [0 , a ]. We first note that h d ( a ) = 0 for all d >
0. When d = g (cid:48) ( a ; a )4 this is in fact theonly zero, which can be seen by using (3.168) and explicitly verifying the inequality v − (cid:0) u min ( a ) (cid:1) > v d (cid:0) u min ( a ) (cid:1) (3.183)for ( a, d ) = ( , g (cid:48) ( ; )4 ).On the other hand, (3.166) implies that h d has at least two zeroes for 0 < d < g (cid:48) ( a ; a )4 . Furthermore,we claim that h d has precisely two zeroes for 0 < d ≤ d upon choosing d > u ∈ (0 , a ) we have v (cid:48)(cid:48) d ( u ) < | v (cid:48) d ( u ) | + v d ( u ) ≥ ≤ ˜ u ≤ a { (cid:12)(cid:12) v (cid:48)− (˜ u ) (cid:12)(cid:12) } (3.184)by restricting the size of d > (cid:15) > h d is strictly decreasing on [0 , u min ( a ) + (cid:15) ] for all d ≤ d ≤ g (cid:48) ( a ; a )4 . This is possible because Corollary 3.22 allows us to enforce h (cid:48) d < u = u min ( a ) when a = .Let us now define the critical value d c = sup { d ≤ d ≤ g (cid:48) ( a ; a )4 : h d = 0 has two distinct solutions on [0 , a ] } (3.185)and assume for the moment that d c < g (cid:48) ( a ; a ) /
4. The preparations above show that there exists u min ( a )
0. As a consequence of themonotonicity ∂ d v d ( u ) <
0, this means that for all sufficiently small δ >
0, the function h d with d = d c − δ must have at least three zeroes. This yields a contradiction, which implies that d c = g (cid:48) ( a ; a ) / u AD ( d ) ∈ [0 , a ] to be the left-most root of h d ( u ) = 0 for 0 < d ≤ d c . Thestatements (i)-(v) follow readily from the observations above together with the monotonicity ∂ d v d ( u ) < u ∈ (0 , a ). 31 emma 3.25. Fix 0 < a ≤ . Then there exists a constant 0 < d − < g (cid:48) ( a ; a )4 together with twocontinuous functions ( u B , u C ) : [0 , d − ] → [0 , a ] × [0 , a ] (3.186)that satisfy the following properties.(i) We have ( u B , u C )(0) = (0 , a ) and u B ( d − ) = u C ( d − ).(ii) The function u B is strictly increasing, while the function u C is strictly decreasing.(iii) For any 0 < d ≤ d − the identity v + ( u ) = v d ( u ) holds for ∈ { B, C } .(iv) If v + ( u ) = v d ( u ) for some 0 < d ≤ d − and 0 ≤ u ≤ a then in fact u ∈ { u B ( d ) , u C ( d ) } .(v) For any 0 < d < d − we have v (cid:48) + (cid:0) u B ( d ) (cid:1) < v (cid:48) d (cid:0) u B ( d ) (cid:1) , v (cid:48) + (cid:0) u C ( d ) (cid:1) > v (cid:48) d (cid:0) u C ( d ) (cid:1) . (3.187)If a (cid:54) = , then we also have v (cid:48) + (cid:0) u C ( d − ) (cid:1) = v (cid:48) + (cid:0) u B ( d − ) (cid:1) = v (cid:48) d (cid:0) u B ( d − ) (cid:1) = v (cid:48) d (cid:0) u C ( d − ) (cid:1) . (3.188)(vi) For any d > d − the inequality v + ( u ) > v d ( u ) holds for all 0 ≤ u ≤ a .(vii) If a = then we have d − ( ) = together with u B ( ) = u C ( ) = u min ( a ). Proof.
Writing h d ( u ) = v + ( u ) − v d ( u ), we observe first that h d is strictly decreasing on [0 , u min ( a )] because v (cid:48) d > v (cid:48) + < a = we can use Corollary 3.23 to conclude that h d is strictlyincreasing on [ u min ( a ) , a ]. Since h d (cid:0) u min ( a ) (cid:1) = 0 occurs precisely when d = , all the desired statementscan be easily verified.Throughout the remainder of this proof we therefore assume that 0 < a < . Arguing as in the proofof Lemma 3.24, we may pick d > h d has precisely two zeroes on [0 , a ] for every0 < d ≤ d . This allows us to define the critical value d − = sup { d ≤ d : h d = 0 has two distinct solutions on [0 , a ] } . (3.189)Since v − < v + it is clear that d − < g (cid:48) ( a ; a )4 .Let us assume for the moment that h d − has two or more zeroes on [0 , a ]. This implies that thereexists at least one u ∈ (0 , a ) for which h d − ( u ) = 0 and h (cid:48) d − ( u ) = 0. Using Corollary 3.20 it follows that h (cid:48)(cid:48) d − ( u ) > ∂ d v d ( u ) <
0, this meansthat for all sufficiently small δ >
0, the function h d with d = d − − δ must have at least three zeroes.This yields a contradiction, which by continuity shows that h d − ( u ) = 0 has precisely one root on [0 , a ].Upon defining u B ( d ) and u C ( d ) to be the left-most respectively right-most root of h d ( u ) = 0, the desiredproperties (i) - (vii) can be easily verified.Recalling the function G introduced in (2.6), we see that D , G ( u, v ; a, d ) = g (cid:48) ( u ) − d d d g (cid:48) ( v ) − d . (3.190)In order to study the stability and parameter-dependence of the roots constructed in Lemmata 3.24-3.25,it is crucial to understand when the determinant of D , G vanishes. The result below states that thishappens at tangential intersections of v ± and v d . 32 emma 3.26. Fix 0 < a < together with d > v ( u ) = v d ( u ) (3.191)for some ∈ {− , + } and u ∈ [0 , a ]. Then we have det D , G ( u, v d ( u ); a, d ) = 0 if and only if v (cid:48) ( u ) = v (cid:48) d ( u ). Proof.
We first consider the case v (cid:48) ( u ) = 0, which occurs when g (cid:48) ( u ; a ) = 0 and hence u = u min ( a ).Using 0 < a < we see that v − ( u ) < u max ( a ) < v + ( u ) . (3.192)Writing v = v d ( u ) = v ± ( u ), we hence obtain g (cid:48) ( v ; a ) (cid:54) = 0 and hencedet D , G ( u, v ; a, d ) = − dg (cid:48) ( v ; a ) (cid:54) = 0 . (3.193)Since v (cid:48) d ( u ) = 1, the desired equivalence indeed holds for this case.Assuming now that v (cid:48) ( u ) (cid:54) = 0 and hence g (cid:48) ( u ; a ) (cid:54) = 0, we again write v = v d ( u ) = v ( u ) and use(3.113) to compute g (cid:48) ( v ; a ) = − [ v (cid:48) ( u )] − g (cid:48) ( u ; a ) . (3.194)In particular, we finddet D , G ( u, v ; a, d ) = ( g (cid:48) ( u ; a ) − d ) (cid:0) − [ v (cid:48) ( u )] − g (cid:48) ( u ; a ) − d (cid:1) − d = − [ v (cid:48) ( u )] − g (cid:48) ( u ; a ) − dg (cid:48) ( u ; a ) + 2 d [ v (cid:48) ( u )] − g (cid:48) ( u ; a )= 2 dg (cid:48) ( u ; a )[ v (cid:48) ( u )] − (cid:104) − g (cid:48) ( u ; a )2 d − v (cid:48) ( u ) (cid:105) = 2 dg (cid:48) ( u ; a )[ v (cid:48) ( u )] − (cid:104) v (cid:48) d ( u ) − v (cid:48) ( u ) (cid:105) , (3.195)from which the statement follows.In order to characterize the dependence of d − ( a ) on a , we introduce the function G sn ( u, v, d ; a ) = (cid:0) G ( u, v ; a, d ) , G ( u, v ; a, d ) , det D , G ( u, v ; a, d ) (cid:1) T (3.196)and symbolically write[ D , , G sn ( u, v, d ; a )] − = (cid:2) det D , , G sn ( u, v, d ; a ) (cid:3) − ∗ ∗ ∗∗ ∗ ∗ γ ( u, v, d ; a ) γ ( u, v, d ; a ) γ ( u, v, d ; a ) . (3.197)For any 0 < a ≤ , we use the functions defined in Lemma 3.25 to introduce the notation ω ( a ) = (cid:16) u B (cid:0) d − ( a ) (cid:1) , v + (cid:0) u B ( d − ( a )) (cid:1) , d − ( a ) (cid:17) = (cid:16) u C (cid:0) d − ( a ) (cid:1) , v + (cid:0) u C ( d − ( a )) (cid:1) , d − ( a ) (cid:17) , (3.198)which corresponds to the critical point where the branches u B and u C collide. Corollary 3.27.
Upon fixing 0 < a < , the following two statements are equivalent.(a) The identity G sn ( u, v, d ; a ) = 0 holds for some d > u, v ) ∈ [0 , that has v ≥ u .(b) We have ( u, v, d ) = ω ( a ) or ( u, v, d ) = ( a, a, g (cid:48) ( a ; a )4 ) .33 roof. As a preparation, we note that det D , G ( a, a ; a, d ) = 0 if and only if d = g (cid:48) ( a ; a )4 . In addition, itis easy to check that det D , G (0 , a, d ) > D , G (1 , a, d ) > d ≥ b ) → ( a ) can now be verified directly using Lemma 3.26 and (3.188). In addition,we only need to establish the reverse implication under the additional assumption that v > u .Let us therefore assume that ( a ) holds with g ( u ) = − g ( v ) (cid:54) = 0, which allows us to write0 < u < a < v < . (3.199)In particular, Lemma 3.13 implies that v = v − ( u ) or v = v + ( u ). Lemma 3.26 together with (3.187) and(3.182) now imply that (b) must hold. Lemma 3.28.
We have the identities γ (0 , ,
0; 0) = − ,γ (0 , ,
0; 0) = 0 . (3.200)In addition, the identity γ ( ω ( a ); a ) = 0 (3.201)holds for any 0 < a < , together with the inequalitiesdet D , , G sn ( ω ( a ); a ) < ,γ ( ω ( a ); a ) − γ ( ω ( a ); a ) < ,γ ( ω ( a ); a ) γ ( ω ( a ); a ) ≥ . (3.202) Proof.
Let us assume for the moment that G sn ( u, v, d ; a ) = 0, which directly implies( g (cid:48) ( u ; a ) − d ) g (cid:48) ( v ; a ) = 2 dg (cid:48) ( u ; a ) . (3.203)In view of the identity D , , G sn ( u, v, d ; a ) = g (cid:48) ( u ; a ) − d d v − u )2 d g (cid:48) ( v ; a ) − d u − v )( g (cid:48) ( v ; a ) − d ) g (cid:48)(cid:48) ( u ; a ) ( g (cid:48) ( u ; a ) − d ) g (cid:48)(cid:48) ( v ; a ) − g (cid:48) ( u ; a ) + g (cid:48) ( v ; a )) , (3.204)we can use (3.203) to computedet D , , G sn ( u, v, d ; a ) = 2( v − u ) (cid:104) g (cid:48) ( u ; a )( g (cid:48) ( u ; a ) − d ) g (cid:48)(cid:48) ( v ; a ) − g (cid:48) ( v ; a )( g (cid:48) ( v ; a ) − d ) g (cid:48)(cid:48) ( u ; a ) (cid:105) = 2( v − u ) (cid:104) dg (cid:48) ( u ; a ) g (cid:48) ( v ; a ) g (cid:48)(cid:48) ( v ; a ) − g (cid:48) ( v ; a )( g (cid:48) ( v ; a ) − d ) g (cid:48)(cid:48) ( u ; a ) (cid:105) . (3.205)Expanding the subdeterminants of (3.204) and reusing (3.203), we also find γ ( u, v, d ; a ) = 2 d ( g (cid:48) ( u ; a ) − d ) g (cid:48)(cid:48) ( v ; a ) − ( g (cid:48) ( v ; a ) − d ) g (cid:48)(cid:48) ( u ; a ) ,γ ( u, v, d ; a ) = 2 d ( g (cid:48) ( v ; a ) − d ) g (cid:48)(cid:48) ( u ; a ) − ( g (cid:48) ( u ; a ) − d ) g (cid:48)(cid:48) ( v ; a ) ,γ ( u, v, d ; a ) = 0 , (3.206)which allows us to explicitly verify (3.200). 34et us now fix 0 < a < . For any ( u, v ) ∈ [0 , and d > h (cid:0) u, v, d (cid:1) = d g (cid:48)(cid:48) ( u ; a ) − g (cid:48)(cid:48) ( u ; a ) g (cid:48) ( v ; a ) − g (cid:48) ( u ; a ) g (cid:48) ( v ; a ) g (cid:48)(cid:48) ( v ; a )= dg (cid:48) ( v ; a ) (cid:104) g (cid:48) ( v ; a )( g (cid:48) ( v ; a ) − d ) g (cid:48)(cid:48) ( u ; a ) − d g (cid:48) ( u ; a ) g (cid:48) ( v ; a ) g (cid:48)(cid:48) ( v ; a ) (cid:105) , (3.207)which allows us to rewrite (3.205) in the formdet D , , G sn ( u, v, d ; a ) = − d ( v − u ) g (cid:48) ( v ; a ) h ( u, v, d ) . (3.208)Using (3.132) and (3.133) we observe that v (cid:48)(cid:48) + ( u ) − v (cid:48)(cid:48) d ( u ) = h (cid:0) u, v + ( u ) , d (cid:1) . (3.209)In particular, Corollary 3.19 and (3.188) imply that h (cid:0) ω ( a ) (cid:1) > , (3.210)which shows that det D , , G sn ( ω ( a ); a ) < , (3.211)as desired.Let us now write ( u, v, d ) = ω ( a ) and compute γ (cid:0) ω ( a ); a (cid:1) − γ (cid:0) ω ( a ); a (cid:1) = g (cid:48) ( u ) (cid:0) g (cid:48) ( u ) − d ) g (cid:48)(cid:48) ( v ) − g (cid:48) ( v ) (cid:0) g (cid:48) ( v ) − d ) g (cid:48)(cid:48) ( u )= ( v − u ) − det DG sn ( u, v, d ; a ) ≤ . (3.212)In addition, since u > u min ( a ) and hence v (cid:48) d ( u ) = v (cid:48) + ( u ) >
0, we have g (cid:48) ( u ) < d . This allows us to define α = √ d (cid:0) d − g (cid:48) ( u ; a ) (cid:1) / g (cid:48)(cid:48) ( v ; a ) ,β = √ d (cid:0) d − g (cid:48) ( v ; a ) (cid:1) / g (cid:48)(cid:48) ( u ; a ) (3.213)and compute γ ( ω ( a ); a ) γ ( ω ( a ); a ) = (4 d )(4 d ) g (cid:48)(cid:48) ( u ; a ) g (cid:48)(cid:48) ( v ; a ) + (4 d ) g (cid:48)(cid:48) ( u ; a ) g (cid:48)(cid:48) ( v ; a ) − d ( g (cid:48) ( u ; a ) − d ) g (cid:48)(cid:48) ( v ; a ) − d ( g (cid:48) ( v ; a ) − d ) g (cid:48)(cid:48) ( u ; a ) = 2 αβ + α + β ≥ , (3.214)as desired. Corollary 3.29.
The map a (cid:55)→ ω ( a ) is C ∞ -smooth on (0 , ) and we have d (cid:48)− ( a ) > Proof.
Since d − < g (cid:48) ( a ; a )4 , we may use Corollary 3.27, together with the first inequality in (3.202) toapply the implicit function and establish the C ∞ -smoothness of ω .Since g (cid:48) ( a ; a ) / ↓ a ↓
0, a squeezing argument shows that lim a ↓ ω ( a ) = (0 , , γ and γ implies that γ ( ω ( a ); a ) < γ ( ω ( a ); a ) ≤ < a < . In particlar, writing ( u, v, d ) = ω ( a ), we may use the inequalities D g ( u ; a ) < , D g ( v ; a ) < d (cid:48)− ( a ) = − [det D , , G sn ( ω ( a ); a )] − (cid:2) γ ( ω ( a ); a ) D g ( u ; a ) + γ ( ω ( a ); a ) D g ( v ; a ) (cid:3) > . (3.217) Proof of Proposition 3.1.
Items (i), (iii), (iv) follow from Lemmata 3.24-3.25 and the observations in theproof of Corollary 3.27. Item (ii) follows from Corollary 3.29, while the expansions in (v) follow fromPropositions 3.6 and 3.9.
Proof of Proposition 3.2.
The identity g (1 − u, − a ) = − g ( u, a ) implies that G (1 − u, − v ; 1 − a, d ) = − G ( u, v ; a, d ) . (3.218)This allows us to extend the solutions constructed in Lemmata 3.24-3.25 to the entire interval 0 ≤ a ≤ ∈ { A, B, C } the sign of det D , G ( u , v ; a, d ) is constant onΩ − on account of Corollary 3.27. Using (ii) the eigenvalues of these matrices can be explicitly computedat d = 0, which yields the desired stability properties.To obtain the strict ordering in (iii), we fix 0 < a ≤ and 0 < d < d − ( a ). We observe that v + ( u ) ≥ v − ( u ) for u ∈ [0 , a ], with equality only at u = u min ( a ) for a = . Since v d (0) = 0 this impliesthat u A ( a, d ) < u B ( a, d ) and v A ( a, d ) < v B ( a, d ). By construction, we also have u B ( a, d ) < u C ( a, d ). Inaddition, it cannot be the case that v B ( a, ˜ d ) = v C ( a, ˜ d ) for any 0 < ˜ d < d − ( a ) since then G = 0 impliesthat also u B ( a, ˜ d ) = u C ( a, ˜ d ). Using the general identity D a,d ( u , v ) = − [ D , G ( u , v ; a, d )] − D , G ( u , v ; a, d ) (3.219)we see that ∂ d v ( a,
0) = − g (cid:48) ( v ( a, a ) (cid:0) u ( a, − v ( a, (cid:1) , (3.220)which yields ∂ d v B ( a,
0) = 1 g (cid:48) (1; a ) < − ag (cid:48) (1; a ) = ∂ d v C ( a, . (3.221)These observations allow us to conclude v B ( a, d ) < v C ( a, d ), as desired. Proof of Proposition 3.3.
Arguing in a similar fashion as the proof of Proposition 3.2, the statementsfollow from Lemma 3.24 and Corollary 3.27.
Our goal here is to establish the existence of bichromatic wave solutions to the Nagumo LDE (2.1) andto obtain detailed results concerning their speed. In particular, we establish Theorems 2.2 and 2.3, whichare the main results of this paper.Upon introducing the standard discrete Laplacian[∆ + u ]( ξ ) = u ( ξ + 1) + u ( ξ − − u ( ξ ) (4.1)36ogether with the off-diagonal matrix J = (cid:18) (cid:19) , (4.2)the travelling wave system (2.5) can be rewritten as − c Φ (cid:48) = d J ∆ + Φ + G (Φ; a, d ) . (4.3)For any ( a, d ) ∈ Ω − , we set out to seek solutions to (4.3) that satisfy the boundary conditionslim ξ →−∞ Φ( ξ ) = (0 , , lim ξ → + ∞ Φ( ξ ) = (cid:0) u B ( a, d ) , v B ( a, d ) (cid:1) . (4.4)The existence of such solutions will be established in § c ≥ § c = 0 and c >
0. We verify these criteria in § a, d ) ∈ Ω − that are close to the cusp( , ) and the corner (1 , Our preparatory work in § ,
0) and ( u B , v B ) are both stable underthe dynamics of ( ˙ u, ˙ v ) = G ( u, v ; a, d ) and all intermediate equilibria are unstable. Using a straightforwardestimate based on the ordering (3.11), the wavespeeds can be shown to be non-negative. Lemma 4.1.
Pick ( a, d ) ∈ Ω − . Then there exists a constant c ∈ R and a non-decreasing functionΦ : R → R that satisfy (4.3)-(4.4). This constant c is unique, while the function Φ is unique up totranslation if c (cid:54) = 0. In the latter case we also have Φ (cid:48) ( ξ ) > (0 ,
0) for all ξ ∈ R . Proof.
These statements follow directly from the main results in [11].
Lemma 4.2.
Pick ( a, d ) ∈ Ω − . Then the constant c defined in Lemma 4.1 satisfies c ≥ Proof.
Suppose that c (cid:54) = 0. We estimate − c Φ (cid:48) v ( ξ ) = d [Φ u ( ξ + 1) + Φ u ( ξ − − v ( ξ )] + g (Φ v ( ξ ); a ) ≤ d [ u − v ( ξ )] + g (Φ v ( ξ ); a ) . (4.5)Since 0 < u B ( a, d ) < a < v B ( a, d ), there exists ξ ∗ ∈ R for which Φ v ( ξ ∗ ) = u B ( a, d ). This yields − c Φ (cid:48) v ( ξ ∗ ) ≤ g ( u ; a ) < , (4.6)from which we conclude c > c ( a, d ) and Φ( a, d ) for the wavespeed and waveprofile defined in Lemma 4.1, we introduce theset T = { ( a, d ) ∈ Ω − : c ( a, d ) > } , (4.7)which corresponds to the set T low used in §
2. In addition, for any ( a, d ) ∈ T we introduce the linearoperators L a,d : W , ∞ ( R ; R ) → L ∞ ( R ; R ) , L adj a,d : W , ∞ ( R ; R ) → L ∞ ( R ; R ) , (4.8)that act as L a,d φ = − c ( a, d ) φ (cid:48) − d J ∆ + φ − DG (Φ( a, d ); a, d ) φ, L adj a,d ψ = c ( a, d ) ψ (cid:48) − d J ∆ + ψ − DG (Φ( a, d ); a, d ) ψ. (4.9)37he results in [28, §
8] imply that there exists Ψ = Ψ( a, d ) ∈ W , ∞ ( R ; R ) with Ψ > (0 ,
0) for which wehave the identities Ker L a,d = span { Φ (cid:48) ( a, d ) } , Ker L adj a,d = span { Ψ( a, d ) } (4.10)together with the normalization (cid:90) ∞−∞ (cid:104) Ψ( ξ ) , Φ (cid:48) ( ξ ) (cid:105) dξ = 1 . (4.11)In particular, [32, Thm. A] implies thatRange L a,d = { f ∈ L ∞ ( R ; R ) : (cid:90) ∞−∞ (cid:104) Ψ( ξ ) , f ( ξ ) (cid:105) dξ = 0 } . (4.12)These ingredients allow us to use the implicit function theorem to show that the pair ( c, Φ) dependssmoothly on the parameters ( a, d ) ∈ T . In addition, we obtain a sign on ∂ a c . Lemma 4.3.
The maps
T (cid:51) ( a, d ) (cid:55)→ c ( a, d ) ∈ (0 , ∞ ) , T (cid:51) ( a, d ) (cid:55)→ Φ( a, d ) ∈ BC ( R ; R ) (4.13)are C ∞ -smooth. In addition, we have ∂ a c ( a, d ) > a, d ) ∈ T . Proof.
The C -smoothness of the maps ( c, Φ) follows from [28, Thm. 2.3]. On account of the smoothnessof the function g , this can readily be extended to the desired C ∞ -smoothness by using the ideas in [28, §
8] to set up an implicit function argument along the lines of [33, Prop. 6.5].Differentiating (4.3) with respect to a , we compute − [ ∂ a c ]Φ (cid:48) − c [ ∂ a Φ (cid:48) ] = d J ∆ + [ ∂ a Φ] + DG (Φ; a, d ) ∂ a Φ + ∂ a G (Φ; a, d ) , (4.15)which gives − [ ∂ a c ]Φ (cid:48) + L a,d ∂ a Φ = ∂ a G (Φ; a, d ) . (4.16)Applying (4.12) and noting that ∂ a g ( u ; a ) = − u (1 − u ) < u ∈ (0 , − ∂ a c = (cid:90) ∞−∞ (cid:104) ∂ a G (Φ( ξ ); a, d ) , Ψ( ξ ) (cid:105) dξ < , (4.17)as desired. Corollary 4.4.
If ( a, d ) ∈ T , then also ( a (cid:48) , d ) ∈ T for all ( a (cid:48) , d ) ∈ Ω − with a (cid:48) ≥ a . Proof of Theorem 2.2.
The statements follow directly from Lemmata 4.1-4.3.
We now set out to derive conditions that guarantee either c ( a, d ) = 0 or c ( a, d ) >
0. To this end, weintroduce the functions u a,d ( v ) = v − g ( v ; a )2 d , v a,d ( u ) = u − g ( u ; a )2 d (4.18)and note that u ( a, d ) = u a,d (cid:0) v ( a, d ) (cid:1) , v ( a, d ) = v a,d (cid:0) u ( a, d ) (cid:1) (4.19)38or each ∈ { A, B, C } and ( a, d ) ∈ Ω − . The local extrema of these functions are located at the criticalpoints γ c ; ± ( a, d ) = ( a + 1) ± √ a − a + 1 − d = u infl ( a ) ± (cid:113) ( g (cid:48) ( u infl ( a ); a ) − d ) . (4.20)One can verify that the functions u a,d and v a,d are strictly decreasing on [ γ c ; − ( a, d ) , γ c ;+ ( a, d )] and strictlyincreasing outside this interval. The following ordering result exploits this characterization and will allowus to establish c ( a, d ) = 0 for a significant portion of the parameter set Ω − . Lemma 4.5.
For any ( a, d ) ∈ Ω − we have the ordering0 ≤ γ c ; − ( a, d ) ≤ a ≤ γ c ;+ ( a, d ) ≤ v B ( a, d ) . (4.21)If ( a, d ) ∈ Ω − then the inequalities in (4.21) are all strict. Proof.
Let us first fix ( a, d ) ∈ Ω − . We note that 2 d < d + ( a ) = g (cid:48) ( a ; a )2 ≤ g (cid:48) ( u infl ( a ); a )2 , which implies that γ c ; ± ( a, d ) are well-defined. In addition, this allows us to compute u (cid:48) a,d ( a ) = 1 − g (cid:48) ( a ; a )2 d ≤ − , (4.22)which means that a ∈ (cid:0) γ c ; − ( a, d ) , γ c ;+ ( a, d ) (cid:1) .In particular, the function v (cid:55)→ u a,d ( v ) is strictly decreasing on [ a, γ c ;+ ( a, d )]. On account of theorderings u a,d (cid:0) v A ( a, d ) (cid:1) = u A ( a, d ) < u B ( a, d ) = u a,d (cid:0) v B ( a, d ) (cid:1) , a < v A ( a, d ) < v B ( a, d ) (4.23)we hence cannot have v B ( a, d ) ≤ γ c ;+ ( a ). The results for the general case ( a, d ) ∈ Ω − now follow bycontinuity. Lemma 4.6.
Consider any ( a, d ) ∈ Ω − for which d ≤ (1 − a ) . Then we have c ( a, d ) = 0. Proof.
Fix any 0 < a <
1, write d ∗ = (1 − a ) and suppose that ( a, d ∗ ) ∈ Ω − . On account of Corollary4.4 it suffices to show that c ( a, d ∗ ) = 0. Assuming to the contrary that c = c ( a, d ∗ ) >
0, we compute0 > − c Φ (cid:48) v ( ξ )= d ∗ [Φ u ( ξ + 1) + Φ u ( ξ − − v ( ξ )] + g (Φ v ( ξ ); a ) > − d ∗ Φ v ( ξ ) + g (Φ v ( ξ ); a )= − d ∗ u a,d ∗ (cid:0) Φ v ( ξ ) (cid:1) . (4.24)Since 0 < γ c ;+ ( a, d ∗ ) < v B ( a, d ∗ ), there exists ξ ∗ ∈ R for which Φ v ( ξ ∗ ) = γ c ;+ ( a, d ∗ ). The key point isthat u a,d ∗ (cid:0) γ c ;+ ( a, d ∗ ) (cid:1) = 0 , (4.25)which allows us to obtain a contradiction by picking ξ = ξ ∗ in (4.24).We now set out to derive a (non-sharp) condition that guarantees c ( a, d ) >
0. The strategy will beto rule out the existence of standing bichromatic waves. Let us therefore pick any ( a, d ) ∈ Ω − \ T , whichimplies c ( a, d ) = 0. Writing (Φ u , Φ v ) = Φ( a, d ) for the corresponding profile, we introduce the sequence( u i , v i ) = (cid:0) Φ u (2 i ) , Φ v (2 i + 1) (cid:1) , (4.26)39 u i , v i ) ( u i + , v i )( u i + , v i + ) u bot u top u B u v bot v top v B v Figure 9: Illustration of the two reflections described by the system (4.30). The values have been modifiedfor illustrative purposes, since the real regions are minuscule.which satisfies the limitslim i →−∞ ( u i , v i ) = (0 ,
0) lim i → + ∞ ( u i , v i ) = ( u B ( a, d ) , v B ( a, d ) (cid:1) , (4.27)together with the difference equation0 = d (cid:2) v i + v i − − u i (cid:3) + g ( u i ; a ) , d (cid:2) u i +1 + u i − v i (cid:3) + g ( v i ; a ) . (4.28)Applying a shift to the first equation, we obtain the implicit system v i +1 = 2 (cid:2) u i +1 − g ( u i +1 ; a )2 d (cid:3) − v i ,u i +1 = 2 (cid:2) v i − g ( v i ; a )2 d (cid:3) − u i , (4.29)which can be written as v i +1 = 2 v a,d ( u i +1 ) − v i ,u i +1 = 2 u a,d ( v i ) − u i . (4.30)In particular, we can obtain ( u i +1 , v i +1 ) by first reflecting ( u i , v i ) horizontally through the curve u = u a,d ( v ) and then vertically through the curve v = v a,d ( u ). Based on this geometric intuition, we setout to construct a rectangle [ u top ( a, d ) , u B ( a, d )] × [ v top ( a, d ) , v B ( a, d )] (4.31)that must contain ( u i +1 , v i +1 ) for some critical i , together with a rectangle[0 , u bot ( a, d )] × [0 , v bot ( a, d )] (4.32)that must contain the intermediate point ( u i +1 , v i ); see Figure 9.40 emma 4.7. There exist continuous functions( v bot , v top ) : Ω − → [0 , (4.33)that satisfy the inequalities0 < v bot ( a, d ) < γ c ; − ( a, d ) < v top ( a, d ) < γ c ;+ ( a, d ) < v B ( a, d ) (4.34)together with the identities u a,d (cid:0) v bot ( a, d ) (cid:1) = u a,d (cid:0) v top ( a, d ) (cid:1) = u B ( a, d ) (4.35)for each ( a, d ) ∈ Ω − . Furthermore, these functions can be continuously extended to Ω − in such a waythat (4.35) holds whenever d > Proof.
Pick any ( a, d ) ∈ Ω − . On account of the identity u a,d (0) = 0 and the inequalities u a,d (cid:0) γ c ;+ ( a, d ) (cid:1) < u a,d (cid:0) v B ( a, d ) (cid:1) = u B ( a, d ) < a = u a,d ( a ) , (4.36)the equation u a,d ( v ) = u B ( a, d ) has three distinct solutions on (0 , v B ( a, d )]. Using the fact that u B ( a,
0) =0, these solutions can be extended continuously to d = 0 by writing v bot ( a,
0) = 0 and v top ( a,
0) = a .The extension to d = d − ( a ) > a, d ) ∈ Ω − with d > u top ( a, d ) = 2 u a,d (cid:0) γ c ;+ ( a, d ) (cid:1) − u B ( a, d ) . (4.37)We note that (cid:0) u top ( a, d ) , γ c ;+ ( a, d ) (cid:1) can be seen as the horizontal reflection of (cid:0) u B ( a, d ) , γ c ;+ ( a, d ) (cid:1) through the curve u = u a,d ( v ). Lemma 4.8.
Pick any ( a, d ) ∈ Ω − for which c ( a, d ) = 0 and consider the sequence { ( u i , v i ) } defined in(4.26). Then there exists i ∈ Z for which (cid:0) u top ( a, d ) , v top ( a, d ) (cid:1) ≤ (cid:0) u i +1 , v i +1 (cid:1) ≤ (cid:0) u B ( a, d ) , v B ( a, d ) (cid:1) , (4.38)while (0 , ≤ (cid:0) u i , v i (cid:1) ≤ (cid:0) u B ( a, d ) , v bot ( a, d ) (cid:1) . (4.39) Proof.
For any i ∈ Z we have the inequalities u i ≤ u i +1 ≤ u B ( a, d ) , (4.40)which implies that we must have u i ≤ u a,d ( v i ) ≤ u i +1 . In particular, we see that u a,d ( v i ) ≤ u B ( a, d ) , (4.41)which implies that v i / ∈ (cid:16) v bot ( a, d ) , v top ( a, d ) (cid:17) . (4.42)In addition, we have u i = 2 u a,d ( v i ) − u i +1 ≥ u a,d ( v i ) − u B ( a, d ) . (4.43)In particular, for every i we either have(0 , ≤ ( u i , v i ) ≤ (cid:0) u B ( a, d ) , v bot ( a, d ) (cid:1) (4.44)or (cid:0) u top ( a, d ) , v top ( a, d ) (cid:1) ≤ ( u i , v i ) ≤ (cid:0) u B ( a, d ) , v B ( a, d ) (cid:1) . (4.45)The limits (4.27) imply that there exists M (cid:29) i ≤ − M and (4.45) holds forall i ≥ M . In particular, there must be jump between these two sets.41or any ( a, d ) ∈ Ω − with d >
0, the inequalities (4.34) imply that the function u a,d is strictly increasingon [0 , v bot ( a, d )]. This allows us to define an inverse u inv a,d : [0 , u B ( a, d )] → [0 , v bot ( a, d )] . (4.46)In addition, we introduce the function v a,d ; r ( u ) = 2 v a,d ( u ) − v B ( a, d ) , (4.47)which can be interpreted as the vertical reflection of the line v = v B ( a, d ) through the curve v = v a,d ( u ).For any ( a, d ) ∈ Ω − with d >
0, these definitions allow us to introduce the notation u bot ( a, d ) = max { u ∈ [0 , u B ( a, d )] : v a,d ; r ( u ) = u inv a,d ( u ) } . (4.48)On account of the inequalities v a,d ; r (0) < u inv a,d (0) < u inv a,d (cid:0) u B ( a, d ) (cid:1) = v bot ( a, d ) < v B ( a, d ) = v a,d ; r (cid:0) u B ( a, d ) (cid:1) , (4.49)one can verify that u bot is well-defined and continous in ( a, d ). Lemma 4.9.
Pick ( a, d ) ∈ Ω − and assume that u bot ( a, d ) < u top ( a, d ) (4.50)holds. Then we have c ( a, d ) > Proof.
Suppose the contrary that c ( a, d ) = 0 and consider the sequence { ( u i , v i ) } defined by (4.26),together with the critical value i ∈ Z that appears in Lemma 4.8. Since the sequence { u i } is non-decreasing, we have u i +1 ≥ u i which implies that u i +1 ≥ u a,d ( v i ) . (4.51)Exploiting 0 ≤ v i ≤ v bot ( a, d ) this gives 0 ≤ v i ≤ u inv a,d (cid:0) u i +1 (cid:1) . (4.52)On the other hand, the inequality v i +1 ≤ v B ( a, d ) yields v i ≥ v a,d ; r ( u i +1 ) . (4.53)Combining this with (4.52), we see that u i +1 ≤ u bot ( a, d ). However, (4.38) implies that u i +1 ≥ u top ( a, d ), which yields the desired contradiction. u top > u bot In Figure 10 we show where one may numerically verify that the scalar inequality (4.50) holds, whichensures that c ( a, d ) >
0. We also plot the curve d = (1 − a ) , below which we have established that c ( a, d ) = 0. Taken together, we feel that these results cover a reasonable portion of the parameter spaceΩ − .Our final task is to analytically verify (without resorting to any numerics) that c ( a, d ) > a, d ) = (cid:0) , (cid:1) and the corner ( a, d ) = (1 , u top > u bot by exploiting the monotonicity of v a,d ; r . Lemma 4.10.
Pick ( a, d ) ∈ Ω − with d >
0. Then the function v a,d ; r is strictly increasing on [0 , u B ( a, d )].42 - a ) Lemma 4.9 yields c > Ω - ∖ Figure 10: The darkest region contains all pairs ( a, d ) where the assumption u bot ( a, d ) < u top ( a, d ) wasverified numerically. We also include the boundaries of the sets T and Ω − as computed numerically bythe procedure described in § Proof.
We note first that v (cid:48) a,d ( u ) = 1 − g (cid:48) ( u ; a )2 d > u ∈ [0 , u min ( a )]. In particular, we only have toconsider the case u B ( a, d ) > u min ( a ), which cannot occur for a = .Let us first assume that 0 < a < . Using (3.187) or (3.188) we may conclude that v (cid:48) a,d (cid:0) u B ( a, d ) (cid:1) ≥ v (cid:48) + (cid:0) u B ( a, d ) (cid:1) > . (4.54)Since u B ( a, d ) ≤ a ≤ γ c ;+ ( a, d ) this implies that u B ( a, d ) ≤ γ c ; − ( a, d ). In particular, v a,d and hence v a,d ; r are strictly increasing on [0 , u B ( a, d )].It remains to consider a ∈ ( , v B (1 − a, d ) > u max (1 − a ), we may use Corollary 3.4 toconclude u B ( a, d ) = 1 − v B (1 − a, d ) < − u max (1 − a ) = u min ( a ) . (4.55) Corollary 4.11.
Consider any ( a, d ) ∈ Ω − with d > v a,d ; r (cid:0) u top ( a, d ) (cid:1) > v bot ( a, d ) . (4.56)Then we have u top ( a, d ) > u bot ( a, d ). Proof.
This follows from the uniform bound u inv a,d ≤ v bot ( a, d ) and the fact that v a,d ; r is strictly increasing.We now set out to verify the explicit condition (4.56) for the boundary points (cid:0) a, d − ( a ) (cid:1) with a ∼ a = . Using the continuity of u top and u bot , this means that c ( a, d ) > a, d ) ∈ Ω − that aresufficiently close to these critical boundary points. Lemma 4.12.
We have the expansions u B (cid:0) a, d − ( a ) (cid:1) = (1 − a ) + (1 − a ) + O (cid:0) (1 − a ) (cid:1) ,v B (cid:0) a, d − ( a ) (cid:1) = 1 − (1 − a ) + O (cid:0) (1 − a ) (cid:1) (4.57)as a ↑
1. 43 roof.
Exploiting Corollary 3.4 together with the symmetry d ( a ) = d (1 − a ), we obtain u B (cid:0) a, d − ( a ) (cid:1) = 1 − v B (cid:0) − a, d − (1 − a ) (cid:1) , v B ( a, d − ( a )) = 1 − u B (cid:0) − a, d − (1 − a ) (cid:1) . (4.58)The desired expansions hence follow from Proposition 3.6. Lemma 4.13.
We have the expansions u top (cid:0) a, d − ( a ) (cid:1) = ( a − + O (cid:0) (1 − a ) (cid:1) ,v a,d − ( a ); r (cid:16) u top (cid:0) a, d − ( a ) (cid:1)(cid:17) = 1 + O (cid:0) − a (cid:1) (4.59)as a ↑ Proof.
These expansions can be found by substition of (4.57) into the definitions (4.37) and (4.47).On account of the identity γ c ; − (1 ,
0) = and the inequality v bot ( a, d ) ≤ γ c ; − ( a, d ), we see thatCorollary 4.11 implies that u top (cid:0) a, d − ( a ) (cid:1) > u bot (cid:0) a, d − ( a ) (cid:1) (4.60)whenever 1 − a > Lemma 4.14.
The inequality (4.56) holds for ( a, d ) = ( , ). Proof.
Writing ( a cp , d cp ) = ( , ), we can explicitly compute γ c ;+ ( a cp , d cp ) = 12 + 16 √ , (4.61)which together with the expressions (3.26) and (4.37) yields u top ( a cp , d cp ) = 12 − √ √ . (4.62)Using (4.47) we obtain v a cp ,d cp ; r (cid:0) u top ( a cp , d cp ) (cid:1) = 12 − √ √ ∼ . . (4.63)In particular, we have v a cp ,d cp ; r (cid:0) u top ( a cp , d cp ) (cid:1) > a cp ≥ γ c ; − ( a cp , d cp ) ≥ v bot ( a cp , d cp ) , (4.64)as desired. Proof of Theorem 2.3.
Item (i), (ii) and (iii) follow from Lemma 4.6, Corollary 4.4 and Lemma 4.14respectively. Item (iv) follows from Lemma 4.5, together with Lemma 4.9 and the continuity of thefunctions u bot and u top . Indeed, the expression (2.21) can be rewritten asΓ( a ) = u top (cid:0) a, d − ( a ) (cid:1) − u bot (cid:0) a, d − ( a ) (cid:1) . (4.65)Finally, item (v) follows directly from (4.60). 44 eferences [1] D. G. Aronson and H. F. Weinberger (1975), Nonlinear diffusion in population genetics, combustion,and nerve pulse propagation. In: Partial differential equations and related topics . Springer, pp.5–49.[2] P. W. Bates and A. Chmaj (1999), A Discrete Convolution Model for Phase Transitions.
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