RENDICONTI DEL SEMINARIO MATEMATICO

72 M. Francadynamical system of the form (8) through (4) with l = q = p ∗ . In this section we willalways set l = q = p ∗ in (4) so we will leave the subscript unsaid, to simplify the notation.Since (8) is C 1 and uniformly continuous in the t variable, O admits local unstableand stable manifolds, denoted respectively by Wǫ,loc u (τ) and W ǫ,loc s (τ), see [30], [14].From Proposition 3 we know that, if Q u ∈ Wǫ,loc u (τ), then lim t→−∞x τ (Q u , t) = O andthe corresponding solution u(r) of (3) is a regular solution, while if Q s ∈ Wǫ,loc s (τ),then lim t→∞ x τ (Q s , t) = O and the corresponding solution v(r) of (3) is a solutionwith fast decay. Using the flow it is possible to extend the local manifolds to globalmanifolds Wǫ u(τ) and W ǫ s (τ). As usual we commit the following abuse of notation:we denote by Wǫ u(τ) and W ǫ s (τ) just the branches of the manifolds that depart fromthe origin and get into R 2 + . From [30] we also know that the leaves are C1 and varycontinuously in the C 1 topology with respect to τ and ǫ. Observe that for ǫ = 0, bothin the regular and in the singular perturbation case, the manifold Wǫ u(τ) and W ǫ s(τ)coincide and are the image of the homoclinic trajectory. We fix a segment L which istransversal to W0 u(τ) ≡ W 0 s (τ) and which intersects it in a point, say U. Using a continuityargument, we deduce that, for ǫ > 0 small enough, Wǫ u(τ) and W ǫ s (τ) continueto cross L transversally in points ξ s (τ,ǫ) and ξ u (τ,ǫ) close to U. We want to find intersectionsQ between Wǫ u(τ) and W ǫ s(τ); then the trajectory xτ (Q, t) corresponds toa regular solution u(r) having fast decay. Then it is easily proved that x τ (Q, t) ∈ R 2 +for any t so it is a monotone decreasing G.S. with fast decay.Let us rewrite (8) as ẋ = f(x,τ + t,ǫ). From now on we restrict our attentionto the singularly perturbed system since the other can be treated similarly, see [30] and[14]. We define a Melnikov function which measures the distance with sign betweenξ s (τ,ǫ) and ξ u (τ,ǫ) along L.M(τ) = d dǫ[ξ − (τ,ǫ) − ξ + (τ,ǫ) ] ⌊ ǫ=0 ∧f(U,τ)where “∧” denotes the standard wedge product in R 2 . Then define{M(τ) for ǫ = 0h(τ,ǫ) =ξ − (τ,ǫ)−ξ + (τ,ǫ)ǫ∧ f(U,τ) for ǫ ̸= 0.We point out that the vector ξ − (τ,ǫ) − ξ + (τ,ǫ) belongs to the transversal segment L,so we have h(τ,ǫ) = 0 ⇐⇒ ξ − (τ,ǫ) − ξ + (τ,ǫ) = 0 for ǫ ̸= 0.Suppose M(τ 0 ) = 0 and M ′ (τ 0 ) ̸= 0, then, using the implicit function theorem,we construct a C 1 function ǫ → τ(ǫ) defined on a neighborhood of ǫ = 0, suchthat τ(0) = τ 0 , for which we have ξ − (τ(ǫ),ǫ) = ξ + (τ(ǫ),ǫ). Therefore we have ahomoclinic solution of the system (8).Following [30] and [14] we find that(20) M(τ) = −φ ′ (τ)φ(τ) − n p∫ +∞−∞|x 1 (t)| σdt = −Cφ ′ (τ)φ(τ) − n pσwhere x 1 (t) = (x 1 (t), y 1 (t)) is a homoclinic trajectory of (8) where φ ≡ 1, so C > 0 isa computable positive constant. Note that M(τ) is closely related to the first term in the