RENDICONTI DEL SEMINARIO MATEMATICO

74 M. FrancaW u (τ)ON8XN 9ẋ=0E(τ)W s (τ)N 1N300000000000001111111111111000000000000011111111111110000000000000111111111111100000000000001111111111111000000000000011111111111110000000000000111111111111100000000000001111111111111N00000000000001111111111111600000000000001111111111111000000000000011111111111110000000000000111111111111100000000000001111111111111N4N7N 2U−N5U +ξ u (τ)sξ (τ)YFigure 3: A sketch of the set E(τ), when k(r) = K(r ǫ ) has 5 maxima and 4 minima.The solid line represents B(τ), and it is obtained joining segments of W u (τ) (dottedline), and of W s (τ) (dashed line).First observe that if Q τ ∈ Wǫ u(τ)∩W ǫ s(τ) then xτ (Q τ , t) ∈ Wǫ u(τ +t)∩W ǫ s(τ +t) for any t. So the number of intersection between Wǫ u(τ) and W ǫ s (τ) does not dependon τ. Therefore there are 9 functions τ i (ǫ) such that ξ u (τ(ǫ),ǫ) = ξ s (τ(ǫ),ǫ) fori = 1,··· , 9 and 9 points N i (τ) of intersection between stable and unstable manifolds.We denote by N 1 (τ), the first point met following Wǫ s(τ) from the origin towards R2 + ,by N 2 (τ) the second, and so on. Let us denote by B 0 (τ) the branch of Wǫ s (τ) betweenthe origin and N 1 (τ), by B 1 (τ) the branch of Wǫ u(τ) between N1 (τ) and N 2 (τ), byB 2 (τ) the branch of Wǫ s(τ) between N2 (τ) and N 3 (τ), and so on till the branch ofWǫ u(τ) between N9 (τ) and the origin which is denoted by B 9 (τ). Finally we denote byB(τ) = ∪i=0 9 Bi (τ), and by E(τ) the bounded open subset enclosed by B(τ). The keyobservation is that B(τ) is contained in R 2 + for any τ, and in {x| y < 0 < x} when φis uniformly positive, see [14] for a detailed proof.Observe that E(τ)\ (Wǫ u(τ) ∪ W ǫ s (τ)) contains uncountably many points andtake Q in it. The trajectory x τ (Q, t) is forced to stay in the interior of E(τ + t) forany t, therefore it corresponds to a S.G.S. with slow decay. With a careful analysison the phase portrait it is possible to find points Q ∈ B(τ)\ Wǫ s(τ) such that xτ (Q, t)is forced to stay in the interior of E(τ + t) for any t > 0, and P ∈ B(τ)\ Wǫ s(τ)such that x τ (P, t) has to cross the y axis for some t > 0. Therefore they correspondrespectively to G.S. with slow decay and to crossing solutions. Analogously we findQ, P ∈ B(τ)\ Wǫ u(τ) such that xτ (Q, t) ∈ E(τ + t) for any t < 0 and x τ (P, t) hasto cross the y axis for some t < 0, which correspond respectively to S.G.S. with fastdecay and to solutions of the Dirichlet problem in the exterior of a ball, see [14] for