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By Richard Haberman

This article is designed for engineers, scientists, and mathematicians with a historical past in basic traditional differential equations and calculus.

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We will next explain how to introduce such a set and such a mapping. The assumption that Z is compactly embedded in X together with the exponential decay rate of eB0t actually implies that the radius of the essential spectrum of the period map Π0 (ω0 , 0) (defined by the linearized equation around p0 (t), see (17)) is strictly less than 1. In [13], this property is essential [13, Hypothesis (H3) of the introduction]. It implies that there exists 0 < ρ0 < 1, such that, outside the ball of center 0 and radius ρ0 in the complex plane, the spectrum σ (Π0 (ω0 , 0)) consists only in a finite number of eigenvalues.

As in [11], we consider the vertical mean value operator M ∈ L(L2 (Qε ), L2 (Ω)), given by, Mu = 1 εh ∀u ∈ L2 (Qε ). u(x, y)dy, Qε (135) We still denote M the corresponding operator from (L2 (Qε ))4 into (L2 (Ω))4 given by MU = ε1h Qε U(x, y)dy. We briefly recall the needed comparison results (proved in [10, 11]). An elementary computation shows [11] that, w − Mw L2 (Qε ) ≤Cε w H 1 (Qε ) , w − Mw H 1 (Qε ) ≤Cε w D(Aε ) , ∀w ∈ H 1 (Qε ), ∀w ∈ D(Aε ). (136) From the inequalities (136), we deduce that, for any W ∈ D(Bε ), W − MW Xε ≤ Cε W D(Bε ) , (137) and that, for any f ∗ ∈ L2 (Qε ), ∗ −1 ∗ A−1 ε f − Aε M f H 1 (Qε ) ≤ Cε f ∗ L2 (Qε ) .

Also we can apply it to perturbations of dynamical systems, which are not necessarily generated by an evolutionary equation. K. Hale and G. Raugel make that the modified Poincar´e method is in general a more powerful and better method than the integral equation method of the previous section. Here we want to compare the hypotheses of the modified Poincar´e method with the ones of the integral equation method in the frame of the semilinear equations (4) and (5). For sake of simplicity, we assume that f0 = fε ≡ f in (4) and (5).

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