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The above formula implies that the zeros of the Melnikov function form two lines on the torus. As already noticed by C. Sim´o [Sim94], the averaged slopes of those lines approach γ when ε → 0. 3. Normal Form and Local Manifolds As we have seen during the analysis of the Melnikov function, the size of the splitting depends essentially on the widths of the analyticity strip (r1 , r2 ) of the angular variables θ1 , θ2 , as well as on the width of the analyticity strip of the separatrix (x0 (t), y0 (t)).
Delshams, V. Gelfreich, A. 2). Suppose for a moment that the mean value of the function H u (T, θ1 , θ2 ; ε) with respect to the angle variables is equal to zero. 2)√and of Lemma √4 of Sect. 7. In the last lemma one√has to replace r1 and r2 by r1 − ε and r2 − ε, respectively, and take ρ = π/2 − ε. 5). The exponentially small upper bound for the error is proved for p > 5/2. So what we have for 5/2 < p ≤ 3 is a very sharp upper bound for the splitting. To use Lemma 4 we have to prove that the mean value of the function H u (T, θ1 , θ2 ; ε) with respect to the angle variables is equal to zero.
Gelfreich, A. 3) and continued by 2 log γ-periodicity. The following lemma gives the exponentially small upper bound for the function F for real values of the variables. √ Lemma 4. Let F satisfy the conditions of Lemma 3. 4) on the real values of its arguments. The constant depends continuously on r1 and r2 on r1 > 0 and r2 > 0. Proof. 1). 5) we separate it into two parts. The first one contains non-resonant terms, that is, all the terms such that |k1 + γk2 | ≥ 1/2. We easily obtain the upper estimate e−|k1 |r1 −|k2 |r2 −ρ|k1 +γk2 |/ε |k1 +γk2 |≥1/2 < e−ρ/(2ε) e−|k1 |r1 −|k2 |r2 = |k1 |+|k2 |=0 2(e−r1 + er2 − e−r1 −r2 )e−ρ/(2ε) .