By A. Jaffe (Chief Editor)
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Additional info for Communications in Mathematical Physics - Volume 189
In order to facilitate the reading, the proof has been split in several parts. 1. Rearranging the initial Hamiltonian. Let us start by expanding K1 in Taylor series with respect to x and y, aij (θ, ε)xi y j , K1 (x, y, θ, ε) = i+j≥2 where aij (θ, ε) is a periodic function of θ, that can be expanded in Fourier series, akij (ε)ek·θ aij (θ, ε) = √ −1 . k∈Z 2 Moreover, the following bounds hold: aij ρ ≤ A1 , ri+j |akij | ≤ A1 −|k1 |ρ1 −|k2 |ρ2 e , ri+j 0 < ε ≤ ε0 . Let us write K1 in the following form: akij (ε)(xy)j xi−j ek·θ K1 (x, y, θ, ε) = i+j≥2 √ −1 .
189, 35 – 71 (1997) Communications in Mathematical Physics c Springer-Verlag 1997 Exponentially Small Splitting of Separatrices Under Fast Quasiperiodic Forcing ` Amadeu Delshams1 , Vassili Gelfreich23 , Angel Jorba2 , Tere M. Seara1 1 Departament de Matem` atica Aplicada I, Universitat Polit`ecnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain. es 2 Departament de Matem` atica Aplicada i An`alisi, Universitat de Barcelona, Gran via 585, 08007 Barcelona, Spain. Petersburg Academy of Aerospace Instrumentation, Bolshaya Morskaya 67, 190000, St.
In the normal form coordinates the stable whisker is given by the equation X = 0 and the unstable one by the equation Y = 0. Let λ = H (0, ε). 1). 4) T ≥ T0 . 5) 4. Extension Theorem By the Normal Form Theorem, the unstable manifold is O εp−1 -close to the unperturbed separatrix. The next theorem extends this local approximation to a global one. , up to a distance to the singularity T = ±π/2 of the same order as the loss of domain in the angular variables. Besides, the extension time t + T will be chosen big enough in order that the unperturbed separatrix reaches again the domain of convergence of the normal form.
Communications in Mathematical Physics - Volume 189 by A. Jaffe (Chief Editor)