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A. Gonzalez-enriquez, A. Haro, R. De LA Llave's Singularity theory for non-twist KAM tori PDF

By A. Gonzalez-enriquez, A. Haro, R. De LA Llave

ISBN-10: 0821890182

ISBN-13: 9780821890189

During this monograph the authors introduce a brand new way to examine bifurcations of KAM tori with fastened Diophantine frequency in parameter-dependent Hamiltonian platforms. it truly is in line with Singularity thought of serious issues of a real-valued functionality which the authors name the capability. the capability is built in this kind of approach that: nondegenerate serious issues of the aptitude correspond to curl invariant tori (i.e. with nondegenerate torsion) and degenerate severe issues of the aptitude correspond to non-twist invariant tori. for that reason, bifurcating issues correspond to non-twist tori

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The main assumption is the existence of an approximate solution (ζ0 ; t0 , K0 ), with sufficiently small error, which satisfies a nondegeneracy condition. To state rigorously our KAM theorem, we first make this nondegeneracy condition explicit . 51 52 6. 3), respectively. 5) where M f is the local moment map of f . 2, we introduce the following general nondegeneracy condition. 1. 5). 6) Q(t,K) = ⎝ DK Z MK (I −Rω Tˆ ) Dt Z −DK Z MK Rω (B − Tˆ Rω B) where, for typographical simplicity, we have not written the dependence on (t, K), and DK Z and Dt Z are evaluated at (t, K).

Let f ∈ Symp(A0 , A), ω ∈ Rn and K : Tn → A0 . The torsion of K with respect to f and ω is the n×n-matrix given by ¯ , T = T (f,K) (f,K) 26 3. 7). ¯ is invertible, we say that K is twist with respect to f and ω. Otherwise If T (f,K) we say that K is non-twist with respect to f and ω. 8. Assume that An is endowed with the standard symplectic form ω 0 . Let U ⊂ Rn be open and simply connected. Let f0 : Tn × U → An be the integrable symplectomorphism with frequency map ω ˆ (y) = ω + ∇y A(y): f0 (x, y) = x+ω ˆ (y) .

Statement of the KAM theorem Many of our estimates will involve some quantities that depend in a polynomial way on specific bounds associated to a given (t, K) ∈ Ξ×Emb(Tnρ , B0 , C 1 ). 6), respectively. 2. Let ω ∈ Dn (γ, τ ), for some γ > 0 and τ ≥ n. Let m ≥ n > 0, r ≥ 2 be fixed integer numbers. Let Ξ ⊂ Rm be open and let f ∈ C r (Ξ, Symp(B0 , B, C 2 )). Let ρ0 > 0 be fixed and such that γ ρτ0 < 1. 1), with Z ∈ C r (Ξ × Emb(Tnρ0 /2 , B0 , C 0 ), Rm ). Assume that there exist positive constants β0 ≤ β1 ≤ · · · ≤ βr such that for ζ ∈ Ξ fixed and any ρ ∈ [ρ0 /2, ρ0 ] and for k = 0, .

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Singularity theory for non-twist KAM tori by A. Gonzalez-enriquez, A. Haro, R. De LA Llave


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