Existence of divergent Birkhoff normal forms of Hamiltonian functions

Existence of Divergent Birkhoff Normal Forms of Hamiltonian Functions

where κ = 0, . . . , n, and λj is pure imaginary precisely when 1 ≤ j ≤ κ, and λ1, −λ1, . . . , λn, −λn are eigenvalues of Hzz(0)J with z = (x, y) and Jxj = yj = −J yj. One says that λ1, . . . , λn are non-resonant, if λ ·α ≡ λ1α1 + · · ·+λnαn 6= 0 for all multi-indices of integers α 6= 0. The Birkhoff normal form says that under the non-resonance condition on λ, there is a formal symplectic tr...

Birkhoff Normal Form and Hamiltonian PDEs

These notes are based on lectures held at the Lanzhou university (China) during a CIMPA summer school in july 2004 but benefit from recent devellopements. Our aim is to explain some perturbations technics that allow to study the long time behaviour of the solutions of Hamiltonian perturbations of integrable systems. We are in particular interested with stability results. Our approach is centere...

Birkhoff Normal Forms and KAM Theory for Gumowski-Mira Equation

By using the KAM theory we investigate the stability of equilibrium solutions of the Gumowski-Mira equation: xn+1=(2ax n)/(1+x n2)-xn-1, n=0,1,…, where x-1, x0∈(-∞,∞), and we obtain the Birkhoff normal forms for this equation for different equilibrium solutions.

Birkhoff Normal Forms in Semi-classical Inverse Problems

The purpose of this note is to apply the recent results on semi-classical trace formulæ [17], and on quantum Birkhoff normal forms for semi-classical Fourier Operators [12] to inverse problems. We show how the classical Birkhoff normal form can be recovered from semi-classical spectral invariants. In fact the full quantum Birkhoff normal form of the quantum Hamiltonian near a closed orbit, and ...

Symmetry and integrability in Hamiltonian normal forms