Perturbations of stable invariant tori for hamiltonian systems

L. H. Eliasson

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For a positive integer n and R>0, we set $B_{R}^{n} = \{x \in ℝ^{n} | {∥x∥}_{\infty} < R\}$ . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H j) of the completely integrable Hamiltonian $h (r) = \frac{1}{2} r_{1}^{2} + . . . \frac{1}{2} r_{n - 1}^{2} + r_{n}$ on $𝕋^{n} \times B_{R}^{n}$ , with unstable orbits for which we can estimate the time of drift in the action space. These functions H j are analytic on a fixed complex neighborhood V of $𝕋^{n} \times B_{R}^{n}$ , and setting $ε_{j} : = {∥h - H_{j}∥}_{C^{0} (V)}$ the time of drift of these orbits is smaller than (C(1/ɛ j)1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly...

Whiskered and low dimensional tori in nearly integrable Hamiltonian systems.

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Stability and instability for Gevrey quasi-convex near-integrable hamiltonian systems

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Galerkin averaging method and Poincaré normal form for some quasilinear PDEs

Dario Bambusi (2005)

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We use the Galerkin averaging method to construct a coordinate transformation putting a nonlinear PDE in Poincaré normal form up to finite order. We also give a rigorous estimate of the remainder showing that it is small as a differential operator of very high order. The abstract theorem is then applied to a quasilinear wave equation, to the water wave problem and to a nonlinear heat equation. The normal form is then used to construct approximate solutions whose difference from true...