Whiskered and low dimensional tori in nearly integrable Hamiltonian systems.

de la Llave, R.; Wayne, C.E.

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Displaying similar documents to “Whiskered and low dimensional tori in nearly integrable Hamiltonian systems.”

Diffusion times and stability exponents for nearly integrable analytic systems

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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...

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The response of an oscillator to a small amplitude periodic excitation is discussed. In particular, sufficient conditions are formulated for the perturbed oscillator to have an invariant torus in the phase cylinder. When such an invariant torus exists, some perturbed solutions are in the basin of attraction of this torus and are thus entrained to the dynamical behavior of the perturbed system on the torus. In particular, the perturbed solutions in the basin of attraction of the invariant...

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