Normal forms, stability and splitting of invariant manifolds II. Finitely differentiable Hamiltonians
Tóm tắt
This paper is a sequel to “Normal forms, stability and splitting of invariant manifolds I. Gevrey Hamiltonians”, in which we gave a new construction of resonant normal forms with an exponentially small remainder for near-integrableGevrey Hamiltonians at a quasiperiodic frequency, using a method of periodic approximations. In this second part we focus on finitely differentiable Hamiltonians, and we derive normal forms with a polynomially small remainder. As applications, we obtain a polynomially large upper bound on the stability time for the evolution of the action variables and a polynomially small upper bound on the splitting of invariant manifolds for hyperbolic tori.
Tài liệu tham khảo
Abraham, R. and Robbin, J., Transversal Mappings and Flows, New York: Benjamin, 1967.
Bounemoura, A. and Fischler, S., A Diophantine Duality Applied to the KAM and Nekhoroshev Theorems, Math. Z., 2013 (to appear).
Bounemoura, A. and Niederman, L., Generic Nekhoroshev Theory without Small Divisors, Ann. Inst. Fourier (Grenoble), 2012, vol. 62, no. 1, pp. 277–324.
Bounemoura, A., Nekhoroshev Estimates for Finitely Differentiable Quasi-Convex Hamiltonians, J. Differential Equations, 2010, vol. 249, no. 11, pp. 2905–2920.
Bounemoura, A., Optimal Stability and Instability for Near-Linear Hamiltonians, Ann. Henri Poincaré, 2012, vol. 13, no. 4, pp. 857–868.
Bounemoura, A., Normal Forms, Stability and Splitting of Invariant Manifolds: I. Gevrey Hamiltonians, Regul. Chaotic Dyn., 2013, vol. 18, no. 3, pp. 237–260.
Delshams, A. and Huguet, G., Geography of Resonances and Arnold Diffusion in a priori Unstable Hamiltonian Systems, Nonlinearity, 2009, vol. 22, no. 8, pp. 1997–2077.
Lochak, P., Marco, J.-P., and Sauzin, D., On the Splitting of Invariant Manifolds in Multidimensional Near-Integrable Hamiltonian Systems, Mem. Amer. Math. Soc., 2003, vol. 163, no. 775, 145 pp.