Arnold diffusion for smooth convex systems of two and a half degrees of freedom

V. Kaloshin, K. Zhang, Nonlinearity 28 (2015) 2699–2720.

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Abstract
In the present note we announce a proof of a strong form of Arnold diffusion for smooth convex Hamiltonian systems. Let ${\mathbb T}^2$ be a 2-dimensional torus and B2 be the unit ball around the origin in ${\mathbb R}^2$ . Fix ρ > 0. Our main result says that for a 'generic' time-periodic perturbation of an integrable system of two degrees of freedom $H_0(p)+\varepsilon H_1(\theta,p,t),\quad \ \theta\in {\mathbb T}^2,\ p\in B^2,\ t\in {\mathbb T}={\mathbb R}/{\mathbb Z}$ , with a strictly convex H0, there exists a ρ-dense orbit (θε, pε, t)(t) in ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ , namely, a ρ-neighborhood of the orbit contains ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ . Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are the usage of crumpled normally hyperbolic invariant cylinders from [9], flower and simple normally hyperbolic invariant manifolds from [36] as well as their kissing property at a strong double resonance. This allows us to build a 'connected' net of three-dimensional normally hyperbolic invariant manifolds. To construct diffusing orbits along this net we employ a version of the Mather variational method [41] equipped with weak KAM theory [28], proposed by Bernard in [7].
Publishing Year
Date Published
2015-06-30
Journal Title
Nonlinearity
Volume
28
Issue
8
Page
2699-2720
IST-REx-ID

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Kaloshin V, Zhang K. Arnold diffusion for smooth convex systems of two and a half degrees of freedom. Nonlinearity. 2015;28(8):2699-2720. doi:10.1088/0951-7715/28/8/2699
Kaloshin, V., & Zhang, K. (2015). Arnold diffusion for smooth convex systems of two and a half degrees of freedom. Nonlinearity, 28(8), 2699–2720. https://doi.org/10.1088/0951-7715/28/8/2699
Kaloshin, Vadim, and K Zhang. “Arnold Diffusion for Smooth Convex Systems of Two and a Half Degrees of Freedom.” Nonlinearity 28, no. 8 (2015): 2699–2720. https://doi.org/10.1088/0951-7715/28/8/2699.
V. Kaloshin and K. Zhang, “Arnold diffusion for smooth convex systems of two and a half degrees of freedom,” Nonlinearity, vol. 28, no. 8, pp. 2699–2720, 2015.
Kaloshin V, Zhang K. 2015. Arnold diffusion for smooth convex systems of two and a half degrees of freedom. Nonlinearity. 28(8), 2699–2720.
Kaloshin, Vadim, and K. Zhang. “Arnold Diffusion for Smooth Convex Systems of Two and a Half Degrees of Freedom.” Nonlinearity, vol. 28, no. 8, IOP Publishing, 2015, pp. 2699–720, doi:10.1088/0951-7715/28/8/2699.

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