Arnold diffusion in Hamiltonian systems: a priori unstable case. (English) Zbl 1179.37081

The authors consider the Hamiltonian \[ H(u,v,t)= h_1(p)+ h_2(x,y)+ P(u,v,t), \] where \(u=(q,x)\), \(v= (p,y)\), \(p\in\mathbb{R}\), \(q\in\mathbb{T}\), \((x,y)\in \mathbb{T}^N\times \mathbb{R}^n\) and \(P\) is a time-one periodic perturbation. The following assumptions about \(H\) are made:
(1) \(H\in C^r(\mathbb{T}^{n+1}, \mathbb{R}^{n+1},\mathbb{T})\);
(2) \(h_1+ h_2\) is a convex function in \(v\), so the Hessian \(\partial^2_{vv}(h_1+ h_2)\) is positive definite;
(2) \(H\) is a priori hyperbolic in the sense that the Hamiltonian flow \(\Phi^t_{h_2}\), determined by \(h_2\), has a nondegenerate fixed point at \((x,y)= (0,0)\) and the function \(h_2(x,0): \mathbb{T}^n\to \mathbb{R}\) has a strict maximum at \(x= 0\text{\,mod\,}2\pi\). It is also assumed that \(h_2(0,0)= 0\).
The authors’ main result is the following: If the Hamiltonian \(H\) satisfies the above assumptions, then there exists \(\varepsilon> 0\) and \(k>0\) and a residual set \(S_{\varepsilon,k}\subset B_{\varepsilon,k}\) such that for each \(P\in S_{\varepsilon,k}\) there exist orbits of the Hamiltonian flow that connect the region with \(p<A\) to the region with \(p> B\) for any two real \(A\) and \(B\) with \(A<B\). \(B_{\varepsilon,k}\) is the ball centered at the origin with radius \(\varepsilon\) in the function space \[ C^r(\{u,v,t)\in \mathbb{T}^{n+1}\times \mathbb{R}^{n+1}\times \mathbb{T}:\| v\|\leq K\}\to \mathbb{R}. \] The result is a generalization of Arnold’s result from 1964, where \(n= 1\). Thus Arnold diffusion exists in a prioric hyperbolic and time-periodic Hamiltonian systems with multiple degrees of freedom.
The authors employ variational arguments to construct diffusion orbits and apply Mather’s theory to a modified Lagrangian.


37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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