Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems (Q2918666)

The author considers Hamiltonian systems with Hamiltonian of the form \(H= H_0+\varepsilon H_1 + O(\varepsilon^2)\), where \(H_0\) is integrable and satisfies suitable hypothesis -- in particular, it must be a priori unstable -- and proves that for a \(C^r\)-generic set of time-periodic perturbations \(H_1\), \(r\in \mathbb{N}\cup\{\infty,\omega\}\) large enough, if \(\varepsilon\) is small enough, the system \(H\) exhibits Arnold diffusion, that is, a drift in some variables of the system which is independent of the perturbative parameter~\(\varepsilon\). This drift takes place in small but independent-of-\(\varepsilon\) regions free of strong resonances.NEWLINENEWLINEThe Hamiltonian under consideration is of the form NEWLINE\[NEWLINE H(y,x,v,u,t,\varepsilon) = H_0(y,v,u)+ \varepsilon H_1(y,x,v,u,t) + \varepsilon^2 H_2(y,x,v,u,t,\varepsilon) NEWLINE\]NEWLINE where \(x\in \mathbb{T}^n\), \(t\in \mathbb{T}\), \(y \in \overline{D} \subset \mathbb{R}^n\), being \(D\) is an open subset with compact closure, \(n\geq 1\) and \((u,v) \in U \subset \mathbb{R}^2\), with \(U\) open.NEWLINENEWLINEThe unperturbed integrable Hamiltonian \(H_0\) must be of the form NEWLINE\[NEWLINE H_0(y,v,u) = F(y,f(v,u)) NEWLINE\]NEWLINE and the function \(f\) must have a non-degenerate saddle point \((v,u) = (0,0)\) unique on a compact connected component of \( \gamma = \{ (v,u) \in U\mid f(v,u) = f(0,0)\}. \) The separatrices of the Hamiltonian system \((U, dv\wedge du,f)\) are precisely the set~\(\gamma\) and form a figure-eight, \(\gamma = \gamma^+\cup \gamma^-\). Moreover, defining \(E(y) = H_0(y,0,0)\), the matrix \(\partial^2 E / \partial y^2\) must be non-degenerate for all \(y\in \overline{D}\).NEWLINENEWLINEThe region \(Q\) where the diffusion takes place is defined as follows. Let \(\nu = \partial E / \partial y\) and \(\overline{\nu} = (-\nu,1)^\top\), the frequency vector. Given a constant \(C>0\) and \(k\in \mathbb{Z}^{n+1}\), \(k\neq 0\), a resonance region \(S_0^k=\{ y\in D\mid \langle k,\overline{\nu}(y)\rangle = 0\}\) is called \(C\)-strong if \(|k| \leq C\). Then \(Q\) is a connected domain whose closure lies in a connected component of \(D\setminus (\cup_{0<|k|\leq C} S_0^k)\).NEWLINENEWLINEFinally, an assumption on \(H_1\) is needed, which is formulated in terms of the splitting potential obtained from \(H_1\) when one writes the separatrix map associated to the loop \(\gamma\). This condition is satisfied for an open and dense set in \(C^r\)NEWLINENEWLINEIf all the above hypotheses are satisfied, then, for any \(\alpha \geq \alpha_0(n,r)\) there exist \(\varepsilon_0, c_d, c_v>0\) such that for any \(0<\varepsilon < \varepsilon_0\) and any broken line \(\chi \subset Q\), with intervals of length~\(\varepsilon^{1/8}\), the Hamiltonian~\(H\) possesses a trajectory \((y(t),x(t),v(t),u(t))\), \(t\in [0,T]\) such that the curve \(y(t)\), \(t\in[0,T]\) lies in a \(c_d |\log\varepsilon|^{\alpha} \varepsilon^{1/4}\)-neighborhood of \(\chi\). Moreover, the drifting time~\(T\) satisfies \(c_v T \varepsilon/|\log \varepsilon| < \text{length}(\chi)\).NEWLINENEWLINEThe proof of the theorem is based on the use of the separatrix map; see [the author, Physica D 116, No. 1--2, 21--43 (1998; Zbl 1038.37049)]).