Dynamics around the double resonance (Q2400463)

A Lagrangian system with two degrees of freedom of the following form is considered: \[ L(x,\dot x,t)= {1\over 2} \langle A\dot x,\dot x\rangle+ V(x)+ \sqrt{\varepsilon} R_\varepsilon(x,\dot x,\theta(t)), \] where \((x,\theta)\in \mathbb{T}^2\times \mathbb{T}\), \(A\) is a \(2\times 2\) positive definite matrix of constants, \(V\in C^r(\mathbb{T}^2,\mathbb{R})\) with \(r\geq 5\), \(R_\varepsilon\in C^{r-1}(\mathbb{T}^2\times \mathbb{R}^2\times \mathbb{T},\mathbb{R})\) is a small perturbation and either \(\theta(t)= t\) or \(\theta(t)={t\over Q\sqrt{\varepsilon}}\), where \(Q\) is a positive constant. It is further assumed that \[ \sqrt{\varepsilon} \| R_\varepsilon\|_{C^{r-1}}\leq C\varepsilon^k\text{ for }|\dot x|< \varepsilon^{-k}\text{ and }(x,\theta)\in \mathbb{T}^2\times\mathbb{T}\text{ for }k\in \Biggl(0, {1\over 2}\Biggr). \] The author considers conditions that enable two different Aubry sets to be connected by orbits of the Euler-Lagrange flow. The way this is accomplished -- that two Aubry sets \(A(c')\) and \(A(c'')\) are dynamically connected -- is to establish a continuous path in the first cohomology space \(\Gamma:[0,1]\to H'(M,\mathbb{R})\) that joins \(c'\) to \(c''\) with \(\Gamma(0)= c'\) and \(\Gamma(1)= c''\). (\(M\) is the closed manifold representing the phase space of the system.) Once two cohomology classes \(c'\) and \(c''\) are connected by such a transition chain, a global connecting orbit is constructed using a sequence of local connecting orbits. The author's main result is that there is a residual set in \(C^r(\mathbb{T}^2,\mathbb{R})\) for which functions \(V\) in that set permit the existence of a transition chain. Then, by applying his results to the nearly integrable Hamiltonian \(H(p,q)= h(p)+ \varepsilon P(p,q)\) with \((p,q)\in\mathbb{R}^3\times \mathbb{T}^3\), \(\partial^2 h(p)\) positive definite, and both \(h\) and \(P\), \(C^r\) differentiable with \(r\geq 6\), the author shows that there is a transition chain passing through a small neighborhood of double resonances.