Homoclinic solutions of quasiperiodic Lagrangian systems
Let \(M^m\) be a smooth connected manifold, \(\omega\in \mathbb{R}^n\) a fixed nonresonant frequency vector, and \(L: P= TM\times \mathbb{T}^n\to \mathbb{R}\), \(L= L(x, v, \theta)\), a Lagrangian function which determines the Lagrangian quasiperiodic system (1) \({d\over dt} L_v(x, \dot x, \theta)- L_x(x, \dot x, \theta)= 0\), \(\dot\theta= \omega\). A quasiperiodic solution of this system is given by a \(C^1\) map \(f: \mathbb{T}^n\to M\) such that the corresponding map \(g: \mathbb{T}^n\to TM\), \(g(\theta)= (f(\theta), f'(\theta)\omega)\) is of class \(C^1\) and the graph of \(g\) is an invariant manifold of (1). A solution of the system (1), \(z= z(t)\in M\), \(\theta= \phi+ \omega t\in \mathbb{T}^n\), \(t\in \mathbb{R}\), (\(\phi= \theta\bigl|_{t= 0}\) is the initial phase) is called homoclinic to the quasiperiodic solution \(f\), if \(\text{dist}((z(t), \dot z(t))\), \(g(\phi+ \omega t))\to 0\) as \(t\to \pm\infty\). Using variational methods, the authors prove the existence of one or more solutions homoclinic to a quasiperiodic solution of the Lagrangian system (1) provided the Lagrangian \(L\) satisfies some additional technical assumptions. The result is applied to quasiperiodically perturbed Lagrangian systems.
scientific article; zbMATH DE number 775644