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Homoclinic solutions of quasiperiodic Lagrangian systems. (English) Zbl 0827.34037
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.

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
34C27 Almost and pseudo-almost periodic solutions to ordinary differential equations