# zbMATH — the first resource for mathematics

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.

##### MSC:
 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