We consider time-dependent perturbations of integrable and near-integrable Hamiltonians. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of biasymptotically quasi-periodic solutions. That is, orbits converging to some quasi-periodic solutions in the future (as \(t\to+\infty\)) and the past (as \(t\to-\infty\)).

Concerning the proof, thanks to the implicit function theorem, we prove the existence of a family of orbits converging to some quasi-periodic solutions in the future and another family of motions converging to some quasi-periodic solutions in the past. Then, we look at the intersection between these two families when \(t=0\). Under suitable hypotheses on the Hamiltonian’s regularity and the perturbation’s smallness, it is a large set, and each point gives rise to biasymptotically quasi-periodic solutions.

我们考虑可积和近可积哈密顿量的时间相关扰动。假设随着时间趋于无穷大，扰动以多项式方式快速衰减，我们证明了偏渐近准周期解的存在。也就是说，轨道收敛于未来（如\(t\to+\infty\)）和过去（如\(t\to-\infty\)）的一些准周期解。

在证明方面，借助隐函数定理，我们证明了存在一族在未来收敛于某些准周期解的轨道和另一族在过去收敛于某些准周期解的运动。然后，我们看看当\(t=0\)时这两个族之间的交集。在哈密顿量的正则性和扰动的小性的适当假设下，它是一个大集合，并且每个点都会产生偏渐近的准周期解。