On quasi-periodic perturbations of elliptic equilibrium points
On quasi-periodic perturbations of elliptic equilibrium points
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DOI：
10.1137/s0036141094276913
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发表时间：
1996-11-01
影响因子：
2
通讯作者：
Simo, C
中科院分区：
文献类型：
--
作者：
Jorba, A;Simo, C
这项工作重点关注普通 dx = (A + epsilon Q(t, epsilon))x + epsilon g(t, epsilon) + h(x, t, epsilon) 的准周期时间相关扰动，其中 A 是椭圆形，h 是 O(x(2))。结果表明，在关于 epsilon 的解析性、非共振和非简并性的适当假设下，存在一个 Cantorian 集合 epsilon，使得当所有 epsilon 都是 epsilon 的一个元素时，存在一个准周期解，使得当 epsilon 运动时它会转向航空。该准周期解具有与扰动相同的一组基本频率。此外，集合 [0, epsilon(0)] \epsilon in [0, epsilon(0)] 的相对度量在 epsilon(0) 中呈指数小。还考虑了g = 0，h = 0（准周期Floquet定理）的情况。最后，研究了哈密顿量情况。在这种情况下，平衡点附近的大多数不变环面不会被破坏，而只会以准周期的方式轻微变形和“震动”。这种准周期性“振动”具有与扰动相同的基本频率。
This work focuses on quasi-periodic time-dependent perturbations of ordinary dx = (A + epsilon Q(t, epsilon))x + epsilon g(t, epsilon) + h(x, t, epsilon), where A is elliptic and h is O(x(2)). It is shown that, under suitable hypothesis of analyticity, nonresonance and nondegeneracy with respect to epsilon, there exists a Cantorian set epsilon such that fur all epsilon is an element of epsilon there exists a quasi-periodic solution such that it goes to aero when epsilon does. This quasi-periodic solution has the same set of basic frequencies as the perturbation. Moreover, the relative measure of the Set [0, epsilon(0)] \epsilon in [0, epsilon(0)] is exponentially small in epsilon(0). The case g = 0, h = 0 (quasi-periodic Floquet. theorem) is also considered.Finally, the Hamiltonian case is studied. In this situation, most of the invariant tori that are near the equilibrium point are not destroyed but only slightly deformed and ''shaken'' in a quasi-periodic way. This quasi-periodic ''shaking'' has the same basic frequencies as the perturbation.