Integrability and strong normal forms for non-autonomous systems in a neighbourhood of an equilibrium

Integrability and strong normal forms for non-autonomous systems in a neighbourhood of an equilibrium

The paper deals with the problem of existence of a convergent “strong” normal form in the neighbourhood of an equilibrium, for a finite dimensional system of differential equations with analytic and time-dependent non-linear terms. The problem can be solved either under some non-resonance hypotheses on the spectrum of the linear part or if the non-linear term is assumed to be (slowly) decaying in time. This paper “completes” a pioneering work of Pustyl’nikov in which, despite under weaker non-resonance hypotheses, the nonlinearity is required to be asymptotically autonomous. The result is obtained as a consequence of the existence of a strong normal form for a suitable class of real-analytic Hamiltonians with non-autonomous perturbations.

本文研究具有解析且依赖于时间的非线性项的有限维微分方程系统在平衡点附近收敛的“强”范式的存在性问题。该问题可以在关于线性部分的谱的一些非共振假设下解决，或者如果假设非线性项在时间上（缓慢地）衰减也可解决。本文“完善”了普斯蒂尔尼科夫的一项开创性工作，在他的工作中，尽管非共振假设较弱，但要求非线性是渐近自治的。这个结果是作为一类具有非自治扰动的实解析哈密顿量存在强范式的一个推论而得到的。