arXiv subjects

Jiayin Du

Publications and source records attributed to Jiayin Du.

4 recordsLinked to original sources

arXiv2024

Melnikov's persistence for completely degenerate Hamiltonian systems

In this paper, we study the Melnikov's persistence for completely degenerate Hamiltonian systems with the following Hamiltonian \begin{equation*} H(x,y,u,v)=h(y)+g(u,v)+\varepsilon P(x,y,u,v),~~~(x,y,u,v)\in \mathbb{T}^n\times{G}\times \mathbb{R}^d\times \mathbb{R}^d, \end{equation*} where $n\geq2$ and $d\geq1$ are positive integers, $G\subset\mathbb{R}^n$, $g=o(|u|^2+|v|^2)$ admits complete degeneracy and certain transversality, and $\varepsilon P$ is the small perturbation. This is a try in studying lower-dimensional invariant tori in the normal complete degeneracy. Under Rüssmann-like non-degenerate condition and transversality condition, we apply the homotopy invariance of topological degree to remove the first order terms about $u$ and $v$ and employ the quasi-linear KAM iterative procedure to derive the persistence of lower-dimensional invariant tori.

math.DS↗

arXiv2023

Kolmogorov's Theorem for Degenerate Hamiltonian Systems with Continuous Parameters

In this paper, we investigate Kolmogorov type theorems for small perturbations of degenerate Hamiltonian systems. These systems are index by a parameter $ξ$ as $ H(y,x,ξ) = \langleω(ξ),y\rangle + \varepsilon P(y,x,ξ,\varepsilon) $ where $\varepsilon>0$. We assume that the frequency map, $ω$, is continuous with respect to $ξ$. Additionally, the perturbation function, $P(y,x,\cdot, \varepsilon)$, maintains Hölder continuity about $ξ$. We prove that persistent invariant tori retain the same frequency as those of the unperturbed tori, under certain topological degree conditions and a weak convexity condition for the frequency mapping. Notably, this paper presents, to our understanding, pioneering results on the KAM theorem under such conditions-with only assumption of continuous dependence of frequency mapping $ω$ on the parameter.

math.DS↗

arXiv2023

An Infinite-dimensional KAM Theorem with Normal Degeneracy

In this paper, we consider a classical Hamiltonian normal form with degeneracy in normal direction. In previous results, one needs to assume that the perturbation satisfies certain non-degenerate conditions in order to remove the degeneracy in the normal form. In stead of that, we introduce a topological degree condition and a weak convexity condition, which are easy to be verified, and we prove the persistence of lower dimensional tori without any restriction on perturbation but only smallness and analyticity.

math.DS↗

arXiv2023

KAM theorem on modulus of continuity about parameter

In this paper, we study the Hamiltonian systems $ H\left( {y,x,ξ,\varepsilon } \right) = \left\langle {ω\left( ξ\right),y} \right\rangle + \varepsilon P\left( {y,x,ξ,\varepsilon } \right) $, where $ ω$ and $ P $ are continuous about $ ξ$. We prove that persistent invariant tori possess the same frequency as the unperturbed tori, under certain transversality condition and weak convexity condition for the frequency mapping $ ω$. As a direct application, we prove a KAM theorem when the perturbation $P$ holds arbitrary Hölder continuity with respect to parameter $ ξ$. The infinite dimensional case is also considered. To our knowledge, this is the first approach to the systems with the only continuity in parameter beyond Hölder's type.

math.DS↗