arXiv subjects

Jiamin Xing

Publications and source records attributed to Jiamin Xing.

3 recordsLinked to original sources

arXiv2025

Averaging method for quasi-periodic response solutions

In this paper, we present an averaging method for obtaining quasi-periodic response solutions in perturbed, real analytic, quasi-periodic systems with Diophantine frequency vectors. Under the assumptions that the averaged system possesses a non-degenerate equilibrium and that the eigenvalues of its linearized matrix are pairwise distinct, we show that the original system admits a quasi-periodic response solution for parameters in a Cantorian set. The proof relies on KAM techniques. It is worth mentioning that our results do not require the equilibrium to be hyperbolic, meaning that the eigenvalues of the linearized matrix of the averaged system may be purely imaginary. Furthermore, the proposed averaging method is applicable to second-order systems, and a higher-order averaging framework is also established.

math.DS↗

arXiv2023

Global Weinstein Type Theorem on Multiple Rotating Periodic Solutions for Hamiltonian Systems

This paper concerns the existence of multiple rotating periodic solutions for $2n$ dimensional convex Hamiltonian systems. For the symplectic orthogonal matrix $Q$, the rotating periodic solution has the form of $z(t+T)=Qz(t)$, which might be periodic, anti-periodic, subharmonic or quasi-periodic according to the structure of $Q$. It is proved that there exist at least $n$ geometrically distinct rotating periodic solutions on a given $Q$ invariant convex energy surface under a pinching condition. As a result, it is proved that if the symmetric energy surface admits a nonsymmetric periodic solution, it has infinitely many periodic orbits. In order to prove the result, we introduce a new index on rotating periodic orbits.

math.DS↗

arXiv2018

Rotating Quasi-periodic Solutions of Second Order Hamiltonian Systems with Sub-quadratic Potential

This paper concerns the existence of multiple rotating quasi-periodic solutions for second order Hamiltonian systems with sub-quadratic potential. Such solutions have the form $x(t+T)=Qx(t)$ for some orthogonal matrix $Q$. To deal with such quasi-periodic solutions, we introduce the $\mathcal{Q}(s)$ index which is a development of the well known $S^1$ index. Applying the $\mathcal{Q}(s)$ index, we give an estimate of the number for rotating quasi-periodic orbits with a fixed period.

math.DS↗