Article
Mathematics
Xiaomei Yang, Junxiang Xu
Summary: In this paper, the persistence of multi-dimensional degenerate hyperbolic lower dimensional invariant tori with prescribed frequencies in reversible systems is proven using KAM techniques and topological degree theory.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Xiaomei Yang, Junxiang Xu
Summary: This paper investigates a class of degenerate reversible systems with Bruno non-degeneracy conditions, and proves the persistence of a lower dimensional invariant torus, whose frequency vector is only a small dilation of the prescribed one.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Xiaomei Yang, Junxiang Xu, Shunjun Jiang
Summary: This paper considers small perturbations of an integrable reversible system with a degenerated lower dimensional invariant torus. By employing KAM technique and stability theory, the persistence of the degenerate lower dimensional invariant torus is proved, without requiring extra conditions on the perturbations except for smallness. This result extends the partial result of Hamiltonian systems to reversible systems, as shown by Xu and You (Regul Chaotic Dyn 25(6):616-650, 2020).
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Tianqi Jing, Wen Si
Summary: This paper investigates the persistence of completely degenerate lower-dimensional invariant tori in a reversible system and proves the existence of such tori under certain conditions using the Kolmogorov-ArnoldMoser method. This is believed to be the first result on the persistence of lower-dimensional invariant tori in completely degenerate reversible systems.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Applied
Alex Haro, J. M. Mondelo
Summary: This paper presents a methodology for computing invariant tori in Hamiltonian systems by combining flow map methods, parameterization methods, and symplectic geometry. The methods reduce dimensionality and cost, and are applied to the computation of invariant tori and their invariant bundles around equilibrium points in the Restricted Three Body Problem. The invariant bundles are important for dynamical organization and have applications in space mission design.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2021)
Article
Mathematics, Applied
Ru Qu, DongFeng Zhang
Summary: This paper investigates the persistence of degenerate lower-dimensional tori in reversible systems and proves that under certain conditions, the system still possesses a lower-dimensional torus with a specific frequency.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Adrian P. Bustamante, Cristel Chandre
Summary: We investigate the critical surfaces for the existence of invariant tori in Hamiltonian systems with two and three degrees of freedom. Two methods, renormalization-group transformations and conjugation in configuration space, are used and compared to compute the critical surfaces. We discover the presence of cusps in the critical surface of three-dimensional invariant tori in Hamiltonian systems with three degrees of freedom, while the critical surface of two-dimensional invariant tori in Hamiltonian systems with two degrees of freedom is expected to be smooth.
SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Xiaocai Wang, Xiaofei Cao
Summary: This paper focuses on the persistence of lower-dimensional tori in reversible systems with hyperbolic-type degenerate equilibrium points under small perturbations. Through KAM iteration and the Topological degree theorem, the authors prove that the invariant torus with given frequency persists under small perturbations.
ACTA APPLICANDAE MATHEMATICAE
(2021)
Article
Mathematics, Applied
Xiaocai Wang, Xiaofei Cao, Xuqing Liu
Summary: This paper focuses on the persistence of lower-dimensional tori in reversible systems with high dimensional degenerate equilibrium under small perturbations. By applying an improved KAM iteration and Topological degree theory, we prove that the invariant torus with given frequency persists under small perturbations. Our result is a generalization of the work by X. Wang et al [On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2022)
Article
Mathematics, Applied
Dongfeng Zhang, Ru Qu
Summary: This paper focuses on the persistence of degenerate lower-dimensional invariant tori with a normal degenerate equilibrium point in reversible systems. Using the Herman method and the topological degree theory, it is proved that under certain conditions, the invariant torus persists under small perturbations, and this result also holds for reversible systems that are Gevrey smooth.
REGULAR & CHAOTIC DYNAMICS
(2022)
Article
Mathematics, Applied
Yingte Sun
Summary: This paper considers a class of nonlinear beam equations and proves the existence of many quasi-periodic solutions with non-resonant frequencies.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Xiaomei Yang, Junxiang Xu
Summary: This paper proves the persistence of degenerate hyperbolic lower-dimensional invariant tori in Hamiltonian systems that satisfy the Bruno non-degeneracy conditions and have a frequency vector that is a small dilation of the prescribed one. The proof is based on the stability of real roots of approximating real odd-order polynomials.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Junxiang Xu
Summary: In this paper, we develop KAM techniques to prove the persistence of lower dimensional elliptic-type degenerate invariant tori with prescribed frequencies in Hamiltonian systems. The proof is based on a formal KAM theorem and the Leray-Schauder continuation theorem.
Article
Mathematics
Xinyu Guan, Jianguo Si, Wen Si
Summary: This study investigates the existence of parabolic invariant tori for a class of quasi-periodically forced analytic skew-product maps. Different conditions are considered for different scenarios, and it is shown that parabolic invariant tori exist under certain conditions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
N. Kallinikos, R. S. MacKay, T. Syndercombe
Summary: This study applies a method to establish regions in phase space where no invariant tori pass through a given direction field. The implications for stable orbit locations of planets around binary stars are deduced. The lessons learned from this problem are expected to be useful for applications in other contexts such as flux surfaces for magnetic fields, guiding centre motion in magnetic fields, and classical models of chemical reaction dynamics.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics, Applied
Dongfeng Zhang, Junxiang Xu
Summary: This paper investigates a nonlinear quasi-periodic system and proves that it has a quasi-periodic solution with basic frequencies omega = (1, alpha), which tends to zero as epsilon approaches zero.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Xiaomei Yang, Junxiang Xu
Summary: This paper investigates a class of degenerate reversible systems with Bruno non-degeneracy conditions, and proves the persistence of a lower dimensional invariant torus, whose frequency vector is only a small dilation of the prescribed one.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2021)
Article
Mathematics, Applied
Hui Zhang, Junxiang Xu
Summary: This study deals with the singularly perturbed Choquard equation on the plane, with critical exponential growth in terms of the Trudinger-Moser inequality. Taking a local condition on the potential into consideration, the multiplicity and concentration of positive solutions are demonstrated using penalization technique and variational methods.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Mathematics
Hui Zhang, Junxiang Xu
Summary: This paper investigates the existence of ground states for the singularly perturbed Gross-Pitaevskii equation with small ε, and describes the concentration phenomena of ground states as ε approaches 0. The relationship between the number of positive solutions and the profile of the potential V is also explored.
REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS
(2021)
Article
Mathematics, Applied
Xiaomei Yang, Junxiang Xu, Shunjun Jiang
Summary: This paper considers small perturbations of an integrable reversible system with a degenerated lower dimensional invariant torus. By employing KAM technique and stability theory, the persistence of the degenerate lower dimensional invariant torus is proved, without requiring extra conditions on the perturbations except for smallness. This result extends the partial result of Hamiltonian systems to reversible systems, as shown by Xu and You (Regul Chaotic Dyn 25(6):616-650, 2020).
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Junxiang Xu, Qi Li, Jun Wang
Summary: This paper considers a class of 3-dimensional real analytic nonlinear quasi-periodic systems with a small perturbation parameter, and proves that the system has a small response solution for many sufficiently small parameters using the Leray-Schauder Continuation Theorem and the technique of outer parameter.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Dongfeng Zhang, Junxiang Xu
Summary: In this paper, we study the properties of a linear quasi-periodic system, including the analytic properties of quasi-periodic functions, the conditions of basic frequencies, and stability. We also apply these results to the study of quasi-periodic Schrodinger equations.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2022)
Article
Mathematics, Applied
Zhichao Ma, Junxiang Xu
Summary: This paper discusses quasi-periodic non-twist mappings with self-intersection property, which are dependent on a small parameter. Without assuming any twist condition, it is proven that for many sufficiently small parameters, the mapping has an invariant curve. As an application, this result is used to study the Lagrange stability of second-order systems.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Junxiang Xu
Summary: In this paper, we develop KAM techniques to prove the persistence of lower dimensional elliptic-type degenerate invariant tori with prescribed frequencies in Hamiltonian systems. The proof is based on a formal KAM theorem and the Leray-Schauder continuation theorem.
Article
Physics, Mathematical
Zhichao Ma, Junxiang Xu
Summary: In this paper, we prove the Lagrange stability of asymptotic linear Duffing equations under weaker nonlinear assumptions by employing Moser's non-twist theorem. We also avoid certain assumptions that are typically required to ensure the twist condition.
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Mathematics
Xiaomei Yang, Junxiang Xu
Summary: In this paper, the persistence of multi-dimensional degenerate hyperbolic lower dimensional invariant tori with prescribed frequencies in reversible systems is proven using KAM techniques and topological degree theory.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Xiaomei Yang, Junxiang Xu
Summary: This paper proves the persistence of degenerate hyperbolic lower-dimensional invariant tori in Hamiltonian systems that satisfy the Bruno non-degeneracy conditions and have a frequency vector that is a small dilation of the prescribed one. The proof is based on the stability of real roots of approximating real odd-order polynomials.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Song Ni, Junxiang Xu
Summary: This paper investigates a class of degenerate systems with a quasi-periodic perturbation of diophantine frequency in n dimensions. Assuming an equilibrium at the origin for the unperturbed system, which is degenerate in one direction, the study proves the existence of a small response solution for sufficiently small perturbations in the perturbed quasi-periodic system through the use of KAM iteration. The proof relies on the idea of reducibility and the introduction of parameters in the KAM technique.
QUALITATIVE THEORY OF DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Jia Li, Junxiang Xu
Summary: This paper investigates a linear almost periodic hamiltonian system with a constant matrix of different eigenvalues and an analytic almost periodic function. Without any non-degeneracy condition, it is proved that the linear system can be reduced for most sufficiently small parameter values through an almost periodic symplectic mapping.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2021)
Article
Mathematics, Applied
Yanling Shi, Junxiang Xu
Summary: This paper considers a one dimensional nonlinear wave equation with Dirichlet boundary condition, proving the existence of many quasi-periodic solutions with Liouvillean frequency. The proof is based on an infinite dimensional KAM Theorem.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2021)
