Article
Physics, Mathematical
Zhichao Ma, Junxiang Xu
Summary: In this paper, a one-dimensional completely degenerate oscillator under an analytically e-dependent quasi-periodic perturbation is studied, where the frequencies satisfy a Diophantine condition. By the KAM method, two possible results are shown: 1. For all sufficiently small e and all initial values ? E T1, there exists a family of analytically (e,?)-parameterized response solutions, which corresponds to the persistence of the resonant Lagrangian torus of the equivalent Hamiltonian system. 2. For all sufficiently small e, there exists a response solution, and for an uncountable number of sufficiently small e, there exists another response solution. In this case, the resonant Lagrangian torus of the equivalent Hamiltonian system is destroyed and it splits into a hyperbolic or hyperbolic-type degenerate lower dimensional torus for all sufficiently small e, and another (possibly elliptic) lower dimensional torus for an uncountable number of sufficiently small e.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
Qi Li, Yixian Gao, Yong Li
Summary: This paper discusses the existence of quasi-periodic solutions for completely resonant quintic beam equations on one-dimensional tori. A novel Birkhoff normal form is developed to reformulate the problem into a nearly integrable system depending on the angle variables. Symplectic transformations, Floquet theory, and the averaging method are utilized to reduce the influence of the angle variables. The proof mainly relies on an infinite-dimensional KAM theorem.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
Zhichao Ma, Ru Qu, Junxiang Xu
Summary: This paper investigates quasi-periodic perturbations of 2-dimensional degenerate systems. It is demonstrated that if the equilibrium point of the unperturbed system is hyperbolic-type degenerate, then the perturbed system has a small response solution. The proof relies on the topological degree theory and some KAM techniques developed in [3].
JOURNAL OF DIFFERENTIAL EQUATIONS
(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
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
Physics, Mathematical
Chuanfang Ge, Jiansheng Geng
Summary: In this paper, we investigate two dimensional completely resonant, derivative, quintic nonlinear beam equations with reversible structure. Due to the absence of external parameters or potentials in this reversible system, Birkhoff normal form reduction is necessary before applying Kolmogorov-Arnold-Moser (KAM) theorem. As an application of KAM theorem, the existence of partially hyperbolic, small amplitude, quasi-periodic solutions of the reversible system is proven.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Operations Research & Management Science
Mircea Balaj
Summary: The paper discusses existence results for scalar and vector quasi-equilibrium problems, where the constraint set depends on the current point. It establishes these results by replacing the standard equilibrium condition with a weaker one.
OPERATIONS RESEARCH LETTERS
(2021)
Article
Mathematics, Applied
Xiaomei Yang, Junxiang Xu
Summary: This paper focuses on a specific type of quasi-periodic systems with a small parameter, whose unperturbed part has a degenerate equilibrium point. The existence of response solutions for many sufficiently small parameters is proven, based on formal KAM techniques and the LeraySchauder Continuation Theorem.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
(2023)
Article
Astronomy & Astrophysics
A. G. Suvorov, H. J. Kuan, K. D. Kokkotas
Summary: The study suggests that the precursor flare of GRB 211211A event may have resulted from the resonant shattering of one star's crust prior to coalescence, generating seismic aftershocks and low-frequency torsional modes. This interpretation is supported by the computed torsional mode properties, showing similarities between the precursor and specific torsional modes. Potential candidates for the modulations in the precursor include global or discrete Alfven modes.
ASTRONOMY & ASTROPHYSICS
(2022)
Article
Mathematics, Applied
Guangzhao Zhou, Yuan Zhang, Wen Si
Summary: This paper investigates the existence and quantitative properties of completely degenerate quasi-periodically forced skew-product maps of specific forms, and proves that under certain conditions, these skew-product maps not only have weak-hyperbolic invariant tori, but also weak-elliptic invariant tori. The number of invariant tori is also examined in both cases. These results address situations that have not been discussed in existing literature.
COMMUNICATIONS ON PURE AND APPLIED ANALYSIS
(2023)
Article
Mathematics, Applied
Xinyu Guan, Wen Si
Summary: In this paper, an almost-periodic tori bifurcation theory for 2-dimensional degenerate Hamiltonian vector fields is developed. The universal unfolding of completely degenerate Hamiltonian N(x, y) = x2y+yl and partially degenerate Hamiltonian M(x, y) = x2 + yl can persist under small almost-periodic time-dependent perturbation and certain non-resonant conditions on almost-periodic frequency omega. The main results of this study show that infinite-dimensional degenerate umbilical tori or normally parabolic tori can bifurcate according to a generalized umbilical catastrophe or generalized cuspoid catastrophe under small almost-periodic perturbation.
JOURNAL OF APPLIED ANALYSIS AND COMPUTATION
(2023)
Article
Engineering, Electrical & Electronic
Bo Wang, Minghe Tian, Yong Yu, Qinghua Dong, Dianguo Xu
Summary: The article introduces a method of combining active disturbance rejection control with quasi-resonant controllers to solve the problem of speed fluctuations in PMSM drives, and analyzes the stability and anti-disturbance capability.
IEEE TRANSACTIONS ON TRANSPORTATION ELECTRIFICATION
(2022)
Article
Physics, Mathematical
Min Zhang, Yi Wang, Jie Rui
Summary: This paper investigates the existence of one-dimensional quasi-periodic solutions for a nonlinear Schrodinger equation under Dirichlet boundary conditions. The authors verify the existence of quasi-periodic solutions for the equation. By applying infinitely many symplectic transformations, the Hamiltonian of the linear part of the equation can be simplified to an autonomous system. Utilizing the measure estimation of small divisors, a symplectic change of coordinate transformation is found to transform the Hamiltonian of the equation into a nice Birkhoff normal form. The existence of a class of small-amplitude quasi-periodic solutions for the above equation is then verified using an abstract KAM (Kolmogorov-Arnold-Moser) theorem.
JOURNAL OF MATHEMATICAL PHYSICS
(2023)
Article
Engineering, Aerospace
Daniel Villegas-Pinto, Nicola Baresi, Slim Locoche, Daniel Hestroffer
Summary: Motivated by the upcoming exploration of cislunar space, this study investigates dynamical substitutes for Earth-Moon's resonant Near-Rectilinear Halo Orbits (NRHOs) under the Elliptic-Circular Restricted Four-Body Problem formulation. By incorporating the Sun's influence and the Moon's eccentricity, the study replaces resonant periodic NRHOs of the Earth-Moon Circular Restricted Three-Body Problem with two-dimensional quasi-periodic tori that better represent the dynamics of satellites near the Moon. The study presents the steps and algorithms to compute these dynamical structures and assess their usefulness for spacecraft missions, focusing on specific resonant orbits and their advantageous properties.
ADVANCES IN SPACE RESEARCH
(2023)
Article
Mathematics, Applied
Zechang Zheng, Zhongrong Lu, Jike Liu, Yanmao Chen
Summary: Solving quasi-periodic responses of nonlinear dynamical systems with multiple unknown frequencies is a challenging task. This paper proposes a new phase condition that is unconditionally valid for such solutions, providing a theoretical basis for solving these problems. The effectiveness of the phase condition is demonstrated through numerical examples using finite difference and harmonic balance methods.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)