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
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
Yin Chen, Jiansheng Geng
Summary: We prove an abstract infinite dimensional KAM theorem, which could be applied to demonstrate the existence and linear stability of small-amplitude quasi-periodic solutions for higher dimensional Kirchhoff equations with periodic boundary conditions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
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
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
Xuefeng Zhao, Yong Li
Summary: We study the iso-manifold persistence in formulism and show that unperturbed tori can give rise to invariant tori in the perturbed system while maintaining the ratio of certain frequency components. We also consider the iso-manifold Melnikov persistence.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Applied
Rafael de la Llave, Lu Xu
Summary: In this paper, we prove the persistence result of whiskered tori for a dynamical system that preserves an exact presymplectic form. The iterative procedure used in the proof offers an efficient numerical method with low storage requirements and quadratic convergence. This approach is particularly useful for systems perturbed by quasi-periodic systems.
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Renato Calleja, Alessandra Celletti, Joan Gimeno, Rafael de la Llave
Summary: We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. The paper presents the background, assumptions, and numerical methods used to compute the attractors close to the breakdown threshold. The authors hope that their work can stimulate further research in computer-assisted proofs.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
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
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
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
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
Jiansheng Geng, Shuaishuai Xue
Summary: In this study, a Whitney smooth family of small-amplitude quasi-periodic solutions is obtained in the two-dimensional nonlinear Schrödinger equation using an infinite dimensional KAM theorem.
JOURNAL OF FUNCTIONAL ANALYSIS
(2022)
Article
Physics, Atomic, Molecular & Chemical
L. I. Kolesnikova, L. Yu. Rusin, M. B. Sevryuk
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B
(2015)
Article
Mathematics, Applied
Mikhail B. Sevryuk
REGULAR & CHAOTIC DYNAMICS
(2016)
Article
Mathematics, Applied
Mikhail B. Sevryuk
REGULAR & CHAOTIC DYNAMICS
(2017)
Article
Mathematics, Applied
Vincenzo Aquilanti, Andrea Lombardi, Mikhail B. Sevryuk
REGULAR & CHAOTIC DYNAMICS
(2014)
Article
Mathematics, Applied
Mikhail B. Sevryuk
REGULAR & CHAOTIC DYNAMICS
(2014)
Article
Physics, Atomic, Molecular & Chemical
L. I. Kolesnikova, L. Yu Rusin, M. B. Sevryuk
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B
(2013)
Article
Physics, Atomic, Molecular & Chemical
E. V. Ermolova, L. Yu Rusin, M. B. Sevryuk
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B
(2014)
Article
Mathematics
Mikhail B. Sevryuk
Summary: This paper presents examples of Hamiltonian and reversible systems with smooth d-parameter families of invariant n-tori carrying conditionally periodic motions. The cases of isotropic, coisotropic, and atropic tori in non-compact and compact phase spaces are considered. Additionally, an example of an analytic Hamiltonian system with an isolated invariant N-torus carrying conditionally periodic motions is presented for any N >= 3.
INDAGATIONES MATHEMATICAE-NEW SERIES
(2021)
Article
Chemistry, Physical
Vyacheslav M. Akimov, Vladimir M. Azriel', Ekaterina V. Ermolova, Dmitrii B. Kabanov, Lyubov' I. Kolesnikova, Lev Yu. Rusin, Mikhail B. Sevryuk
Summary: This paper investigates the detailed dynamics of direct three-body ion-ion recombination reactions, considering the main aspects of non-central ion encounters. The reactions are simulated using the quasiclassical trajectory method with diabatic semiempirical potential energy surfaces. The recombination mechanisms are studied through visualization of randomly selected trajectories, and a comparison is made between trajectories with identical initial conditions for different systems.
PHYSICAL CHEMISTRY CHEMICAL PHYSICS
(2022)
Article
Multidisciplinary Sciences
Vyacheslav M. Akimov, Vladimir M. Azriel, Ekaterina V. Ermolova, Dmitrii B. Kabanov, Lev Yu. Rusin, Mikhail B. Sevryuk
Summary: By simulating the dynamics of bimolecular recombination reactions under different conditions, it was found that the collision energy has important effects on the recombination cross section and the energy distribution of products. The type of halide ion also plays a significant role in the dynamics.
RENDICONTI LINCEI-SCIENZE FISICHE E NATURALI
(2022)
Article
Chemistry, Physical
Vyacheslav M. Akimov, Vladimir M. Azriel, Ekaterina Ermolova, Dmitrii B. Kabanov, Lyubov' Kolesnikova, Lev Yu Rusin, Mikhail B. Sevryuk
Summary: The direct three-body recombination reactions Cs+ + X- + R -> CsX + R (X = F, I and R = Ar, Xe) were studied using the quasiclassical trajectory method, revealing the distinct features of different recombination pairs and the superior efficiency of xenon in accepting excess energy from ion pairs. The resulting energy distributions of the recombination products show equilibrium for rotational energy in CsF and CsI molecules, but strong non-equilibrium for vibrational energy.
PHYSICAL CHEMISTRY CHEMICAL PHYSICS
(2021)
Article
Multidisciplinary Sciences
Vyacheslav M. Akimov, Vladimir M. Azriel, Lyubov I. Kolesnikova, Lev Yu. Rusin, Mikhail B. Sevryuk
RENDICONTI LINCEI-SCIENZE FISICHE E NATURALI
(2019)
Article
Physics, Atomic, Molecular & Chemical
V. M. Azriel', V. M. Akimov, E. V. Ermolova, D. B. Kabanov, L. I. Kolesnikova, L. Yu. Rusin, M. B. Sevryuk
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B
(2018)
Article
Physics, Atomic, Molecular & Chemical
V. M. Azriel', V. M. Akimov, E. V. Ermolova, L. I. Kolesnikova, L. Yu. Rusin, M. B. Sevryuk
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B
(2018)
Article
Mathematics, Applied
Mikhail B. Sevryuk
MOSCOW MATHEMATICAL JOURNAL
(2017)
