Article
Mathematics, Applied
Erman Cineli, Viktor L. Ginzburg
Summary: This paper focuses on the behavior under iterations of the filtered and local Floer homology of a Hamiltonian on a symplectically aspherical manifold. The authors prove that the supertrace of a generator of the cyclic group action on the filtered Floer homology is equal to the Euler characteristic of the un-iterated Hamiltonian's homology. They also establish the Lefschetz index of fixed points in the local homology, and analogs of the classical Smith inequality for iterated local homology and equivariant versions of these results.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2021)
Article
Mathematics
Lei Zhao
Summary: In this paper, a time-periodically forced Kepler problem is examined, showing the existence of infinitely many periodic orbits in the system. These orbits can experience double collisions with the attractive center and accumulate at it.
MATHEMATISCHE ANNALEN
(2023)
Article
Mathematics, Applied
Gabriele Benedetti, Jungsoo Kang
Summary: We study the relationship between positive symplectic homology and the existence of periodic orbits for Hamiltonian systems. We provide upper bounds for positive symplectic homology and discuss their applications in cotangent bundles.
JOURNAL OF FIXED POINT THEORY AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Joontae Kim, Seongchan Kim, Myeonggi Kwon
Summary: This article applies Floer theory to study symmetric periodic Reeb orbits, defines positive equivariant wrapped Floer homology, and obtains a lower bound on the number of geometrically distinct symmetric periodic Reeb orbits on a certain class of real contact manifolds through careful analysis of index iterations.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2022)
Article
Mathematics, Applied
Wenmin Gong
Summary: This paper establishes the existence of periodic orbits in a-atoroidal free homotopy classes for Hamiltonian systems in the twisted disc bundle, under certain conditions. The proof relies on the invariance of Floer homology and computation of Floer homology for the cotangent bundle. As an application, it is shown that twisted geodesic flows associated with a small magnetic field have periodic orbits on almost every energy level.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2021)
Article
Mathematics, Interdisciplinary Applications
H. I. Alrebdi, Konstantinos E. Papadakis, Juan F. Navarro, Euaggelos E. Zotos
Summary: This article explores the dynamics and geometry of the invariant manifolds that determine escapes from a multiwell potential. It presents the network of symmetric and asymmetric solutions of the system, and extracts valuable information about the periodic solutions, such as their locations, multiplicity, and linear stability.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics
Yi Xie
Summary: This paper introduces the annular instanton Floer homology and constructs a spectral sequence for links in a thickened annulus, with applications in detecting the unlink and distinguishing braids from other tangles using the annular Khovanov homology.
ADVANCES IN MATHEMATICS
(2021)
Article
Physics, Mathematical
U. Frauenfelder, J. Weber
Summary: This article discusses the generalization of the Conley-Zehnder index from ODEs to delay equations, and examines the equivalence between the Morse index and the clockwise normalized Conley-Zehnder index. By utilizing nonlocal Lagrangian and Hamiltonian action functionals, the regularized 1-periodic solutions of gravitational free fall are represented and analyzed in different ways, with a focus on the nonlocal aspects and the new phenomenon arising compared to the local case.
RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS
(2021)
Article
Mathematics
Yoshihiro Sugimoto
Summary: This paper addresses an open problem known as the (generic) Conley conjecture, which is related to the number of periodic orbits of Hamiltonian diffeomorphisms on closed symplectic manifolds. The generic Conley conjecture states that Hamiltonian diffeomorphisms typically have infinitely many simple contractible periodic orbits. The proof provided in this paper relies on applications of the Birkhoff-Moser fixed point theorem and Floer homology theory.
ARCHIV DER MATHEMATIK
(2021)
Article
Mathematics
Sara Venkatesh
Summary: We investigate the reduced symplectic cohomology of disk subbundles in negative symplectic line bundles and its relation to the spectrum of a quantum action on quantum cohomology. By analyzing the quantum cohomology decomposed into generalized eigenspaces, we demonstrate the properties of eigenspaces in the reduced symplectic cohomology of disk and annulus subbundles. These computations lead to statements about local closed-string mirror symmetry.
ADVANCES IN MATHEMATICS
(2021)
Article
Mathematics, Applied
Linh Truong
Summary: We calculate the knot Floer filtration induced by a cable of the meridian of a knot in the manifold obtained by large integer surgery along the knot. Furthermore, we show that a knot concordance invariant of Hom can be alternatively defined in terms of filtered maps on the Heegaard Floer homology groups induced by the two-handle attachment cobordism of surgery along a knot.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Interdisciplinary Applications
Carlos B. Briozzo
Summary: This work presents an impulsive controller for Hamiltonian linear time-periodic systems, which sends the system state vector periodically to a target subspace for control. By minimizing the system energy on the center manifold, it ensures damping of oscillatory motions crucial for controllability.
CHAOS SOLITONS & FRACTALS
(2021)
Article
Mathematics
Yi Xie
Summary: We established a rank inequality between the instanton knot homology and Khovanov homology, and constructed a spectral sequence to relate the two.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2021)
Article
Mathematics, Applied
Yocelyn Perez Rothen, Claudio Vidal
Summary: The aim of this study is to analytically prove the existence of symmetric periodic solutions of a specific family of Hamiltonian systems and characterize their stability. Additionally, a first-order analytical approach for these symmetric periodic solutions is obtained. These families of periodic solutions are different from the ones found in existing literature.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
P. A. Patsis, T. Manos, L. Chaves-Velasquez, Ch Skokos, I Puerari
Summary: The evolution of phase space near complex unstable periodic orbits in two galactic type potentials was investigated. Weakly chaotic orbits associated with complex unstable periodic orbits were found to reinforce the morphological features studied, and should be considered as structure-supporting.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
