scholarly journals Symplectic deformations of Floer homology and non-contractible periodic orbits in twisted disc bundles

2019 ◽  
Vol 23 (01) ◽  
pp. 1950084
Author(s):  
Wenmin Gong
Keyword(s):  
Magnetic Field ◽  
Periodic Orbits ◽  
Homotopy Class ◽  
Linear Growth ◽  
Floer Homology ◽  
Hamiltonian Function ◽  
Universal Cover ◽  
Zero Section ◽  
Free Homotopy Class ◽  
The Given

In this paper, we establish the existence of periodic orbits belonging to any [Formula: see text]-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact [Formula: see text]-form [Formula: see text] admitting a primitive with at most linear growth on the universal cover is sufficiently small. The proof relies on showing the invariance of Floer homology under symplectic deformations and on the computation of Floer homology for the cotangent bundle endowed with its canonical symplectic form. As a consequence, we also prove that, for any non-trivial atoroidal free homotopy class and any positive finite interval, if the magnitude of a magnetic field admitting a primitive with at most linear growth on the universal cover is sufficiently small, the twisted geodesic flow associated to the magnetic field has a periodic orbit on almost every energy level in the given interval whose projection to the underlying manifold represents the given free homotopy class. This application is carried out by showing the finiteness of the restricted Biran–Polterovich–Salamon capacity.

scholarly journals Floer homology for magnetic fields with at most linear growth on the universal cover

2012 ◽  
Vol 262 (7) ◽  
pp. 3062-3090 ◽  
Author(s):  
Urs Frauenfelder ◽  
Will J. Merry ◽  
Gabriel P. Paternain
Keyword(s):  
Magnetic Fields ◽  
Linear Growth ◽  
Floer Homology ◽  
Universal Cover

Canonical transformations

2020 ◽  
pp. 317-337
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Harmonic Oscillator ◽  
Canonical Transformation ◽  
Hamiltonian Function ◽  
Harmonic Oscillators ◽  
Nonlinear Coupling ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Anharmonic Oscillations ◽  
The Given

This chapter addresses the canonical transformation defined by the given generating function, the rotation in the phase space as a canonical transformation, and themovement of the system as a canonical transformation. The chapter also discusses using the canonical transformations for solving the problems of the anharmonic oscillations and using the canonical transformation to diagonalize the Hamiltonian function of an anisotropic charged harmonic oscillator in a magnetic field. Finally, the chapter addresses the canonical variables which reduce the Hamiltonian function of the harmonic oscillator to zero and using them for consideration of the system of the harmonic oscillators with the weak nonlinear coupling.

Canonical transformations

2020 ◽  
pp. 59-66
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Magnetic Field ◽  
Harmonic Oscillator ◽  
Canonical Transformation ◽  
Hamiltonian Function ◽  
Harmonic Oscillators ◽  
Nonlinear Coupling ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Anharmonic Oscillations ◽  
The Given

This chapter addresses the canonical transformation defined by the given generating function, the rotation in the phase space as a canonical transformation, and themovement of the system as a canonical transformation. The chapter also discusses using the canonical transformations for solving the problems of the anharmonic oscillations and using the canonical transformation to diagonalize the Hamiltonian function of an anisotropic charged harmonic oscillator in a magnetic field. Finally, the chapter addresses the canonical variables which reduce the Hamiltonian function of the harmonic oscillator to zero and using them for consideration of the system of the harmonic oscillators with the weak nonlinear coupling.

Формирование изображений электрических сигналов преобразователя магнитного поля при гистерезисной интерференции для контроля металлов в импульсных магнитных полях

2020 ◽  
pp. 38-45
Author(s):  
В.В. Павлюченко ◽  
Е.С. Дорошевич
Keyword(s):  
Magnetic Field ◽  
Aluminum Plate ◽  
Thickness Variation ◽  
Magnetic Carrier ◽  
Electric Voltage ◽  
Total Strength ◽  
Magnetic Field Transducer ◽  
The Given ◽  
The Magnetic Field

Based on the developed methods of hysteresis interference, the calculated dependences U(x) of the electric voltage taken from the magnetic field transducer on the x coordinate were obtained. A magnetic carrier with an arctangent characteristic was exposed to a series of bipolar pulses of the magnetic field of a linear inductor of one, two, three, four, five and fifteen pulses. An algorithm is presented for the sequence of changes in the magnitude of the total strength of the magnetic field pulses on the surface of an aluminum plate, which provides the same amplitude of hysteresis oscillations of the electric voltage and makes it possible to obtain a linear difference dependence U(x) for wedge-shaped and flat aluminum samples. The results obtained make it possible to increase the accuracy and efficiency of control of the thickness of the object and its thickness variation in the given directions, as well as the defects of the object.

scholarly journals Tracking Control of a Class of Chaotic Systems

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 568 ◽  
Author(s):  
Anqing Yang ◽  
Linshan Li ◽  
Zuoxun Wang ◽  
Rongwei Guo
Keyword(s):  
Control Problem ◽  
Periodic Orbits ◽  
Chaotic System ◽  
Tracking Control ◽  
Reference System ◽  
Chaotic Systems ◽  
Asymptotic Tracking ◽  
The Given

This paper investigates the asymptotic tracking control problem of the chaotic system. Firstly, a reference system is presented, the output of which can asymptotically track a given command. Then, a both physically implementable and simple controller is designed, by which the given chaotic system synchronizes the reference system, and thus the output of such chaotic systems can asymptotically track the given command. It should be pointed out that the output of the given chaotic system can asymptotically track arbitrary desired periodic orbits. Finally, several illustrative examples are taken as example to show the validity and effectiveness of the obtained results.

scholarly journals Typical representatives of free homotopy classes in multi-punctured plane

2019 ◽  
Vol 11 (03) ◽  
pp. 623-659
Author(s):  
Maxim Arnold ◽  
Yuliy Baryshnikov ◽  
Yuriy Mileyko
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Probability Measure ◽  
Homotopy Class ◽  
Piecewise Linear ◽  
Homotopy Classes ◽  
Simple Method ◽  
Monte Carlo Techniques ◽  
Free Homotopy Class

We show that a uniform probability measure supported on a specific set of piecewise linear loops in a nontrivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii’s theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled using standard Markov Chain Monte Carlo techniques, thus providing a simple method for approximating shortest loops.

scholarly journals Relative normal modes for nonlinear Hamiltonian systems

2003 ◽  
Vol 133 (3) ◽  
pp. 665-704 ◽  
Author(s):  
Juan-Pablo Ortega
Keyword(s):  
Periodic Orbits ◽  
Energy Levels ◽  
Relative Equilibrium ◽  
Normal Modes ◽  
Hamiltonian Function ◽  
Isotropy Subgroup ◽  
Continuous Symmetry ◽  
Highly Nonlinear ◽  
Spatio Temporal ◽  
Energy And Momentum

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a stable equilibrium. The estimate obtained is very precise in the sense that it provides a lower bound for the number of relative periodic orbits at each prescribed energy and momentum values neighbouring the stable relative equilibrium in question and with any prefixed (spatio-temporal) isotropy subgroup. Moreover, it is easily computable in particular examples. It is interesting to see how, in our result, the existence of non-trivial relative periodic orbits requires (generic) conditions on the higher-order terms of the Taylor expansion of the Hamiltonian function, in contrast with the purely quadratic requirements of the Weinstein–Moser theorem, which emphasizes the highly nonlinear character of the relatively periodic dynamical objects.

scholarly journals Calculations of magnetic field in dynamo sheets taking into account their texture

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 1034-1038
Author(s):  
Witold Mazgaj ◽  
Zbigniew Szular ◽  
Paweł Szczurek
Keyword(s):  
Magnetic Field ◽  
Field Strength ◽  
Magnetic Measurements ◽  
Iron Particles ◽  
Easy Magnetization ◽  
Anisotropic Properties ◽  
Steel Sheets ◽  
The Given ◽  
Dynamo Steel ◽  
Magnetization Processes

Abstract Magnetic measurements have shown that the most dynamo steel sheets have certain anisotropic properties, which are due to the presence of textures in these sheets. These anisotropic properties have been taken into account usually in a simplified way assuming that iron particles of the dynamo sheets have only one axis of the easy magnetization. In the proposed approach, these particles are treated as grains which have three axes of the easy magnetization, and therefore the magnetization processes can be considered along each of these axes. These processes depend on the actual value and on the direction of the field strength and also on textures occurring in the given dynamo sheet. A method which allows calculations of the field distribution as a function of the assumed changes of external currents is described in this paper.

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