Dynamical Systems and Topology of Magnetic Fields in Conducting Medium

2021 ◽  
Author(s):  
V. Z. Grines ◽  
E. V. Zhuzhoma ◽  
O. V. Pochinka
Keyword(s):  
Dynamical Systems ◽  
Magnetic Fields ◽  
Conducting Medium ◽  
And Topology

scholarly journals Dynamical Systems and Topology of Magnetic Fields in Conducting Medium

2017 ◽  
Vol 63 (3) ◽  
pp. 455-474
Author(s):  
V Z Grines ◽  
E V Zhuzhoma ◽  
O V Pochinka
Keyword(s):  
Dynamical Systems ◽  
Magnetic Fields ◽  
Chaotic Dynamics ◽  
Vector Fields ◽  
Field Model ◽  
Physical Models ◽  
Hyperbolic Dynamics ◽  
And Topology ◽  
Magnetic Field Model ◽  
Basic Sets

We discuss application of contemporary methods of the theory of dynamical systems with regular and chaotic hyperbolic dynamics to investigation of topological structure of magnetic fields in conducting media. For substantial classes of magnetic fields, we consider well-known physical models allowing us to reduce investigation of such fields to study of vector fields and Morse-Smale diffeomorphisms as well as diffeomorphisms with nontrivial basic sets satisfying the A axiom introduced by Smale. For the point-charge magnetic field model, we consider the problem of separator playing an important role in the reconnection processes and investigate relations between its singularities. We consider the class of magnetic fields in the solar corona and solve the problem of topological equivalency of fields in this class. We develop a topological modification of the Zeldovich funicular model of the nondissipative cinematic dynamo, constructing a hyperbolic diffeomorphism with chaotic dynamics that is conservative in the neighborhood of its transitive invariant set.

scholarly journals Dynamical systems and topology optimization

2010 ◽  
Vol 42 (2) ◽  
pp. 179-192 ◽  
Author(s):  
Anders Klarbring ◽  
Bo Torstenfelt
Keyword(s):  
Topology Optimization ◽  
Dynamical Systems ◽  
And Topology

scholarly journals Classification of perturbations of the hydrogen atom by small static electric and magnetic fields

2007 ◽  
Vol 463 (2083) ◽  
pp. 1771-1790 ◽  
Author(s):  
K Efstathiou ◽  
D.A Sadovskií ◽  
B.I Zhilinskií
Keyword(s):  
Dynamical Systems ◽  
Hydrogen Atom ◽  
Magnetic Fields ◽  
Parameter Space ◽  
Field Limit ◽  
Electric And Magnetic Fields ◽  
Orthogonal Field

We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalizing with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1 : 1 resonance corresponds to the orthogonal field limit, studied earlier by Cushman & Sadovskií (Cushman & Sadovskií 2000 Physica 142 , 166–196). We describe the structure of the 1 : 1 zone, where the system may have monodromy of different kinds, and consider briefly the 1 : 2 zone.

scholarly journals A Surface of Section for Hydrogen in Crossed Electric and Magnetic Fields

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 185
Author(s):  
Korana Burke ◽  
Kevin Mitchell
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Hydrogen Atom ◽  
Magnetic Fields ◽  
Geometric Approach ◽  
Electron Dynamics ◽  
Surface Of Section ◽  
Electric And Magnetic Fields ◽  
Global Surface ◽  
Global Surface Of Section

A well defined global surface of section (SOS) is a necessary first step in many studies of various dynamical systems. Starting with a surface of section, one is able to more easily find periodic orbits as well as other geometric structures that govern the nonlinear dynamics of the system in question. In some cases, a global surface of section is relatively easily defined, but in other cases the definition is not trivial, and may not even exist. This is the case for the electron dynamics of a hydrogen atom in crossed electric and magnetic fields. In this paper, we demonstrate how one can define a surface of section and associated return map that may fail to be globally well defined, but for which the dynamics is well defined and continuous over a region that is sufficiently large to include the heteroclinic tangle and thus offers a sound geometric approach to studying the nonlinear dynamics.

Randall Dougherty and Alexander S. Kechris. The complexity of antidifferentiation. Advances in mathematics, vol. 88 (1991), pp. 145–169. - Ferenc Beleznay and Matthew Foreman. The collection of distal flows is not Borel. American journal of mathematics, vol. 117 (1995), pp. 203–239. - Ferenc Beleznay and Matthew Foreman. The complexity of the collection of measure-distal transformations. Ergodic theory and dynamical systems, vol. 16 (1996), pp. 929–962. - Howard Becker. Pointwise limits of subsequences and sets. Fundamenta mathematicae, vol. 128 (1987), pp. 159–170. - Howard Becker, Sylvain Kahane, and Alain Louveau. Some complete sets in harmonic analysis. Transactions of the American Mathematical Society, vol. 339 (1993), pp. 323–336. - Robert Kaufman. PCA sets and convexity Fundamenta mathematicae, vol. 163 (2000), pp. 267–275). - Howard Becker. Descriptive set theoretic phenomena in analysis and topology. Set theory of the continuum, edited by H. Judah, W. Just, and H. Woodin, Mathematical Sciences Research Institute publications, vol. 26, Springer-Verlag, New York, Berlin, Heidelberg, etc., 1992, pp. 1–25.

2001 ◽  
Vol 7 (3) ◽  
pp. 385-388
Author(s):  
Gabriel Debs
Keyword(s):  
New York ◽  
Dynamical Systems ◽  
Harmonic Analysis ◽  
Ergodic Theory ◽  
Set Theory ◽  
American Mathematical Society ◽  
And Topology ◽  
Mathematical Sciences ◽  
Descriptive Set ◽  
The Continuum

scholarly journals Dynamo–Action in Accretion Disks

1993 ◽  
Vol 157 ◽  
pp. 185-192
Author(s):  
W. Deinzer
Keyword(s):  
Magnetic Fields ◽  
Accretion Disks ◽  
Cataclysmic Variables ◽  
T Tauri Stars ◽  
Conducting Medium ◽  
Dynamo Action ◽  
T Tauri ◽  
Thin Disks ◽  
Excitation Conditions

Accretion disks are approximated by thin tori and the generation of magnetic fields by torus–dynamos is investigated. Solutions for the general α2ω–dynamo embedded into vacuum and into an ideally conducting medium are presented. Whereas the former solutions are qualitatively similar to those obtained for thin disks, there is a mode for the latter peculiar to torus–geometry. Excitation conditions for torus–dynamos are compared to those realized in accretion disks in cataclysmic variables, around T Tauri stars and in AGN's.

The group Gn2 and invariants of free links valued in free groups

2015 ◽  
Vol 24 (13) ◽  
pp. 1541010 ◽  
Author(s):  
V. O. Manturov ◽  
S. Kim
Keyword(s):  
Dynamical Systems ◽  
Free Product ◽  
Equivalence Relation ◽  
Braid Group ◽  
Knot Theory ◽  
Free Groups ◽  
Group Element ◽  
Group Presentation ◽  
End Points ◽  
And Topology

In this paper, we define an invariant of free links valued in a free product of some copies of [Formula: see text]. In [Non-Reidemeister knot theory and its applications in dynamical systems, geometry, and topology, preprint (2015), arXiv:1501.05208v1] the second author constructed a connection between classical braid group and group presentation generated by elements corresponding to horizontal trisecants. This approach does not apply to links nor tangles because it requires that when counting trisecants, we have the same number of points at each level. For general tangles, trisecants passing through one component twice may occur. Free links can be obtained from tangles by attaching two end points of each component. We shall construct an invariant of free links and free tangles valued in groups as follows: we associate elements in the groups with 4-valent vertices of free tangles (or free links). For a free link with enumerated component, we “read” all the intersections when traversing a given component and write them as a group element. The problem of “pure crossings” of a component with itself by using the following statement: if two diagrams with no pure crossings are equivalent then they are equivalent by a sequence of moves where no intermediate diagram has a pure crossing. This statement is a result of a sort that an equivalence relation within a subset coincides with the equivalence relation induced from a larger set and it is interesting by itself.

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