Dynamical systems and topology of magnetic fields in conducting medium

V. Z. Grines ; E. V. Zhuzhoma ; O. V. Pochinka

Contemporary Mathematics. Fundamental Directions, Differential and functional differential equations, Tome 63 (2017) no. 3, pp. 455-474

Résumé

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.