Dynamical Systems and Topology of Magnetic Fields in Conducting Medium
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.