Realistic examples of chaotic magnetic fields created by wires

In this paper we study some examples of magnetic fields with hyperbolic periodic orbits and erratic behavior. The remarkable fact is that the fields under consideration are originated by configurations which consist of two thin wires. We present a rigorous proof of the chaoticity of these magnetic fields, in the sense that they possess KAM islands and homoclinic tangles, and we also provide numerical simulations. In particular we illustrate, contrary to folk wisdom, that magnetic lines originated by current filaments can show a very complex nature. Finally, we propose a simple experimental verification of these results, as well as possible fields of application. Copyright c © EPLA, 2007 Introduction. – Magnetic fields created by current distributions usually appear in physics, both in theory and applications. The most suitable approach is the dynamical systems one, i.e. visualizing the magnetic field B as a vector field in R, the orbits of B being called magnetic lines. This viewpoint goes back to Faraday and is important in multidisciplinary research, e.g. biomedical engineering [1], electrical engineering [2], spectroscopy [3] and medical applications as magnetic resonance imaging [4]. In most applications the configurations of wires possess some Euclidean symmetry, e.g. rotational symmetry [5]. In general it is very difficult to get closed analytical expressions for the magnetic fields originated by wires, see [5]. For this reason the qualitative theory of dynamical systems is an effective tool in order to study the most relevant properties of the magnetic lines, as well as their action on charged particles, without actually integrating the Biot-Savart law [6]. In [7] and [8] the symmetries and first integrals of magnetic fields originated by certain current distributions were studied. It was proved that charged particles subjected to these fields verify the non-swallowing property, i.e. they cannot approach the wires indefinitely. The phase portraits of the fields were also described. In [8] it was posed the major unanswered question of finding (a)E-mail: [email protected] (b)E-mail: [email protected] a configuration of wires giving rise to a chaotic (nonintegrable) magnetic field. Let us briefly comment on the contexts in which chaotic magnetic fields have already been obtained. Examples of divergence-free chaotic dynamical systems were found in [9], but these examples are not realistic. Chaotic magnetic lines have also been described in the divertor region when analyzing tokamak dynamics in the context of plasma physics, but these magnetic fields are not created by wires. Indeed the divertor region is sometimes modelled by thin wires, see e.g. [10,11], which give rise to a non-chaotic portrait (the configuration has some Euclidean symmetry). Chaos appears when a vacuum field perturbation is added to the system, but the source of this perturbation is not a current distribution. In some cases the perturbation is modelled by a complicated structure of wires or a surface current density (this is called ergodic limiter in the specialized literature, see [12,13]), but the final field, which is chaotic, is in the tokamak region and hence part of it is created by a plasma current density. Apart from plasma physics (tokamaks) [14,15] chaotic magnetic lines have also appeared in other parts of physics, e.g. coronal structures [16] and force-free fields [17], although, of course, these fields are not created by wires and possess chaotic sources. Physicists and engineers have widely believed that chaos is not possible when the magnetic field is created by few thin wires. In fact the wrong idea that magnetic lines produced by current filaments form closed loops is