Stable and unstable solutions of the nonlinear dynamo problem

Abstract

A local potential approach to nonlinear dynamo models which allows the use of variational techniques to investigate the problem of stability is introduced. The method applies at least to quasi-kinematic dynamo models, i. e. to models which include the back-reaction of the magnetic field on the fluid motion in a simplified way. A special application leads to a previously investigated one-dimensional dynamo model which shows a coexistence of a periodic solution (limit cycle) with two stable steady solutions of opposite polarities. The inclusion of some small-amplitude noise leads to interesting transition phenomena which may be of relevance to explain the behaviour of astrophysical dynamos. A simple dynamical system with a two-dimensional phase-space is used for illustration.