Important aspects of activity of space plasmas can be described in terms of transitions from stable to unstable states. Therefore, it was necessary to deal with the stability properties of selected equilibria, which has been a major aspect in this part of the book.

However, to obtain a deeper physical understanding of the dynamic properties of a given system it is desirable to have available a complete overview of all equilibrium states and their stability properties for every choice of a suitable (control) parameter. Such information is provided by bifurcation diagrams. They are particularly useful to assess the qualitative behaviour of nonlinear systems. Here we can give only an elementary introduction aimed at clarifying notions that will be used later. For rigorous treatments the reader should consult the literature (e.g., Berge et al., 1986; Guckenheimer and Holmes, 1983).

Statistical mechanics and nonlinear dynamics provide additional techniques that have been applied to space plasma activity.

Bifurcation

For illustration of bifurcations let us begin with a set of simple examples shown in Fig. 12.1. A point mass moves in a potential U(x, λ) subject to the force –∂U/∂x, λ being the control parameter. On the right, the figure also shows the bifurcation diagrams, which are the plots of the equilibrium positions versus λ. At bifurcation points the solution structure (here manifested by the number of solutions for a given value of λ) as well as stability undergoes qualitative changes.