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Modelling the solar dynamo

Bifurcation Structure d=0

We first look at the bifurcation structure for d = 0. When this is the case the system can be written in cylindrical polar coordinates as

This rules out the possibility of chaotic solutions to the equations, since the model is now essentially only 2-dimensional.

Kirk (1991) finds the bifurcation structure shown in the sketch below:

This sketch illustrates the changes we expect in the bifurcation diagram when the values of a and c are altered. The line of the saddle node bifurcation at μ=4/(27c2) will be far from the origin for small c. Given this value of c we can make the secondary Hopf bifurcation line at λ=-2a/3c as far from the origin as we like by increasing a. In this way changing the values of a and c allows us to make the regions in phase space separated by the bifurcation curves as small or as large as we please.

Continuing to examine the case when d = 0, we label these regions and examine the types of solution we expect to find in each.

In regions 1, 2, 7, 8, 11 and 12 the field free steady states are stable, so any seed magnetic field decays away. The initial conditions of the system will determine which of the three solutions to 0 = μ - z2 + cz3 the system reaches.

There are two ways in which periodic solutions are obtained. At the primary Hopf bifurcation a limit cycle bifurcates from the fixed points, giving rise to periodic orbits (for suitable initial conditions.) At the saddle-node bifurcation a pair of periodic orbits are created, one of which is stable. Hence periodic solutions are observed in regions 3, 4, 9, 10, and 13, an example of which is shown in below.

At the secondary Hopf bifurcation a torus bifurcates from the periodic orbit. This gives rise to quasiperiodic behaviour in regions 5 and 6, and example of which is illustrated below.


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