UK Nonlinear News, November 1997

Recent Theses

Symmetry breaking and chaos in nonlinear mean-field dynamos

Mathematics Department, University of Manchester

John Martin Brooke

Supervisor: David Moss

This thesis describes interesting dynamical behaviour encountered in models designed to simulate the generation of magnetic fields in rotating astrophysical objects. The focus of attention is on bifurcations involving the breaking of pure symmetry about the equator (parity symmetry). The equations of the governing PDE's have two invariant subspaces representing solutions that are symmetric or antisymmetric about the equator.

The mean-field dynamo equations are solved in toroidal geometry with a Keplerian rotation law, vacuum boundary conditions and an alpha-quenching nonlinearity. With symmetric equatorial symmetry imposed there is a transition to chaos via the Ruelle-Takens route, which demonstrates the phenomenon of torus-doubling. Similar behaviour is observed with imposed antisymmetry and the torus-doubling occurs at a very similar absolute value of the dynamo number D. However in the latter case the torus-doubling shows a hysterisis loop around the bifurcation point.

When the parity is allowed to evolve freely, three regimes were examined. In the alpha-squared regime there were two metastable solutions with pure symmetric and antisymmetric parity and the dynamo evolved to either according to the initial conditions. In the alpha squared-omega regime the solutions for D positive were dominated by a steady symmetric mode which showed the field configuration of the most easily excited mode in linear theory. For D negative the first stable solution was oscillating with pure symmetric parity. This lost stability to an intermittent solution at the same parameter value at which the solution with imposed symmetric parity underwent period-doubling. In the alpha-omega regime this attractor coexisted with a steady mixed-parity solution.

The intermittent transition was modelled by a 1-dimensional mapping. It was shown that the form of this mapping can be derived from a consideration of on-off intermittency from a periodic attractor in an invariant manifold which represents one of the two possible equatorial symmetries. Thus the tranverse stability of a periodic, rather than chaotic, attractor in an invariant manifold is the focus of investigation. From this mapping a scaling law of epsilon^{-1} was derived for the average length of the laminar intervals the bifurcation parameter epsilon passes through zero. This prediction of the mapping was confirmed in the solutions of the mean-field dynamo PDE.

An explanation was given of the observation that three separate dynamical phenomena, symmetry breaking, intermittency and period-doubling occur at the same bifurcation point. This was done by considering the symmetry group of a time-periodic axisymmetric dynamo. As the attractor in the invariant manifold representing equatorial symmetry loses stability, the spatio-temporal symmetries select certain transverse perturbations and amplify them exponentially via a resonance mechanism. The theory predicts that with a subharmonic bifurcation the spatio-temporal symmetries should alternate in a periodic orbit related to the period of the now-unstable orbit in the invariant manifold.

These predictions were confirmed in four different dynamo models by two methods. Firstly techniques of phase dispersion analysis (PDA) were used to confirm the predictions of resonances and period-doubling. Secondly symmetry detectives were constructed to reveal the orbits of the spatio-temporal symmetries. Thus this form of intermittent behaviour, here termed icicle intermittency, persists across different dynamo models.

The analysis of symmetry breaking was compared to other work on symmetry breaking in dynamo models. The relevance of this form of intermittency to the grand minima observed in stellar cycles was discussed and a dynamo model in a torus with boundary conditions appropriate to a stellar convective zone was shown to exhibit a hysteritic symmetry breaking transition to chaos via intermittency.

Source: J.M. Brooke (

Recent Theses at Heriot-Watt University


J. Taylor: Numerical Analysis of Fast and Slow Transients in Gas Transmission Networks
C. Nikolopoulos: Mathematical modelling of Modulated - Temperature Differential Scanning Calorimetry
A R Soheili: Numerical Analysis of Coagulation-Fragmentation Equations
G. A. Afrouzi: Some Problems in Elliptic Equations Involving Indefinite Weight Functions.
R. M. Dunwell: The Becker-D\"{o}ring Cluster Equations.
R. P. K. C. Malmini: Gonihedric 3D Ising Models

MSc (in Mathematics of Nonlinear Models, joint with Edinburgh University):

A. J. Adam: The effect of noise on a deterministic model of a chemical reaction scheme.
M. A. Gray: Numerical approximation of pressure transients in oil reservoirs.
G. J. D. Petrie: Windscreen heating.
B. K. Tope: Implied volatilities between caps and swaptons.
D. C. Vicary: Gravothermal oscillations in globular star clusters.
A. J. Waugh: A model of council house allocation.

Source: Chris Eilbeck (

<< Move to UK Nonlinear News Issue 10 Index Page (November 1997).
Last Updated: 28th October 1997.