6.2
Construction of the Model
The model considered in this chapter is an extension of that derived in Tobias et al. (1995). In that paper, a third-order model was derived using a Poincar´e-Birkhoff normal form for a saddle-node–Hopf bifurcation, to obtain a system exhibiting generic and therefore robust behaviour. This normal form was chosen since it has a bifurcation structure that gives qualitatively similar behaviour to that observed in stars as solutions along a cut through parameter space are examined.
Considering stellar magnetic activity observations we expect, qualitatively speaking, that, as the evolu- tion of a star is tracked backwards in time (i.e. as its rotation rate increases), periodic cyclic solutions will bifurcate from a steady free-field state in a supercritical Hopf bifurcation. These regular cyclic solutions will, in turn, give way to trajectories lying on a two-torus after a supercritical secondary Hopf bifurcation, reflecting periodic solutions with amplitude modulation in time. Finally, this activity should become chaot- ically modulated to account for those stars with irregular activity. Indeed, such a bifurcation structure is mirrored in mean-field PDE models as the non-dimensional measure of rotation rate (the dynamo number D) is increased (Tobias, 1996, Pipin, 1999, Bushby, 2005).
In the model of Tobias et al. (1995), the magnetic field was decomposed in the usual way into its toroidal part, represented byx, and its poloidal part, represented byy. The third coordinate of the system, z, represents all the hydrodynamics of the system, including as differential rotation and convection. Though a consideration of normal-form theory, the basic system is taken to be given by
˙ z = µ−z2− x2+y2, ˙ x = (λ+az)x−ωy, (6.1) ˙ y = (λ+az)y+ωx.
Forµ > 0 the equations (6.1) have have two fixed points, P+ and P−, given by the solutions to x= 0, y = 0, z=±√µ. These correspond to field-free, purely hydrodynamic, solutions where the flows are statistically steady and arise from the saddle-node bifurcation atµ= 0. Thus the parameterµcontrols the hydrodynamics of the system, so is related to effects such as thermal forcing and rotation. The term
(x2+y2)in thez-equation, being quadratic in the magnetic field, represents the back reaction of the Lorentz˙ force on the field. Its coefficient has been chosen to be less than zero so that the secondary Hopf bifurcation is supercritical.
By settingz = 0, we see thatλgives the growth-rate (i.e. strength of the dynamo action) ofxandy andωthe basic cycle frequency (the location of the bifurcation curves in the model is independent ofω). In a more complicated PDE model these features would be linked with the dynamo number.
For the system of equations (6.1) the secondary Hopf bifurcation is found to be degenerate and, to break this degeneracy, a cubic term must be added to the model. A cubic term,cz3, was added to thez˙equation and to takec < 0 so that solutions on thez-axis remain finite. This inclusion introduces another fixed point to the system, again on thez-axis and and associated additional line of saddle-node bifurcations (at µ= 4/27c2).
6.2 Construction of the Model 91
The system derived thus far is axisymmetry essentially two-dimensional – it may be written in cylindri- cal polars as:
˙ z = µ−z2−r2+cz3, ˙ r = λr+azr, ˙ φ = ω.
The addition of a symmetry-breaking term would add physical realism to the system and making the system fully three-dimensional would allow for chaotic dynamics. For these reasons the authors chose to add a cubic term to thex˙ equation to break the normal form axisymmetry. However, the exact choice of term is arbitrary and the term chosen in Tobias et al. (1995) is one proportional to x2+y2z, the motivation being to preserve the invariance of thez-axis. Thus the system of ODEs now takes the form
˙ z=µ−z2−(x2+y2) +cz3, ˙ x= (λ+az)x−ωy+dz(x2+y2), (6.2) ˙ y= (λ+az)y+ωx.
See Tobias et al. (1995) for further details of the model’s derivation.
In order to demonstrate the type of behaviour that such a model yields, Tobias et al. (1995) fixed all parameters except forλandµand chose a parameterized path through theλ−µplane to demonstrate the bifurcation structure of the model. In summary, the showed that, as the controlling parameter was increased, purely hydrodynamic solutions lost stability in a primary Hopf bifurcation to oscillatory solutions. In turn these gave way to quasiperiodic solutions, where the basic cycle is modulated on a longer timescale and solutions lie on a two-torus in phase-space. Further increase in the parameter led to a breakdown of the torus and a transition to chaos. The solution then took the form of active periods, interspersed chaotically with minima. Such solutions are associated with close-approach to an invariant manifold and near heteroclinicity.
However, as noted by Ashwin et al. (2004), a limitation of the model is that the choice of term to break the normal form axisymmetry in Tobias et al. (1995) results in a loss of equivalence of the system under the transformationx → −x, y → −y which corresponds toB → −B. In this chapter we choose an alternative term, which does not suffer from the above disadvantage, to break the axial symmetry. Again, the exact choice of cubic term is arbitrary; available terms are, for example,x3,xz2,xy2,xyz,(x+y)z2 etc. Similarly the choice made in Tobias et al. (1995) of including this term in thex˙ equation was arbitrary, they˙equation would also be suitable. In view of these considerations we choose to add a term proportional to(x3−3xy2)to thex-equation and one proportional to˙ (3x2y−y3)to they-equation. Thus the model˙
becomes ˙ z=µ−z2−(x2+y2) +cz3, ˙ x=λx−ωy+azx+d(x3−3xy2), (6.3) ˙ y=λy+ωx+azy+d(3x2y−y3).