Abstract. Suppose a smooth dynamical system has an invariant subspace and a parameter that leaves the dynamics in the invariant subspace invariant while changing the normal dynamics. Then we say the parameter is a normal parameter, and much is understood of how attractors can change with normal parameters. Unfortunately, normal parameters do not arise very often in practise.We consider the behaviour of attractors near invariant subspaces on varying a parameter that does not preserve the dynamics in the invariant subspace but is otherwise generic, in a smooth dynamical system. We refer to such a parameter as "non-normal". If there is chaos in the invariant subspace that is not structurally stable, this has the effect of "blurring out" blowout bifurcations over a range of parameter values that we show can have positive measure in parameter space.Associated with such blowout bifurcations are bifurcations to attractors displaying a new type of intermittency that is phenomenologically similar to on-off intermittency, but where the intersection of the attractor by the invariant subspace is larger than a minimal attractor. The presence of distinct repelling and attracting invariant sets leads us to refer to this as "in-out" intermittency. Such behaviour cannot appear in systems where the transverse dynamics is a skew product over the system on the invariant subspace.We characterise in-out intermittency in terms of its structure in phase space and in terms of invariants of the dynamics obtained from a Markov model of the attractor. This model predicts a scaling of the length of laminar phases that is similar to that for on-off intermittency but which has some differences.Finally, we discuss some other bifurcation effects associated with non-normal parameters, in particular a bifurcation to riddled basins.

Abstract. Recent analyses of helioseismological observations seem to suggest the presence of two new phenomena connected with the dynamics of the solar convective zone. Firstly, there are present torsional oscillations with periods of about 11 years, which penetrate significantly into the solar convection zone and secondly, oscillatory regimes exist near the base of the convection which are markedly different from those observed near the top, having either significantly reduced periods or being non-periodic. Recently spatiotemporal fragmentation/bifurcation has been proposed as a possible dynamical mechanism to account for such observed multi-mode behaviours in different parts of the solar convection zone. Evidence for this scenario was produced in the context of an axisymmetric mean field dynamo model operating in a spherical shell, with a semi-open outer boundary condition and a zero order angular velocity obtained by the inversion of the MDI data, in which the only nonlinearity was the action of the Lorentz force of the dynamo generated magnetic field on the solar angular velocity. Here we make a detailed study of the robustness of this model with respect to plausible changes to its main ingredients, including changes to the α and η profiles as well as the inclusion of a nonlinear α quenching. We find that spatiotemporal fragmentation is present in this model for different choices of the rotation data and as the details of the model are varied. Taken together, these results give strong support to the idea that spatiotemporal fragmentation is likely to occur in general dynamo settings.

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Abstract. Analyses of recent helioseismic data have produced ample evidence for substantial dynamical variation of the differential rotation within the solar convection zone. Given the inevitable difficulties in resolving the precise nature of variations in deeper layers, much effort has recently gone into determining theoretically the expected modes of behaviour, using nonlinear dynamo models. Two important limitations of these models are that they have so far included only one form of nonlinearity, and as yet they have not taken into account the density stratification in the solar convection zone. Here we address both of these issues by studying the effects of including density stratification, as well as including an α-quenching nonlinearity in addition to the previously studied effects of the Lorentz force on the differential rotation. We find that observationally important features found in the earlier uniform density models remain qualitatively unchanged, although there are quantitative differences. This is important as it provides more realistic theoretical predictions to be compared with and guide observations, especially in the deeper regions where the uncertainties in the inversions are larger. However the presence of an effective alpha-quenching nonlinearity significantly reduces the amplitudes of the oscillations.

Abstract. Increasing evidence is becoming available about not only the surface differential rotation of rapidly rotating cool stars but, in a small number of cases, also about temporal variations, which possibly are analogous to the solar torsional oscillations. Given the present difficulties in resolving the precise nature of such variations, due to both the short length and poor resolution of the available data, theoretical input is vital to help assess the modes of behaviour that might be expected, and will facilitate interpretation of the observations. Here we take a first step in this direction by studying the variations in the convection zones of such stars, using a two dimensional axisymmetric mean field dynamo model operating in a spherical shell in which the only nonlinearity is the action of the azimuthal component of the Lorentz force of the dynamo generated magnetic field on the stellar angular velocity. We consider three families of models with different depths of dynamo-active regions. For moderately supercritical dynamo numbers we find torsional oscillations that penetrate all the way down to the bottom of the convection zones, similar to the case of the Sun. For larger dynamo numbers we find fragmentation in some cases and sometimes there are other dynamical modes of behaviour, including quasi-periodicity and chaos. We find that the largest deviations in the angular velocity distribution caused by the Lorentz force are of the order of few percent, implying that the original assumed "background" rotation field is not strongly distorted.