Symmetry-breaking instabilities of convection in squares

Convection in an infinite fluid layer is often modelled by considering a finite box with periodic boundary conditions in the two horizontal directions. The translational invariance of the problem implies that any solution can be translated horizontally by an arbitrary amount. Some solutions travel, but those solutions that are invariant under reflections in both horizontal directions cannot travel, since motion in any horizontal direction is balanced by an equal and opposite motion elsewhere. Equivariant bifurcation theory allows us to understand the steady and time-dependent ways in which a pattern can travel when a mirror symmetry of the pattern is broken in a bifurcation. Here we study symmetry-breaking instabilities of convection with a square planform. A pitchfork bifurcation leads to squares that travel uniformly, while a Hopf bifurcation leads to a new class of oscillations in which squares drift to and fro but with no net motion of the pattern. Two types of travelling squares are possible after a pitchfork bifurcation, and three or more oscillatory solutions are created in a Hopf bifurcation. One of the three oscillations, alternating pulsating waves, has been observed in recent numerical simulations of convection in the presence of a magnetic field. We also present a low-order model of three-dimensional compressible convection that contains these symmetry-breaking instabilities. Our analysis clarifies the relationship between several types of time-dependent patterns that have been observed in numerical simulations of convection

Global bifurcations in the Takens-Bogdanov normal form with D_4 symmetry near the O(2) limit

The dynamics of the normal form of the Takens-Bogdanov bifurcation with D_4 symmetry is governed by a one-dimensional map near the gluing bifurcation and near the O(2) integrable limit, rather than the three-dimensional map one would expect. This great simplification allows a quantitative description of the bifurcation sequence through which stability is transfered between invariant subspaces

Travelling and standing waves in magnetoconvection

The problem of Boussinesq magnetoconvection with periodic boundary conditions is studied using standard perturbation techniques. It is fbund that either travelling waves or standing waves can be stable at the onset of oscillatory convection, depending on the parameters of the problem. When travelling waves occur, a steady shearing flow is present that is quadratic in the amplitude of the convective flow. The weakly nonlinear predictions are confirmed by comparison with numerical solutions of the full partial differential equations at Rayleigh numbers 10% above critical. Modulated waves (through which stability is transferred between travelling and standing waves) are found near the boundary between the regions in parameter space where travelling waves and standing waves are preferred

Convergence properties of the 8, 10 and 12 mode representations of quasipatterns

Spatial Fourier transforms of quasipatterns observed in Faraday wave experiments suggest that the patterns are well represented by the sum of 8, 10 or 12 Fourier modes with wavevectors equally spaced around a circle. This representation has been used many times as the starting point for standard perturbative methods of computing the weakly nonlinear dependence of the pattern amplitude on parameters. We show that nonlinear interactions of n such Fourier modes generate new modes with wavevectors that approach the original circle no faster than a constant times n^-2, and that there are combinations of modes that do achieve this limit. As in KAM theory, small divisors cause difficulties in the perturbation theory, and the convergence of the standard method is questionable in spite of the bound on the small divisors. We compute steady quasipattern solutions of the cubic Swift-Hohenberg equation up to 33rd order to illustrate the issues in some detail, and argue that the standard method does not converge sufficiently rapidly to be regarded as a reliable way of calculating properties of quasipatterns

Cycling chaotic attractors in two models for dynamics with invariant subspaces

Nonergodic attractors can robustly appear in symmetric systems as structurally stable cycles between saddle-type invariant sets. These saddles may be chaotic giving rise to 'cycling chaos'. The robustness of such attractors appears by virtue of the fact that the connections are robust within some invariant subspace. We consider two previously studied examples and examine these in detail for a number of effects: (i) presence of internal symmetries within the chaotic saddles, (ii) phase-resetting, where only a limited set of connecting trajectories between saddles are possible and (iii) multistability of periodic orbits near bifurcation to cycling attractors. The first model consists of three cyclically coupled Lorenz equations and was investigated first by Dellnitz et al. (1995). We show that one can find a 'false phase-resetting' effect here due to the presence of a skew product structure for the dynamics in an invariant subspace; we verify this by considering a more general bi-directional coupling. The presence of internal symmetries of the chaotic saddles means that the set of connections can never be clean in this system, that is, there will always be transversely repelling orbits within the saddles that are transversely attracting on average. Nonetheless we argue that 'anomalous connections' are rare. The second model we consider is an approximate return mapping near the stable manifold of a saddle in a cycling attractor from a magnetoconvection problem previously investigated by two of the authors. Near resonance, we show that the model genuinely is phase-resetting, and there are indeed stable periodic orbits of arbitrarily long period close to resonance, as previously conjectured. We examine the set of nearby periodic orbits in both parameter and phase space and show that their structure appears to be much more complicated than previously suspected. In particular, the basins of attraction of the periodic orbits appear to be pseudo-riddled in the terminology of Lai (2001)

Chaos in models of double convection

ln certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosotutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear

Solar magnetoconvection

In recent years the study of how magnetic fields interact with thermal convection in the Sun has made significant advances. These are largely due to the rapidly increasing computer power and its application to more physically relevant parameters regimes and to more realistic physics and geometry in numerical models. Here we present a survey of recent results following one line of investigations and discuss and compare the results of these with observed phenomena

Spatial period-multiplying instabilities of hexagonal Faraday waves

A recent Faraday wave experiment with two-frequency forcing reports two types of `superlattice' patterns that display periodic spatial structures having two separate scales. These patterns both arise as secondary states once the primary hexagonal pattern becomes unstable. In one of these patterns (so-called `superlattice-II') the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2sqrt{3} from the original scale of the hexagons. In contrast, the time-averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale (sqrt{3} larger than the original scale) and apparently has 60 degree rotation symmetry. We present a symmetry-based approach to the analysis of this bifurcation. Taking as our starting point only the observed instantaneous symmetry of the superlattice-II pattern presented in and the subharmonic nature of the secondary instability, we show (a) that the superlattice-II pattern can bifurcate stably from standing hexagons; (b) that the pattern has a spatio-temporal symmetry not reported in [1]; and (c) that this spatio-temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time-averaged pattern, but not for the apparent 60 degree rotation symmetry. The approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns, and does not rely on the primary pattern having small amplitude

Phase resetting effects for robust cycles between chaotic sets

In the presence of symmetries or invariant subspaces, attractors in dynamical systems can become very complicated owing to the interaction with the invariant subspaces. This gives rise to a number of new phenomena including that of robust attractors showing chaotic itinerancy. At the simplest level this is an attracting heteroclinic cycle between equilibria, but cycles between more general invariant sets are also possible. This paper introduces and discusses an instructive example of an ODE where one can observe and analyse robust cycling behaviour. By design, we can show that there is a robust cycle between invariant sets that may be chaotic saddles (whose internal dynamics correspond to a Rossler system), and/or saddle equilibria. For this model, we distinguish between cycling that include phase resetting connections (where there is only one connecting trajectory) and more general non-phase resetting cases where there may be an infinite number (even a continuum) of connections. In the non-phase resetting case there is a question of connection selection: which connections are observed for typical attracted trajectories? We discuss the instability of this cycling to resonances of Lyapunov exponents and relate this to a conjecture that phase resetting cycles typically lead to stable periodic orbits at instability whereas more general cases may give rise to `stuck on' cycling. Finally, we discuss how the presence of positive Lyapunov exponents of the chaotic saddle mean that we need to be very careful in interpreting numerical simulations where the return times become long; this can critically influence the simulation of phase-resetting and connection selection

Destabilization by noise of tranverse perturbations to heteroclinic cycles: a simple model and an example from dynamo theory

We show that transverse perturbations from structurally stable heteroclinic cycles can be destabilized by surprisingly small amounts of noise, even when each individual fixed point of the cycle is stable to transverse modes. A condition that favours this process is that the linearization of the dynamics in the transverse direction be characterized by a non-normal matrix. The phenomenon is illustrated by a simple two-dimensional switching model and by a simulation of a convectively driven dynamo