arXiv subjects

A. M. Rucklidge

Publications and source records attributed to A. M. Rucklidge.

24 records · Page 2Linked to original sources

arXiv2003

Boundary effects and the onset of Taylor vortices

It is well established that the onset of spatially periodic vortex states in the Taylor-Couette flow between rotating cylinders occurs at the value of Reynold's number predicted by local bifurcation theory. However, the symmetry breaking induced by the top and bottom plates means that the true situation should be a disconnected pitchfork. Indeed, experiments have shown that the fold of the disconnected branch can occur at more than double the Reynold's number of onset. This leads to an apparent contradiction: why should Taylor vortices set in so sharply at the value Reynold's number predicted by the symmetric theory, given such large symmetry-breaking effects caused by the boundary conditions? This paper offers a generic explanation. The details are worked out using a Swift-Hohenberg pattern formation model that shares the same qualitative features as the Taylor-Couette flow. Onset occurs via a wall mode whose exponential tail penetrates further into the bulk of the domain as the driving parameter increases. In a large domain of length L, we show that the wall mode creates significant amplitude in the centre at parameter values that are O(L^{-2}) away from the value of onset in the problem with ideal boundary conditions. We explain this as being due to a Hamiltonian Hopf bifurcation in space, which occurs at the same parameter value as the pitchfork bifurcation of the temporal dynamics. The disconnected anomalous branch remains O(1) away from the onset parameter since it does not arise as a bifurcation from the wall mode.

nlin.PS↗

arXiv2002

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.

nlin.PS↗

arXiv2001

Secondary instabilities of hexagons: a bifurcation analysis of experimentally observed Faraday wave patterns

We examine three experimental observations of Faraday waves generated by two-frequency forcing, in which a primary hexagonal pattern becomes unstable to three different superlattice patterns. We use the symmetry-based approach developed by Tse et al. to analyse the bifurcations involved in creating the three new patterns. Each of the three examples reveals a different situation that can arise in the theoretical analysis.

nlin.PS↗

arXiv2000

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 [1]. 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 [1] 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.

nlin.PS↗

arXiv1997

Cycling chaos: its creation, persistence and loss of stability in a model of nonlinear magnetoconvection

We examine a model system where attractors may consist of a heteroclinic cycle between chaotic sets; this `cycling chaos' manifests itself as trajectories that spend increasingly long periods lingering near chaotic invariant sets interspersed with short transitions between neighbourhoods of these sets. This behaviour can be robust (i.e., structurally stable) for systems with symmetries and provides robust examples of non-ergodic attractors in such systems; we examine bifurcations of this state. We discuss a scenario where an attracting cycling chaotic state is created at a blowout bifurcation of a chaotic attractor in an invariant subspace. This is a novel scenario for the blowout bifurcation and causes us to introduce the idea of set supercriticality to recognise such bifurcations. The robust cycling chaotic state can be followed to a point where it loses stability at a resonance bifurcation and creates a series of large period attractors. The model we consider is a 9th order truncated ODE model of three-dimensional incompressible convection in a plane layer of conducting fluid subjected to a vertical magnetic field and a vertical temperature gradient. Symmetries of the model lead to the existence of invariant subspaces for the dynamics; in particular there are invariant subspaces that correspond to regimes of two-dimensional flows. Stable two-dimensional chaotic flow can go unstable to three-dimensional flow via the cross-roll instability. We show how the bifurcations mentioned above can be located by examination of various transverse Liapunov exponents. We also consider a reduction of the ODE to a map and demonstrate that the same behaviour can be found in the corresponding map.

chao-dyn↗

arXiv1997

Instabilities of periodic orbits with spatio-temporal symmetries

Motivated by recent analytical and numerical work on two- and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional pulsating waves (PW) and three-dimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z_n, n=2 (PW) and n=4 (APW). These symmetries force the Poincare return map M to be the nth iterate of a map G: M=G^n. The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. When the bifurcation breaks these reflections, the map G has a "two-symmetry", as analysed by Lamb (1996). This leads to a doubling of the marginal Floquet multiplier and the possibility of bifurcation to two distinct types of drifting solutions.

patt-sol↗