279 research outputs found

### Pattern formation in large domains

Pattern formation is a phenomenon that arises in a wide variety of physical, chemical
and biological situations. A great deal of theoretical progress has been made in
understanding the universal aspects of pattern formation in terms of amplitudes of
the modes that make up the pattern. Much of the theory has sound mathematical
justification, but experiments and numerical simulations over the last decade have
revealed complex two-dimensional patterns that do not have a satisfactory theoretical
explanation. This paper focuses on quasi-patterns, where the appearance of small
divisors causes the standard theoretical method to fail, and ends with a discussion
of other outstanding problems in the theory of two-dimensional pattern formation
in large domains

### 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.Comment: 14 pages LaTeX, Birkhauser style, 5 figures, submitted to the
proceedings of the conference on Bifurcations, Symmetry and Patterns, held in
Porto, June 200

### Resonance bifurcations of robust heteroclinic networks

Robust heteroclinic cycles are known to change stability in resonance
bifurcations, which occur when an algebraic condition on the eigenvalues of the
system is satisfied and which typically result in the creation or destruction
of a long-period periodic orbit. Resonance bifurcations for heteroclinic
networks are more complicated because different subcycles in the network can
undergo resonance at different parameter values, but have, until now, not been
systematically studied. In this article we present the first investigation of
resonance bifurcations in heteroclinic networks. Specifically, we study two
heteroclinic networks in $\R^4$ and consider the dynamics that occurs as
various subcycles in each network change stability. The two cases are
distinguished by whether or not one of the equilibria in the network has real
or complex contracting eigenvalues. We construct two-dimensional Poincare
return maps and use these to investigate the dynamics of trajectories near the
network. At least one equilibrium solution in each network has a
two-dimensional unstable manifold, and we use the technique developed in [18]
to keep track of all trajectories within these manifolds. In the case with real
eigenvalues, we show that the asymptotically stable network loses stability
first when one of two distinguished cycles in the network goes through
resonance and two or six periodic orbits appear. In the complex case, we show
that an infinite number of stable and unstable periodic orbits are created at
resonance, and these may coexist with a chaotic attractor. There is a further
resonance, for which the eigenvalue combination is a property of the entire
network, after which the periodic orbits which originated from the individual
resonances may interact. We illustrate some of our results with a numerical
example.Comment: 46 pages, 20 figures. Supplementary material (two animated gifs) can
be found on
http://www.maths.leeds.ac.uk/~alastair/papers/KPR_res_net_abs.htm

### Analysis of the shearing instability in nonlinear convection and magnetoconvection

Numerical experiments on two-dimensional convection with or without a vertical magnetic field reveal a bewildering variety of periodic and aperiodic oscillations. Steady rolls can develop a shearing instability, in which rolls turning over in one direction grow at the expense of rolls turning over in the other, resulting in a net shear across the layer. As the temperature difference across the fluid is increased, two-dimensional pulsating waves occur, in which the direction of shear alternates. We analyse the nonlinear dynamics of this behaviour by first constructing appropriate low-order sets of ordinary differential equations, which show the same behaviour, and then analysing the global bifurcations that lead to these oscillations by constructing one-dimensional return maps. We compare the behaviour of the partial differential equations, the models and the maps in systematic two-parameter studies of both the magnetic and the non-magnetic cases, emphasising how the symmetries of periodic solutions change as a result of global bifurcations. Much of the interesting behaviour is associated with a discontinuous change in the leading direction of a fixed point at a global bifurcation; this change occurs when the magnetic field is introduced

### Numerical simulations of sunspots

The origin, structure and evolution of sunspots are investigated using a numerical model. The compressible MHD equations are solved with physical parameter values that approximate the top layer of the solar convection zone. A three dimensional (3D) numerical code is used to solve the set of equations in cylindrical geometry, with the numerical domain in the form of a wedge. The linear evolution of the 3D solution is studied by perturbing an axisymmetric solution in the azimuthal direction. Steady and oscillating linear modes are obtained

### Bifurcations 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. This leads to the possibility of a doubling of the marginal Floquet multiplier and of bifurcation to two distinct types of drifting solutions. We conclude by proposing a systematic way of analysing steady-state bifurcations of periodic orbits with discrete spatio-temporal symmetries, based on applying the equivariant branching lemma to the irreducible representations of the spatio-temporal symmetry group of the periodic orbit, and on the normal form results of Lamb (1996). This general approach is relevant to other pattern formation problems, and contributes to our understanding of the transition from ordered to disordered behaviour in pattern-forming systems

### Towards Convectons in the Supercritical Regime: Homoclinic Snaking in Natural Doubly Diffusive Convection

Fluids subject to both thermal and compositional variations can undergo
doubly diffusive convection when these properties both affect the fluid density
and diffuse at different rates. A variety of patterns can arise from these
buoyancy-driven flows, including spatially localised states known as
convectons, which consist of convective fluid motion localised within a
background of quiescent fluid. We consider these states in a vertical slot with
the horizontal temperature and solutal gradients providing competing effects to
the fluid density while allowing the existence of a conduction state. In this
configuration, convectons have been studied with specific parameter values
where the onset of convection is subcritical, and the states have been found to
lie on a pair of secondary branches that undergo homoclinic snaking in a
parameter regime below the onset of linear instability. In this paper, we show
that convectons persist into parameter regimes in which the primary bifurcation
is supercritical and there is no bistability, despite coexistence between the
stable conduction state and large-amplitude convection. We detail this
transition by considering spatial dynamics and observe how the structure of the
secondary branches becomes increasingly complex owing to the increased role of
inertia at low Prandtl numbers

### The effect of symmetry breaking on the dynamics near a structurally stable heteroclinic cycle between equilibria and a periodic orbit

The effect of small forced symmetry breaking on the dynamics near a structurally stable heteroclinic
cycle connecting two equilibria and a periodic orbit is investigated. This type of system is known
to exhibit complicated, possibly chaotic dynamics including irregular switching of sign of various
phase space variables, but details of the mechanisms underlying the complicated dynamics have
not previously been investigated. We identify global bifurcations that induce the onset of chaotic
dynamics and switching near a heteroclinic cycle of this type, and by construction and analysis
of approximate return maps, locate the global bifurcations in parameter space. We find there is a
threshold in the size of certain symmetry-breaking terms below which there can be no persistent
switching. Our results are illustrated by a numerical example

### Near-onset dynamics in natural doubly diffusive convection

Doubly diffusive convection is considered in a vertical slot where horizontal
temperature and solutal variations provide competing effects to the fluid
density while allowing the existence of a conduction state. In this
configuration, the linear stability of the conductive state is known, but the
convection patterns arising from the primary instability have only been studied
for specific parameter values. We have extended this by determining the nature
of the primary bifurcation for all values of the Lewis and Prandtl numbers
using a weakly nonlinear analysis. The resulting convection branches are
extended using numerical continuation and we find large-amplitude steady
convection states can coexist with the stable conduction state for sub- and
supercritical primary bifurcations. The stability of the convection states is
investigated and attracting travelling waves and periodic orbits are identified
using time-stepping when these steady states are unstable

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