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

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

Multi-parameter models of innovation diffusion on complex networks

A model, applicable to a range of innovation diffusion applications with a strong peer to peer component, is developed and studied, along with methods for its investigation and analysis. A particular application is to individual households deciding whether to install an energy efficiency measure in their home. The model represents these individuals as nodes on a network, each with a variable representing their current state of adoption of the innovation. The motivation to adopt is composed of three terms, representing personal preference, an average of each individual's network neighbours' states and a system average, which is a measure of the current social trend. The adoption state of a node changes if a weighted linear combination of these factors exceeds some threshold. Numerical simulations have been carried out, computing the average uptake after a sufficient number of time-steps over many realisations at a range of model parameter values, on various network topologies, including random (Erdos-Renyi), small world (Watts-Strogatz) and (Newman's) highly clustered, community-based networks. An analytical and probabilistic approach has been developed to account for the observed behaviour, which explains the results of the numerical calculations

Database analysis of children and adolescents with Bipolar Disorder consuming a micronutrient formula

<p>Abstract</p> <p>Background</p> <p>Eleven previous reports have shown potential benefit of a 36-ingredient micronutrient formula (known as EMPowerplus) for the treatment of psychiatric symptoms. The current study asked whether children (7-18 years) with pediatric bipolar disorder (PBD) benefited from this same micronutrient formula; the impact of Attention-Deficit/Hyperactivity Disorder (ADHD) on their response was also evaluated.</p> <p>Methods</p> <p>Data were available from an existing database for 120 children whose parents reported a diagnosis of PBD; 79% were taking psychiatric medications that are used to treat mood disorders; 24% were also reported as ADHD. Using Last Observation Carried Forward (LOCF), data were analyzed from 3 to 6 months of micronutrient use.</p> <p>Results</p> <p>At LOCF, mean symptom severity of bipolar symptoms was 46% lower than baseline (effect size (ES) = 0.78) (<it>p </it>< 0.001). In terms of responder status, 46% experienced >50% improvement at LOCF, with 38% still taking psychiatric medication (52% drop from baseline) but at much lower levels (74% reduction in number of medications being used from baseline). The results were similar for those with both ADHD and PBD: a 43% decline in PBD symptoms (ES = 0.72) and 40% in ADHD symptoms (ES = 0.62). An alternative sample of children with just ADHD symptoms (n = 41) showed a 47% reduction in symptoms from baseline to LOCF (ES = 1.04). The duration of reductions in symptom severity suggests that benefits were not attributable to placebo/expectancy effects. Similar findings were found for younger and older children and for both sexes.</p> <p>Conclusions</p> <p>The data are limited by the open label nature of the study, the lack of a control group, and the inherent self-selection bias. While these data cannot establish efficacy, the results are consistent with a growing body of research suggesting that micronutrients appear to have therapeutic benefit for children with PBD with or without ADHD in the absence of significant side effects and may allow for a reduction in psychiatric medications while improving symptoms. The consistent reporting of positive changes across multiple sites and countries are substantial enough to warrant a call for randomized clinical trials using micronutrients.</p

Chaos in magnetoconvection

The partial differential equations (PDEs) for two-dimensional incompressible convection in a strong vertical magnetic field have a codimension-three bifurcation when the parameters are chosen so that the bifurcations to steady and oscillatory convection coincide and the limit of narrow rolls is taken. The third-order set of ordinary differential equations (ODEs) that govern the behaviour of the PDEs near this bifurcation are derived using perturbation theory. These ODEs are the normal form of the codimension-three bifurcation; as such, they prove to be an excellent predictor of the behaviour of the PDEs. This is the first time that a detailed comparison has been made between the chaotic behaviour of a set of PDEs and that of the corresponding set of model ODEs, in a parameter regime where the ODEs are expected to provide accurate approximations to solutions of the PDEs. Most significantly, the transition from periodic orbits to a chaotic Lorenz attractor predicted by the ODEs is recovered in the PDEs, making this one of the few situations in which the nature of chaotic oscillations observed numerically in PDEs can be established firmly. Including correction terms obtained from the perturbation calculation enables the ODEs to track accurately the bifurcations in the PDEs over an appreciable range of parameter values. Numerical calculations suggest that the T-point (where there are heteroclinic connections between a saddle point and a pair of saddle-foci), which is associated with the transition from a Lorenz attractor to a quasi-attractor in the normal form, is also found in the PDEs. Further numerical simulations of the PDEs with square rolls confirm the existence of chaotic oscillations associated with a heteroclinic connection between a pair of saddle-foci

Modeling the Subsurface Structure of Sunspots

While sunspots are easily observed at the solar surface, determining their subsurface structure is not trivial. There are two main hypotheses for the subsurface structure of sunspots: the monolithic model and the cluster model. Local helioseismology is the only means by which we can investigate subphotospheric structure. However, as current linear inversion techniques do not yet allow helioseismology to probe the internal structure with sufficient confidence to distinguish between the monolith and cluster models, the development of physically realistic sunspot models are a priority for helioseismologists. This is because they are not only important indicators of the variety of physical effects that may influence helioseismic inferences in active regions, but they also enable detailed assessments of the validity of helioseismic interpretations through numerical forward modeling. In this paper, we provide a critical review of the existing sunspot models and an overview of numerical methods employed to model wave propagation through model sunspots. We then carry out an helioseismic analysis of the sunspot in Active Region 9787 and address the serious inconsistencies uncovered by \citeauthor{gizonetal2009}~(\citeyear{gizonetal2009,gizonetal2009a}). We find that this sunspot is most probably associated with a shallow, positive wave-speed perturbation (unlike the traditional two-layer model) and that travel-time measurements are consistent with a horizontal outflow in the surrounding moat.Comment: 73 pages, 19 figures, accepted by Solar Physic

Nearly inviscid Faraday waves

Many powerful techniques from Hamiltonian mechanics are available for the study of ideal hydrodynamics. This article explores some of the consequences of including small viscosity in a study of surface gravity-capillary waves excited by the vertical vibration of a container. It is shown that in this system, as in others, the addition of small viscosity provides a singular perturbation of the ideal fluid system, and that as a result its effects are nontrivial. The relevance of existing studies of ideal fluid problems is discussed from this point of view

How Distinctive are ADHD and RD? Results of a Double Dissociation Study

The nature of the comorbidity between Attention-Deficit/Hyperactivity Disorder (ADHD) and Reading Disability (RD) was examined using a double dissociation design. Children were between 8 and 12 years of age and entered into four groups: ADHD only (n = 24), ADHD+RD (n = 29), RD only (n = 41) and normal controls (n = 26). In total, 120 children participated in the study; 38 girls and 82 boys. Both ADHD and RD were associated with impairments in inhibition and lexical decision, although inhibition and lexical decision were more severely impaired in RD than in ADHD. Visuospatial working memory deficits were specific to children with only ADHD. It is concluded that there was overlap on lexical decision and to a lesser extent on inhibition between ADHD and RD. In ADHD, impairments were dependent on IQ, which suggest that the overlap in lexical decision and inhibition is different in origin for ADHD and RD. The ADHD only group was specifically characterized by deficits in visuospatial working memory. Hence, no double dissociation between ADHD and RD was found on executive functioning and lexical decision