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UNIVERSITY OF SURREY An Isaac Newton Institute Satellite Meeting on Theoretical Aspects of Pattern Formation 19-23 September 2005 |
| Blömker, D |
| Stochastic modulation equations |
| We consider as an example the Swift-Hohenberg equation on a large (but still bounded) domain near its change of stability. This equation is a toy model for the Rayleigh-Bénard convection, and it is well known that sufficiently close to the bifurcation, solutions can be approximated by a periodic wave, which is modulated by the solutions of a Ginzburg-Landau equation. Noise, for instance induced by thermal fluctuations, is natural for physical models like these. We discuss how noise in the equation effects the approximation, and give rigorous error estimates. This approximation also extends to long time behaviour given by invariant measures, and has applications in pattern formation below the threshold of instability. |
| Cross, M |
| Pattern formation and dynamics in Rayleigh-Bénard convection |
| I will discuss some outstanding challenges in understanding the formation and dynamics of two dimensional patterns, partially motivated by our numerical simulations of Rayleigh-Bénard convection in realistic experimental geometries. I will discuss the search for a reduced amplitude equation description, wavenumber selection, coarsening, and the spatiotemporal chaos observed at threshold in rotating convection. |
| Ercolani, N |
| Defect formation in convective patterns as seen from variational models with twist |
| This talk will survey some recent work related to variational models of defect formation in pattern forming systems. The particular model we consider is the Regularized Cross-Newell Phase Diffusion Equation. Our particular focus will be on special (symmetry based) extensions of this model which incorporate twist; i.e., the variational vector fields we consider will only be locally (not globally) gradient fields. We show that the energy minimizers for models allowing twist differ from those which restrict variations to be globally gradient. Analytical construction of test functions for the extended model shows many features that are consistent with physical and numerical experiments on Rayleigh-Bénard Convection and the Swift-Hohenberg equation. The work discussed represents joint works with R. Indik, A.C. Newell, T. Passot and S. Venkataramani. |
| Kapral, R |
| Spiral waves in chaotic systems |
| Spiral waves are commonly seen in excitable and oscillatory media. Perhaps somewhat surprisingly, they also persist in chaotic media. The structure and dynamics of spiral waves will be described as the medium changes from simple oscillatory dynamics to being strongly chaotic. In some regimes the spiral waves exhibit unusual dynamics. In other regimes defect-mediated turbulence is found with statistical features that differ those in oscillatory media. Spiral waves persist in very strongly chaotic regimes even when it is difficult to define a local phase and eventually break up through mechanisms that differ from those identified in excitable or oscillatory media. |
| Knobloch, E |
| Dynamics of nearly inviscid Faraday waves in almost circular containers |
| In the nearly inviscid regime parametrically driven surface gravity-capillary waves couple to a streaming flow driven in oscillatory viscous boundary layers at rigid walls and the free surface; this flow in turn interacts with the waves responsible for the boundary layers in the first place. In small domains the resulting system is described in the weakly nonlinear regime by a pair of amplitude equations coupled to a Navier-Stokes-like equation for the streaming flow with boundary conditions determined by matching to the boundary layers. Properties of this novel pattern-forming system will be described with emphasis on the dynamics in circular and elliptical domains. Among the new dynamical behavior that results are relaxation oscillations involving abrupt transitions between standing and quasiperiodic oscillations, and exhibiting 'canards'. |
| Lamb, J |
| On low speed travelling waves of the Kuramoto-Sivashinsky equation |
| We discuss travelling wave solutions of the Kuramoto-Sivashinsky equation ∂tu+u∂xu+∂xx+∂xxxxu=0. They are described by the Michelson system ∂tttx=c2-x2/2-∂tx, with c representing the wave speed. The Michelson system has a fold-Hopf bifurcation at c=0, corresponding to zero wave speed limit. We discuss this bifurcation as a codimension one phenomenon in the context of reversible volume preserving vector fields in R3. It turns out that this bifurcation point is an accumulation point of so-called T-point homoclinic bifurcations. We discuss how these bifurcations can be studied using Lin's method. Joint work with Kevin Webster (Imperial), Marco-Antonio Teixeira (Campinas) and Juergen Knobloch (Ilmenau). |
| Lega, J |
| Dynamics and growth of bacterial colonies |
| I will present a model for the dynamics and growth of bacterial colonies on soft agar plates. This model consists of a set of advection-reaction-diffusion equations coupled to a hydrodynamic equation for the horizontal velocity field of the mixture of water and bacteria near the surface of the plate. I will show numerical simulations illustrating how the form of the colony, as described by this model, is affected by the initial amount of nutrients and initial wetness of the agar, and compare the results with experiments. Towards the end of the presentation, I will indicate how this model raises interesting questions related to the dynamics of fronts in the presence of nonlinear diffusion, and briefly mention more general issues related to pattern formation in bioconvection and bacterial biofilms. Partly joint work with Thierry Passot. |
| Riecke, H |
| Complex patterns in non-Boussinesq convection |
| I will present results from numerical computations of the full Navier-Stokes equations for non-Boussinesq convection using water and various gases as working fluids, respectively. I will focus on our results for rotating systems for which weakly nonlinear theory predicts a supercritical Hopf bifurcation to oscillating (`whirling') hexagons. Our computations confirm this prediction in the weakly non-Boussinesq case. The resulting defect chaos state is found to be quite well described by the two-dimensional cubic complex Ginzburg-Landau equation. The whirling hexagons constitute therefore one of only a few physically realistic systems that are described by this generic equation in a regime with complex dynamics. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and the dynamics are characterized by spatially localized bursts in the oscillation amplitude. In this regime the coupling of the oscillation amplitude to the slow phase modes of the underlying hexagon pattern becomes significant. To analyze such complex patterns we have started to measure various geometric properties of the contour lines characterizing the patterns. We are first applying this approach to spiral-defect chaos in Boussinesq and non-Boussinesq convection. |
| Schatz, M |
| Transient amplification of rivulets in a thin liquid film: experiment and theory |
| The role played by transient disturbances in flow instabilities is poorly understood for many important problems in hydrodynamics. We present experimental and theoretical results on transient behavior in the temperature-induced surface-tension-driven spreading of a thin liquid film on a horizontal solid substrate. Perturbations with well-defined spatial and temporal characteristics are applied via distributed optical heating of the film prior to instability onset; the corresponding perturbation-induced variations in film thickness are characterized by interferometry. The subsequent evolution of rivulets arising from contact line instability is measured using image time series. Comparison of the initial disturbance to the final disturbance enables quantitative measurement of transient amplification rates; these rates are compared to the predictions of generalized stability theory that accounts both for the initial conditions of the experiments (i.e., the specific structure of the imposed perturbations) and for the non-normal character of the linear operator that governs the evolution of small disturbances. |
| Uecker, H |
| Stability, instability, and transient dynamics in inclined film problems |
| The flow of a liquid film over an inclined plane is governed by a Navier-Stokes problem with a free boundary. Depending on the parameters of the problem the basic Nusselt solution with a parabolic flow profile and a flat free surface is linearly stable or unstable. First we show that in the stable case small amplitude long wave perturbations decay in a universal manner governed by the Burgers equation. Then we discuss (transient) dynamics of surface waves in the unstable case, which are governed by a Korteweg-de Vries--Kuramoto-Sivashinsky equation. Finally we give an outlook on wave patterns which occur for the inclined film flow over wavy bottoms. |
| Wulff, C |
| A Hamiltonian analogue of the meandering transition |
| In this talk we compare the generic bifurcation behaviour of relative equilibria in dissipative and Hamiltonian systems with Euclidean symmetry. In particular we present a Hamiltonian analogue of the well-known meandering transition from rotating waves to modulated rotating and travelling waves in dissipative systems. In the dissipative case this transition is caused by varying external parameters such that a Hopf bifurcation in a corotating frame occurs. It is a well-known bifurcation of spiral waves in reaction-diffusion systems. In the Hamiltonian case the conserved quantities of the system like angular, linear momentum and energy are bifurcation parameters. We will see, that in contrast to the dissipative case, modulated traveling waves are the typical scenario near rotating waves in the energy-momentum parameter space. We will also discuss stability of Hamiltonian rotating waves. |
| Zelik, S |
| Multipulse structures and Sinai-Bunimovich space-time chaos in dissipative PDEs |
| We consider a general semilinear parabolic PDE in Rn which allows at least one pulse equilibrium. Under the natural assumptions on this pulse, we prove that the weak interaction between infinitely many well-separated shifted copies of the initial pulse can be described by the appropriate lattice system of ODEs. We also find an asymptotical form of that equations as the distance between pulses tends to infinity and compute it explicitly for a number of equations of mathematical physics — Ginzburg-Landau, Swift-Hohenberg equations, etc. Finally, applying this result to the 1D Swift-Hohenberg equation with a small space-time periodic external force, we construct a special multipulse structure such that weak pulse interaction in it is described by a lattice of ODEs of Sinai-Bunimovich type and, thus, verify the existence of Sinai-Bunimovich space-time chaos in that equation. |