Theoretical Aspects of Pattern Formation University of Surrey

Transcription

1 Theoretical Aspects of Pattern Formation University of Surrey September 2005 Isaac Newton Institute

2 Programme Monday 19 September 12:30-14:30 Registration and Lunch at the Wates House 14:30-15:30 M Schatz (Georgia Tech) Transient amplification of rivulets in a thin liquid film: experiment and theory 15:30-16:00 Tea 16:00-17:00 P Ashwin (Exeter) Local and global instabilities of patterns 17:00-18:00 M Cross (Caltech) Pattern formation and dynamics in Rayleigh-Bénard convection Tuesday 20 September IAS Seminar 10:00-11:00 G Ahlers (UCSB) The legacy of Henri Bénard and Lord Rayleigh 11:00-11:30 Coffee 11:30-12:30 CE Wayne (Boston) Long-time asymptotics and coherent structures in Navier-Stokes flows in R 2 and R 3 12:30-14:30 Lunch at the Wates House 14:30-15:30 D Barkley (Warwick) Pattern formation in turbulent flow 15:30-16:30 A Scheel (Minneapolis) Following coherent structures 16:30-17:00 Tea 17:00-18:00 A Newell (Arizona) Patterns on plants 19:00-22:00 Conference dinner at the Lakeside Restaurant

3 Wednesday 21 September 9:00-10:00 E Knobloch (Berkeley) Dynamics of nearly inviscid Faraday waves in almost circular containers 10:00-10:30 Coffee 10:30-11:30 R Kapral (Toronto) Spiral waves in chaotic systems 11:30-12:30 C Wulff (Surrey) A Hamiltonian analogue of the meandering transition 12:30-14:30 Lunch at the Wates House 14:30-17:30 Poster session Thursday 22 September 9:00-10:00 S Zelik (Stuttgart) Multipulse structures and Sinai-Bunimovich space-time chaos in dissipative PDEs 10:00-10:30 Coffee 10:30-11:30 H Uecker (Karlsruhe) Stability, instability, and transient dynamics in inclined film problems 11:30-12:30 J Lamb (Imperial) On low speed travelling waves of the Kuramoto-Sivashinsky equation 12:30-14:30 Lunch at the Wates House 14:30-15:30 M Silber (Northwestern) Controlling Faraday waves with multi-frequency forcing 15:30-16:00 Tea 16:00-17:00 N Ercolani (Arizona) Defect formation in convective patterns as seen from variational models with twist Friday 23 September 9:00-10:00 H Riecke (Northwestern) Complex patterns in non-boussinesq convection 10:00-10:30 Coffee 10:30-11:30 D Blömker (Aachen) Stochastic modulation equations 11:30-12:30 J Lega (Arizona) Dynamics and growth of bacterial colonies 12:30-14:30 Lunch at the Wates House

4 Abstracts IAS Seminar Ahlers, G The legacy of Henri Bénard and Lord Rayleigh This talk will trace the major breakthroughs in the study of Rayleigh-Bénard convection since the experiments of Henri Bénard a century ago. A few selected topics of current interest will be presented in somewhat more detail. These include the effect of thermal noise on the bifurcation to convection and spatiotemporal chaos. Newell, A Patterns on plants Plant patterns, namely the tiling of plant surfaces into regular polygonal shapes, and plant phyllotaxis, the arrangements of leaves and stickers, has fascinated natural scientists for over four hundred years. In this lecture, I will outline some of the ideas of Patrick Shipman and myself (about to appear in J. Theor. Bio.) which provide a rational explanation for many observations including the prevalence of Fibonacci sequences of spiral families. Scheel, A Following coherent structures Spiral waves, sources and sinks, and line defects, are examples of localized structures embedded in a background of waves with characteristic wavenumbers. We will discuss how these structures change as system parameters are varied. Although coherent structures often appear to be very robust, the generic persistence already turns out to be a delicate question due to essential spectrum caused by the background waves. We show how methods from dynamical systems can be used to prove robustness and bifurcation theorems, hence enabling a pathfollowing type exploration of large-scale nonequilibrium systems. We also present examples where our approach exhibits structural barriers to pathfollowing: coherent structures can suddenly disappear. Wayne, CE Long-time asymptotics and coherent structures in Navier-Stokes flows in R 2 and R 3 I will describe how one can combine ideas from dynamical systems theory and kinetic theory to describe the long-time behavior of solutions of the Navier-Stokes equations. In two dimensions this leads to a very complete description of the behavior of solutions whose initial vorticity is at least slightly localized. In three dimensions it gives a better understanding of the existence of the Burgers vortex and its variants.

5 Abstracts 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 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=c 2 -x 2 /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 R 3. 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.

6 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 R n 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.

7 Poster Session Presenter A Comanici M Cristina Depassier Y Kyrychko A Nepomnyashchy O Nekhamina J Scheel N Venkov Title Hopf bifurcation of relative equilibria in spherical geometry On the transition from pulled to pushed fronts for the extended Fisher Kolmogorov equation t u = xx u - γ xxxx u + f(u) Pattern formation in a delayed predator-prey model with non-monotonic functional response Pattern formation under feedback control in models of directional solidification Period-adding bifurcations and moving waves in excitable systems subject to inhomogeneous boundary conditions Scaling laws for rotating Rayleigh-Bénard convection Neural field model with axonal time-delay: generation of travelling Turing patterns

8 General Information access: Each participant will be provided with a guest account that allows you to connect to the internet from any computer in the following labs ( 30AA04 translates into room 30 on the 4th floor in the AA building): 30AA04 34BB04 34aBB04 32BB03 34BB03 ( 5 computers only) (swan lab) (duck lab) (penguin lab) (whale lab) The BB labs are open from 7:30am until 19:00pm. Some of the BB labs will be used for computer inductions during the week, so you may have to use the smaller lab in 30AA04. We have reserved 34BB04: M-F 8-9am and M-W, F 12-2pm 34BB03: Thursday 2-4pm for our exclusive use. You can also use the wireless network on AA04, BB03 and BB04 with your laptop: Network name/ssid: SCSECM01 Network settings obtained via DHCP Open your browser, go to any website, and type in your login information on the login page that will appear in your browser. Library access: To use the library, you need to sign in at the registration desk. Copy machines can be found on every floor. To use them, you need to purchase a re-chargable copy card from the Lending Services desk on the ground Floor which costs 2.00 and gives 20 photocopying credits. Additional credits may be added to the copy card at the re-charging machines on the ground floor and 1st floor. No refunds will be given for unused credits. Lunch: Lunch will be served in the Wates House from 12:30-14:30.