Spatio-Temporal Pattern Formation

Spatio-temporal patterns appear almost everywhere in nature, and their description and understanding still raise important and basic questions. However, if one looks back 20 or 30 years, definite progress has been made in the modeling of insta­ bilities, analysis of the dynamics in their vicinity, pattern formation and stability, quantitative experimental and numerical analysis of patterns, and so on. Universal behaviors of complex systems close to instabilities have been determined, leading to the wide interdisciplinarity of a field that is now referred to as nonlinear science or science of complexity, and in which initial concepts of dissipative structures or synergetics are deeply rooted. In pioneering domains related to hydrodynamics or chemical instabilities, the interactions between experimentalists and theoreticians, sometimes on a daily basis, have been a key to progress. Everyone in the field praises the role played by the interactions and permanent feedbacks between ex­ perimental, numerical, and analytical studies in the achievements obtained during these years. Many aspects of convective patterns in normal fluids, binary mixtures or liquid crystals are now understood and described in this framework. The generic pres­ ence of defects in extended systems is now well established and has induced new developments in the physics of laser with large Fresnel numbers. Last but not least, almost 40 years after his celebrated paper, Turing structures have finally been ob­ tained in real-life chemical reactors, triggering anew intense activity in the field of reaction-diffusion systems.

1 Introduction.- 2 Instabilities and Patterns in Hydrodynamical Systems.- 2.1 Rayleigh-Bénard instability.- 2.2 Taylor-Couette instability.- 2.3 Liquid crystal instabilities.- 3 Instabilities and Patterns in Reaction-Diffusion Systems.- 3.1 Chemical instabilities.- 3.2 Defect microstructures in irradiated materials.- 3.3 Plastic deformation and dislocation patterns.- 4 Generic Aspects of Pattern-Forming Instabilities.- 4.1 Phenomenology.- 4.2 Reaction-diffusion dynamics and stability.- 4.3 Reduced dynamics and amplitude equations.- 4.4 Spatial patterns: selection and stability.- 4.4.1 Isotropic systems.- 4.4.2 Anisotropic systems.- 4.5 Phase dynamics of periodic patterns.- 4.5.1 Isotropic systems.- 4.5.2 Anisotropic systems.- 5 The Hopf Bifurcation and Related Spatio-Temporal Patterns.- 5.1 The generic aspects of oscillatory media.- 5.1.1 The complex Ginzburg-Landau equation.- 5.1.2 Phase dynamics and spiral waves.- 5.2 Real chemical systems and the complex Ginzburg-Landau equation.- 5.2.1 Determination of the CGLE parameters in real systems.- 5.2.2 CGLE parameters of the BZ reaction.- 5.3 The effect of natural forcings on chemical oscillators.- 5.3.1 The effect of convection on chemical waves.- 5.3.2 The effect of vertical gradients on chemical oscillations.- 5.4 Conclusions.- 6 The Turing Instability and Associated Spatial Structures.- 6.1 The Turing mechanism.- 6.2 The search for Turing structures.- 6.2.1 Convectively driven chemical patterns.- 6.2.2 Double diffusion and chemical fingers.- 6.3 At last, genuine Turing structures?.- 6.4 The interaction between Turing and Hopf instabilities.- 6.4.1 Amplitude equations for Turing-Hopf modes.- 6.4.2 Pattern selection for codimension-2 Turing-Hopf bifurcations.- 6.4.3 Defects, defect bifurcations and localized structures.- 7 Defects and Defect Bifurcations.- 7.1 Generic existence of defects.- 7.2 Examples of defects.- 7.2.1 Codimension-1 defects.- 7.2.2 Codimension-2 defects.- 7.3 Defects and disorder.- 7.4 Bifurcation of defects.- 8 The Effect of External Fields.- 8.1 Spatial forcing of stationary patterns.- 8.1.1 Resonant forcings.- 8.1.2 Near-resonant forcings and commensurate incommensurate transitions.- 8.2 Temporal forcing of a Hopf bifurcation.- 8.3 Temporal forcing of one-dimensional wave patterns.- 8.3.1 Pattern selection and defects.- 8.3.2 Experimental observations.- 8.4 Temporal forcing of two-dimensional wave patterns.- 8.4.1 Isotropic systems.- 8.4.2 Anisotropic systems.- 8.5 Spatial forcing of wave patterns.- 8.6 Flow field effects on pattern forming instabilities.- 8.7 The effect of noise on wave patterns.- 8.8 Conclusions.- 9 Fronts.- 9.1 One-dimensional aspects.- 9.1.1 The leading-edge approach.- 9.1.2 Breakdown of the leading-edge approach.- 9.1.3 Envelope fronts.- 9.1.4 Multiple fronts.- 9.2 Two-dimensional aspects.- 9.2.1 Propagation of roll patterns.- 9.2.2 Propagation of hexagonal patterns.- 10 Pattern Formation: Generic versus Nongeneric Aspects.- 10.1 Kinetic coefficients.- 10.2 Nongradient dynamics.- 10.3 Experimental set-ups.- 11 Microstructures in Irradiated Materials.- 11.1 Particle irradiation of metals and alloys.- 11.1.1 A rate theory model for microstructure evolution under irradiation.- 11.1.2 Dislocation loop dynamics.- 11.1.3 The linear stability analysis.- 11.1.4 The weakly nonlinear regime.- 11.1.5 Numerical analysis.- 11.1.6 Conclusions.- 11.2 Laser induced deformation of surfaces.- 11.2.1 Thin film dynamics under laser irradiation.- 11.2.2 Linear stability analysis.- 11.2.3 Weakly nonlinear analysis.- 11.2.4 Finite size effects.- 11.2.5 Conclusions.- 12 Plastic Instabilities.- 12.1 Dislocation dynamics and rate equations.- 12.2 Stability analysis and bifurcations.- 12.3 Nonlinear analysis.- 12.4 Multiple slip.- 13 Afterword.- 14 Appendices.- 14.1 Bifurcations and normal forms.- 14.1.1 Bifurcations.- 14.1.2 Stability of equilibria.- 14.1.3 Lyapounov functions.- 14.1.4 Typical bifurcation examples.- 14.1.5 The Hopf bifurcation theorem.- 14.1.6 The Center Manifold Theorem.- 14.2 More about dynamical models.- 14.2.1 Proctor-Sivashinsky.- 14.2.2 Ginzburg-Landau.- 14.3 The Brusselator: A toy model for pattern formation in RD systems.- 14.3.1 The Turing instability.- 14.3.2 The Hopf Instability.- 14.3.3 Amplitude equations for Turing-Hopf modes.- 14.4 Resonant forcings of nonlinear oscillators.- 14.4.1 Parametric forcing.- 14.4.2 Strong resonances in spatially extended systems.- 14.4.3 Stationary solutions and resonance horns.- 14.4.4 From oscillations to excitability.