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Modeling spatio-temporal pattern formation in complex systems far from thermal equilibrium has long been considered as a challenge. Fortunately, during the last decade, a unified framework, which allows the study of generic aspects of pattern formation phenomena, emerged from intensive theoretical and experimental research. It has been successfully applied in several domains, from hydrodynamics to chemistry and nonlinear optics, and, more recently, to materials instabilities. It is shown here, how the methods of nonlinear dynamics may be used to establish conditions for instability in dynamical models for surface processing. Their application to the determination of selected patterns in post-bifurcation regimes is illustrated in a dynamical description of laser-induced thin film deformation.

A dynamical model of the Swift-Hohenberg type is proposed to describe the formation of twelvefold quasipattern as observed, for instance, in optical systems. The model incorporates the general mechanisms leading to quasipattern formation and does not need external forcing to generate them. Besides quadratic nonlinearities, the model takes into account an angular dependence of the nonlinear couplings between spatial modes with different orientations. Furthermore, the marginal stability curve presents other local minima than the one corresponding to critical modes, as usual in optical systems. Quasipatterns form when one of these secondary minima may be associated with harmonics built on pairs of critical modes. The model is analyzed numerically and in the framework of amplitude equations. The results confirm the importance of harmonics to stabilize quasipatterns and assess the applicability of the model to other systems with similar generic properties. © 2014 American Physical Society., We acknowledge useful discussions with P. Colet, and financial support from FEDER and MINECO (Spain), through Grant No. FIS2012-30634 INTENSE@COSYP, and from Comunitat Autónoma de les Illes Balears.

In this paper, we analyze the onset of phase-dominant dynamics in a uniformly forced system. The study is based on the numerical integration of the Swift–Hohenberg equation and adresses the characterization of phase disorder detected from gradient computational operators as complex entropic form (CEF). The transition from amplitude to phase dynamics is well characterized by means of the variance of the CEF phase component.

Recent computer simulations on dislocation patterning have provided remarkable results in accordance with empirical laws. Moreover, several analytical models on dislocation dynamics have provided qualitative insight on dislocation patterning. However, a model, based on partial differential equations, which gives a dynamical evolution of dislocation patterns in function of measurable variables still missing. Here, we give a re-formulation of a model proposed some years ago. From this formulation, we obtained that the onset of a dislocation instability is related to the applied stress. The analytical and numerical results reported are partial and studies on this direction are under development.

This volume is devoted to the presentation of the main contributions to the workshop ‘‘From oscillations to excitability: A case study in spatially extended systems,’’ organized by the authors in Nice in June 1993. It gives an overview of the current research on spatiotemporal patterns in a wide range of systems that display self‐oscillatory or excitable behavior. It tries to give a better understanding of the transition from the oscillatory to the excitable regime and of its effect on the properties of spiral waves, and to fill the gap between the theories and concepts used to describe both regimes in the so‐called ‘‘active media.’’

Spatio-temporal pattern formation in physico-chemical systems far from thermal equilibrium has long been a puzzling phenomenon. Until the last decade, understanding pattern selection and stability mechanisms was considered as a challenge. Fortunately, thanks to intensive theoretical and experimental research, a unified framework is now available to study pattern formation phenomena. It has been successfully applied to several systems, in different fields, such as hydrodynamics, chemistry, and nonlinear optics. They are now being applied to various types of materials instabilities, and will hopefully lead to a better understanding of phenomena such as the formation and evolution of defect microstructures in plastically deformed or irradiated materials, the formation and symmetries of regular deformation patterns in surfaces and thin films under laser irradiation, the role and the control of instabilities in surface modification technologies, etc. In this context, defect microstructures appear as the result of defect motion and nonlinear interactions, which naturally destabilize uniform distributions. The applicability of the methods of nonlinear dynamics to materials instabilities is analyzed, and an appropriate methodology is proposed. The importance of nonlinear analysis beyond instability thresholds in the determination of pattern selection and stability is emphasized. Several examples are discussed, with references to relevant reviews and technical publications.

It is now well documented that, despite their complexity, the dynamics of physicochemical systems driven away from thermal equilibrium may be reduced, close to bifurcation points, to much simpler forms describing the universal properties of the spatio-temporal patterns. One of the most celebrated instabilities in this framework corresponds to the Hopf bifurcation that induces oscillations of the limit cycle type, which are of particular interest in nonlinear chemical systems, nonlinear optics and biology. The description of real oscillatory media is based on a combination of generic aspects that depend only on the characteristics of the bifurcation and of the symmetries of the problem and of nongeneric aspects that arise through experimental set-ups, boundary or geometrical effects, and so forth. Hence, it is interesting to discuss both aspects with the simplest description of oscillatory media close to a supercritical Hopf bifurcation, namely the complex Ginzburg-Landau equation.

As emphasized earlier in this book, stationary spatial structures may be induced by the interplay between reaction and diffusion in nonlinear chemical systems, and this pattern formation mechanism was proposed by A. Turing as early as 1952 [6.1]. Since then, various kinetic models have been dedicated to the justification of this hypothesis and to the existence of pattern-forming instabilities in these systems [6.2]. Theoretical analysis has shown the existence of universal behaviors that do not depend on the details of the kinetics but on more general properties such as, for example, the presence of autocatalytic interactions between molecules of different mobilities. Here also, the full dynamics may be reduced to much simpler forms near the instability, which capture the dominant effects of the asymptotic behavior of the system and allows a simplified analysis of pattern selection and stability [6.3].

Besides generic properties that only depend on the symmetries of the system, the type of bifurcation encountered, the structure of the dynamics, and so forth, non-equilibrium patterns may however also present specific properties induced by particular dependences of the kinetic coefficients on the control parameters or driving fields, or geometrical effects induced by particular boundary conditions or experimental set-ups. In this chapter I will review some of these aspects that are more specifically related to hydrodynamical or chemical systems.

As discussed in previous chapters, the spontaneous nucleation of spatio-temporal patterns in systems driven far from thermal equilibrium by external constraints is the subject of intensive experimental and theoretical research. These studies, which were at first considered related to fundamental problems, appear now to be also of technological importance, as discussed in previous sections. Despite the complexity of the dynamics which gives rise to this phenomenon, great progress has been achieved in the understanding of pattern formation and stability near instability points where the reduction of the dynamics leads to amplitude equations for the patterns. According to the restricted number of bifurcation types and amplitude equation structures, the way similarities occur between the behaviors of systems of different nature, is reminiscent of the universality classes of phase transitions. It is nevertheless essential to understand how instabilities arising at the microscale are able to organize the system at the macroscale and to determine the stability and selection properties of the patterns for each particular case.

Pattern forming instabilities in spatially extended dissipative systems driven away from equilibrium have been the focus of a large activity for many years. The goal of this chapter is to present some theoretical concepts that have been developed to understand and describe these dissipative structures [1] from a macroscopic point of view. Although these methods present generic features we shall only be concerned with chemical patterning and shall not discuss here instabilities in hydrodynamics, liquid crystals and nonlinear optics that all present similar types of organization because the latter have been the subject of recent well-documented reviews [2–5]. Moreover, we essentially consider the self-organization of structures discarding the spatial patterning resulting from boundary conditions.

Close to a supercritical Hopf bifurcation, oscillatory media may be described, by the complex Ginzburg–Landau equation. The most important spatiotemporal behaviors associated with this dynamics are reviewed here. It is shown, on a few concrete examples, how real chemical oscillators may be described by this equation, and how its coefficients may be obtained from the experimental data. Furthermore, the effect of natural forcings, induced by the experimental realization of chemical oscillators in batch reactors, may also be studied in the framework of complex Ginzburg–Landau equations and its associated phase dynamics. We show, in particular, how such forcings may locally transform oscillatory media into excitable ones and trigger the formation of complex spatiotemporal patterns.

One of the most natural and still intriguing behavior of complex physico-chemical systems driven sufficiently far from thermal equilibrium is their ability to undergo symmetry-breaking instabilities leading to the spontaneous formation of coherent structures over macroscopic time and space scales (Nicolis and Prigogine, 1977). Such a behavior has been widely studied in various fields including physics, biophysics, chemistry and materials science. The question of why order appears spontaneously and which patterns are selected among a large manifold of possibilities remains a major theme of experimental and theoretical research (Swinney and Gollub, 1981, Nicolis and Baras, 1983, Hlavacek, 1985). Even though this research was at first fundamental in nature, it now appears more and more to be of technological importance.

SCOPUS: ar.j, info:eu-repo/semantics/published

A kinetic equation for the photon density matrix of a monomode laser model with explicit heat baths is derived in the framework of the Prigogine-Resibois theory of nonequilibrium statistical mechanics. Approximate solutions are discussed and are found to agree with other theories. The phase transition behaviour and the creation of order which is due to the nonlinear structure of the macroscopic kinetic equations may be interpreted within the more recent theories of far-from-thermal-equilibrium open systems while further study is needed to fill completely the gap between microscopic and macroscopic description.

The effects of inhomogeneous fluctuations on instabilities in various nonlinear chemical models are studied in terms of concepts developed in the theory of equilibrium phase transitions., SCOPUS: NotDefined.j, info:eu-repo/semantics/published

We want to present in this note some topics in the study of stationary twodimensional nonequilibrium structures. Two experiments which have stimulated a reniewed interest in the theory of spatial patterns are described. The problem of pattern selection both in isotropic and anisotropic systems is reviewed. The dynamics of the phase is also presented.

When driven away from equilibrium by the effect of an external constraint, many nonlinear systems present spatio-temporal order beyond a threshold value for the control parameter [1]. At this point, the homogeneous steady state becomes unstable versus inhomogeneous or oscillatory perturbations and various spatial or temporal patterns may nucleate spontaneously in the medium. This has long been a puzzling phenomenon from experimental and theoretical point of views and numerous questions related to the pattern formation, selection and stability mechanisms in physical, chemical and biological systems remain still unanswered. Due to the complexity of the dynamics a complete analytical solution of the kinetic equations governing the behavior of the system is usually impossible. However near threshold great simplifications occur as a consequence of the separation between the time and space scales of the stable and unstable modes [2], The evolution of the system may effectively be reduced to its slow mode dynamics which leads to amplitude equations for the spatio-temporal patterns. This important simplification which reduces the complete set of nonlinear kinetic equations to the kinetics of the unstable modes only is valid and may be justified by multiple scale analysis in the vicinity of the instability [3].

The spontaneous appearance of spatial structures far from thermal equilibrium has long been a puzzling phenomenon both from the experimental and theoretical points of view. Numerous questions related to pattern formation and pattern selection in physical, chemical and biological systems still remain unanswered [1, 2]. Contrary to the case of convective patterns, the spatial structures which appear in chemically active media have often been considered as spurious effects, bad experimentation and aroused at first little interest. For a long time, the activity remained therefore on the theoretical level, in the framework of the theories of dissipative structures and nonequilibrium phase transitions. Fortunately chemical selforganization is now observed in an evergrowing class of systems and its relationship to actual problems from morphogenesis to materials sciences is now well accepted. Moreover, an enormous amount of computational and experimental work is now being reported where chemically active media exhibit multiple steady states, periodic solutions, wave phenomena and mosaic-like patterns and quantitative comparison between theoretical and experimental data will soon be available [3–4].

The spontaneous nucleation of spatial patterns far from thermal equilibrium has long been a puzzling phenomenon both from experimental and theoretical point of views. Numerous questions related to pattern formation and pattern selection in physical, chemical and biological systems remain still unanswered. However, great progress in the understanding of these phenomena have been achieved in the framework of reaction-diffusion equations which are believed to accurately describe many nonequilibrium systems (I). It has also been suggested that, like in equilibrium phase transitions and hydrodynamic instabilities, when chemical spatial or temporal structures appear through the breaking of a continuous symmetry(e.g. translational or rotational symmetry for spatial structures, phase or gauge symmetry for temporal oscillations) they are particularly sensitive to small external fields or to Internal fluctuations (2). Effectively, in this case, long range fluctuations may spontaneously develop, leading to topological defects In the structure (e.g. dislocations in hydrodynamical patterhs (3), chemical waves in oscillating or excitable media (4)). A stochastic analysis leads to the evaluation of the probability of such fluctuations and consequently to their statistics.

The existence of macroscopic noise-sustained structures in nonlinear optics is theoretically predicted and numerically observed, in the regime of convective instability. The advection-like term, necessary to turn the instability to convective for the parameter region where advection overwhelms the growth, can stem from pump beam tilting or birefringence induced walk-off. The growth dynamics of both noise-sustained and deterministic patterns is exemplified by means of movies. This allows to observe the process of formation of these structures and to confirm the analytical predictions. The amplification of quantum noise by several orders of magnitude is predicted. The qualitative analysis of the near- and far-field is given. It suffices to distinguish noise-sustained from deterministic structures; quantitative informations can be obtained in terms of the statistical properties of the spectra. © 1998 Optical Society of America.