Явление буферности в обобщенном уравнении Свифта-Хоэнберга

The generalization of the Swift-Hohenberg equation is considerexl in this artide. We establish that the number of steady-state solutions unrestricctexHy grows up when the delay tends to infinity and other parameters are correxctly fixed.Рассматривается специальным образом обобщенное уравнение Свифта-Хо-энберга с нулевыми граничными условиями типа Дирихле на концах конечного отрезка. Устанавливается, что при увеличении длины l упомянутого отрезка и при фиксированной достаточно малой надкритичности е количество сосуществующих устойчивых состояний равновесия у этой краевой задачи неограниченно растет, т.е. наблюдается явление буферности

Dissipative solitons in pattern-forming nonlinear optical systems : cavity solitons and feedback solitons

Many dissipative optical systems support patterns. Dissipative solitons are generally found where a pattern coexists with a stable unpatterned state. We consider such phenomena in driven optical cavities containing a nonlinear medium (cavity solitons) and rather similar phenomena (feedback solitons) where a driven nonlinear optical medium is in front of a single feedback mirror. The history, theory, experimental status, and potential application of such solitons is reviewed

Dynamics of Patterns

Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction

Spatio-temporal dynamics of lasers and photorefractive oscillators under rocking: phase-bistable patterns and localized structures

El objectiu de aquesta tesi es l’estudi teòric, analític i numèric, de la dinàmica espaciotemporal d’oscil·ladors òptics no lineals sotmesos a un forçament bicromàtic (rocking). Aquest tipus d’injecció té la característica principal de trencar la invariància de fase (qualsevol fase del camp complex) del sistema lliure (sense forçament) i genera un sistema que és biestable en fase, ja que únicament dues fases (separades per ¼) són permeses per a les solucions estacionàries homogènies. Aquest canvi en la naturalesa del sistema provoca l’aparició d’una nova dinàmica caracteritzada per la presència d’un nou tipus d’estructures espacials en el pla transversal bidimensional: patrons biestables de fase en els quals dominis d’ambdues fases conviuen separades per parets de domini (Ising si la intensitat s’anul·la en elles o Bloch, en cas contrari). Aquests dominis poden evolucionar a patrons homogenis (d’una de les dues fases) o uns altres, més complexos, que els efectes de curvatura condueixen a la creació de patrons laberíntics segons els valors dels paràmetres del sistema. A més, poden existir estructures localitzades (dominis de grandària mínima estables) en la forma de solitons de cavitat d’anell fosc. Altres mètodes de trencament de la simetria de fase han sigut usats per a controlar la dinàmica de molts sistemes. Un dels més populars és la ressonància paramètrica, i.e. injectar un camp la freqüència del qual és aproximadament el doble de la freqüència natural de oscil·lació del sistema. No obstant això, aquests mètodes són menys versàtils que el rocking, el qual pot aplicar-se a una àmplia gamma de sistemes com el làser, que són insensibles a la ressonància paramètrica. De fet, s’han fet múltiples propostes teòriques i experimentals d’aplicació del rocking a diferents sistemes (òptics i no òptics). En el domini d’aquesta tesi, ens centrarem en la influència del rocking en dos sistemes que han sigut estudiats profusament en la literatura, donat el seu gran interès tant des del punt de vista fonamental compràctic:làsers i oscil·ladors fotorrefractius. Al llarg d’aquesta tesi, estudiarem detalladament la influència del rocking en aquests sistemes. Com és usual en el camp de la ciència no lineal, és convenient deduir equacions que descriguen el comportament d’aquests sistemes prop dels punts (punts crítics) on emergeixen les solucions estacionàries del sistema. Aquestes equacions (anomenades de paràmetre d’ordre) tenen una forma aparentment simple i són capaces de descriure multitud de sistemes no lineals, físics, químics, biològics.. (l’única diferència és el significat dels diferents paràmetres, però l’estructura matemàtica és la mateixa), per la qual cosa posseeixen un caràcter universal. Així mateix, analitzarem l’estabilitat de les solucions trobades i realitzarem simulacions numèriques dels diferents models teòrics. Es presentaran els següents resultats: A partir de les equacions de MB amb injecció rocking, es deduirà una equació de paràmetre d’ordre per a làsers de classe C amb desintonia positiva de la cavitat i s’estudiaran numèricament els patrons del sistema. Per a làsers de classe B, s’obtindrà un model reduït de dues equacions i s’analitzarà la seua dinàmica temporal i la influència de la desintonia de la injecció rocking. També esmostraran patrons espacials obtinguts a partir de la simulació de les equacions de MB.Es desenvoluparà un model unificat (vàlid per a desintonies de la cavitat positives i negatives) per a làsers de dos nivells (classe C i A) i oscil·ladors fotrorefractius, proporcionant els dominis d’estabilitat dels estats biestables en fase i estudiant numèricament els patrons espacials que apareixen. S’analitzarà la dinàmica temporal d’un làser bidireccional amb injecció rocking i es presentaran alguns resultats preliminars de patrons espacialsThe objective of this thesis is the theoretical, analytical and numerical, study of the spatio-temporal dynamics of optical oscillators under bichromatic forcing (rocking). This kind of injection possesses the feature of breaking the phase invariance (any phase of the complex field is possible) of the free-running system and generates a phase-bistable system in which two only phases are allowed for the homogeneous stationary solutions. This change in the nature of the system enables a new dynamics characterized by the presence of a new kind of spatial structures in the bidimensional transverse plane: bistable phase patterns in which both phases coexist separated by domain walls (Ising if they have null intensity or Bloch if it is different from zero). These domains can evolve either to homogeneous patterns (in which only one phase is present) or to more complex ones, in which curvature effects lead to the emergence of labyrinthic patterns depending on the value of the parameters of the system. Moreover, localized structures (stable minimum-size domains) as dark-ring cavity solitons can exist. In the scope of this thesis, we have focused on the influence of rocking in two systems which have been studied profusely in the literature, as they are very interesting both from a fundamental and a practical point of views: lasers and photorefractive oscillators. Along this thesis, we will study the influence of rocking in those systems in detail. As it is usual in nonlinear science, is convenient to derive equations describing the behaviour of those systems close to (critical) points where the stationary solutions emerge. These equations (called order parameter equations) are relatively simple and are able to describe a large number of nonlinear systems: physical, chemical, biological.. (the meaning ot the parameters being the only difference , but the mathematical structure is the same). Moreover, we will analyze the stability of the solutions and we will perform numerical simulations of the theoretical models. The following results will be presented: Starting from the MB equations under rocking injection, an order parameter equation will be derived for class C lasers with positive cavity detuning and the patterns of the system will be studied numerically. A reduced model of two equations will be obtained for class B lasers and its temporal dynamics and the influence of the detuning of rocking injection will be studied. We will also show spatial patterns obtained from simulations of the MB equations. A unified model (valid for positive and negative cavity detunings) for two level lasers (class C and A) and photorefractive oscillators will be developed, providing the stability domains of the phase bistable states and studying numerically the spatial patterns that arise from the system. The temporal dynamics of a bidirectional laser under rocking injection will be analyzed and some preliminary results regarding spatial patterns will be given

Principles and theory of protein-based pattern formation

Biological systems perform functions by the orchestrated interplay of many small components without a "conductor." Such self-organization pervades life on many scales, from the subcellular level to populations of many organisms and whole ecosystems. On the intracellular level, protein-based pattern formation coordinates and instructs functions like cell division, differentiation and motility. A key feature of protein-based pattern formation is that the total numbers of the involved proteins remain constant on the timescale of pattern formation. The overarching theme of this thesis is the profound impact of this mass-conservation property on pattern formation and how one can harness mass conservation to understand the underlying physical principles. The central insight is that changes in local densities shift local reactive equilibria, and thus induce concentration gradients which, in turn, drive diffusive transport of mass. For two-component systems, this dynamic interplay can be captured by simple geometric objects in the (low-dimensional) phase space of chemical concentrations. On this phase-space level, physical insight can be gained from geometric criteria and graphical constructions. Moreover, we introduce the notion of regional (in)stabilities, which allows one to characterize the dynamics in the highly nonlinear regime reveals an inherent connection between Turing instability and stimulus-induced pattern formation. The insights gained for conceptual two-component systems can be generalized to systems with more components and several conserved masses. In the minimal setting of two diffusively coupled "reactors," the full dynamics can be embedded in the phase-space of redistributed masses where the phase space flow is organized by surfaces of local reactive equilibria. Building on the phase-space analysis for two component systems, we develop a new approach to the important open problem of wavelength selection in the highly nonlinear regime. We show that two-component reaction–diffusion systems always exhibit uninterrupted coarsening (the continual growth of the characteristic length scale) of patterns if they are strictly mass conserving. Selection of a finite wavelength emerges due to weakly broken mass-conservation, or coupling to additional components, which counteract and stop the competition instability that drives coarsening. For complex dynamical phenomena like wave patterns and the transition to spatiotemporal chaos, an analysis in terms of local equilibria and their stability properties provides a powerful tool to interpret data from numerical simulations and experiments, and to reveal the underlying physical mechanisms. In collaborations with different experimental labs, we studied the Min system of Escherichia coli. A central insight from these investigations is that bulk-surface coupling imparts a strong dependence of pattern formation on the geometry of the spatial confinement, which explains the qualitatively different dynamics observed inside cells compared to in vitro reconstitutions. By theoretically studying the polarization machinery in budding yeast and testing predictions in collaboration with experimentalists, we found that this functional module implements several redundant polarization mechanisms that depend on different subsets of proteins. Taken together, our work reveals unifying principles underlying biological self-organization and elucidates how microscopic interaction rules and physical constraints collectively lead to specific biological functions.Biologische Systeme führen Funktionen durch das orchestrierte Zusammenspiel vieler kleiner Komponenten ohne einen "Dirigenten" aus. Solche Selbstorganisation durchdringt das Leben auf vielen Skalen, von der subzellulären Ebene bis zu Populationen vieler Organismen und ganzen Ökosystemen. Auf der intrazellulären Ebene koordiniert und instruieren proteinbasierte Muster Funktionen wie Zellteilung, Differenzierung und Motilität. Ein wesentliches Merkmal der proteinbasierten Musterbildung ist, dass die Gesamtzahl der beteiligten Proteine auf der Zeitskala der Musterbildung konstant bleibt. Das übergreifende Thema dieser Arbeit ist es, den tiefgreifenden Einfluss dieser Massenerhaltung auf die Musterbildung zu untersuchen und Methoden zu entwickeln, die Massenerhaltung nutzen, um die zugrunde liegenden physikalischen Prinzipien von proteinbasierter Musterbildung zu verstehen. Die zentrale Erkenntnis ist, dass Änderungen der lokalen Dichten lokale reaktive Gleichgewichte verschieben und somit Konzentrationsgradienten induzieren, die wiederum den diffusiven Transport von Masse antreiben. Für Zweikomponentensysteme kann dieses dynamische Wechselspiel durch einfache geometrische Objekte im (niedrigdimensionalen) Phasenraum der chemischen Konzentrationen erfasst werden. Auf dieser Phasenraumebene können physikalische Erkenntnisse durch geometrische Kriterien und grafische Konstruktionen gewonnen werden. Darüber hinaus führen wir den Begriff der regionalen (In-)stabilität ein, der es erlaubt, die Dynamik im hochgradig nichtlinearen Regime zu charakterisieren und einen inhärenten Zusammenhang zwischen Turing-Instabilität und stimulusinduzierter Musterbildung aufzuzeigen. Die für konzeptionelle Zweikomponentensysteme gewonnenen Erkenntnisse können auf Systeme mit mehr Komponenten und mehreren erhaltenen Massen verallgemeinert werden. In der minimalen Fassung von zwei diffusiv gekoppelten "Reaktoren" kann die gesamte Dynamik in den Phasenraum umverteilter Massen eingebettet werden, wobei der Phasenraumfluss durch Flächen lokaler reaktiver Gleichgewichte organisiert wird. Aufbauend auf der Phasenraumanalyse für Zweikomponentensysteme entwickeln wir einen neuen Ansatz für die wichtige offene Fragestellung der Wellenängenselektion im hochgradig nichtlinearen Regime. Wir zeigen, dass "coarsening" (das stetige wachsen der charakteristischen Längenskala) von Mustern in Zweikomponentensystemen nie stoppt, wenn sie exakt massenerhaltend sind. Die Selektion einer endlichen Wellenlänge entsteht durch schwach gebrochene Massenerhaltung oder durch Kopplung an zusätzliche Komponenten. Diese Prozesse wirken der Masseumverteilung, die coarsening treibt, entgegen und stoppen so das coarsening. Bei komplexen dynamischen Phänomenen wie Wellenmustern und dem Übergang zu raumzeitlichen Chaos bietet eine Analyse in Bezug auf lokale Gleichgewichte und deren Stabilitätseigenschaften ein leistungsstarkes Werkzeug, um Daten aus numerischen Simulationen und Experimenten zu interpretieren und die zugrunde liegenden physikalischen Mechanismen aufzudecken. In Zusammenarbeit mit verschiedenen experimentellen Labors haben wir das Min-System von Escherichia coli untersucht. Eine zentrale Erkenntnis aus diesen Untersuchungen ist, dass die Kopplung zwischen Volumen und Oberfläche zu einer starken Abhängigkeit der Musterbildung von der räumlichen Geometrie führt. Das erklärt die qualitativ unterschiedliche Dynamik, die in Zellen im Vergleich zu in vitro Rekonstitutionen beobachtet wird. Durch die theoretische Untersuchung der Polarisationsmaschinerie in Hefezellen, kombiniert mit experimentellen Tests theoretischer Vorhersagen, haben wir herausgefunden, dass dieses Funktionsmodul mehrere redundante Polarisationsmechanismen implementiert, die von verschiedenen Untergruppen von Proteinen abhängen. Zusammengenommen beleuchtet unsere Arbeit die vereinheitlichenden Prinzipien, die der intrazellulären Selbstorganisation zugrunde liegen, und zeigt, wie mikroskopische Interaktionsregeln und physikalische Bedingungen gemeinsam zu spezifischen biologischen Funktionen führen

Collective Behaviour of Polar Active Matter in Two Dimensions

Self-organization is a common way of forming functional structures in biology. It involves biochemical signalling pathways, triggered by a host of external conditions, that alter the mechnical properties of the individual constituents. These changes then propagate up in scale and, via changes in the self-organization, alter the biological function. In this work, we investigate self-organization and pattern formation due to self-propulsion in biological systems. The aim is to understand and map the collective dynamics in terms of mechanical properties of single constituents. We study dense ensembles of self-propelled vesicles that act as models for motile cells and ensembles of self-propelled semiflexible filaments that mimic actin filaments and microtubules in motility assays. Both systems are made of polar and active objects that possess extended shapes with an associated flexibility. We explore the collective dynamics in both systems as a function of activity, flexibility, and interactions between objects. Epithelial tissue serves as barrier for tissues and organs. To achieve this function, epithelial cells are typically tightly-packed, spatially well-ordered, and non-motile. However, a set of conditions can turn epithelial cells motile. During vertebrate embryonic development, wound healing, and cancer metastasis, cells become motile to rearrange the tissue, heal the wound, or travel away from the primary tumor, respectively. In vitro experiments on motile cell monolayers furthermore revealed a jamming transition in which an initially motile, fluid-like tissue undergoes a dynamic arrest. We study such motility transitions of dense cell monolayers in a minimal model approach. We go beyond existing models by including finite extension and flexibility of cells. To this end, we develop a novel computational model of cells as active vesicles that incorporates cell motility, cell-cell adhesions, compressibility, and flexibility. Increasing motility strength and decreasing cell-cell adhesions, area compression modulus, and bending rigidity lead to fluidization of the monolayer. In between the jammed and completely fluid- like states, we identify an active turbulence regime where cell motion is dominated by the formation of vortices. We thus uncover deformability-driven motility transitions and predict an active turbulent state for motile cell monolayers. In a second part, we study the collective behaviour of self-propelled semiflexible filaments by introducing self-propulsion as a constant magnitude force acting tangentially along the bonds of each filament. The combination of polymer properties, excluded-volume interactions, and self-propulsion leads to distinct phases as a function of rigidity, activity, and aspect ratio of individual filaments. We identify a transition from a free-swimming phase to a frozen steady state wherein strongly propelled filaments form spirals at a regime of low rigidity and high aspect ratio. Filaments form clusters of various sizes depending on rigidity and activity. In particular, we observe that filaments form small and transient clusters at low rigidities while stiffer filaments organize into giant clusters. However, as activity increases further, the clustering of filaments displays a reentrant phase behaviour where giant clusters melt, due to the strong propulsion forces bending the filaments. Our results highlight the role of mechanical properties and the finite extent of the constituents on the collective motion patterns. Cells and filaments display different symmetry properties at high densities due to structural differences. Filaments show an effective nematic symmetry, which results in an active turbulence regime characterized by half-integer topological defects. Cells, with polar symmetry, exhibit an active turbulence phase dominated by vortices

Intermittency and Self-Organisation in Turbulence and Statistical Mechanics

There is overwhelming evidence, from laboratory experiments, observations, and computational studies, that coherent structures can cause intermittent transport, dramatically enhancing transport. A proper description of this intermittent phenomenon, however, is extremely difficult, requiring a new non-perturbative theory, such as statistical description. Furthermore, multi-scale interactions are responsible for inevitably complex dynamics in strongly non-equilibrium systems, a proper understanding of which remains a main challenge in classical physics. As a remarkable consequence of multi-scale interaction, a quasi-equilibrium state (the so-called self-organisation) can however be maintained. This special issue aims to present different theories of statistical mechanics to understand this challenging multiscale problem in turbulence. The 14 contributions to this Special issue focus on the various aspects of intermittency, coherent structures, self-organisation, bifurcation and nonlocality. Given the ubiquity of turbulence, the contributions cover a broad range of systems covering laboratory fluids (channel flow, the Von Kármán flow), plasmas (magnetic fusion), laser cavity, wind turbine, air flow around a high-speed train, solar wind and industrial application

Self-organization in heterogeneous biological systems

Self-organization is an ubiquitous and fundamental process that underlies all living systems. In cellular organisms, many vital processes, such as cell division and growth, are spatially and temporally regulated by proteins -- the building blocks of life. To achieve this, proteins self-organize and form spatiotemporal patterns. In general, protein patterns respond to a variety of internal and external stimuli, such as cell shape or inhomogeneities in protein activity. As a result, the dynamics of intracellular pattern formation generally span multiple spatial and temporal scales. This thesis addresses the underlying mechanisms that lead to the formation of heterogeneous patterns. The main themes of this work are organized into three parts, which are summarized below. The first part deals with the general problem of mass-conserving reaction-diffusion dynamics in spatially non-uniform systems. In section 1 of chapter II, we study the dynamics of the E. coli Min protein system -- a paradigmatic model for pattern formation. More specifically, we consider a setup with a fixed spatial heterogeneity in a control parameter, and show that this leads to complex multiscale pattern formation. We develop a coarse-graining approach that enables us to explain and reduce the dynamics to the "hydrodynamic variables'' at large length and time scales. In another project, we consider a system where spatial heterogeneities are not imposed externally, but self-generated by the dynamics via a mechanochemical feedback loop between geometry and reaction-diffusion system (section 2 of chapter II). We show that the resulting dynamics can be explained from the phase-space geometry of the reaction-diffusion system. The second part focuses on how patterns in realistic cell geometries are controlled by shape and biochemical cues. We examine axis selection of PAR polarity patterns in C. elegans, where we show that spatial variations in the bulk-surface ratio and a tendency of the system to minimize the pattern interface yield robust long-axis polarization of PAR protein patterns (section 1 of chapter III). In a second project, we develop a theoretical model that explains the localization of the B. subtilis Min protein system (section 2 of chapter 3). We show that a biochemical cue -- which acts as a template for pattern formation -- guides and stabilizes Min patterns. In the third part, we study the coupling between lipid membranes and curvature-generating proteins. We demonstrate that myosin-VI motor proteins cooperatively bind to saddle-shaped regions of lipid membranes, and thereby induce large-scale membrane remodeling (section 1 of chapter IV). To understand the dynamics, we develop a coarse-grained geometric model and show that the emergence of regular spatial structures can be explained by a "push-pull'' mechanism: protein binding destabilizes the membrane shape at all length scales, and this is counteracted by line tension. Inspired by this system, we then investigate a general model for the dynamics of growing protein-lipid interfaces (section 2 of chapter IV). A key feature of the model is that the protein binding kinetics is explicitly coupled to the morphology of the interface. We show that such a coupling leads to turbulent dynamics and a roughening transition of the interface that is characterized by universal scaling behaviour