Stationary localized structures and the effect of the delayed feedback in the Brusselator model

The Brusselator reaction-diffusion model is a paradigm for the understanding of dissipative structures in systems out of equilibrium. In the first part of this paper, we investigate the formation of stationary localized structures in the Brusselator model. By using numerical continuation methods in two spatial dimensions, we establish a bifurcation diagram showing the emergence of localized spots. We characterize the transition from a single spot to an extended pattern in the form of squares. In the second part, we incorporate delayed feedback control and show that delayed feedback can induce a spontaneous motion of both localized and periodic dissipative structures. We characterize this motion by estimating the threshold and the velocity of the moving dissipative structures.Comment: 18 pages, 11 figure

Extended patchy ecosystems may increase their total biomass through self-replication

Patches of vegetation consist of dense clusters of shrubs, grass, or trees, often found to be circular characteristic size, defined by the properties of the vegetation and terrain. Therefore, vegetation patches can be interpreted as localized structures. Previous findings have shown that such localized structures can self-replicate in a binary fashion, where a single vegetation patch elongates and divides into two new patches. Here, we extend these previous results by considering the more general case, where the plants interact non-locally, this extension adds an extra level of complexity and shrinks the gap between the model and real ecosystems, where it is known that the plant-to-plant competition through roots and above-ground facilitating interactions have non-local effects, i.e. they extend further away than the nearest neighbor distance. Through numerical simulations, we show that for a moderate level of aridity, a transition from a single patch to periodic pattern occurs. Moreover, for large values of the hydric stress, we predict an opposing route to the formation of periodic patterns, where a homogeneous cover of vegetation may decay to spot-like patterns. The evolution of the biomass of vegetation patches can be used as an indicator of the state of an ecosystem, this allows to distinguish if a system is in a self-replicating or decaying dynamics. In an attempt to relate the theoretical predictions to real ecosystems, we analyze landscapes in Zambia and Mozambique, where vegetation forms patches of tens of meters in diameter. We show that the properties of the patches together with their spatial distributions are consistent with the self-organization hypothesis. We argue that the characteristics of the observed landscapes may be a consequence of patch self-replication, however, detailed field and temporal data is fundamental to assess the real state of the ecosystems.Comment: 38 pages, 12 figures, 1 tabl

Analýza charakteru pohybu Bělousovova–Žabotinského modelu v závislosti na parametru

The main aim of this thesis is to detect dynamical properties of the Györgyi-Field model of the Belousov-Zhabotinsky chemical reaction. The corresponding three-variable model given as a set of nonlinear ordinary differential equations depends on one parameter, the flow rate. As certain values of this parameter can give rise to chaos, the analysis was performed in order to identify different dynamics regimes. Dynamical properties were qualified and quantified using classical and also new techniques. Namely, phase portraits, bifurcation diagrams, the Fourier spectra analysis, the 0-1 test for chaos, and approximate entropy. The correlation between approximate entropy and the 0-1 test for chaos was observed and described in detail. Moreover, the three-stage system of nested subintervals of flow rates, for which in every level the 0-1 test for chaos and approximate entropy was computed, is showing the same pattern. The study leads to an open problem whether the set of flow rate parameters has Cantor like structure.Hlavním cílem této práce je zkoumat dynamické vlastnosti Györgyi-Fieldova modelu Belousov-Žabotinského chemické reakce. Odpovídající model o třech proměnných je zadaný soustavou nelineárních obyčejných diferenciálních rovnic závislých na jednom parametru, který označuje průtok. Protože některé hodnoty tohoto parametru mohou vést k chaosu, byla provedena analýza za účelem identifikace různých dynamických režimů. Dynamické vlastnosti byly kvalifikovány a kvantifikovány klasickými i novými technikami, jako jsou fázové portréty, bifurkační diagram, Fourierova spektrální analýza, 0-1 test na chaos a aproximační entropie. Podrobně byla zkoumána korelace mezi aproximační entropií a 0-1 testem chaosu. Navíc byl zkonstruován systém tří vnořených podintervalů parametru průtoků, pro který byl v každé úrovni vypočítán 0-1 test na chaos a aproximační entropie, které měly stejnou strukturu. Studie vede k otevřenému problému, zda množina parametrů průtoku má strukturu podobnou Cantorově množině.470 - Katedra aplikované matematikyvýborn

Models of self-organization in biological development

Bibliography: p. 297-320.In this thesis we thus wish to consider the concept of self-organization as an overall paradigm within which various theoretical approaches to the study of development may be described and evaluated. In the process, an attempt is made to give a fair and reasonably comprehensive overview of leading modelling approaches in developmental biology, with particular reference to self-organization. The work proceeds from a physical or mathematical perspective, but not unduly so - the major mathematical derivations and results are relegated to appendices - and attempts to fill a perceived gap in the extant review literature, in its breadth and attempted impartiality of scope. A characteristic of the present account is its markedly interdisciplinary approach: it seeks to place self-organization models that have been proposed for biological pattern formation and morphogenesis both within the necessary experimentally-derived biological framework, and in the wider physical context of self-organization and the mathematical techniques that may be employed in its study. Hence the thesis begins with appropriate introductory chapters to provide the necessary background, before proceeding to a discussion of the models themselves. It should be noted that the work is structured so as to be read sequentially, from beginning to end; and that the chapters in the main text were designed to be understood essentially independently of the appendices, although frequent references to the latter are given. In view of the vastness of the available information and literature on developmental biology, a working knowledge of embryological principles must be assumed. Consequently, rather than attempting a comprehensive introduction to experimental embryology, chapter 2 presents just a few biological preliminaries, to 'set the scene', outlining some of the major issues that we are dealing with, and sketching an indication of the current status of knowledge and research on development. The chapter is aimed at furnishing the necessary biological, experimental background, in the light of which the rest of the thesis should be read, and which should indeed underpin and motivate any theoretical discussions. We encounter the different hierarchical levels of description in this chapter, as well as some of the model systems whose experimental study has proved most fruitful, some of the concepts of experimental embryology, and a brief reference to some questions that will not be addressed in this work. With chapter 3, we temporarily move away from developmental biology, and consider the wider physical and mathematical concepts related to the study of self-organization. Here we encounter physical and chemical examples of spontaneous structure formation, thermodynamic considerations, and different approaches to the description of complexity. Mathematical approaches to the dynamical study of self-organization are also introduced, with specific reference to reaction-diffusion equations, and we consider some possible chemical and biochemical realizations of self-organizing kinetics. The chapter may be read in conjunction with appendix A, which gives a somewhat more in-depth study of reaction-diffusion equations, their analysis and properties, as an example of the approach to the analysis of self-organizing dynamical systems and mathematically-formulated models. Appendix B contains a more detailed discussion of the Belousov-Zhabotinskii reaction, which provides a vivid chemical paradigm for the concepts of symmetry-breaking and self-organization. Chapter 3 concludes with a brief discussion of a model biological system, the cellular slime mould, which displays rudimentary development and has thus proved amenable to detailed study and modelling. The following two chapters form the core of the thesis, as they contain discussions of the detailed application of theoretical concepts and models, largely based on self-organization, to various developmental situations. We encounter a diversity of models which has arisen largely in the last quarter century, each of which attempts to account for some aspect of biological pattern formation and morphogenesis; an aim of the discussion is to assess the extent of the underlying unity of these models in terms of the self-organization paradigm. In chapter 4 chemical pre-patterns and positional information are considered, without the overt involvement of cells in the patterning. In chapter 5, on the other hand, cellular interactions and activities are explicitly taken into account; this chapter should be read together with appendix C, which contains a brief introduction to the mathematical formulation and analysis of some of the models discussed. The penultimate chapter, 6, considers two other approaches to the study of development; one of these has faded away, while the other is still apparently in the ascendant. The assumptions underlying catastrophe theory, the value of its applications to developmental biology and the reasons for its decline in popularity, are considered. Lastly, discrete approaches, including the recently fashionable cellular automata, are dealt with, and the possible roles of rule-based interactions, such as of the so-called L-systems, and of fractals and chaos are evaluated. Chapter 7 then concludes the thesis with a brief assessment of the value of the self-organization concept to the study of biological development

Nonlinear optical waves in disordered ferroelectrics

This thesis describes an experimental, numerical and theoretical investigation of nonlinear optical phenomena in disordered photorefractive ferroelectrics in proximity of their phase-transition temperature. The work addresses different physical issues that find in nonlinear optics a common fertile research arena and are closely related to each other in the considered systems. Nonlinear wave dynamics in the spatial domain, where self-interaction of propagating waves generally results into non-spreading localized wavepackets such as spatial solitons, is extended in photorefractive ferroelectrics to non-equilibrium regimes characterized by stochastic instabilities and large material fluctuations. We discover the emergence of rogue waves, localized perturbations of abnormal intensity, whose understanding is challenging in various physical contexts and resides in the general problem of long-tail statistical distributions in complex systems. We identify their origin in spatiotemporal soliton dynamics in a saturable nonlinearity which can support scale-invariant waveforms. Properties and predictability of the observed extreme events are investigated, and, in particular, we demonstrate their active control through the spatial incoherence scale of the optical field. Moreover, we report how their emergence is sustained by turbulent transitions to an incoherent and disordered optical state triggered by modulational instability. The onset of strong turbulence for propagating optical waves has remained unobserved up to now and our results demonstrate a new experimental setting for its study. When the functional form of the nonlinearity is turned into a nonlocal one due to diffusive fields, this setting also exploits photonics to address fundamental physical problems and access to otherwise hidden phenomena. The natural spreading of waves during propagation, representing the wavelength-defined ultimate limit to spatial resolution, can be eliminated and reversed leading to diffraction cancellation and anti-diffraction of light. Since these behaviors on modifying the nature of underlying Schrödinger equation, we are the first to demonstrate how nonlinearity can make the spatial light distribution behave as the wavefunction of a quantum particle with negative mass. All these findings have roots in the nonlinear optical response of critical disordered ferroelectric crystals, which are also extremely interesting from the condensed matter point of view. In fact, competition of different microscopic structural phases and the associated polar-domain dynamics at the nanoscale results into non-ergodic dipolar-glass behaviors giving giant responses such as giant polarization, piezoelectricity and electro-optic effect. Disordered ferroelectrics crystals are investigated electro-optically across their ferroelectric phase-transition, where we report the observation of an anomalous electro-optic effect compatible with ultracold dipolar reorientation. In compounds presenting spatial inhomogeneity in their chemical composition, we discover a new ferroelectric phase of matter in which polar domains spontaneously coordinate into a mesoscopic coherent polarization super-crystals. This phase mimics standard solid-state structures but on scales that are thousands of times larger and represent the first spontaneous three-dimensional photonic crystal

Dynamics of Macrosystems; Proceedings of a Workshop, September 3-7, 1984

There is an increasing awareness of the important and persuasive role that instability and random, chaotic motion play in the dynamics of macrosystems. Further research in the field should aim at providing useful tools, and therefore the motivation should come from important questions arising in specific macrosystems. Such systems include biochemical networks, genetic mechanisms, biological communities, neutral networks, cognitive processes and economic structures. This list may seem heterogeneous, but there are similarities between evolution in the different fields. It is not surprising that mathematical methods devised in one field can also be used to describe the dynamics of another. IIASA is attempting to make progress in this direction. With this aim in view this workshop was held at Laxenburg over the period 3-7 September 1984. These Proceedings cover a broad canvas, ranging from specific biological and economic problems to general aspects of dynamical systems and evolutionary theory

Self-Replicating Spots in the Brusselator Model and Extreme Events in the One-Dimensional Case with Delay

We consider the paradigmatic Brusselator model for the study of dissipative structures in far from equilibrium systems. In two dimensions, we show the occurrence of a self-replication phenomenon leading to the fragmentation of a single localized spot into four daughter spots. This instability affects the new spots and leads to splitting behavior until the system reaches a hexagonal stationary pattern. This phenomenon occurs in the absence of delay feedback. In addition, we incorporate a time-delayed feedback loop in the Brusselator model. In one dimension, we show that the delay feedback induces extreme events in a chemical reaction diffusion system. We characterize their formation by computing the probability distribution of the pulse height. The long-tailed statistical distribution, which is often considered as a signature of the presence of rogue waves, appears for sufficiently strong feedback intensity. The generality of our analysis suggests that the feedback-induced instability leading to the spontaneous formation of rogue waves in a controllable way is a universal phenomenon.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Education Reform at the "Edge of Chaos": Constructing ETCH (An Education Theory Complexity Hybrid) for an Optimal Learning Education Environment

EDUCATION REFORM AT THE "EDGE OF CHAOS":CONSTRUCTING ETCH (AN EDUCATION THEORY COMPLEXITY HYBRID) FOR AN OPTIMAL LEARNING EDUCATION ENVIRONMENT AbstractCurrently, the theoretical foundation that inspires educational theory, which in turn shapes the systemic structure of institutions of learning, is based on three key interconnected, interacting underpinnings -mechanism, reductionism, and linearity. My dissertation explores this current theoretical underpinning including its fallacies and inconsistencies, and then frames an alternative educational theoretical base - a hybrid complex adaptive systems theory model for education - that more effectively meets the demands to prepare students for the 21st century. My Education Theory Complexity Hybrid (ETCH) differs by focusing on the systemic, autopoietic nature of schools, the open, fluid processes of school systems as a dissipative structure, and nonlinearity or impossibility of completely predicting the results of any specific intervention within a school system.. In addition, I show how ETCH principles, when applied by educational system leaders, permit them to facilitate an optimal learning environment for a student-centered complex adaptive system.ETCH is derived from Complexity Theory and is a coherent, valid, and verifiable systems' framework that accurately aligns the education system with its goal as a student-centered complex adaptive system. In contrast to most dissertations in the School Leadership Program, which are empirical studies, mine explores this new theoretical orientation and illustrates the power of that orientation through a series of examples taken from my experiences in founding and operating the Lancaster Institute for Learning, a private state-licensed alternative high school in eastern Pennsylvania