The mathematical description of complex spatiotemporal behaviour observed in dissipative continuous systems is a major challenge for modern research in applied mathematics. While the behaviour of low-dimensional systems, governed by the dynamics of a finite number of modes is well understood, systems with large or unbounded spatial domains show intrinsic infinite-dimensional behaviour --not a priori accessible to the methods of finite dimensionaldynamical systems. The purpose of the four contributions in this book is to present some recent and active lines of research in evolution equations posed in large or unbounded domains. One of the most prominent features of these systems is the propagation of various types of patterns in the form of waves, such as travelling and standing waves and pulses and fronts. Different approaches to studying these kinds of phenomena are discussed in the book. A major theme is the reduction of an original evolution equation in the form of a partial differential equation system to a simpler system of equations, either a system of ordinary differential equation or a canonical system of PDEs. The study of the reduced equations provides insight into the bifurcations from simple to more complicated solutions and their stabilities. .
This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems further the understanding of mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. However, the authors and editors have made an effort to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers. The eighteen surveys collected here do not aspire to encyclopedic completeness, but present selected paradigms. The surveys are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and partial differential equations. Application areas include the dynamics of neural networks, fluid flows, nonlinear optics, and many others.While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to namejust a few, are ubiquitous dynamical concepts throughout the articles.
Includes a paper that studies bifurcations of stationary and time-periodic solutions to reaction-diffusion systems. This title develops a center-manifold and normal form theory for radial dynamics which allows for a complete description of radially symmetric patterns.
Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.
Applied mathematics is a central connecting link between scientific observations and their theoretical interpretation. Nonlinear analysis has surely contributed major developments which nowadays shape the face of applied mathematics. At the beginning of the millennium, all sciences are expanding at increased speed. Technological, ecological, economical and medical problem solving is a central issue of every modern society. Mathematical models help to expose fundamental structures hidden in these problems and serve as unifying tools to deepen our understanding. What are the new challenges applied mathematics has to face with the increased diversity of scientific problems? In which direction should the classical tools of nonlinear analysis be developed further? How do new available technologies influence the development of the field? How can problems be solved which have been beyond reach in former times? It is the aim of this book to explore new developments in the field by way of discussion of selected topics from nonlinear analysis.
This is an introductory textbook about nonlinear dynamics of PDEs, with a focus on problems over unbounded domains and modulation equations. The presentation is example-oriented, and new mathematical tools are developed step by step, giving insight into some important classes of nonlinear PDEs and nonlinear dynamics phenomena which may occur in PDEs. The book consists of four parts. Parts I and II are introductions to finite- and infinite-dimensional dynamics defined by ODEs and by PDEs over bounded domains, respectively, including the basics of bifurcation and attractor theory. Part III introduces PDEs on the real line, including the Korteweg-de Vries equation, the Nonlinear Schrödinger equation and the Ginzburg-Landau equation. These examples often occur as simplest possible models, namely as amplitude or modulation equations, for some real world phenomena such as nonlinear waves and pattern formation. Part IV explores in more detail the connections between such complicated physical systems and the reduced models. For many models, a mathematically rigorous justification by approximation results is given. The parts of the book are kept as self-contained as possible. The book is suitable for self-study, and there are various possibilities to build one- or two-semester courses from the book.
This book is an expanded version of a Master Class on the symmetric bifurcation theory of differential equations given by the author at the University of Twente in 1995. The notes cover a wide range of recent results in the subject, and focus on the dynamics that can appear in the generic bifurcation theory of symmetric differential equations. This text covers a wide range of current results in the subject of bifurcations, dynamics and symmetry. The style and format of the original lectures has largely been maintained and the notes include over 70 exercises.
In the last few decades, complex dynamical systems have received widespread public attention and emerged as one of the most active fields of mathematical research. Starting where other monographs in the subject end, Progress in Holomorphic Dynamics advances the theoretical aspects and recent results in complex dynamical systems, with particular emphasis on Siegel discs. Organized into four parts, the papers in this volume grew out of three workshops: two hosted by the Georg-August-Universität Göttingen and one at the "Mathematisches Forschungsinstitut Oberwolfach." Part I addresses linearization. The authors review Yoccoz's proof that the Brjuno condition is the optimal condition for linearizability of indifferent fixed points and offer a treatment of Perez-Marco's refinement of Yoccoz's work. Part II discusses the conditions necessary for the boundary of a Siegel disc to contain a critical point, builds upon Herman's work, and offers a survey of the state-of-the-art regarding the boundaries of Siegel discs. Part III deals with the topology of Julia sets with Siegel discs and contains a remarkable highlight: C.L. Petersen establishes the existence of Siegel discs of quadratic polynomials with a locally connected boundary. Keller, taking a different approach, explains the relations between locally connected "real Julia sets" with Siegel discs and the abstract concepts of kneading sequences and itineraries. Part IV closes the volume with four papers that review the different directions of present research in iteration theory. It includes discussions on the relations between commuting rational functions and their Julia sets, interactions between the iteration of polynomials and the iteration theory of entire transcendental functions, a deep analysis of the topology of the limbs of the Mandelbrot set, and an overview of complex dynamics in higher dimensions.
The subject “Systems sciences and cybernetics” is the outcome of the convergence of a number of trends in a larger current of thought devoted to the growing complexity of (primarily social) objects and arising in response to the need for globalized treatment of such objects. This has been magnified by the proliferation and publication of all manner of quantitative scientific data on such objects, advances in the theories on their inter-relations, the enormous computational capacity provided by IT hardware and software and the critical revisiting of subject-object interaction, not to mention the urgent need to control the efficiency of complex systems, where “efficiency” is understood to mean the ability to find a solution to many social problems, including those posed on a planetary scale. The result has been the forging of a new, academically consolidated scientific trend going by the name of Systems Theory and Cybernetics, with a comprehensive, multi-disciplinary focus and therefore apt for understanding realities still regarded to be inescapably chaotic. This subject entry is subdivided into four sections. The first, an introduction to systemic theories, addresses the historic development of the most commonly used systemic approaches, from new concepts such as the so-called “geometry of thinking” or the systemic treatment of “non-systemic identities” to the taxonomic, entropic, axiological and ethical problems deriving from a general “systemic-cybernetic” conceit. Hence, the focus in this section is on the historic and philosophical aspects of the subject. Moreover, it may be asserted today that, beyond a shadow of a doubt, problems, in particular problems deriving from human interaction but in general any problem regardless of its nature, must be posed from a systemic perspective, for otherwise the obstacles to their solution are insurmountable. Reaching such a perspective requires taking at least the following well-known steps: a) statement of the problem from the determinant variables or phenomena; b) adoption of theoretical models showing the interrelationships among such variables; c) use of the maximum amount of – wherever possible quantitative – information available on each; d) placement of the set of variables in an environment that inevitably pre-determines the problem. That epistemology would explain the substantial development of the systemic-cybernetic approach in recent decades. The articles in the second section deal in particular with the different methodological approaches developed when confronting real problems, from issues that affect humanity as a whole to minor but specific questions arising in human organizations. Certain sub-themes are discussed by the various authors – always from a didactic vantage –, including: problem discovery and diagnosis and development of the respective critical theory; the design of ad hoc strategies and methodologies; the implementation of both qualitative (soft system methodologies) and formal and quantitative (such as the “General System Problem Solver” or the “axiological-operational” perspective) approaches; cross-disciplinary integration; and suitable methods for broaching psychological, cultural and socio-political dynamisms. The third section is devoted to cybernetics in the present dual meaning of the term: on the one hand, control of the effectiveness of communication and actions, and on the other, the processes of self-production of knowledge through reflection and the relationship between the observing subject and the observed object when the latter is also observer and the former observed. Known as “second order cybernetics”, this provides an avenue for rethinking the validity of knowledge, such as for instance when viewed through what is known as “bipolar feedback”: processes through which interactions create novelty, complexity and diversity. Finally, the fourth section centres around artificial and computational intelligence, addressing sub-themes such as “neural networks”, the “simulated annealing” that ranges from statistical thermodynamics to combinatory problem-solving, such as in the explanation of the role of adaptive systems, or when discussing the relationship between biological and computational intelligence.
