Martin Golubitsky – författare

This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob­ lems and we hope to convert the reader to this view. In this preface we will discuss what we feel are the strengths of the singularity theory approach. This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it. Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems. Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V. I. Arnold. In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory. Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence. In this volume our emphasis is on singularity theory, with group theory playing a subordinate role. In Volume II the emphasis will be more balanced. Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation. As we use the term, bifurcation theory is the study of equations with multiple solutions.

Bifurcation theory studies how the structure of solutions to equations changes as parameters are varied. The nature of these changes depends both on the number of parameters and on the symmetries of the equations. Volume I discusses how singularity-theoretic techniques aid the understanding of transitions in multiparameter systems. This volume focuses on bifurcation problems with symmetry and shows how group-theoretic techniques aid the understanding of transitions in symmetric systems. Four broad topics are covered: group theory and steady-state bifurcation, equicariant singularity theory, Hopf bifurcation with symmetry, and mode interactions. The opening chapter provides an introduction to these subjects and motivates the study of systems with symmetry. Detailed case studies illustrate how group-theoretic methods can be used to analyze specific problems arising in applications.

This volume contains a number of mini-review articles authored by speakers and attendees at the IMA workshop on Pattern Formation in Continuous and Coupled Systems. Pattern formation has been studied intensively for most of this century by both experimentalists and theoreticians. This workshop focused on new directions in the patterns literature. The goals were to continue communication between these groups, and to familiarize a larger audience with some of the newer directions in the field. Systems that generate new types of pattern such as discrete coupled systems, systems with global coupling, and combustion experiments were stressed, as were new types of pattern. The mini-reviews in this volume are intended to be pointers to the current literature for researchers at all levels and therefore include extensive bibliographies. They are also intended to discuss why certain subjects are currently exciting and worthy of additional research.

Mathematical symmetry and chaos come together to form striking, beautiful colour images throughout this impressive work, which addresses how the dynamics of complexity can produce familiar universal patterns.The book, a richly illustrated blend of mathematics and art, was widely hailed in publications as diverse as the New York Review of Books, Scientific American, and Science when first published in 1992. This much-anticipated second edition features many new illustrations and addresses the progress made in the mathematics and science underlying symmetric chaos in recent years; for example, the classifications of attractor symmetries and methods for determining the symmetries of higher dimensional analogues of images in the book. In particular, the concept of patterns on average and their occurrence in the Faraday fluid dynamics experiment is described in a revised introductory chapter.The ideas addressed in this book have been featured at various conferences on intersections between art and mathematics, including the annual Bridges conference, and in lectures to art students at the University of Houston.

This IMA Volume in Mathematics and its Applications PATTERN FORMATION IN CONTINUOUS AND COUPLED SYSTEMS is based on the proceedings of a workshop with the same title, but goes be­ yond the proceedings by presenting a series of mini-review articles that sur­ vey, and provide an introduction to, interesting problems in the field. The workshop was an integral part of the 1997-98 IMA program on "EMERG­ ING APPLICATIONS OF DYNAMICAL SYSTEMS." I would like to thank Martin Golubitsky, University of Houston (Math­ ematics) Dan Luss, University of Houston (Chemical Engineering), and Steven H. Strogatz, Cornell University (Theoretical and Applied Mechan­ ics) for their excellent work as organizers of the meeting and for editing the proceedings. I also take this opportunity to thank the National Science Foundation (NSF), and the Army Research Office (ARO), whose financial support made the workshop possible. Willard Miller, Jr., Professor and Director v PREFACE Pattern formation has been studied intensively for most of this cen­ tury by both experimentalists and theoreticians, and there have been many workshops and conferences devoted to the subject. In the IMA workshop on Pattern Formation in Continuous and Coupled Systems held May 11-15, 1998 we attempted to focus on new directions in the patterns literature.

Bifurcation theory studies how the structure of solutions to equations changes as parameters are varied. The nature of these changes depends both on the number of parameters and on the symmetries of the equations. Volume I discusses how singularity-theoretic techniques aid the understanding of transitions in multiparameter systems. This volume focuses on bifurcation problems with symmetry and shows how group-theoretic techniques aid the understanding of transitions in symmetric systems. Four broad topics are covered: group theory and steady-state bifurcation, equicariant singularity theory, Hopf bifurcation with symmetry, and mode interactions. The opening chapter provides an introduction to these subjects and motivates the study of systems with symmetry. Detailed case studies illustrate how group-theoretic methods can be used to analyze specific problems arising in applications.

This book has been written in a frankly partisian spirit-we believe that singularity theory offers an extremely useful approach to bifurcation prob­ lems and we hope to convert the reader to this view. In this preface we will discuss what we feel are the strengths of the singularity theory approach. This discussion then Ieads naturally into a discussion of the contents of the book and the prerequisites for reading it. Let us emphasize that our principal contribution in this area has been to apply pre-existing techniques from singularity theory, especially unfolding theory and classification theory, to bifurcation problems. Many ofthe ideas in this part of singularity theory were originally proposed by Rene Thom; the subject was then developed rigorously by John Matherand extended by V. I. Arnold. In applying this material to bifurcation problems, we were greatly encouraged by how weil the mathematical ideas of singularity theory meshed with the questions addressed by bifurcation theory. Concerning our title, Singularities and Groups in Bifurcation Theory, it should be mentioned that the present text is the first volume in a two-volume sequence. In this volume our emphasis is on singularity theory, with group theory playing a subordinate role. In Volume II the emphasis will be more balanced. Having made these remarks, Iet us set the context for the discussion of the strengths of the singularity theory approach to bifurcation. As we use the term, bifurcation theory is the study of equations with multiple solutions.

In recent years, there has been an explosion of interest in network-based modeling in many branches of science. This book synthesizes some of the common features of many such models, providing a general framework analogous to the modern theory of nonlinear dynamical systems. How networks lead to behavior not typical in a general dynamical system and how the architecture and symmetry of the network influence this behavior are the book's main themes.Dynamics and Bifurcation in Networks: Theory and Applications of Coupled Differential Equations is the first book to describe the formalism for network dynamics developed over the past 20 years. In it, the authors introduce a definition of a network and the associated class of "admissible" ordinary differential equations, in terms of a directed graph whose nodes represent component dynamical systems and whose arrows represent couplings between these systems. They also develop connections between network architecture and the typical dynamics and bifurcations of these equations and discuss applications of this formalism to various areas of science, including gene regulatory networks, animal locomotion, decision-making, homeostasis, binocular rivalry, and visual illusions.This book will be of interest to scientific researchers in any area that uses network models, which includes many parts of biology, physics, chemistry, computer science, electrical and electronic engineering, psychology, and sociology.

This book collects selected, peer-reviewed works presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering (ICMASC), which took place at the University of Porto, Portugal, in hybrid mode on June 20–22, 2024. This book covers advancements in mathematical analysis and modeling, with an emphasis on mathematical biology. It addresses topics such as partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, and computational mathematics, to name a few. Applications range from tumor and tissue properties, infection spatial models, cancer immunotherapy, and others.The conference was dedicated to Professor J. A. Tenreiro Machado, who unexpectedly passed away in October 2021. Prof. Tenreiro was a Professor at the Polytechnic Institute of Porto, where he served as the Dean of the Institute. Additionally, he held a visiting position at the Lodz Institute of Technology, Poland, for many years. Prof. Tenreiro Machado authored 1,160 publications, including 11 books.

Pattern formation in physical systems is one of the major research frontiers of mathematics. A central theme of this book is that many instances of pattern formation can be understood within a single framework: symmetry.The book applies symmetry methods to increasingly complex kinds of dynamic behavior: equilibria, period-doubling, time-periodic states, homoclinic and heteroclinic orbits, and chaos. Examples are drawn from both ODEs and PDEs. In each case the type of dynamical behavior being studied is motivated through applications, drawn from a wide variety of scientific disciplines ranging from theoretical physics to evolutionary biology. An extensive bibliography is provided.

Sunyer 2001! Pattern formation in physical systems is one of the major research frontiers of mathematics. A central theme of The Symmetry Perspective is that many instances of pattern formation can be understood within a single framework: symmetry. The symmetries of a system of nonlinear ordinary or partial differential equations can be used to analyze, predict, and understand general mechanisms of pattern-formation. The symmetries of a system imply a 'catalogue' of typical forms of behavior, from which the actual behavior is 'selected'. A central theme is the distinction between phase space and physical space. The theory of nonlinear dynamical systems is largely discussed in terms of trajectories in an abstract phase space. The connection between the variables in the theory and the variables that are being observed can often be lost. Experimentalists typically observe a time series of measurements, whereas the abstract theory works with geometric objects such as attractors, homoclinic orbits, or invariant measures. Symmetries therefore provide an important route between the abstract theory and experimental observations.The book applies symmetry methods to increasingly complex kinds of dynamic behavior: equilibria, period-doubling, time-periodic states, homoclinic and heteroclinic orbits, and chaos. Examples are drawn from both ODEs and PDEs. In each case the type of dynamical behavior being studied is motivated through applications, drawn from a wide variety of scientific disciplines ranging from theoretical physics to evolutionary biology. An extensive bibliography is provided.