Dynamical Systems And Bifurcation Theory
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| Author | : V.I. Arnold |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 279 |
| Release | : 2013-12-01 |
| Genre | : Mathematics |
| ISBN | : 3642578845 |
Bifurcation theory and catastrophe theory are two well-known areas within the field of dynamical systems. Both are studies of smooth systems, focusing on properties that seem to be manifestly non-smooth. Bifurcation theory is concerned with the sudden changes that occur in a system when one or more parameters are varied. Examples of such are familiar to students of differential equations, from phase portraits. Understanding the bifurcations of the differential equations that describe real physical systems provides important information about the behavior of the systems. Catastrophe theory became quite famous during the 1970's, mostly because of the sensation caused by the usually less than rigorous applications of its principal ideas to "hot topics", such as the characterization of personalities and the difference between a "genius" and a "maniac". Catastrophe theory is accurately described as singularity theory and its (genuine) applications. The authors of this book, previously published as Volume 5 of the Encyclopaedia, have given a masterly exposition of these two theories, with penetrating insight.
| Author | : Yuri Kuznetsov |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 648 |
| Release | : 2013-03-09 |
| Genre | : Mathematics |
| ISBN | : 1475739788 |
Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.
| Author | : Vladimir I. Arnol'd |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 120 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 3642969372 |
| Author | : John Guckenheimer |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 475 |
| Release | : 2013-11-21 |
| Genre | : Mathematics |
| ISBN | : 1461211409 |
An application of the techniques of dynamical systems and bifurcation theories to the study of nonlinear oscillations. Taking their cue from Poincare, the authors stress the geometrical and topological properties of solutions of differential equations and iterated maps. Numerous exercises, some of which require nontrivial algebraic manipulations and computer work, convey the important analytical underpinnings of problems in dynamical systems and help readers develop an intuitive feel for the properties involved.
| Author | : Maoan Han |
| Publisher | : World Scientific |
| Total Pages | : 476 |
| Release | : 1997-11-29 |
| Genre | : Mathematics |
| ISBN | : 9814501093 |
Dynamical bifurcation theory is concerned with the changes that occur in the global structure of dynamical systems as parameters are varied. This book makes recent research in bifurcation theory of dynamical systems accessible to researchers interested in this subject. In particular, the relevant results obtained by Chinese mathematicians are introduced as well as some of the works of the authors which may not be widely known. The focus is on the analytic approach to the theory and methods of bifurcations. The book prepares graduate students for further study in this area, and it serves as a ready reference for researchers in nonlinear sciences and applied mathematics.
| Author | : Shangjiang Guo |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 295 |
| Release | : 2013-07-30 |
| Genre | : Mathematics |
| ISBN | : 1461469929 |
This book provides a crash course on various methods from the bifurcation theory of Functional Differential Equations (FDEs). FDEs arise very naturally in economics, life sciences and engineering and the study of FDEs has been a major source of inspiration for advancement in nonlinear analysis and infinite dimensional dynamical systems. The book summarizes some practical and general approaches and frameworks for the investigation of bifurcation phenomena of FDEs depending on parameters with chap. This well illustrated book aims to be self contained so the readers will find in this book all relevant materials in bifurcation, dynamical systems with symmetry, functional differential equations, normal forms and center manifold reduction. This material was used in graduate courses on functional differential equations at Hunan University (China) and York University (Canada).
| Author | : Shouhong Wang |
| Publisher | : World Scientific |
| Total Pages | : 391 |
| Release | : 2005-06-27 |
| Genre | : Science |
| ISBN | : 9814480592 |
This book covers comprehensive bifurcation theory and its applications to dynamical systems and partial differential equations (PDEs) from science and engineering, including in particular PDEs from physics, chemistry, biology, and hydrodynamics.The book first introduces bifurcation theories recently developed by the authors, on steady state bifurcation for a class of nonlinear problems with even order nondegenerate nonlinearities, regardless of the multiplicity of the eigenvalues, and on attractor bifurcations for nonlinear evolution equations, a new notion of bifurcation.With this new notion of bifurcation, many longstanding bifurcation problems in science and engineering are becoming accessible, and are treated in the second part of the book. In particular, applications are covered for a variety of PDEs from science and engineering, including the Kuramoto-Sivashinsky equation, the Cahn-Hillard equation, the Ginzburg-Landau equation, reaction-diffusion equations in biology and chemistry, the Benard convection problem, and the Taylor problem. The applications provide, on the one hand, general recipes for other applications of the theory addressed in this book, and on the other, full classifications of the bifurcated attractor and the global attractor as the control parameters cross certain critical values, dictated usually by the eigenvalues of the linearized problems. It is expected that the book will greatly advance the study of nonlinear dynamics for many problems in science and engineering.
| Author | : David Ruelle |
| Publisher | : Elsevier |
| Total Pages | : 196 |
| Release | : 2014-05-10 |
| Genre | : Mathematics |
| ISBN | : 1483272184 |
Elements of Differentiable Dynamics and Bifurcation Theory provides an introduction to differentiable dynamics, with emphasis on bifurcation theory and hyperbolicity that is essential for the understanding of complicated time evolutions occurring in nature. This book discusses the differentiable dynamics, vector fields, fixed points and periodic orbits, and stable and unstable manifolds. The bifurcations of fixed points of a map and periodic orbits, case of semiflows, and saddle-node and Hopf bifurcation are also elaborated. This text likewise covers the persistence of normally hyperbolic manifolds, hyperbolic sets, homoclinic and heteroclinic intersections, and global bifurcations. This publication is suitable for mathematicians and mathematically inclined students of the natural sciences.
| Author | : Jack K. Hale |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 577 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461244269 |
In recent years, due primarily to the proliferation of computers, dynamical systems has again returned to its roots in applications. It is the aim of this book to provide undergraduate and beginning graduate students in mathematics or science and engineering with a modest foundation of knowledge. Equations in dimensions one and two constitute the majority of the text, and in particular it is demonstrated that the basic notion of stability and bifurcations of vector fields are easily explained for scalar autonomous equations. Further, the authors investigate the dynamics of planar autonomous equations where new dynamical behavior, such as periodic and homoclinic orbits appears.
| Author | : Mohd Hafiz Mohd |
| Publisher | : Springer Nature |
| Total Pages | : 239 |
| Release | : 2019-10-11 |
| Genre | : Mathematics |
| ISBN | : 9813298324 |
This book is the result of ​Southeast Asian Mathematical Society (SEAMS) School 2018 on Dynamical Systems and Bifurcation Analysis (DySBA). It addresses the latest developments in the field of dynamical systems, and highlights the importance of numerical continuation studies in tracking both stable and unstable steady states and bifurcation points to gain better understanding of the dynamics of the systems. The SEAMS School 2018 on DySBA was held in Penang from 6th to 13th August at the School of Mathematical Sciences, Universiti Sains Malaysia.The SEAMS Schools are part of series of intensive study programs that aim to provide opportunities for an advanced learning experience in mathematics via planned lectures, contributed talks, and hands-on workshop. This book will appeal to those postgraduates, lecturers and researchers working in the field of dynamical systems and their applications. Senior undergraduates in Mathematics will also find it useful.
