Canard Cycles And Center Manifolds

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Canard Cycles and Center Manifolds

Freddy Dumortier,Robert H. Roussarie
Canard Cycles and Center Manifolds Book PDF

Author : Freddy Dumortier,Robert H. Roussarie
Publisher : American Mathematical Soc.
Page : 100 pages
File Size : 50,8 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804438

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Canard Cycles and Center Manifolds by Freddy Dumortier,Robert H. Roussarie Pdf

In this book, the ``canard phenomenon'' occurring in Van der Pol's equation $\epsilon \ddot x+(x^2+x)\dot x+x-a=0$ is studied. For sufficiently small $\epsilon >0$ and for decreasing $a$, the limit cycle created in a Hopf bifurcation at $a = 0$ stays of ``small size'' for a while before it very rapidly changes to ``big size'', representing the typical relaxation oscillation. The authors give a geometric explanation and proof of this phenomenon using foliations by center manifolds and blow-up of unfoldings as essential techniques. The method is general enough to be useful in the study of other singular perturbation problems.

Canard Cycles

Peter De Maesschalck,Freddy Dumortier,Robert Roussarie
Canard Cycles Book PDF

Author : Peter De Maesschalck,Freddy Dumortier,Robert Roussarie
Publisher : Springer Nature
Page : 408 pages
File Size : 40,6 Mb
Release : 2021-08-07
Category : Mathematics
ISBN : 9783030792336

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Canard Cycles by Peter De Maesschalck,Freddy Dumortier,Robert Roussarie Pdf

This book offers the first systematic account of canard cycles, an intriguing phenomenon in the study of ordinary differential equations. The canard cycles are treated in the general context of slow-fast families of two-dimensional vector fields. The central question of controlling the limit cycles is addressed in detail and strong results are presented with complete proofs. In particular, the book provides a detailed study of the structure of the transitions near the critical set of non-isolated singularities. This leads to precise results on the limit cycles and their bifurcations, including the so-called canard phenomenon and canard explosion. The book also provides a solid basis for the use of asymptotic techniques. It gives a clear understanding of notions like inner and outer solutions, describing their relation and precise structure. The first part of the book provides a thorough introduction to slow-fast systems, suitable for graduate students. The second and third parts will be of interest to both pure mathematicians working on theoretical questions such as Hilbert's 16th problem, as well as to a wide range of applied mathematicians looking for a detailed understanding of two-scale models found in electrical circuits, population dynamics, ecological models, cellular (FitzHugh–Nagumo) models, epidemiological models, chemical reactions, mechanical oscillators with friction, climate models, and many other models with tipping points.

Mathematical Sciences with Multidisciplinary Applications

Bourama Toni
Mathematical Sciences with Multidisciplinary Applications Book PDF

Author : Bourama Toni
Publisher : Springer
Page : 641 pages
File Size : 43,6 Mb
Release : 2016-08-19
Category : Mathematics
ISBN : 9783319313238

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Mathematical Sciences with Multidisciplinary Applications by Bourama Toni Pdf

This book is the fourth in a multidisciplinary series which brings together leading researchers in the STEAM-H disciplines (Science, Technology, Engineering, Agriculture, Mathematics and Health) to present their perspective on advances in their own specific fields, and to generate a genuinely interdisciplinary collaboration that transcends parochial subject-matter boundaries. All contributions are carefully edited, peer-reviewed, reasonably self-contained, and pedagogically crafted for a multidisciplinary readership. Contributions are drawn from a variety of fields including mathematics, statistics, game theory and behavioral sciences, biomathematics and physical chemistry, computer science and human-centered computing. This volume is dedicated to Professor Christiane Rousseau, whose work inspires the STEAM-H series, in recognition of her passion for the mathematical sciences and her on-going initiative, the Mathematics of Planet Earth paradigm of interdisciplinarity. The volume's primary goal is to enhance interdisciplinary understanding between these areas of research by showing how new advances in a particular field can be relevant to open problems in another and how many disciplines contribute to a better understanding of relevant issues at the interface of mathematics and the sciences. The main emphasis is on important methods, research directions and applications of analysis within and beyond each field. As such, the volume aims to foster student interest and participation in the STEAM-H domain, as well as promote interdisciplinary research collaborations. The volume is valuable as a reference of choice and a source of inspiration for a broad spectrum of scientists, mathematicians, research students and postdoctoral fellows.

The Integral Manifolds of the Three Body Problem

Christopher Keil McCord,Kenneth Ray Meyer,Quidong Wang
The Integral Manifolds of the Three Body Problem Book PDF

Author : Christopher Keil McCord,Kenneth Ray Meyer,Quidong Wang
Publisher : American Mathematical Soc.
Page : 91 pages
File Size : 52,7 Mb
Release : 1998
Category : Science
ISBN : 9780821806920

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The Integral Manifolds of the Three Body Problem by Christopher Keil McCord,Kenneth Ray Meyer,Quidong Wang Pdf

The phase space of the spatial three-body problem is an open subset in ${\mathbb R}^{18}$. Holding the ten classical integrals of energy, center of mass, linear and angular momentum fixed defines an eight dimensional submanifold. For fixed nonzero angular momentum, the topology of this manifold depends only on the energy. This volume computes the homology of this manifold for all energy values. This table of homology shows that for negative energy, the integral manifolds undergo seven bifurcations. Four of these are the well-known bifurcations due to central configurations, and three are due to 'critical points at infinity'. This disproves Birkhoff's conjecture that the bifurcations occur only at central configurations.

Two Classes of Riemannian Manifolds Whose Geodesic Flows Are Integrable

Kazuyoshi Kiyohara
Two Classes of Riemannian Manifolds Whose Geodesic Flows Are Integrable Book PDF

Author : Kazuyoshi Kiyohara
Publisher : American Mathematical Soc.
Page : 143 pages
File Size : 47,8 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821806401

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Two Classes of Riemannian Manifolds Whose Geodesic Flows Are Integrable by Kazuyoshi Kiyohara Pdf

In this work, two classes of manifolds whose geodesic flows are integrable are defined, and their global structures are investigated. They are called Liouville manifolds and Kahler-Liouville manifolds respectively. In each case, the author finds several invariants with which they are partly classified. The classification indicates, in particular, that these classes contain many new examples of manifolds with integrable geodesic flow.

Geometric Singular Perturbation Theory Beyond the Standard Form

Martin Wechselberger
Geometric Singular Perturbation Theory Beyond the Standard Form Book PDF

Author : Martin Wechselberger
Publisher : Springer Nature
Page : 143 pages
File Size : 47,7 Mb
Release : 2020-02-21
Category : Mathematics
ISBN : 9783030363994

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Geometric Singular Perturbation Theory Beyond the Standard Form by Martin Wechselberger Pdf

This volume provides a comprehensive review of multiple-scale dynamical systems. Mathematical models of such multiple-scale systems are considered singular perturbation problems, and this volume focuses on the geometric approach known as Geometric Singular Perturbation Theory (GSPT). It is the first of its kind that introduces the GSPT in a coordinate-independent manner. This is motivated by specific examples of biochemical reaction networks, electronic circuit and mechanic oscillator models and advection-reaction-diffusion models, all with an inherent non-uniform scale splitting, which identifies these examples as singular perturbation problems beyond the standard form. The contents cover a general framework for this GSPT beyond the standard form including canard theory, concrete applications, and instructive qualitative models. It contains many illustrations and key pointers to the existing literature. The target audience are senior undergraduates, graduate students and researchers interested in using the GSPT toolbox in nonlinear science, either from a theoretical or an application point of view. Martin Wechselberger is Professor at the School of Mathematics & Statistics, University of Sydney, Australia. He received the J.D. Crawford Prize in 2017 by the Society for Industrial and Applied Mathematics (SIAM) for achievements in the field of dynamical systems with multiple time-scales.

Featured Reviews in "Mathematical Reviews" 1995-1996

Donald G. Babbitt,Jane E. Kister
Featured Reviews in Mathematical Reviews 1995 1996 Book PDF

Author : Donald G. Babbitt,Jane E. Kister
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 55,9 Mb
Release : 2024-07-02
Category : Mathematics
ISBN : 0821895192

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Featured Reviews in "Mathematical Reviews" 1995-1996 by Donald G. Babbitt,Jane E. Kister Pdf

This collection of reprinted 'Featured Reviews' published in Mathematical Reviews (MR) in 1995 and 1996 makes widely available informed reviews of some of the best mathematics published recently. 'Featured Reviews' were introduced in MR at the beginning of 1995 in part to provide some guidance to the current research-level literature. With the exponential growth of publications in mathematical research in the first half-century of MR, it had become essentially impossible for users of MR to identify the most important new research-level books and papers, especially in fields outside of the users' own expertise. This work identifies some of the "best" new publications, papers, and books that are expected to have a significant impact on the area of pure or applied mathematics with which researchers are concerned. All of the papers reviewed here contain interesting new ideas or applications, a deep synthesis of existing ideas, or any combination of these. The volume is intended to lead the user to important new research across all fields covered by MR.

Normal Forms, Bifurcations and Finiteness Problems in Differential Equations

Christiane Rousseau,Gert Sabidussi
Normal Forms Bifurcations and Finiteness Problems in Differential Equations Book PDF

Author : Christiane Rousseau,Gert Sabidussi
Publisher : Springer Science & Business Media
Page : 548 pages
File Size : 48,7 Mb
Release : 2004-02-29
Category : Mathematics
ISBN : 1402019297

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Normal Forms, Bifurcations and Finiteness Problems in Differential Equations by Christiane Rousseau,Gert Sabidussi Pdf

Proceedings of the Nato Advanced Study Institute, held in Montreal, Canada, from 8 to 19 July 2002

Numerical Continuation Methods for Dynamical Systems

Bernd Krauskopf,Hinke M. Osinga,Jorge Galan-Vioque
Numerical Continuation Methods for Dynamical Systems Book PDF

Author : Bernd Krauskopf,Hinke M. Osinga,Jorge Galan-Vioque
Publisher : Springer
Page : 399 pages
File Size : 41,8 Mb
Release : 2007-11-06
Category : Science
ISBN : 9781402063565

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Numerical Continuation Methods for Dynamical Systems by Bernd Krauskopf,Hinke M. Osinga,Jorge Galan-Vioque Pdf

Path following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel's 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects and showcase examples of how numerical bifurcation analysis can be used in concrete applications. Topics that are treated include: interactive continuation tools, higher-dimensional continuation, the computation of invariant manifolds, and continuation techniques for slow-fast systems, for symmetric Hamiltonian systems, for spatially extended systems and for systems with delay. Three chapters review physical applications: the dynamics of a SQUID, global bifurcations in laser systems, and dynamics and bifurcations in electronic circuits.

Recent Trends in Dynamical Systems

Andreas Johann,Hans-Peter Kruse,Florian Rupp,Stephan Schmitz
Recent Trends in Dynamical Systems Book PDF

Author : Andreas Johann,Hans-Peter Kruse,Florian Rupp,Stephan Schmitz
Publisher : Springer Science & Business Media
Page : 628 pages
File Size : 50,8 Mb
Release : 2013-09-24
Category : Mathematics
ISBN : 9783034804516

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Recent Trends in Dynamical Systems by Andreas Johann,Hans-Peter Kruse,Florian Rupp,Stephan Schmitz Pdf

This book presents the proceedings of a conference on dynamical systems held in honor of Jürgen Scheurle in January 2012. Through both original research papers and survey articles leading experts in the field offer overviews of the current state of the theory and its applications to mechanics and physics. In particular, the following aspects of the theory of dynamical systems are covered: - Stability and bifurcation - Geometric mechanics and control theory - Invariant manifolds, attractors and chaos - Fluid mechanics and elasticity - Perturbations and multiscale problems - Hamiltonian dynamics and KAM theory Researchers and graduate students in dynamical systems and related fields, including engineering, will benefit from the articles presented in this volume.

The Real Positive Definite Completion Problem

Wayne Walton Barrett,Charles R. Johnson,Raphael Loewy
The Real Positive Definite Completion Problem Book PDF

Author : Wayne Walton Barrett,Charles R. Johnson,Raphael Loewy
Publisher : American Mathematical Soc.
Page : 69 pages
File Size : 50,9 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821804735

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The Real Positive Definite Completion Problem by Wayne Walton Barrett,Charles R. Johnson,Raphael Loewy Pdf

Given a partial symmetric matrix, the positive definite completion problem asks if the unspecified entries in the matrix can be chosen so as to make the resulting matrix positive definite. Applications include probability and statistics, image enhancement, systems engineering, geophysics, and mathematical programming. The positive definite completion problem can also be viewed as a mechanism for addressing a fundamental problem in Euclidean geometry: which potential geometric configurations of vectors (i.e., configurations with angles between some vectors specified) are realizable in a Euclidean space. The positions of the specified entries in a partial matrix are naturally described by a graph. The question of existence of a positive definite completion was previously solved completely for the restrictive class of chordal graphs and this work solves the problem for the class of cycle completable graphs, a significant generalization of chordal graphs. These are the graphs for which knowledge of completability for induced cycles (and cliques) implies completability of partial symmetric matrices with the given graph.

The Structure of $k$-$CS$- Transitive Cycle-Free Partial Orders

Richard Warren
The Structure of k CS Transitive Cycle Free Partial Orders Book PDF

Author : Richard Warren
Publisher : American Mathematical Soc.
Page : 166 pages
File Size : 53,9 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821806227

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The Structure of $k$-$CS$- Transitive Cycle-Free Partial Orders by Richard Warren Pdf

The class of cycle-free partial orders (CFPOs) is defined, and the CFPOs fulfilling a natural transitivity assumption, called $k$-connected set transitivity ($k$-$CS$-transitivity), are analyzed in some detail. Classification in many of the interesting cases is given. This work generalizes Droste's classification of the countable $k$-transitive trees ($k \geq 2$). In a CFPO, the structure can branch downwards as well as upwards, and can do so repeatedly (though it never returns to the starting point by a cycle). Mostly it is assumed that $k \geq 3$ and that all maximal chains are finite. The main classification splits into the sporadic and skeletal cases. The former is complete in all cardinalities. The latter is performed only in the countable case. The classification is considerably more complicated than for trees, and skeletal CFPOs exhibit rich, elaborate and rather surprising behavior. Features: Lucid exposition of an important generalization of Droste's work Extended introduction clearly explaining the scope of the memoir Visually attractive topic with copious illustrations Self-contained material, requiring few prerequisites

Nonlinear Systems, Vol. 1

Victoriano Carmona,Jesús Cuevas-Maraver,Fernando Fernández-Sánchez,Elisabeth García- Medina
Nonlinear Systems Vol 1 Book PDF

Author : Victoriano Carmona,Jesús Cuevas-Maraver,Fernando Fernández-Sánchez,Elisabeth García- Medina
Publisher : Springer
Page : 424 pages
File Size : 47,9 Mb
Release : 2018-09-15
Category : Science
ISBN : 9783319667669

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Nonlinear Systems, Vol. 1 by Victoriano Carmona,Jesús Cuevas-Maraver,Fernando Fernández-Sánchez,Elisabeth García- Medina Pdf

This book is part of a two volume set which presents the analysis of nonlinear phenomena as a long-standing challenge for research in basic and applied science as well as engineering. It discusses nonlinear differential and differential equations, bifurcation theory for periodic orbits and global connections. The integrability and reversibility of planar vector fields and theoretical analysis of classic physical models are sketched. This first volume concentrates on the mathematical theory and computational techniques that are essential for the study of nonlinear science, a second volume deals with real-world nonlinear phenomena in condensed matter, biology and optics.

Multiple-Time-Scale Dynamical Systems

Christopher K.R.T. Jones,Alexander I. Khibnik
Multiple Time Scale Dynamical Systems Book PDF

Author : Christopher K.R.T. Jones,Alexander I. Khibnik
Publisher : Springer Science & Business Media
Page : 278 pages
File Size : 52,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461301172

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Multiple-Time-Scale Dynamical Systems by Christopher K.R.T. Jones,Alexander I. Khibnik Pdf

Systems with sub-processes evolving on many different time scales are ubiquitous in applications: chemical reactions, electro-optical and neuro-biological systems, to name just a few. This volume contains papers that expose the state of the art in mathematical techniques for analyzing such systems. Recently developed geometric ideas are highlighted in this work that includes a theory of relaxation-oscillation phenomena in higher dimensional phase spaces. Subtle exponentially small effects result from singular perturbations implicit in certain multiple time scale systems. Their role in the slow motion of fronts, bifurcations, and jumping between invariant tori are all explored here. Neurobiology has played a particularly stimulating role in the development of these techniques and one paper is directed specifically at applying geometric singular perturbation theory to reveal the synchrony in networks of neural oscillators.

Structured Population Models in Biology and Epidemiology

Pierre Magal,Shigui Ruan
Structured Population Models in Biology and Epidemiology Book PDF

Author : Pierre Magal,Shigui Ruan
Publisher : Springer Science & Business Media
Page : 314 pages
File Size : 49,7 Mb
Release : 2008-04-30
Category : Mathematics
ISBN : 9783540782728

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Structured Population Models in Biology and Epidemiology by Pierre Magal,Shigui Ruan Pdf

In this new century mankind faces ever more challenging environmental and publichealthproblems,suchaspollution,invasionbyexoticspecies,theem- gence of new diseases or the emergence of diseases into new regions (West Nile virus,SARS,Anthrax,etc.),andtheresurgenceofexistingdiseases(in?uenza, malaria, TB, HIV/AIDS, etc.). Mathematical models have been successfully used to study many biological, epidemiological and medical problems, and nonlinear and complex dynamics have been observed in all of those contexts. Mathematical studies have helped us not only to better understand these problems but also to ?nd solutions in some cases, such as the prediction and control of SARS outbreaks, understanding HIV infection, and the investi- tion of antibiotic-resistant infections in hospitals. Structuredpopulationmodelsdistinguishindividualsfromoneanother- cording to characteristics such as age, size, location, status, and movement, to determine the birth, growth and death rates, interaction with each other and with environment, infectivity, etc. The goal of structured population models is to understand how these characteristics a?ect the dynamics of these models and thus the outcomes and consequences of the biological and epidemiolo- cal processes. There is a very large and growing body of literature on these topics. This book deals with the recent and important advances in the study of structured population models in biology and epidemiology. There are six chapters in this book, written by leading researchers in these areas.