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E-raamat: Numerical Bifurcation Analysis of Maps: From Theory to Software

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(Universiteit Utrecht, The Netherlands), (University of Twente, Enschede, The Netherlands)
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Numerical Bifurcation Analysis of Maps: From Theory to SoftwareSuurem pilt
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This book combines a comprehensive state-of-the-art analysis of bifurcations of discrete-time dynamical systems with concrete instruction on implementations (and example applications) in the free MATLAB® software MatContM developed by the authors. While self-contained and suitable for independent study, the book is also written with users in mind and is an invaluable reference for practitioners. Part I focuses on theory, providing a systematic presentation of bifurcations of fixed points and cycles of finite-dimensional maps, up to and including cases with two control parameters. Several complementary methods, including Lyapunov exponents, invariant manifolds and homoclinic structures, and parts of chaos theory, are presented. Part II introduces MatContM through step-by-step tutorials on how to use the general numerical methods described in Part I for simple dynamical models defined by one- and two-dimensional maps. Further examples in Part III show how MatContM can be used to analyze more complicated models from modern engineering, ecology, and economics.

This book combines a comprehensive treatment of bifurcations of discrete-time dynamical systems with concrete instruction on implementations and applications in the free MATLAB® software MatContM. While self-contained and suitable for independent study, it is also written with users in mind and will be an invaluable reference for practitioners.

Arvustused

'The topic of this book is the study of local and global bifurcations (qualitative changes in dynamics) of discrete-time maps as parameters are varied This book could be used as reference to known results on bifurcations of maps, or as a guide to the software MatcontM. It is clearly written and contains many high-quality figures.' Carlo Laing, zbMATH 'Throughout the whole work, there is an abundance of joyfully complex figures depicting various dynamics via phase portrait sketches and bifurcation structures in parameter space The first half of this book will doubtless be an essential and convenient reference for specialists who already conduct research in this field.' Gavin M. Abernethy, LMS Newsletter 'This book is an excellent compendium of bifurcation results and phenomenology for low-dimensional maps, and would find itself usefully ensconced on the bookshelf next to the computer (running its accompanying software) of any researcher studying dynamical systems.' James Meiss, SIAM Review

Muu info

Combines a systematic analysis of bifurcations of iterated maps with concrete MATLAB® implementations and applications.
Preface xi
PART ONE THEORY
1(216)
1 Analytical Methods
3(27)
1.1 Setting and basic terminology
3(3)
1.2 Center manifold reduction
6(2)
1.3 Normal forms
8(2)
1.4 Approximating ODEs
10(1)
1.5 Simplest bifurcations of planar ODEs
11(14)
1.6 Pontryagin--Melnikov theory
25(5)
2 One-Parameter Bifurcations of Maps
30(20)
2.1 Codim 1 bifurcations of fixed points and cycles
30(13)
2.2 Some global codim 1 bifurcations
43(7)
3 Two-Parameter Local Bifurcations of Maps
50(135)
3.1 Cusp and generalized period-doubling bifurcations
51(3)
3.2 CH (Chenciner bifurcation)
54(7)
3.3 Strong resonances
61(26)
3.4 Fold--flip and fold--Neimark--Sacker bifurcations
87(19)
3.5 Flip--Neimark--Sacker and double Neimark--Sacker bifurcations
106(26)
3.6 Historical perspective
132(2)
Appendices
134(51)
4 Center Manifold Reduction for Local Bifurcations
185(32)
4.1 The homological equation and its solutions
186(4)
4.2 Critical normal form coefficients for local codim 2 bifurcations
190(14)
4.3 Branch switching at local codim 2 bifurcations
204(6)
Appendix: Fifth-order coefficients for flip--Neimark--Sacker and double Neimark--Sacker
210(7)
PART TWO SOFTWARE
217(102)
5 Numerical Methods and Algorithms
219(24)
5.1 Continuation of cycles
219(1)
5.2 Continuation of codimension 1 bifurcation curves
220(4)
5.3 Computation of normal form coefficients
224(5)
5.4 Computation of one-dimensional invariant manifolds of saddle fixed points
229(3)
5.5 Continuation of connecting orbits
232(6)
5.6 Bifurcations of homoclinic orbits
238(3)
5.7 Computation of Lyapunov exponents
241(2)
6 Features and Functionality of MatcontM
243(15)
6.1 General description of MatcontM
244(4)
6.2 The mapfile
248(2)
6.3 Numerical continuation
250(4)
6.4 Calling the Continuer
254(4)
7 MatcontM Tutorials
258(61)
7.1 Tutorial 1: iteration of maps and continuation of fixed points and cycles
258(16)
7.2 Tutorial 2: two-parameter local bifurcation analysis
274(20)
7.3 Tutorial 2: invariant manifolds and connecting orbits
294(14)
7.4 Tutorial 4: computation of Lyapunov exponents
308(11)
PART THREE APPLICATIONS
319(70)
8 The Generalized Henon Map
321(33)
8.1 Introduction
321(3)
8.2 Homoclinic bifurcations and GHM
324(5)
8.3 Bifurcation diagrams of GHM
329(19)
8.4 Interpretation
348(3)
8.5 Discussion
351(3)
9 Adaptive Control Map
354(8)
9.1 Local bifurcations
354(3)
9.2 Numerical continuation
357(1)
9.3 Derivatives for the adaptive control map
358(4)
10 Duopoly Model of Kopel
362(23)
10.1 Description of the model
362(1)
10.2 Fixed points and codim 1 bifurcations
363(2)
10.3 Normal forms of codim 1 bifurcations
365(2)
10.4 Codim 2 bifurcations
367(3)
10.5 Codim 2 normal form coefficients
370(2)
10.6 Numerical analysis using MatcontM
372(10)
10.7 Conclusions
382(3)
11 The SEIR Epidemic Model
385(4)
11.1 The model
385(1)
11.2 Bifurcation diagram
386(3)
References 389(11)
Index 400
Yuri A. Kuznetsov is Associate Professor at Utrecht University and Professor of Numerical Bifurcation Methods at the University of Twente. He has made significant contributions to the theory of codimension two bifurcations of smooth ODEs and iterated maps. His recent work has focussed on efficient numerical continuation and normal form analysis of maps, ODEs and DDEs, and on applications of these methods in ecology, economics, engineering, and neuroscience. He is also the author of the widely-used text and reference Elements of Applied Bifurcation Theory, 3rd edition (2010). Hil G. E. Meijer is Assistant Professor at the University of Twente, Enschede, The Netherlands. He has extensive experience in numerical bifurcation theory and interdisciplinary applications such as modeling Parkinson's disease and epilepsy. He is a co-supervisor of the MatCont software project and has given numerous workshops on its use.
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