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.
Author(s): Yuri A. Kuznetsov, Hil G. E. Meijer
Series: Cambridge Monographs on Applied and Computational Mathematics 34
Publisher: Cambridge University Press
Year: 2019
Language: English
Pages: 423
Contents......Page 8
Preface......Page 12
PART ONE THEORY......Page 16
1.1 Setting and basic terminology......Page 18
1.2 Center manifold reduction......Page 21
1.3 Normal forms......Page 23
1.4 Approximating ODEs......Page 25
1.5 Simplest bifurcations of planar ODEs......Page 26
1.6 Pontryagin–Melnikov theory......Page 40
2.1 Codim 1 bifurcations of fixed points and cycles......Page 45
2.2 Some global codim 1 bifurcations......Page 58
3 Two-Parameter Local Bifurcations of Maps......Page 65
3.1 Cusp and generalized period-doubling bifurcations......Page 66
3.2 CH (Chenciner bifurcation)......Page 69
3.3 Strong resonances......Page 76
3.4 Fold–flip and fold–Neimark–Sacker bifurcations......Page 102
3.5 Flip–Neimark–Sacker and double Neimark–Sacker bifurcations......Page 121
3.6 Historical perspective......Page 147
Appendices......Page 149
4 Center Manifold Reduction for Local Bifurcations......Page 200
4.1 The homological equation and its solutions......Page 201
4.2 Critical normal form coefficients for local codim 2
bifurcations......Page 205
4.3 Branch switching at local codim 2 bifurcations......Page 219
Appendix: Fifth-order coefficients for flip–Neimark–Sacker and double Neimark–Sacker......Page 225
PART TWO SOFTWARE......Page 232
5.1 Continuation of cycles......Page 234
5.2 Continuation of codimension 1 bifurcation curves......Page 235
5.3 Computation of normal form coefficients......Page 239
5.4 Computation of one-dimensional invariant manifolds of saddle fixed points......Page 244
5.5 Continuation of connecting orbits......Page 247
5.6 Bifurcations of homoclinic orbits......Page 253
5.7 Computation of Lyapunov exponents......Page 256
6 Features and Functionality of MatcontM......Page 258
6.1 General description of MatcontM......Page 259
6.2 The mapfile......Page 263
6.3 Numerical continuation......Page 265
6.4 Calling the Continuer......Page 269
7.1 Tutorial 1: iteration of maps and continuation of fixed points and cycles......Page 273
7.2 Tutorial 2: two-parameter local bifurcation analysis......Page 289
7.3 Tutorial 2: invariant manifolds and connecting orbits......Page 309
7.4 Tutorial 4: computation of Lyapunov exponents......Page 323
PART THREE APPLICATIONS......Page 334
8.1 Introduction......Page 336
8.2 Homoclinic bifurcations and GHM......Page 339
8.3 Bifurcation diagrams of GHM......Page 344
8.4 Interpretation......Page 363
8.5 Discussion......Page 366
9.1 Local bifurcations......Page 369
9.2 Numerical continuation......Page 372
9.3 Derivatives for the adaptive control map......Page 373
10.1 Description of the model......Page 377
10.2 Fixed points and codim 1 bifurcations......Page 378
10.3 Normal forms of codim 1 bifurcations......Page 380
10.4 Codim 2 bifurcations......Page 382
10.5 Codim 2 normal form coefficients......Page 385
10.6 Numerical analysis using MatcontM......Page 387
10.7 Conclusions......Page 397
11.1 The model......Page 400
11.2 Bifurcation diagram......Page 401
References......Page 404
Index......Page 415
