| Description |
1 online resource (xiv, 407 pages) : illustrations |
| Series |
Cambridge monographs on applied and computational mathematics ; 34 |
|
Cambridge monographs on applied and computational mathematics ; 34.
|
| Contents |
Cover; Half-title; Series information; Title page; Copyright information; Dedication; Contents; Preface; Part One Theory; 1 Analytical Methods; 1.1 Setting and basic terminology; 1.2 Center manifold reduction; 1.3 Normal forms; 1.4 Approximating ODEs; 1.5 Simplest bifurcations of planar ODEs; 1.5.1 Generic one-parameter local bifurcations in 2D ODEs; 1.5.2 Generic two-parameter local bifurcations in 2D ODEs; 1.6 Pontryagin-Melnikov theory; 2 One-Parameter Bifurcations of Maps; 2.1 Codim 1 bifurcations of fixed points and cycles; 2.1.1 Fold bifurcation; 2.1.2 Period-doubling (flip) bifurcation |
|
2.1.3 Neimark-Sacker bifurcation2.2 Some global codim 1 bifurcations; 2.2.1 Homoclinic tangencies in planar maps; 2.2.2 Quasi-periodic bifurcations of invariant tori; 3 Two-Parameter Local Bifurcations of Maps; 3.1 Cusp and generalized period-doubling bifurcations; 3.1.1 CP (cusp); 3.1.2 GPD (generalized period-doubling); 3.2 CH (Chenciner bifurcation); 3.2.1 Normal forms; 3.2.2 Effect of higher-order terms; 3.3 Strong resonances; 3.3.1 R1 (resonance 1:1); 3.3.2 R2 (resonance 1:2); 3.3.3 R3 (resonance 1:3); 3.3.4 R4 (resonance 1:4); 3.4 Fold-flip and fold-Neimark-Sacker bifurcations |
|
3.4.1 LPPD (fold-flip bifurcation)3.4.2 LPNS (fold-Neimark-Sacker bifurcation); 3.5 Flip-Neimark-Sacker and double Neimark-Sacker bifurcations; 3.5.1 Normal forms; 3.5.2 Bifurcation analysis of symmetric normal forms; 3.5.3 Breaking the symmetries; 3.6 Historical perspective; Appendices; 3.A Proofs for Section 3.1; 3.B Proofs for Section 3.2; 3.C Proofs for Section 3.3; 3.D Proofs for Section 3.4.1; 3.E Proofs for Section 3.4.2; 3.F Proofs for Section 3.5; 4 Center Manifold Reduction for Local Bifurcations; 4.1 The homological equation and its solutions |
|
4.2 Critical normal form coefficients for local codim 2 bifurcations4.3 Branch switching at local codim 2 bifurcations; 4.3.1 Linear branch-switching; 4.3.2 Parameter-dependent center manifold reduction; Appendix: Fifth-order coefficients for flip-Neimark-Sacker and double Neimark-Sacker; Part Two Software; 5 Numerical Methods and Algorithms; 5.1 Continuation of cycles; 5.2 Continuation of codimension 1 bifurcation curves; 5.3 Computation of normal form coefficients; 5.3.1 Symbolic derivatives with respect to phase variables; 5.3.2 Symbolic derivatives with respect to parameters |
|
5.3.3 Recursive formulas for derivatives of the definingsystems for continuation5.3.4 Algorithmic differentiation for directional derivatives; 5.3.5 Numerical computation of the directional derivatives; 5.4 Computation of one-dimensional invariant manifolds of saddle fixed points; 5.4.1 Computing an unstable manifold; 5.4.2 Computing a stable manifold; 5.5 Continuation of connecting orbits; 5.5.1 Continuation of invariant subspaces; 5.5.2 The defining system; 5.5.3 Finding initial data for connecting orbits; 5.6 Bifurcations of homoclinic orbits; 5.7 Computation of Lyapunov exponents |
| Summary |
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 |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Online resource; title from PDF title page (EBSCO, April 1, 2019) |
| Subject |
Cartography -- Mathematical models
|
|
Bifurcation theory.
|
|
SCIENCE -- Earth Sciences -- Geography.
|
|
TECHNOLOGY & ENGINEERING -- Cartography.
|
|
Teoría de bifurcación
|
|
Bifurcation theory
|
|
Cartography -- Mathematical models
|
| Form |
Electronic book
|
| Author |
Meijer, Hil G. E., 1979- author.
|
| ISBN |
9781108695145 |
|
1108695140 |
|
9781108585804 |
|
1108585809 |
|
