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Numerical Bifurcation Analysis of Maps : From Theory to Software

Author: I︠U︡ A Kuznet︠s︡ov; Hil G E Meijer
Publisher: Cambridge : Cambridge University Press, 2019.
Series: Cambridge monographs on applied and computational mathematics, 34.
Edition/Format:   eBook : Document : EnglishView all editions and formats
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  Read more...
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Genre/Form: Electronic books
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: I︠U︡ A Kuznet︠s︡ov; Hil G E Meijer
ISBN: 9781108695145 1108695140 9781108585804 1108585809
OCLC Number: 1090812965
Description: 1 online resource
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
Series Title: Cambridge monographs on applied and computational mathematics, 34.
Responsibility: Yuri A. Kuznetsov, Hil G.E. Meijer.

Abstract:

This book combines a comprehensive treatment of bifurcations of discrete-time dynamical systems with concrete instruction on implementations and applications in the free MATLAB (R) software MatContM.  Read more...

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