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Numerical Methods for Bifurcations of Dynamical Equilibria

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by Willy J. F. Govaerts


2000 / xxii + 362 pages / Softcover / ISBN: 978-0-898714-42-5 / List Price $109.50 / SIAM Member Price $76.65 / Order Code OT66

Dynamical systems arise in all fields of applied mathematics. The author focuses on the description of numerical methods for the detection, computation, and continuation of equilibria and bifurcation points of equilibria of dynamical systems. This subfield has the particular attraction of having links with the geometric theory of differential equations, numerical analysis, and linear algebra.

Several features make this book unique. The first is the systematic use of bordered matrix methods in the numerical computation and continuation of various bifurcations. The second is a detailed treatment of bialternate matrix products and their Jordan structure. Govaerts discusses their use in the numerical methods for Hopf and related bifurcations. A third feature is a unified treatment of singularity theory, with and without a distinguished bifurcation parameter, from a numerical point of view. Finally, numerical methods for symmetry-breaking bifurcations are discussed in detail, up to the fundamental cases covered by the equivariant branching lemma.

Anyone interested in computational methods for ODEs, PDEs, and bifurcation theory will find this volume a great addition to their library. This volume can be used as a text for graduate courses on numerical computation of bifurcations or numerical singularity theory. A basic knowledge of linear algebra, numerical linear algebra, calculus, and differential equations is required for full understanding of the material. Recommended for graduate students.

Preface; Notation; Introduction; Chapter 1: Examples and Motivation; Chapter 2: Manifolds and Numerical Continuation; Chapter 3: Bordered Matrices; Chapter 4: Generic Equilibrium Bifurcations in One-Parameter Problems; Chapter 5: Bifurcations Determined by the Jordan Form of the Jacobian; Chapter 6: Singularity Theory; Chapter 7: Singularity Theory with a Distinguished Bifurcation Parameter; Chapter 8: Symmetry-Breaking Bifurcations; Chapter 9: Bifurcations with Degeneracies in the Nonlinear Terms; Chapter 10: An Introduction to Large Dynamical Systems; Bibliography; Index.



ISBN: 9780898714425

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