Bifurcation Theory

Ale Jan Homburg, Jurgen Knobloch

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English

American Mathematical Society
31 January 2025

Differential calculus & equations

Series: Graduate Studies in Mathematics

This textbook provides a thorough overview of bifurcation theory. Assuming some familiarity with differential equations and dynamical systems, it is suitable for use on advanced undergraduate and graduate level and can, in particular, be used for a graduate course on bifurcation theory.

The book combines a solid theoretical basis with a detailed description of classical bifurcations. It is organized in chapters on local, nonlocal, and global bifurcations; a number of appendices develop the toolbox for the study of bifurcations. The discussed local bifurcations include saddle-node and Hopf bifurcations, as well as the more advanced Bogdanov-Takens and Neimark-Sacker bifurcations. The book also covers nonlocal bifurcations, discussing various homoclinic bifurcations, and it surveys global bifurcations and phenomena, such as intermittency and period-doubling cascades. The book develops a broad range of complementary techniques, both geometric and analytic, for studying bifurcations. Techniques include normal form methods, center manifold reductions, the Lyapunov-Schmidt construction, cross-coordinate constructions, Melnikov's method, and Lin's method.

Full proofs of the results are provided, also for the material in the appendices. This includes proofs of the stable manifold theorem, of the center manifold theorem, and of Lin's method for studying homoclinic bifurcations.

By: Ale Jan Homburg, Jurgen Knobloch
Imprint: American Mathematical Society
Country of Publication: United States
Volume: 246
Dimensions: Height: 254mm, Width: 178mm,
ISBN: 9781470477943
ISBN 10: 1470477947
Series: Graduate Studies in Mathematics
Pages: 646
Publication Date: 31 January 2025
Audience: Professional and scholarly , Undergraduate
Format: Hardback
Publisher's Status: Active

Introductory models Flows and invariant sets Local bifurcations Nonlocal bifurcations Global bifurcations Elements of nonlinear analysis Invariant manifolds and normal forms Lin's method Bibliography Index

Ale Jan Homburg, University of Amsterdam, The Netherlands, and Leiden University, The Netherlands, and Jurgen Knobloch, Technical University Ilmenau, Germany