Introduction to bifurcation theory

Rev. Mod. Phys. 63, 991 – Published 1 October 1991
John David Crawford

Abstract

The theory of bifurcation from equilibria based on center-manifold reduction and Poincaré-Birkhoff normal forms is reviewed at an introductory level. Both differential equations and maps are discussed, and recent results explaining the symmetry of the normal form are derived. The emphasis is on the simplest generic bifurcations in one-parameter systems. Two applications are developed in detail: a Hopf bifurcation occurring in a model of three-wave mode coupling and steady-state bifurcations occurring in the real Landau-Ginzburg equation. The former provides an example of the importance of degenerate bifurcations in problems with more than one parameter and the latter illustrates new effects introduced into a bifurcation problem by a continuous symmetry.

DOI: http://dx.doi.org/10.1103/RevModPhys.63.991

© 1991 The American Physical Society

Authors & Affiliations

John David Crawford*

  • Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712 and Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

  • *Permanent address.

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