Bifurcation Theory

Marty Golubitsky
Mathematical Biosciences Institute, The Ohio State University

Bifurcation Theory

This course will survey how context (distinguished parameter, symmetry, networks) changes the kinds of local bifurcations from an equilibrium that one can expect in systems of differential equations. Specific topics will include:

Singularity theory applied to steady-state bifurcations

  • determinacy

  • universal unfoldings

  • classification by (singularity theory) codimension

Liapunov-Schmidt reduction

  • Hopf bifurcation to periodic solutions

Codimension 1 and 2 dynamic mode interactions

Equivariant bifurcation theory (or symmetry-breaking)

  • elementary representation theory

  • equivariant branching lemma

  • equivariant Hopf theorem

  • spatio-temporal symmetries of periodic solutions

Bifurcations in networks (or synchrony-breaking)

  • balanced colorings

  • quotient networks

  • feed forward networks

As much as possible, theory will be motivated by examples and applications. Detailed proofs can be found in several books and research papers including

  • M. Golubitsky, I.N. Stewart and D.G. Schaeffer. Singularities and Groups in Bifurcation Theory: Vols. I and II. Appl Math Sci 51 and 69, Springer-Verlag, New York, 1985 and 1988.

  • M. Golubitsky, I. Stewart, and A. Torok. Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dynam. Sys. 4 (1) (2005) 78-100.

  • J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Appl Math Sci 42. Springer-Verlag, New York, 2nd edition, 1990.