Journal of Nonlinear Science

, Volume 8, Issue 5, pp 457–490 | Cite as

Singular Hopf Bifurcation in Systems with Fast and Slow Variables

  • B. Braaksma


We study a general nonlinear ODE system with fast and slow variables, i.e., some of the derivatives are multiplied by a small parameter. The system depends on an additional bifurcation parameter. We derive a normal form for this system, valid close to equilibria where certain conditions on the derivatives hold. The most important condition concerns the presence of eigenvalues with singular imaginary parts, by which we mean that their imaginary part grows without bound as the small parameter tends to zero. We give a simple criterion to test for the possible presence of equilibria satisfying this condition. Using a center manifold reduction, we show the existence of Hopf bifurcation points, originating from the interaction of fast and slow variables, and we determine their nature. We apply the theory, developed here, to two examples: an extended Bonhoeffer—van der Pol system and a predator-prey model. Our theory is in good agreement with the numerical continuation experiments we carried out for the examples.


Normal Form Slow Variable Center Manifold Relaxation Oscillation Fast Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag New York Inc. 1998

Authors and Affiliations

  • B. Braaksma
    • 1
  1. 1.Limburgs Universitair Centrum, Universitaire Campus, B-3590 Diepenbeek, Belgium e-mail:

Personalised recommendations