SINGULAR HOPF-BIFURCATION IN SYSTEMS WITH FAST AND SLOW VARIABLES

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 no rmal form for this system, valid close to equilibria where certain con ditions on the derivatives hold. The most important condition concerns the presence of eigenvalues with singular imaginary parts, by which w e mean that their imaginary part grows without bound as the small para meter tends to zero. We give a simple criterion to test for the possib le presence of equilibria satisfying this condition. Using a center ma nifold reduction, we show the existence of Hopf bifurcation points, or iginating from the interaction of fast and slow variables, and we dete rmine their nature. We apply the theory, developed here, to two exampl es: an extended Bonhoeffer-van der Pol system and a predator-prey mode l. Our theory is in good agreement with the numerical continuation exp eriments we carried out for the examples.