Singular Hopf Bifurcation in Systems with Fast and Slow Variables |
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Authors: | B Braaksma |
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Institution: | (1) Limburgs Universitair Centrum, Universitaire Campus, B-3590 Diepenbeek, Belgium e-mail: braaksma@luc.ac.be, BE |
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Abstract: | Summary. 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.
Received October 24, 1996; revised October 31, 1997; accepted November 3, 1997 |
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