Singularly perturbed Hopf bifurcation

Hopf bifurcation is the simplest way in which periodic solutions can emerge from an equilibrium point of an autonomous one-parameter family of ordinary differential equations. The phenomenon occurs if the linearized system has a pair of complex-conjugate simple eigenvalues which cross the imaginary axis transversely for some value of the parameter. In this paper, conditions are derived which ensure persistence of the Hopf bifurcation under singular perturbations of the vector field. The results justify use of reduced-order models in the study of nonlinear oscillations via Hopf bifurcation. Both single parameter and multiparameter singular perturbation problems are considered. In the single parameter case, we show how Fenichel's center manifold theorem for singularly perturbed systems can be used to prove regular degeneration of the bifurcated periodic solutions and to study their stability. In the multiparameter case, we obtain a novel asymptotic formula for the eigenvalues of the perturbed system. This formula is valid regardless of the relative magnitudes of the small parameters, and the results on multiparameter singularly perturbed Hopf bifurcation apply to two time scales as well as multiple time-scale systems.