On Hopf bifurcations in singularly perturbed systems

It has been shown recently that, under some generic assumptions, there exists a Hopf curve λ = λ (ε) for singularly perturbed systems of the form x˙ = f (x, y, λ), εy˙ = g(x, y, λ) near the singular surface defined by det g_v = 0. In this note, we are concerned with the Hopf curve and obtain three results: 1) we prove that the eigenvalue crossing condition for the Hopf curve holds without additional assumption; 2) we provide an improved form of an existing derivative formula for the Hopf curve which is more suitable for practical computations; and 3) we give a quite precise description of the spectrum structure of the linearization along the Hopf curve. All three results (stated in the main theorem) are useful for a better understanding of Hopf bifurcations in singularly perturbed systems. Our analysis is based on a factorization of parameter dependent polynomials (Lemma 2.3).