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On Hopf bifurcations in singularly perturbed systems

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3 Author(s)
Yang, L. ; Dept. of Math. Sci., Tsinghua Univ., Beijing, China ; Tang, Y. ; Du, D.

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 gv = 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).

Published in:

Automatic Control, IEEE Transactions on  (Volume:48 ,  Issue: 4 )

Date of Publication:

April 2003

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