On Hopf bifurcations in singularly perturbed systems

It has been shown recently that, under some generic assumptions, there exists a Hopf curve /spl lambda/ = /spl lambda/ (/spl epsiv/) for singularly perturbed systems of the form x/spl dot/ = f (x, y, /spl lambda/), /spl epsiv/y/spl dot/ = g(x, y, /spl lambda/) near the singular surface defined by det g/sub 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).

[1]  Analysis of Takens-Bogdanov bifurcation by characteristic functions , 1996 .

[2]  Jaroslav Kautsky,et al.  Some necessary and sufficient conditions for eigenstructure assignment , 1985 .

[3]  R. E. Beardmore,et al.  Stability and bifurcation properties of index-1 DAEs , 1998, Numerical Algorithms.

[4]  P. R. D. Silva.  CANARD CYCLES AND HOMOCLINIC BIFURCATION IN A 3 PARAMETER FAMILY OF VECTOR FIELDS ON THE PLANE , 1999 .

[5]  Vaithianathan Venkatasubramanian,et al.  Singularity induced bifurcation and the van der Pol oscillator , 1994 .

[6]  Ricardo Riaza.  On the singularity-induced bifurcation theorem , 2002, IEEE Trans. Autom. Control..

[7]  R. Beardmore.  Double singularity-induced bifurcation points and singular Hopf bifurcations , 2000 .

[8]  H. Schättler,et al.  Local bifurcations and feasibility regions in differential-algebraic systems , 1995, IEEE Trans. Autom. Control..

[9]  Yang Lijun,et al.  An improved version of the singularity-induced bifurcation theorem , 2001 .

[10]  John Guckenheimer,et al.  Towards a Global Theory of Singularly Perturbed Dynamical Systems , 1996 .

[11]  Freddy Dumortier,et al.  Canard Cycles and Center Manifolds , 1996 .

[12]  R. E. Beardmore,et al.  The Singularity-Induced Bifurcation and its Kronecker Normal Form , 2001, SIAM J. Matrix Anal. Appl..

[13]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[14]  S. Baer,et al.  Sungular hopf bifurcation to relaxation oscillations , 1986 .

[15]  M. Stiefenhofer.  Singular perturbation with Limit points in the fast dynamics , 1998 .

[16]  Neil Fenichel.  Geometric singular perturbation theory for ordinary differential equations , 1979 .

[17]  A. Andronov.  Theory of bifurcations of dynamic systems on a plane = Teoriya bifurkatsii dinamicheskikh sistem na ploskosti , 1971 .

[18]  A. Doelman,et al.  Geometric singular perturbation theory and applications , 2002 .