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Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity. (English) Zbl 1154.34039

The authors apply the method for computing normal forms of functional differential equations (FDEs) introduced by T. Faria and L. T. Magalhães [J. Differ. Equations 122, No. 2, 201–224 (1995; Zbl 0836.34069)] to systems of delay-differential equations (DDEs) with a single delay and a Takens-Bogdanov bifurcation (the original paper of Faria and Magalhães had a scalar DDE as an illustrating example).

A puzzling feature of the discussion of the Takens-Bogdanov bifurcation is that the normal form presented in the paper differs qualitatively from the normal form in textbooks, say, [Yu. A. Kuznetsov, Elements of applied bifurcation theory (Applied Mathematical Sciences 112. New York, NY: Springer) (2004; Zbl 1082.37002)]. The unfolding parameters do not unfold the saddle-node bifurcation which should be present near a generic Takens-Bogdanov bifurcation. This peculiarity also shows up in the original paper of Faria and Magalhães.

MSC:
34K17Transformation and reduction of functional-differential equations and systems; normal forms
34K18Bifurcation theory of functional differential equations
References:
[1]Faria, T.; Magalhães, L. T.: Normal forms for retarded functional differential equations and applications to bogdanov – Takens singularity, J. differential equations 122, 201-224 (1995) · Zbl 0836.34069 · doi:10.1006/jdeq.1995.1145
[2]Takens, F.: Singularities of vector fields, Publ. math. Inst. hautes études sci. 43, 47-100 (1974) · Zbl 0279.58009 · doi:10.1007/BF02684366 · doi:numdam:PMIHES_1974__43__47_0
[3]Bogdanov, R. I.: Versal deformations of a singular point on the plane in the case of zero eigenvalues, Funct. anal. Appl. 9, 144-145 (1975) · Zbl 0447.58009 · doi:10.1007/BF01075453
[4]Faria, T.: Normal forms for semilinear functional differential equations in Banach spaces and applications, part II, Discrete contin. Dyn. syst. 7, 155-176 (2001) · Zbl 1060.34511 · doi:10.3934/dcds.2001.7.155
[5]Faria, T.; Magalhães, L. T.: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. differential equations 122, 181-200 (1995) · Zbl 0836.34068 · doi:10.1006/jdeq.1995.1144
[6]Hale, J. K.: Theory of functional differential equations, (1977)
[7]Hale, J. K.; Lunel, S. M.: Introduction to functional differential equations, (1993)
[8]Guckenheimer, J.; Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, (1983)
[9]Chow, S. N.; Hale, J. K.: Methods of bifurcation theory, (1982)