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Bogdanov-Takens singularity in Van der Pol’s oscillator with delayed feedback. (English) Zbl 1124.34048
The paper studies the local bifurcations of the classical Van der Pol equation with nonlinear delayed feedback \[ \ddot x+\varepsilon (x^2-1)\dot x+x =g(x(t-\tau)), \] where \(g(0)=0\). The main bifurcation parameters are \(k:=g'(0)\) and \(\tau\). For \(k=1/\varepsilon\) and \(\tau=\varepsilon\) the authors discover a Takens-Bogdanov bifurcation (plus additional structure because the origin is always a fixed point). Then the authors reduce the system to its normal form using the method of T. Faria and L. Magalhaes [J. Differ. Equations 122, No. 2, 201–224 (1995; Zbl 0836.34069)] for two cases: the generic case \(g''(0)\neq 0\), and the odd case \(g(x)=-g(-x)\) (with \(g'''(0)\neq0\)). The bifurcation diagrams are only sketched in the normal form parameters.

MSC:
34K18 Bifurcation theory of functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms
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