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Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control. (English) Zbl 1141.34044
The authors investigate a van der Pol-Duffing oscillator controlled by parametric delay feedback. The strength of feedback control takes the form of a function exponentially decreasing with the time delay. In the paper, a two-parameter geometrical criterion for the stability and the Hopf bifurcations of the equation are presented. Some weak resonant and non-resonant double Hopf bifurcations are analyzed and the necessary conditions for double Hopf bifurcations are discussed. Computations of normal forms and universal unfoldings at the double Hopf bifurcation points are carried out, and local classification in the neighborhood of double Hopf points is undertaken. Numerical simulations are presented to reveal the dynamical behavior near double Hopf bifurcation points, such as quasi-periodic solutions and chaos. The Neimark-Sacker bifurcation is also detected.

MSC:
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
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