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Triple-zero bifurcation in van der Pol’s oscillator with delayed feedback. (English) Zbl 1273.34079
Consider the scalar differential-delay equation $$\ddot x+\varepsilon(x^2- 1)\dot x+ x=\varepsilon g(x(t-\tau))\tag1$$ with $\varepsilon> 0$, $g(0)= 0$, $g'(0)= k$, $g''(0)\ne 0$, $\tau$ -- $\varepsilon$ and $k$ are bifurcation parameters. (1) exhibits codimension-one and codimension-two bifurcations. For $\tau=\varepsilon= \sqrt{2}$, $k=\sqrt{2}/2$, a codimension-three bifurcation occurs, i.e., triple-zero bifurcation. Using center manifold theory, the authors derive the following normal form with universal unfolding $$\dot z_1= z_2,\quad \dot z_2= z_3,$$ $$\dot z_3= \lambda_1z_1+ \lambda_2z_2+ \lambda_3z_3+ \eta_1z^2_1+ \eta_2z^2_2+ \eta_3z_1z_2+ \eta_4 z_1z_3+ \text{h.o.t.}$$ They determine the bifurcations occurring in the truncated normal form, namely transcritical, Hopf, Takens-Bogdanov, zero-Hopf ones. Finally, numerical simulations are presented.

34K18Bifurcation theory of functional differential equations
Full Text: DOI
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