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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

x ¨+ε(x 2 -1)x ˙+x=εg(x(t-τ))(1)

with ε>0, g(0)=0, g ' (0)=k, g '' (0)0, τε and k are bifurcation parameters. (1) exhibits codimension-one and codimension-two bifurcations. For τ=ε=2, k=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

z ˙ 1 =z 2 ,z ˙ 2 =z 3 ,
z ˙ 3 =λ 1 z 1 +λ 2 z 2 +λ 3 z 3 +η 1 z 1 2 +η 2 z 2 2 +η 3 z 1 z 2 +η 4 z 1 z 3 +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.

MSC:
34K18Bifurcation theory of functional differential equations
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