Triple-zero bifurcation in van der Pol's oscillator with delayed feedback
Consider the scalar differential-delay equation \[ \ddot x+\varepsilon(x^2- 1)\dot x+ x=\varepsilon g(x(t-\tau))\tag{1} \] with \(\varepsilon> 0\), \(g(0)= 0\), \(g'(0)= k\), \(g''(0)\neq 0\), \(\tau\) -- \(\varepsilon\) 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.
scientific article; zbMATH DE number 6160196