Multiple bifurcation analysis in a NDDE arising from van der Pol's equation with extended delay feedback
Consider the neutral differential delay equation \[ \begin{multlined} \ddot x-\mu\ddot x(t- \tau)+ \varepsilon(x^2- 1)\dot x- \mu\varepsilon[x^2(t-\tau)- 1]\dot x(t-\tau)+\\ x-\mu x(t-\tau)=\varepsilon k(1-\mu) x(t)+ g(x)-\mu g(x(t-\tau))\end{multlined}\tag{\(*\)} \] with \(0<\mu< 1\), \(\varepsilon> 0\), \(k> 0\), \(g(0)= g'(0)= 0\), \(g''(0)= 2\kappa_1\), \(g'''(0)= 6\kappa_2\). The authors study fold bifurcation and Hopf bifurcation in \((*)\). Their main focus is on codimension two bifurcations (Bogdanov-Takens, Hopf-fold, Hopf-Hopf). The main tools are center manifold reduction and normal form theory. They establish three periodic solutions near the Bogdanov-Takens point. Numerical solutions show three stable limit cyclies near the Hopf-pitchfork bifurcation point and a stable torus near the Hopf-Hopf bifurcation point. This paper is an important contribution to codimension-two bifurcations in neutral delay-differential equations.
scientific article; zbMATH DE number 6118980