Zero-Hopf bifurcation in the Van der Pol oscillator with delayed position and velocity feedback

Abstract

In this paper, we consider the traditional Van der Pol Oscillator with a forcing dependent on a delay in feedback. The delay is taken to be a nonlinear function of both position and velocity which gives rise to many different types of bifurcations. In particular, we study the Zero-Hopf bifurcation that takes place at certain parameter values using methods of centre manifold reduction of DDEs and normal form theory. We present numerical simulations that have been accurately predicted by the phase portraits in the Zero-Hopf bifurcation to confirm our numerical results and provide a physical understanding of the oscillator with the delay in feedback.

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

“…point ( i ) . profile (2 ,:) , ' Color ' , cm (2 ,:) ) ; end title ( ' Comparison between computed and predicted periodic orbits ') xlabel ( '\ mu_1 ') ; ylabel ( 'x ') ; zlabel ( 'y ') ; view (3) The resulting plot is show in Figure S16.…”

“…where the weak Riemann integral takes values in X and the running assumption of sun-reflexivity justifies the application of j −1 . By using (3) to express the difference T − T 0 of the perturbed and the unperturbed semigroups, one proves that the maximal subspaces of strong continuity X and X are the same for T and T 0 , so there is no need to distinguish them with a subscript. In particular, X is sun-reflexive also with respect to T .…”

“…% % Compare computed and predicted periods figure (3) ; clf ; hold on ; % Plot computed periods on nsbr (1) and nsbr (2) omegas1 = arrayfun ( @ ( p ) p . period , nsbr (1) .…”

Quasi-Periodic Solutions for Differential Equations with an Elliptic-Type Degenerate Equilibrium Point Under Small Perturbations

“…For the case without the perturbation term, the bifurcation and stability of (3.1) have been extensively studied, see [6,18,28,30,33] and the references therein.…”