Nonresonant Hopf-Hopf bifurcation and a chaotic attractor in neutral functional differential equations

An algorithm for calculating the third-order normal form of a nonresonant Hopf-Hopf singularity in a neutral functional differential equation (NFDE) is established. The van der Pol equation with extended delay feedback is investigated as an NFDE of second order. The existence of Hopf-Hopf bifurcation is studied and the unfolding near these critical points is given by applying this algorithm. Periodic solutions and quasi-periodic solutions are found with the aid of the bifurcation diagram, and corresponding numerical illustrations are presented. With the breaking down of the 3-torus, a chaotic attractor appears in this NFDE of second order, following the Ruelle-Takens-Newhouse scenario which usually arises for an ordinary differential equation of order at least 4. This transition is shown via both theoretical and numerical approaches.