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

Abstract

Nonresonant Hopf-Hopf singularity in neutral functional differential equation (NFDE) is considered. An algorithm for calculating the third-order normal form is established by using the formal adjoint theory, center manifold theorem and the traditional normal form method for RFDE. Van der Pol's equation with extended delay feedback is studied as an example. The unfoldings near the Hopf-Hopf bifurcation point is given by applying this algorithm. Periodic solutions, quasi-periodic solutions are found via theoretical bifurcation diagram and numerical illustrations. The Hopf-Hopf bifurcation diagram indicates the possible existence of a chaotic attractor, which is confirmed by a sequence of simulations.