Numerical Hopf normal form for delay-differential equations

A data-driven algorithm to approximate the Poincar & eacute;-Lyapunov constant has been developed to overcome the tedious calculations involved in the center manifold reduction for delay-differential equations. By using a single numerical solution at the critical value of the bifurcation parameter, the flow on the center manifold is recovered by a least-squares fit. This planar system is then used to compute the Poincar & eacute;-Lyapunov constant. The algorithm is tested for the delayed Li & eacute;nard equation, the analytic center manifold reduction forms the basis of the comparison. A performance metric for the method is defined and computed. Numerical results demonstrate that our method works well for identifying the Poincar & eacute;-Lyapunov constant for delay-differential equations.