Numerical third-order Hopf form for ordinary differential equations

Abstract A data-driven approach has been developed to obtain the numerical third-order Hopf form for two-dimensional ODEs. By using a single numerical solution, the flow on the 2D phase-plane is recovered by a least-squares fit. We then calculate the Poincaré-Lyapunov constant. Sensitivity analysis is carried out to determine the accuracy of the calculated Poincaré-Lyapunov constant; a main result is a formula to compute the relative error of the Poincaré-Lyapunov constant from the errors of the identified coefficients in the third-order Hopf form. We also provide a sufficient condition for correctly identifying the criticality of the system (the sign of the Poincaré-Lyapunov constant). A performance metric is defined for our identification algorithm and this metric is computed by Monte Carlo simulations. Our method achieves good accuracy.