Region of Attraction Estimation (Reverse Time Van Der Pol Oscillator)

We propose a method to estimate the region of attraction of a nonlinear ODE based only on measurements of the trajectory - implying that the nonlinear vector field need not be known a priori. This method is based on using trajectory data to determine values of a form of converse Lyapunov function at a finite number of points in the state-space. Least absolute deviations is then used to fit this data to a Sum-of-Squares polynomial whose level sets then become estimates for the region of attraction. This learned Lyapunov function can then be used to predict whether newly generated initial conditions lie in the region of attraction. Extensive numerical testing is used to show that the method correctly predicts whether a new initial condition is within the region of attraction of the nonlinear ODE on over 95% of a generated set of test data.