Arc Length based Maximal Lyapunov Functions and domains of attraction estimation for polynomial nonlinear systems

In the phase space of a dynamical system containing an asymptotically stable equilibrium point, the Arc Length Function (ALF) is defined as sum of length differential elements of phase trajectories starting from state points and ending at the equilibrium point. It is shown that receding from the origin and verging on the Domain of Attraction (DoA) boundary causes ALF to be increased drastically. According to the latter issue and regarding to other properties, it is shown that ALF is a Maximal Lyapunov Function. To this end, a numerical method to approximate the ALF as a polynomial function is proposed. To ensure that the approximated function has positive value and negative derivative inside the desired region, the homotopy continuation method is used. Thus, the approximated ALF presents an ensured Lyapunov behavior to estimate DoA. The efficacy of the proposed method is demonstrated by several simulation examples.