A computational method for the determination of attraction regions

The region of attraction of nonlinear dynamical system can be considered using an analytical R-function that can be written like an infinite series where each term of the series has the homogeneous form of degree n ¿ 2 this function allows to determine and to come near to the region of attraction of a nonlinear system around the point of equilibrium located in the origin. The analytical function and the sequence of this Taylor polynomials are constructed by a recurrence formula using the coefficients of the power series expansion of f at 0. This paper describes a novel computational method using the Software MATHEMATICA for obtaining a solution to this problem, which was proposed by the Russian mathematician, V. I. Zubov. In order to evaluate the method, two examples are treated in which the exact attraction region is found in analytic closed form. Since the construction procedure requires the solution of a linear partial differential equation, there are many cases for which an exact analytic solution is not possible. In some of these cases, however, it is possible to construct an approximate series solution which is always at least as good approximation of the usual quadratic form of Lyapunov functions. The "trajectory reversing method" is presented as a powerful numerical technique for low order systems. Then an analytical procedure based on the same topological approach is developed, and a comparison is made with the classical Zubov method.