On the Lyapunov quantities of two-dimensional autonomous systems of differential equations with a critical point of centre or focus type

Analytical two-dimensional system of differential equations with a weak focus are considered. It is shown how by using a given sequence of humogenous polynomials of even degree one can construct a function with the rate of change along trajectories of system to be a linear combination of them polynomials. The coefficiente of this linear combination are the Lyapunov quantities.