Localizing attractors via a generalized La Salle principle

We are concerned with plane differential systems of the form<F>x˙ = P(x,y), y˙ = Q(x,y)</F>, with<F>P, Q</F> analytic. We propose a formal-numeric method to localize the attractors and the repellers of the system. Such a method consists of looking for a power series solution to a PDE of the type<F>PSHAPE="BUILT" ALIGN="C" STYLE="S"><NU>∂V</NU><DE>∂x</DE></FR></F> + Q<F>SHAPE="BUILT" ALIGN="C" STYLE="S"><NU>∂V</NU><DE>∂y</DE></FR> = μ(V)</F>, with μ is an arbitrary analytic function. When<F>μ(V) = ρ(V − V2)</F>,<F>ρ > 0</F>, the attractors are contained in the set<F>V = 1</F>, the repellers in the set<F>V = 0</F>. [Copyright &y& Elsevier]