Liouvillian Integrability and the Poincaré Problem for Nonlinear Oscillators with Quadratic Damping and Polynomial Forces

Abstract

The upper bound on the degrees of irreducible Darboux polynomials associated to the ordinary differential equations $ x_{tt}+\varepsilon {x_{t}}^{2}+\eta x_{t}+f(x)=0 $ with $ f(x)\in \mathbb {C}[x]\setminus \mathbb {C} $ and e ≠ 0 is derived. The availability of this bound provides the solution of the Poincare problem. Results on uniqueness and existence of Darboux polynomials are presented. The problem of Liouvillian integrability for related dynamical systems is solved completely. It is proved that Liouvillian first integrals exist if and only if η = 0.