Novel algebraic aspects of Liouvillian integrability for two-dimensional polynomial dynamical systems

On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

“…We also observe that there may exist balances and their power solutions such that The following theorem was proved in article [6].…”

“…6) as S(W ). The convex hull of S(W ) is called the Newton polygon of equation(2.6).The boundary of the Newton polygon consists of vertices and edges.…”

Liouvillian integrability of a general Rayleigh-Duffing oscillator

“…Statement (a) of Theorem 1.1 is a consequence of Theorem 1.7 in [4]. In [4] are used the Puiseux series (see [4] for its definition) to find the Liouville integrability. In fact we have checked her results using the methods described in the present work and we have found the same results for the case Proof.…”

Integrability Properties of Cubic Liénard Oscillators with Linear Damping

“…By C[y, w] we denote the ring of polynomials in variables y and w with coefficients in the field C. The zero set of the Darboux polynomial F(y, w) defines an invariant algebraic curve of the corresponding dynamical system. It is known that Darboux polynomials are of great importance if one studies the integrability problem and wants to derive all independent first integrals that are Darboux or Liouvillian functions [11,12]. An effective method of finding and classifying Darboux polynomials is the method of Puiseux series introduced in articles [10,17].…”

“…Second, we use the Darboux theory of integrability, which is a powerful tool for constructing and classifying first integrals of ordinary differential equations. The main objects in the Darboux theory are invariant algebraic curves (or Darboux polynomials) and exponential factors [11,12]. The knowledge of the complete set of these invariants allows one to derive necessary and sufficient conditions of Darboux and Liouvillian integrability.…”