Dynamical and Algebraic Analysis of Planar Polynomial Vector Fields Linked to Orthogonal Polynomials

In the present work, our goal is to establish a study of some families of quadratic polynomial vector fields connected to orthogonal polynomials that relate, via two different points of view, the qualitative and the algebraic ones. We extend those results that contain some details related to differential Galois theory as well as the inclusion of Darboux theory of integrability and the qualitative theory of dynamical systems. We conclude this study with the construction of differential Galois groups, the calculation of Darboux first integral, and the construction of the global phase portraits.

Integrability Properties of Cubic Liénard Oscillators with Linear Damping

We consider a family of cubic Liénard oscillators with linear damping. Particular cases of this family of equations are abundant in various applications, including physics and biology. There are several approaches for studying integrability of the considered family of equations such as Lie point symmetries, algebraic integrability, linearizability conditions via various transformations and so on. Here we study integrability of these oscillators from two different points of view, namely, linearizability via nonlocal transformations and the Darboux theory of integrability. With the help of these approaches we find two completely integrable cases of the studied equation. Moreover, we demonstrate that the equations under consideration have a generalized Darboux first integral of a certain form if and only if they are linearizable.

On the integrability of Hamiltonian systems with d degrees of freedom and homogenous polynomial potential of degree n

We consider Hamiltonian systems with [Formula: see text] degrees of freedom and a Hamiltonian of the form [Formula: see text] where [Formula: see text] is a homogenous polynomial of degree [Formula: see text]. We prove that such Hamiltonian systems with [Formula: see text] odd or [Formula: see text], have a Darboux first integral if and only if they have a polynomial first integral.

EXPONENTIAL FACTORS AND DARBOUX INTEGRABILITY FOR THE RÖSSLER SYSTEM

We characterize all generators of the exponential factors for the Rössler system by using weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations. Moreover, using Darboux polynomials and exponential factors we obtain the necessary and sufficient conditions in order that the Rössler system has a Darboux first integral.