Determining Liouvillian First Integrals for Dynamical Systems in the Plane

Here we present an efficient method to compute Darboux polynomials for polynomial vector fields in the plane. This approach is restricetd to polynomial vector fields presenting a Liouvillian first integral (or, equivalently, to rational first order differential equations (rational 1ODEs) presenting a Liouvillian general solution). The key to obtaining this method was to separate the procedure of solving the (nonlinear) algebraic systems resulting from the equation that transl...

We have already dealt with the problem of solving First Order Differential Equations (1ODEs) presenting elementary functions before in [1, 2]. In this present paper, we have established solid theoretical basis through a relation between the 1ODE we are dealing with and a rational second order ordinary differential equation, presenting a Liouvillian first Integral. Here, we have expanded the results in [3], where we have establish a theoretical background to deal with rational...

Here we present an efficient method for finding and using a nonlocal symmetry admitted by a rational second order ordinary differential equation (rational 2ODE) in order to find a Liouvillian first integral (belonging to a vast class of Liouvillian functions). In a first stage, we construct an algorithm (improving the methodde veloped in [1]) that computes a nonlocal symmetry of a rational 2ODE. In ase cond stage, based on the knowledge of this symmetry, it is possible to con...

In this article we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is in $\tilde{\mathcal{O}}(N^{\omega+1})$, whe...

In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with ...

Here we present a very efficient method to search for Liouvillian first integrals of second order rational ordinary differential equations (rational 2ODEs). This new algorithm can be seen as an improvement to the S-function method we have developed [24]. Here, we show how to further use the knowledge of the S-function to find an integrating factor of a set of first order rational ordinary differential equations (rational 1ODEs) which is shared by the original 2ODE, without ha...

It is known, due to Mordukhai-Boltovski, Ritt, Prelle, Singer, Christopher and others, that if a given rational ODE has a Liouvillian first integral then the corresponding integrating factor of the ODE must be of a very special form of a product of powers and exponents of irreducible polynomials. These results lead to a partial algorithm for finding Liouvillian first integrals. However, there are two main complications on the way to obtaining polynomials in the integrating fa...

Here we present a new approach to search for first order invariants (first integrals) of rational second order ordinary differential equations. This method is an alternative to the Darbouxian and symmetry approaches. Our procedure can succeed in many cases where these two approaches fail. We also present here a Maple implementation of the theoretical results and methods, hereby introduced, in a computational package -- {\it InSyDE}. The package is designed, apart from materia...

Consider a planar polynomial vector field $X$, and assume it admits a symbolic first integral $\mathcal{F}$, i.e. of the $4$ classes, in growing complexity: Rational, Darbouxian, Liouvillian and Riccati. If $\mathcal{F}$ is not rational, it is sometimes possible to reduce it to a simpler class first integral. We will present algorithms to reduce symbolic first integral to a lower complexity class. These algorithms allow to find the minimal class first integral and in particul...

We present an algorithm to solve First Order Ordinary Differential Equations (FOODEs) extending the Prelle-Singer (PS) Method. The usual PS-approach miss many FOODEs presenting Liouvillian functions in the solution (LFOODEs). We point out why and propose an algorithm to solve a large class of these previously unsolved LFOODEs. Although our algorithm does not cover all the LFOODEs, it is an elegant extension mantaining the semi-decision nature of the usual PS-Method.