Liouvillian first integrals of quadratic–linear polynomial differential systems (original) (raw)
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arXiv (Cornell University), 2014
We show that the algebraic curve a 0 (x)(y − r(x)) + p 2 (x)a ′ (x) = 0, where r(x) and p 2 (x) are polynomial of degree 1 and 2 respectively and a 0 (x) is a polynomial solution of the convenient Fucsh's equation, is an invariant curve of the quadratic planar differential system. We study the particular case when a 0 (x) is an orthogonal polynomials. We prove that that in this case the quadratic differential system is Liouvillian integrable.
On Liouvillian integrability of the first–order polynomial ordinary differential equations
Journal of Mathematical Analysis and Applications, 2012
Recently the authors provided an example of an integrable Liouvillian planar polynomial differential system that has no finite invariant algebraic curves, see [8]. In this note we prove that if a complex differential equation of the form y ′ = a 0 (x) + a 1 (x)y + • • • + a n (x)y n with a i (x) polynomials for i = 0, 1,. .. , n, a n (x) ̸ = 0 and n ≥ 2 has a Liouvillian first integral, then it has a finite invariant algebraic curve. So, this result applies to the Riccati and Abel polynomial differential equations. We shall prove that in general this result is not true when n = 1, i.e. for linear polynomial differential equations.
First Integrals and Darboux Polynomials of Homogeneous Linear Differential Systems
1995
This paper studies rational and Liouvillian first integrals of homogeneous linear differential systems Y′=AY over a differential field k. Following [26], our strategy to compute them is to compute the Darboux polynomials associated with the system. We show how to explicitly interpret the coefficients of the Darboux polynomials as functions on the solutions of the system; this provides a correspondence between Darboux polynomials and semi-invariants of the differential Galois groups, which in turn gives indications regarding the possible degrees for Darboux polynomials (particularly in the completely reducible cases). The algorithm is implemented and we give some examples of computations.
Integrability of Planar Polynomial Differential Systems through Linear Differential Equations
Rocky Mountain Journal of Mathematics, 2006
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.
Polynomial inverse integrating factors for quadratic differential systems
Nonlinear Analysis: Theory, Methods & Applications, 2010
We characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals. We also prove that these families of quadratic systems have no limit cycles.
The paper is divided into two parts. In the first one we present a survey about the theory of Darboux for the integrability of polynomial differential equations. In the second part we apply all mentioned results on Darboux theory to study the integrability of real quadratic systems having an invariant conic. The fact that two intersecting straight lines or two parallel straight lines are particular cases of conics allows us to study simultaneously the integrability of quadratic systems having at least two invariant straight lines.
Polynomial differential systems having a given Darbouxian first integral
Bulletin des Sciences Mathématiques, 2004
The Darbouxian theory of integrability allows to determine when a polynomial differential system in C 2 has a first integral of the kind f y], and λ i ∈ C for i = 1, . . . , p. The functions of this form are called Darbouxian functions. Here, we solve the inverse problem, i.e. we characterize the polynomial vector fields in C 2 having a given Darbouxian function as a first integral.
Explicit Construction of First Integrals by Singularity Analysis in Nonlinear Dynamical Systems
2004
The Painlevé and weak Painlevé conjectures have been used widely to identify new integrable nonlinear dynamical systems. For a system which passes the Painlevé test, the calculation of the integrals relies on a variety of methods which are independent from Painlevé analysis. The present paper proposes an explicit algorithm to build first integrals of a dynamical system, expressed as 'quasi-polynomial' functions, from the information provided solely by the Painlevé -Laurent series solutions of a system of ODEs. Restrictions on the number and form of quasi-monomial terms appearing in a quasi-polynomial integral are obtained by an application of a theorem by . The integrals are obtained by a proper balancing of the coefficients in a quasi-polynomial function selected as initial ansatz for the integral, so that all dependence on powers of the time τ = t − t0 is eliminated. Both right and left Painlevé series are useful in the method. Alternatively, the method can be used to show the non-existence of a quasi-polynomial first integral. Examples from specific dynamical systems are given.
EQUADIFF, 2003
We consider the families of quadratic systems in the projective plane with algebraic limit cycles of degree 2 or 4. There are no algebraic limit cycles of degree 3 for a quadratic system. Until the moment, no other families of quadratic systems with an algebraic limit cycle, not birrationally equivalent to the ones that we study, have been found. We prove that none of these systems has a Liouvillian first integral. Our main tool is the characterization of the form of the cofactor of an irreducible invariant algebraic curve, when this curve exists, by means of the study of the singular points of the system. For obtaining this characterization of the form of the cofactor we consider the behavior of the solutions of the system in a neighborhood of a critical point.