Related papers: Pseudo-Abelian integrals: unfolding generic expone…
We study pseudo-Abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.
We consider a multivalued function of the form $H\_{\varepsilon}=P\_{\varepsilon}^{\alpha\_0}\prod^{k}\_{i=1}P\_i^{\alpha\_i}, P\_i\in\mathbb{R}[x,y], \alpha\_i\in\mathbb{R}^{\ast}\_+$, which is a Darboux first integral of polynomial…
We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the…
In this paper we consider the degeneracies of the third type. More exact, the perturbations of the Darboux integrable foliation with a triple point, i.e. the case where three of the curves $\{P_i = 0\}$ meet at one point, are considered.…
We consider Lotka-Volterra systems in three dimensions depending on three real parameters. By using elementary algebraic methods we classify the Darboux polynomials (also known as second integrals) for such systems for various values of the…
We consider infinitesimal perturbations of Hamiltonian differential equations $dH + \varepsilon \omega =0$ on the complex plane $\mathbb{C}^2$, where $H$ is a polynomial of degree $m+1$ and $\omega$ is a non-exact polynomial 1-form of…
In the literature, the existence of Darboux polynomials and additional polynomial first integrals has been considered in the case of Hamiltonian systems. In this article such problem is formulated in the more general framework of Poisson…
We prove an existential finiteness Varchenko-Khovanskii type result for integrals of rational 1-forms over the level curves of Darbouxian integrals.
We study the class of planar polynomial vector fields admitting Darboux first integrals of the type $\prod_{i=1}^r f_i^{\alpha_i}$, where the $\alpha_i$'s are positive real numbers and the $f_i$'s are polynomials defining curves with only…
We apply the Darboux theory of integrability to polynomial ODE's of dimension 3. Using this theory and computer algebra, we study the existence of first integrals for the 3-dimensional Lotka-Volterra systems with polynomial invariant…
We provide the necessary and sufficient conditions of Liouvillian integrability for Li\'{e}nard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Li\'{e}nard…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
We characterize real elliptic differential systems whose solutions can be expressed in terms of holomorphic solutions to an associated holomorphic Pfaffian system $\mathcal H$ on a complex manifold. In particular, these elliptic systems…
We focus on a chaotic differential system in 3-dimension, including an absolute term and a line of equilibrium points. Which describes in the following This system has an implementation in electronic components. The first purpose of this…
A family of solutions of the Jacobi PDEs is investigated. This family is $n$-dimensional, of arbitrary nonlinearity and can be globally analyzed (thus improving the usual local scope of Darboux theorem). As an outcome of this analysis it is…
M.F. Singer [Liouvillian first integrals of differential equations, Trans. Amer. Math. Soc. 333 (1992), 673--688] proved the equivalence between Liouvillian integrability and Darboux integrability for two dimensional polynomial differential…
In this paper, we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of Lotka-Volterra type having a rational first integral of degree 2, under polynomial perturbations of degree $n$.…
We study the existence of real-analytic first integrals and real-analytic integrability for perturbations of integrable systems in the sense of Bogoyavlenskij including non-Hamiltonian ones. We especially assume that there exists a family…
After the nice result introduced by Belotto in [1] concerning the local monomialization of a singular foliation given by n first integrals, this work is a continuation in the same spirit. In this paper, we introduce a important conjecture…
Darboux's theorem and Jouanolou's theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field. These results are given in terms of the degree of the polynomial vector field. Here we show that…