Results 11 to 20 of about 86 (78)
Liouvillian first integrals for Liénard polynomial differential systems [PDF]
Proceedings of the American Mathematical Society, 2010 Summary: We characterize the Liouvillian first integrals for the Liénard polynomial differential systems of the form \(x^{\prime } = y, y^{\prime } = -cx-f(x)y\), with \(c \in \mathbb{R}\) and \(f(x)\) is an arbitrary polynomial. For obtaining this result we need to find all the Darboux polynomials and the exponential factors of these systems.
Llibre, J., Valls, C.
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On the nonexistence of Liouvillian first integrals for generalized Liénard polynomial differential systems [PDF]
Journal of Nonlinear Mathematical Physics, 2021 We consider generalized Lienard polynomial differential systems. In their work, Llibre and Valls have shown that, except in some particular cases, such systems have no Liouvillian first integral. In this letter, we give a direct and shorter proof of this result.
Chèze, Guillaume, Cluzeau, Thomas
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Determining Liouvillian first integrals for dynamical systems in the plane [PDF]
Computer Physics Communications, 2007 Here we present/implement an algorithm to find Liouvillian first integrals of dynamical systems in the plane. In \cite{JCAM}, we have introduced the basis for the present implementation. The particular form of such systems allows reducing it to a single rational first order ordinary differential equation (rational first order ODE).
Avellar, J. +3 more
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Liouvillian first integrals of differential equations [PDF]
Transactions of the American Mathematical Society, 1989 Liouvillian functions are functions that are built up from rational functions using exponentiation, integration, and algebraic functions. We show that if a system of differential equations has a generic solution that satisfies a liouvillian relation, that is, there is a liouvillian function of several variables vanishing on the curve defined by this ...
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Liouvillian first integrals for a class of generalized Liénard polynomial differential systems [PDF]
Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2016 We study the existence of Liouvillian first integrals for the generalized Liénard polynomial differential systems of the form xʹ = y, yʹ = –g(x) – f(x)y, where f(x) = 3Q(x)Qʹ(x)P(x) + Q(x)2Pʹ(x) and g(x) = Q(x)Qʹ(x)(Q(x)2P(x)2 – 1) with P,Q ∈ ℂ[x]. This class of generalized Liénard polynomial differential systems has the invariant algebraic curve (y ...
Llibre, Jaume, Valls, Clàudia
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Liouvillian and Darboux First Integrals of the Self-Assembling Micelle System
Iraqi Journal of Science, 2023 In this paper we prove that the planar self-assembling micelle system has no Liouvillian, polynomial and Darboux first integrals. Moreover, we show that the system has only one irreducible Darboux polynomial with the cofactor being if and only if via the weight homogeneous polynomials and only two irreducible exponential factors and with
Wirya Mohommed Ramadhan +1 more
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Liouvillian first integrals of quadratic–linear polynomial differential systems
Journal of Mathematical Analysis and Applications, 2011 Agraïments: The second author has been partially supported by FCT through CAMGSD, Lisbon. For a large class of quadratic-linear polynomial differential systems with a unique singular point at the origin having non-zero eigenvalues, we classify the ones which have a Liouvillian first integral, and we provide the explicit expression of them.
Llibre, Jaume, Valls, Clàudia
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Liouvillian first integrals of homogeneouspolynomial 3-dimensional vector fields [PDF]
Colloquium Mathematicum, 1996 Summary: Given a three-dimensional vector field \(V\) with coordinates \(V_x\), \(V_y\) and \(V_z\) that are homogeneous polynomials in the ring \(k[x, y, z]\), we give a necessary and sufficient condition for the existence of a Liouvillian first integral of \(V\) which is homogeneous of degree 0.
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Differential Algebra and Liouvillian first integrals of foliations
Journal of Pure and Applied Algebra, 2011 Let \((k,\Delta)\) be a differential field, where \(k\) is a field and \(\Delta\) is a set of derivations of \(k\). A Liouvillian extension of \((k,\Delta)\) is a differential extension \((K,\tilde{\Delta})\) of \( (k,\Delta)\) for which there is a chain of differential extensions \(k=k_{0}\subset k_{1}\subset\dots\subset k_{m}=K\) such that \( k_{i+1}/
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On Liouvillian integrability of the first–order polynomial ordinary differential equations
Journal of Mathematical Analysis and Applications, 2012 The authors prove the following result: Theorem. If a complex differential equation of the form \[ {dy\over dx}= a_0(x)+ a_1(x) y+\cdots+ a_n(x) y^n, \] where \(a_i(x)\), \(i= 0,\dots, n\), are polynomials in \(x\), \(a_n(x)\neq 0\), \(n\geq 2\), has a Liouvillian first integral, then it has a finite invariant algebraic curve.
Giné, Jaume, Llibre, Jaume
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