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Liouvillian first integrals of second order polynomial differential equations

Electronic Journal of Differential Equations, 1999
The author proves the following theorem: If the system \[ \dot x= P(x,y),\quad \dot y= Q(x,y)\quad (P,\;Q\text{ polynomials})\tag{1} \] has a Liouvillian integrating factor of the form: \[ \exp\{\int Udx+ Vdy\},\quad U_y= V_x,\tag{2} \] where \(U\), \(V\) are rational functions of \(x\) and \(y\), then there exists a Darbouxian integrating factor of ...
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Liouvillian integrability of the generalized Duffing oscillators

Analysis and Mathematical Physics, 2021
The generalized Duffing oscillators are given by the following system of ordinary differential equations \[ x_{t}=y,y_{t}=-\alpha y-\epsilon x^{m}-\alpha x,\epsilon \neq 0,m\in N,m\geq 2.\tag{\(\ast\)} \] In this article, the author has solved completely the problem of Liouvillian integrability for the force-free generalized Duffing oscillators.
M. Demina
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The Liouvillian Integrability of Several Oscillators

International Journal of Bifurcation and Chaos, 2019
In this paper, we characterize all the Liouvillian first integrals of a cubic polynomial differential system that contains the van der Pol and the Duffing oscillators. It is also shown that the centers correspond to the Liouville integrable cases.
Giné, Jaume, Valls, Claudia
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Liouvillian integrability of the three-dimensional generalized Hénon–Heiles Hamiltonian

The European Physical Journal Plus, 2020
In this paper, we report about three cases of integrability in sense of Liouville for three-dimensional generalized Henon–Heiles Hamiltonian. This also allow to get explicitly integrals of motions for each case. On the other hand, this paper investigates the phase space structure numerically with Poincare surfaces of section and 3D projections which ...
Idriss El Fakkousy, Jaouad Kharbach, Walid Chatar, Mohamed Benkhali, Abdellah Rezzouk, Mohammed Ouazzani-Jamil   +5 more
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Darboux and Liouvillian Integrability

2017
Darboux and Liouvillian integrability is mainly concerned with algebraic aspects of the integrability of differential systems, which is related to many subjects, such as real and complex analysis, algebraic geometry, differential algebra, differential Galois theory, and so on.
Xiang Zhang
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LIOUVILLIAN INTEGRABILITY OF A DETERMINISTIC BUSINESS CYCLE MODEL

Analysis and Applications, 2013
We introduce a deterministic business cycle model and study its dynamics by determining the existence of nonconstant functions which can be obtained "by quadratures" of elementary functions and are constant on solution curves in some nonempty open set.
C. Valls
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Liouvillian integrability of some quadratic Liénard polynomial differential systems

Rendiconti del Circolo Matematico di Palermo Series 2, 2018
A class of Liénard polynomial differential systems is studied. All Liouvillian first integrals are described. Also, the author gives examples of Liénard polynomial differential systems possesing transcendental first integrals.
C. Valls
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Liouvillian and analytic integrability in a modified Philips model

Chaos, Solitons & Fractals, 2012
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
C. Valls
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On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2020
We present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly
M. Demina, C. Valls
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The Darboux Polynomials and Integrability of Polynomial Levinson-Smith Differential Equations

International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 2023
We provide the necessary and sufficient conditions of Liouvillian integrability for nondegenerate near infinity polynomial Levinson–Smith differential equations.
M. Demina
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