Integrability of a class of N–dimensional Lotka–Volterra and Kolmogorov systems

Llibre, Jaume; Ramírez, Rafael; Ramírez, Valentín

doi:10.1016/j.jde.2020.02.001

Cited by 7 publications

(2 citation statements)

References 24 publications

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“…For general polynomial differential systems this problem is very difficult to solve. During the past three decades many mathematicians investigated the integrability of different classes of polynomial differential systems, such as Liénard systems [5,13,15,16], Lotka-Volterra systems [7,8,[10][11][12], and quasi-homogeneous polynomial systems [1-4, 6, 17], etc.…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

Llibre

Tian

2021

Acta Appl Math

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This paper study the type of integrability of the differential systems with separable variables ẋ = h (x) f (y), ẏ = g (y), where h, f and g are polynomials. We provide a criterion for the existence of generalized analytic first integrals of such differential systems. Moreover we characterize the polynomial integrability of all such systems.In the particular case h (x) = (ax + b) m we provide necessary and sufficient conditions in order that this subclass of systems has a generalized analytic first integral. These results extend known results from [5] and [13]. Such differential systems of separable variables are important due to the fact that after a blow-up change of variables any planar quasi-homogeneous polynomial differential system can be transformed into a special differential system of separable variables ẋ = xf (y), ẏ = g (y), with f and g polynomials.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

Llibre

Tian

2021

Acta Appl Math

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“…More precisely, we deduce the general forms of such systems that admit Hamilton-Poisson formulations. We recall here that the Liouville, Darboux integrability, respectively, of Kolmogorov and Lotka-Volterra systems and their dynamical behavior have been widely investigated (see, e.g., [19][20][21][22][23][24][25] and references therein).…”

Section: A Particular Case: Polynomial Kolmogorov Systemsmentioning

confidence: 99%

Integrable Deformations and Dynamical Properties of Systems with Constant Population

Lăzureanu¹

2021

Mathematics

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In this paper we consider systems of three autonomous first-order differential equations x˙=f(x),x=(x,y,z),f=(f1,f2,f3) such that x(t)+y(t)+z(t) is constant for all t. We present some Hamilton–Poisson formulations and integrable deformations. We also analyze the case of Kolmogorov systems. We study from some standard and nonstandard Poisson geometry points of view the three-dimensional Lotka–Volterra system with constant population.

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Generalized Cartesian–Nambu Vector Fields

Llibre

Ramírez

2023

Dynamics Through First-Order Differential Equations in the Configuration Space

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Integrability of a class of N–dimensional Lotka–Volterra and Kolmogorov systems

Cited by 7 publications

References 24 publications

Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

Generalized Analytic Integrability of a Class of Polynomial Differential Systems in $\mathbb{C}^{2}$

Integrable Deformations and Dynamical Properties of Systems with Constant Population

Generalized Cartesian–Nambu Vector Fields

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