Local integrability and linearizability of three-dimensional Lotka–Volterra systems

Aziz, Waleed; Christopher, Colin

doi:10.1016/j.amc.2012.10.051

Cited by 17 publications

(18 citation statements)

References 25 publications

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“…Recently (Aziz, 2014, Aziz andChristopher, 2012) investigated local integrability and linearizability of a quadratic threedimensional Lotka-Volterra and a particular case in a general quadratic three-dimensional differential systems.…”

Section: Fritmentioning

confidence: 99%

The The 1:-1:1 Resonance Integrable Problem for a Cubic Lotka-Volterra Systems.

2019

ZJPAS

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This paper is devoted to investigate the integrability and linearizability problems around a singular point at the origin of a cubic three-dimensional Lotka-Volterra differential system with-resonance. A complete set of necessary conditions for both integrability and linearizability are given. For sufficiency part, we show that the system has two analytic first integrals at the origin. In particular, we use Darboux method of integrability, linearizable node and transformation technique to show that the system admits two independent first integrals.

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Section: Fritmentioning

confidence: 99%

The The 1:-1:1 Resonance Integrable Problem for a Cubic Lotka-Volterra Systems.

2019

ZJPAS

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“…It is well-known (see e.g. [73,56]) that, for system (2), the Bautin ideal B is generated by the first three focus quantities of this system [7]. Moreover, the center variety V(B) ⊂ R 6 of the ideal B has four irreducible components, namely…”

Section: Introductionmentioning

confidence: 99%

A hybrid symbolic-numerical approach to the center-focus problem

Mahdi

Pessoa

Hauenstein

2017

Journal of Symbolic Computation

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“…The purpose of this paper is to extend some recent work [2,3] on the local integrability in C 3 at the origin of the three-dimensional LotkaVolterra equations…”

Section: Introductionmentioning

confidence: 99%

On the integrability of some three-dimensional Lotka-Volterra equations with rank-1 resonances

Aziz

Christopher

2014

Publicacions Matemàtiques

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Abstract:We investigate the local integrability in C 3 of some three-dimensional Lotka-Volterra equations at the origin with (p : q : r)-resonance,Recent work on this problem has centered on the case where the resonance is of "rank-2". That is, there are two independent linear dependencies of p, q and r over Q. Here, we consider some situations where there is only one such dependency. In particular, we give necessary and sufficient conditions for integrability for the case of (i, −i, λ)-resonance with λ / ∈ iR (after a scaling, this is just the case p + q = 0 with q/r / ∈ R), and also the case of (i − 1, −i − 1, 2)-resonance (a subcase of p + q + r = 0) under the additional assumption that a = k = 0.Our necessary and sufficient conditions for integrability are given via the search for two independent first integrals of the form x α y β z γ (1 + O(x, y, z)). However, a new feature in the case of rank-1 resonance is that there is a distinguished choice of analytic first integral, and hence it makes sense to seek conditions for just one (analytic) first integral to exist. We give necessary and sufficient conditions for just one first integral for the two families of systems mentioned above, except that for the second family some of the cases of sufficiency have been left as conjectural.2010 Mathematics Subject Classification: 34C20.

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Local integrability and linearizability of three-dimensional Lotka–Volterra systems

Cited by 17 publications

References 25 publications

The The 1:-1:1 Resonance Integrable Problem for a Cubic Lotka-Volterra Systems.

The The 1:-1:1 Resonance Integrable Problem for a Cubic Lotka-Volterra Systems.

A hybrid symbolic-numerical approach to the center-focus problem

On the integrability of some three-dimensional Lotka-Volterra equations with rank-1 resonances

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