Integrability and Linearizability of Three Dimensional Vector Fields

Aziz, Waleed

doi:10.1007/s12346-014-0113-0

Cited by 9 publications

(16 citation statements)

References 10 publications

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“…The integrability of some Lotka-Voltera families was considered by several authors, see for example [15,16,18,31,34,36]. Similar technics applied to some more general families like the ones in [5,23]. Some general properties of these systems are studied in the works [12,13].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The polynomial L is called a cofactor of the invariant algebraic surface = 0. Note that from relation (5) we have that any cofactor has at most degree one because the polynomial vector field has degree two. There are other functions which satisfy (5), for example there are invariant analytic surfaces.…”

Section: Preliminariesmentioning

confidence: 99%

“…There are other functions which satisfy (5), for example there are invariant analytic surfaces. Then, the degree of the cofactor L is imposed rather than following from equation (5). Exponential factors also satisfy (5) and are related to the multiplicity of the invariant surfaces, see [32,33].…”

Section: Preliminariesmentioning

confidence: 99%

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Three-Dimensional Lotka–Volterra Systems with 3:−1:2-Resonance

et al. 2021

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We study the local integrability at the origin of a nine parametric family of three dimensional Lotka-Volterra differential systems with 3:-1:2-resonance. We present the necessary and sufficient conditions on the parameters of the family that guarantee the existence of two independent local first integrals at the origin of coordinates. Additionally we check the cases where the origin is linearizable.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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Three-Dimensional Lotka–Volterra Systems with 3:−1:2-Resonance

et al. 2021

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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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“…Related work on the integrability of LotkaVolterra and other three dimensional systems can be found in [1,4,6,7,8,9,10,11,12,13,14].…”

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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Integrability and Linearizability of Three Dimensional Vector Fields

Cited by 9 publications

References 10 publications

Three-Dimensional Lotka–Volterra Systems with 3:−1:2-Resonance

Three-Dimensional Lotka–Volterra Systems with 3:−1:2-Resonance

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

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

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