On a Hamiltonian version of a three-dimensional Lotka–Volterra system

Tudoran, Răzvan M.; Gîrban, Anania

doi:10.1016/j.nonrwa.2012.01.025

Cited by 22 publications

(23 citation statements)

References 16 publications

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“…Other first integrals are known (Llibre & Valls 2011;Tudoran & Gîrban 2012). The relation between the integrability (partial or complete) of a differential system and its having the Painlevé property (PP), i.e., the property that each of its solutions x = x(τ ) has a one-valued continuation to the complex τ -plane (without branch points of any order), is a bit murky.…”

Section: (B) Previous Results Reformulatedmentioning

confidence: 99%

The integration of three-dimensional Lotka–Volterra systems

Maier

2013

Proc. R. Soc. A.

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The general solutions of many three-dimensional Lotka-Volterra systems, previously known to be at least partially integrable, are constructed with the aid of special functions. Examples include certain ABC and May-Leonard systems. The special functions used are elliptic and incomplete beta functions. In some cases, the solution is parametric, with the independent and dependent variables expressed as functions of a 'new time' variable. This auxiliary variable satisfies a nonlinear third-order differential equation of a generalized Schwarzian type, and results of Carton-LeBrun on the equations of this type that have the Painlevé property are exploited, so as to produce solutions in closed form. For several especially difficult Lotka-Volterra systems, the solutions are expressed in terms of Painlevé transcendents. An appendix on incomplete beta functions and closed-form expressions for their inverses is included.

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Section: (B) Previous Results Reformulatedmentioning

confidence: 99%

The integration of three-dimensional Lotka–Volterra systems

Maier

2013

Proc. R. Soc. A.

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“…when such differential systems have first integrals (see for instance [1,2,4,5,6,7,8,17,18,22]),or • in their periodic orbits (see for example [9,10,11,13,16,20,24,25,26]). …”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On the dynamics of a class of Kolmogorov systems

Llibre

Salhi

2013

Applied Mathematics and Computation

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“…For a three dimensional Euclidean space, the Hamilton's equation (41) takes the particular formẋ [12,22,23,52,53]. Here, J is the Poisson vector corresponding to the skewsymmetric Poisson tensor [39].…”

Section: Poisson Manifolds In Three Dimensionsmentioning

confidence: 99%