Inverse Jacobi multipliers

Berrone, Lucio R.; Giacomini, Héctor

doi:10.1007/bf02871926

Cited by 54 publications

(58 citation statements)

References 26 publications

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“…In the context of Darboux integrability, we usually consider the corresponding reciprocals: inverse integrating factors, and inverse Jacobi multipliers [5]. A function M is an inverse Jacobi multiplier for the vector field X if it satisfies the equation…”

Section: Definitionsmentioning

confidence: 99%

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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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Section: Definitionsmentioning

confidence: 99%

“…The computations were carried out in Maple up to terms in (xyz) 5 . The proof of sufficiency is again handled case by case.…”

Section: Systems With (I − 1 : −I − 1 : 2)-resonancementioning

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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“…For understanding ODE's, this tool was intensively studied in the usual Euclidean space R n by mathematicians (as can be seen in the bibliography of [3], [23]- [26]). For all those interested in historical aspects an excellent survey can be found in [1].…”

Section: Introductionmentioning

confidence: 99%

“…(1.2) is that it does not always admit solutions (p. 269 of[13]). (ii) In the terminology of p. 89 of[1], a function h satisfying (1.3) is called an inverse multiplier. (iii) A first result given by (1.2) is the characterization of last multipliers for divergencefree vector fields: m ∈ C ∞ (M) is a last multiplier for the divergenceless vector field A if and only if m is a first integral of A.…”

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confidence: 99%

Last multipliers on Lie algebroids

Crâşmăreanu

Hreƫcanu

2009

Proc Math Sci

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In this paper we extend the theory of last multipliers as solutions of the Liouville's transport equation to Lie algebroids with their top exterior power as trivial line bundle (previously developed for vector fields and multivectors). We define the notion of exact section and the Liouville equation on Lie algebroids. The aim of the present work is to develop the theory of this extension from the tangent bundle algebroid to a general Lie algebroid (e.g. the set of sections with a prescribed last multiplier is still a Gerstenhaber subalgebra). We present some characterizations of this extension in terms of Witten and Marsden differentials.

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“…After Jacobi's untimely death in 1851 his last multiplier had been used to find first integrals, e.g. for a certain second-order ordinary differential equation [34], but it was when Sophus Lie found a link with the symmetries that bear his name [36] that many authors studied the Jacobi last multiplier in different contexts: from searching for its generalization [13]; to exploiting all the possible relationships it has with first integrals and Lie symmetries ( [9], [56]); from determining a Lagrangian formulation for systems of second-order ordinary differential equations ( [58], [38], [29]); to searching for steady compressible flows of perfect gases [51] or applying it to the statistical mechanics of dissipative systems [14]; from identifying the time-dependent probability density as its analogue in Quantum Mechanics [33]; to the ubiquitous relationship between inverse Jacobi last multiplier and limit cycles [2]; from finding first integrals of various systems ( [18], [54], [50]); to deducing nonlocal Lie symmetries [48].…”

Section: Introductionmentioning

confidence: 99%