Darboux integrating factors: Inverse problems

Christopher, Colin; Llibre, Jaume; Pantazi, Chara; Walcher, Sebastian

doi:10.1016/j.jde.2010.10.013

Cited by 9 publications

(32 citation statements)

References 20 publications

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“…, d r ) of F. We will call these the trivial vector fields admitting the given integrating factor. Nontrivial vector fields exist for certain exponents, as the next result shows (see [26], [10] and [12]). …”

Section: Then the Vector Fieldmentioning

confidence: 63%

“…Up to some point, one can use this to reduce y-degrees via Lemma 17. A precise statement can be found in [12], Proposition 8, but for our purposes the following version will suffice.…”

Section: Then the Vector Fieldmentioning

confidence: 99%

“…(This is a special case of an example discussed in [12], which we present here step-by-step to illustrate the approach and computations.) By Corollary 9 we know that every vector field admitting f has the form…”

Section: Then the Vector Fieldmentioning

confidence: 99%

“…These inverse problems on the affine plane have been discussed systematically in recent work by the authors; see [10], [11], [12]. Since the original papers contain (unavoidably, it seems) quite technical sections, it is sensible to present the straightforward basic ideas in a survey, and to illustrate them by examples.…”

Section: Introductionmentioning

confidence: 99%

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Inverse Problems in Darboux’ Theory of Integrability

et al. 2012

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Abstract. The Darboux theory of integrability for planar polynomial differential equations is a classical field, with connections to Lie symmetries, differential algebra and other areas of mathematics. In the present paper we introduce the concepts, problems and inverse problems, and we outline some recent results on inverse problems. We also prove a new result, viz. a general finiteness theorem for the case of prescribed integrating factors. A number of relevant examples and applications is included.

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Section: Then the Vector Fieldmentioning

confidence: 63%

“…Up to some point, one can use this to reduce y-degrees via Lemma 17. A precise statement can be found in [12], Proposition 8, but for our purposes the following version will suffice.…”

Section: Then the Vector Fieldmentioning

confidence: 99%

Section: Then the Vector Fieldmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Inverse Problems in Darboux’ Theory of Integrability

et al. 2012

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“…Erugin ideas were developed in particular in [17]. We observe that such kind of problem has recently been developed in R 2 or C 2 mainly restricted to polynomial differential equations (see for instance [5,6,7,27,39,41,42,43]). …”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

2014

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Abstract. This paper is on the so called inverse problem of ordinary differential systems, i.e. the problem of determining the differential systems satisfying a set of given properties. More precisely, we characterize under very general assumptions the ordinary differential systems in R N which have a given set of either M ≤ N , or M > N partial integrals, or M < N first integrals, or M ≤ N partial and first integrals. Moreover, for such systems we determine the necessary and sufficient conditions for the existence of N − 1 independent first integrals. For the systems with M < N partial integrals we provide sufficient conditions for the existence of a first integral.We give two relevant applications of the solutions of these inverse problems to constrained Lagrangian and constrained Hamiltonian systems. Additionally we provide a particular solution of the inverse problem in dynamics, and give a generalized solution of the problem of integration of the equation of motion in the classical Suslov problem on SO(3).

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A Survey on the Inverse Integrating Factor

García

Grau

2010

Qual. Theory Dyn. Syst.

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The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996. From that moment on, many research articles are devoted to the study of the properties of the inverse integrating factor and its relation with limit cycles and their bifurcations. This paper is a summary of all the results about this topic. We include a list of references together with the corresponding related results aiming at being as much exhaustive as possible. The paper is, nonetheless, self-contained in such a way that all the main results on the inverse integrating factor are stated and a complete overview of the subject is given. Each section contains a different issue to which the inverse integrating factor plays a role: the integrability problem, relation with Lie symmetries, the center problem, vanishing set of an inverse integrating factor, bifurcation of limit cycles from either a period annulus or from a monodromic ω-limit set and some generalizations.2000 AMS Subject Classification: 34C07, 37G15, 34-02.

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Darboux integrating factors: Inverse problems

Cited by 9 publications

References 20 publications

Inverse Problems in Darboux’ Theory of Integrability

Inverse Problems in Darboux’ Theory of Integrability

Inverse Approach in Ordinary Differential Equations: Applications to Lagrangian and Hamiltonian Mechanics

A Survey on the Inverse Integrating Factor

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