On the vanishing set of inverse integrating factors

Giacomini, Héctor; Berrone, Lucio R.

doi:10.1007/bf02969478

Cited by 17 publications

(46 citation statements)

References 12 publications

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“…Nevertheless, the idea has merit, and in [3] Berrone and Giacomini showed that, under mild additional hypotheses, the separatrices of hyperbolic saddle-points lying in U are contained in V −1 (0), and extended this result by showing that if is a compact limit set all of whose critical points are hyperbolic saddle-points, then under mild conditions…”

Section: Remark 13mentioning

confidence: 96%

Integral invariants and limit sets of planar vector fields

García

Shafer

2005

Journal of Differential Equations

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We show that if a planar system of differential equations admits an inverse integrating factor V defined in a neighborhood of a singular point with exactly one zero eigenvalue then V vanishes along any separatrix of the singular point. Additionally we prove that if K is a compact -or -limit set that contains a regular point (or has an elliptic or parabolic sector if not), and if V is defined on a neighborhood of K, then V vanishes at at least one point of K (and on all of K if V is real analytic or Morse).

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Section: Remark 13mentioning

confidence: 96%

Integral invariants and limit sets of planar vector fields

García

Shafer

2005

Journal of Differential Equations

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“…Lately it was shown that the zero-set of V often contains the separatrices of critical points in U. More precisely, in [3], Berrone and Giacomini proved that, if p 0 is a hyperbolic saddle point of system (1), then any inverse integrating factor V defined in a neighborhood of p 0 vanishes on all four separatrices of p 0 , provided V (p 0 ) = 0. We emphasize here that this result does not hold, in general, for nonhyperbolic singularities (see [11] for example).…”

Section: Theoremmentioning

confidence: 99%

The inverse integrating factor and the Poincaré map

García

Giacomini

Grau

2010

Trans. Amer. Math. Soc.

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Abstract. This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincaré return map in terms of the inverse integrating factor. In particular, we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it. We also apply this result to study the homoclinic loop bifurcation. We only consider homoclinic loops whose critical point is a hyperbolic saddle and whose Poincaré return map is not the identity. A local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it. Our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable, that is, the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method.

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“…. , p. A (multi-valued) function of the form f λ 1 1 · · · f λ p p exp(g/ h) is called Darboux. For a definition of Liouvillian function see Singer [17], roughly speaking a Liouvillian function comes from the integral of a Darboux function.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…A good way to study integrable vector fields is through the inverse integrating factor V , for more details see [2]. If X is a real vector field and V : U → R is an inverse integrating factor of X on the open subset U of R 2 , then V becomes very important because V −1 (0) contains a lot of information about the skeleton or separatrices of the phase portrait of X in U , see [1,9,10,13].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

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On the remarkable values of the rational first integrals of polynomial vector fields

Ferragut

Llibre

2007

Journal of Differential Equations

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The remarkable values for polynomial vector fields in the plane having a rational first integral were introduced by Poincaré. He was mainly interested in their algebraic aspects. Here we are interested in their dynamic aspects; i.e. how they contribute to the phase portrait of the system, to its separatrices, to its singular points, etc. The relationship between remarkable values and dynamics mainly takes place through the inverse integrating factor.

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On the vanishing set of inverse integrating factors

Cited by 17 publications

References 12 publications

Integral invariants and limit sets of planar vector fields

Integral invariants and limit sets of planar vector fields

The inverse integrating factor and the Poincaré map

On the remarkable values of the rational first integrals of polynomial vector fields

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