A generalization of Françoise's algorithm for calculating higher order Melnikov functions

Jébrane, Ahmed; Mardešić, Pavao; Pelletier, Michèle

doi:10.1016/s0007-4497(02)01138-7

Cited by 20 publications

(23 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This kind of graphic has also been treated in [16,17]. To study the perturbation of Hamiltonian systems with a homoclinic loop or a double homoclinic loop, which is a topic not addressed by the present paper, see [18] and the references therein.…”

Section: Theoremmentioning

confidence: 99%

“…In [5] it is shown that if λ = 0, m 1 < 0, m 1 = −1 and m 2 > 0, then the ellipse defined by 1 + m 1 x 2 + m 1 m 2 y 2 = 0, and which we denote by γ, is a hyperbolic limit cycle of system (18). Moreover, with the described values of the parameters, we have that the origin of coordinates is a strong focus whose boundary of the focal region is γ.…”

Section: Examplementioning

confidence: 99%

“…In this example we are not concerned with the multiplicity of the limit cycle but with the Poincaré return map associated to it. We are going to use the ordinary differential equation stated in (7) to give an implicit expression for the Poincaré map associated to γ in system (18). We can parameterize the ellipse by ( √ m 2 cos(s), sin(s))/ √ −m 1 m 2 with s ∈ [0, 2π) and we perform the corresponding change to curvilinear coordinates: (x, y) → (s, n) with…”

Section: Examplementioning

confidence: 99%

“…We observe that the point n = √ m 2 corresponds to the focus point at the origin of system (18). This differential equation can be written in the following Pfaffian form:…”

Section: Examplementioning

confidence: 99%

See 3 more Smart Citations

The inverse integrating factor and the Poincaré map

García

Giacomini

Grau

2010

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

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.

show abstract

Section: Theoremmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

Section: Examplementioning

confidence: 99%

“…We observe that the point n = √ m 2 corresponds to the focus point at the origin of system (18). This differential equation can be written in the following Pfaffian form:…”

Section: Examplementioning

confidence: 99%

See 2 more Smart Citations

The inverse integrating factor and the Poincaré map

García

Giacomini

Grau

2010

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…We will call M k (t) the principal Poincaré-Pontryagin function and say that k is its order. It is also called the generating function in [3,6], Melnikov function in [8][9][10] and variation function in [14].…”

Section: Introductionmentioning

confidence: 99%

Principal Poincaré–Pontryagin function associated to polynomial perturbations of a product of(d+1)straight lines

Uribe¹

2009

Journal of Differential Equations

View full text Add to dashboard Cite

In this paper, we study small polynomial perturbations of a Hamiltonian vector field with Hamiltonian F formed by a product of (d + 1) real linear functions in two variables. We assume that the corresponding lines are in a general position in R 2 . That is, the lines are distinct, non-parallel, no three of them have a common point and all critical values not corresponding to intersections of lines are distinct. We prove in this paper that the principal Poincaré-Pontryagin function M k (t), associated to such a perturbation and to any family of ovals surrounding a singular point of center type, belongs to the C[t, 1/t]-module generated by Abelian integrals and some integrals I * i, j (t), with 1 i < j d defined in the paper. Moreover, I * i, j (t) are not Abelian integrals.They are iterated integrals of length two.

show abstract

A note on higher order Melnikov functions

Jébrane

Żołądek

2005

Qual. Th. Dyn. Syst.

View full text Add to dashboard Cite

A generalization of Françoise's algorithm for calculating higher order Melnikov functions

Cited by 20 publications

References 3 publications

The inverse integrating factor and the Poincaré map

The inverse integrating factor and the Poincaré map

Principal Poincaré–Pontryagin function associated to polynomial perturbations of a product of(d+1)straight lines

A note on higher order Melnikov functions

Contact Info

Product

Resources

About