Quasi-homogeneous normal forms

Algaba, Antonio; Freire, Emilio; Gamero, E.; García, Cristóbal

doi:10.1016/s0377-0427(02)00660-x

Cited by 44 publications

(51 citation statements)

References 10 publications

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“…There is a formal close-to-identity change of coordinates, which conjugates X to the Lienard equation (1). It is not difficult to check that…”

Section: 2mentioning

confidence: 99%

Analytic invariants associated with a parabolic fixed point in 2

Gelfreich¹,

Naudot²

2008

Ergod. Th. Dynam. Sys.

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It is well known that in a small neighbourhood of a parabolic fixed point a real-analytic diffeomorphism of (ℝ2,0) embeds in a smooth autonomous flow. In this paper we show that the complex-analytic situation is completely different and a generic diffeomorphism cannot be embedded in an analytic flow in a neighbourhood of its parabolic fixed point. We study two analytic invariants with respect to local analytic changes of coordinates. One of the invariants was introduced earlier by one of the authors. These invariants vanish for time-one maps of analytic flows. We show that one of the invariants does not vanish on an open dense subset. A complete analytic classification of the maps with a parabolic fixed point in ℂ2 is not available at the present time.

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“…There is a formal close-to-identity change of coordinates, which conjugates X to the Lienard equation (1). It is not difficult to check that…”

Section: 2mentioning

confidence: 99%

Analytic invariants associated with a parabolic fixed point in 2

Gelfreich¹,

Naudot²

2008

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

show abstract

“…If we select the type t = (1, 1), we are using in fact the Taylor expansion, but in general, each term in the above expansion involves monomials with different degrees. We give some definitions and properties, see for instance [1,2].…”

Section: Preliminary Definitions and Resultsmentioning

confidence: 99%

“…In this case α (1) 6 , α (2) 6 and α (3) 6 do not vanish simultaneously. x 4 + 1 9 xy 3 − 1 9S y 4 − S 9 x 4 + 1 9 x 3 y + 2 9 y 4 , which has an inverse integrating factor given by V A where A = 9 2 and V = 1 + 2 9 (x − 1 S y) + 1 81 (x + 1 S y) 2 .…”

Section: Smentioning

confidence: 96%

“…(i) If 2b 4 + a 3 = 0, we obtain α (1) 12 = 12(a 4 + 5b 5 ) and α (2) 12 = 2a(a 4 + 5b 5 ). Therefore if we impose a 4 + 5b 5 = 0, we obtain the Hamiltonian case described in (a).…”

Section: Theorem 43 System (43) Is Analytically Integrable In a Neimentioning

confidence: 99%

“…3 ) and, in this case, α(1) 6 and α(2) 6 do not vanish simultaneously. (iii) If d 0 = 0 and d 3 = 0 doing the change x → y, y → x, dt → −dt, d i → −d 3−i for i = 0,1, 2, 3 and c i → c 5−i for i = 0, 1, 2, 3, 4, 5, this case is transformed into (ii.1.1) or (ii.1.2).…”

mentioning

confidence: 92%

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