Bifurcation of 2-periodic orbits from non-hyperbolic fixed points

Cima, Anna; Gasull, Armengol; Mañosa, Vı́ctor

doi:10.1016/j.jmaa.2017.08.029

Cited by 1 publication

(1 citation statement)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then the following statements hold: Notice that the above result deals with orientation reversing local real analytic diffeomorphisms f. For such a maps, and in the discrete setting, it is also interesting to study the bifurcation of 2−periodic orbits. A new definition of cyclicity, the named 2−cyclicity and some new constants, the so called stability constants, appear ( [4]) and the first part of Theorem A can also be used to obtain some algebraic properties for them.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A Note on the Lyapunov and Period Constants

Cima

Gasull

Mañosas

2020

Qual. Theory Dyn. Syst.

View full text Add to dashboard Cite

It is well known that the number of small amplitude limit cycles that can bifurcate from the origin of a weak focus or a non degenerated center for a family of planar polynomial vector fields is governed by the structure of the so called Lyapunov constants, that are polynomials in the parameters of the system. These constants are essentially the coefficients of the odd terms of the Taylor development at zero of the displacement map. Although many authors use that the coefficients of the even terms of this map belong to the ideal generated by the previous odd terms, we have not found a proof in the literature. In this paper we present a simple proof of this fact based on a general property of the composition of one-dimensional analytic reversing orientation diffeomorphisms with themselves. We also prove similar results for the period constants. These facts, together with some classical tools like the Weirstrass preparation Theorem, or the theory of extended Chebyshev systems, are used to revisit some classical results on cyclicity and criticality for polynomial families of planar differential equations.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%