Bifurcations and nondegenerated conditions of higher degree and a new simple proof of the Hopf-Neimark-Sacker bifurcation theorem (Q1306858)

This paper deals with some local bifurcation cases generated by a smooth parameter variation in sufficiently smooth maps, when a multiplier modulus (eigenvalue) of a fixed point crosses through the value one. The study is made for one and two-dimensional maps. The authors' purpose is to present new proofs related to the crossing through a multiplier \(+1\) (fold, transcritical, and pitchfork bifurcations), \(-1\) (flip bifurcation), and a pair of complex multipliers \(\exp(\pm i\varphi)\) (Neimark's bifurcation), \(i^2=-1\), in cases of ``nondegenerated conditions'', i.e. bifurcation in a family with a nonhyperbolic fixed point. With respect to previous papers they give some new such conditions which generalize the classical ones. In the case of Neimark's bifurcation, it seems that the study is made with the implicit hypothesis of nonoccurrence of exceptional cases [cf. \textit{Ph. Holmes} and \textit{R. F. Williams}, Arch. Ration. Mech. Anal. 90, 115-194 (1985; Zbl 0593.58027) and its references]. Such cases appear when the multiplier angle \(\varphi\) is commensurable with \(2\pi\), \(\varphi=2k\pi/q\), and lead to a problem of denominators which cancel, and small denominators near this angle value, during the process of establishment of a normal form. The theorem is limited to the generation of only one invariant closed curve (results concerning the birth of several invariant closed curves have been published from 1974 on).