Abstract
In this paper we prove that the time part of the germ of an analytic family of vector fields with a Hopf bifurcation is rigid in the parameter. Time parts are associated with the temporal invariant of the analytic classification. Because the eigenvalues at zero are complex conjugate, time parts usually unfold in the hyperbolic direction, where the singular points are linearizable. We first identify the time part of a generic conformal family and prove that any weak holomorphic conjugacy between two time parts yields a biholomorphism analytic in the parameter. The existence of Fatou coordinates in both the Siegel and in the Poincaré domains plays a fundamental role in the proof of this result.
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Notes
A time form dt for a vector field \(X_{\varepsilon }\) is a 1-form such that \(i_{X_{\varepsilon }}dt=1,\) where \(i_{X_{\varepsilon }}\) is the interior product on 1-forms: \(i_{X_{\varepsilon }}dt=dt(X_{\varepsilon }).\)
References
A. Algaba, M. Reyes, Isochronous centres and foci via commutators and normal forms. Proc. A R. Soc. Edinb. 138A(2), 1–13 (2008)
V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations. A Series of Comprehensive Studies in Mathematics, vol. 250 (Springer, Berlin, 1988)
W. Arriagada-Silva, Characterization of the generic unfolding of a weak focus. J. Differ. Equ. 253, 1692–1708 (2012)
W. Arriagada, Temporally normalizable generic unfoldings of order-1 weak foci. J. Dyn. Cont. Syst. 21(2), 239–256 (2015)
W. Arriagada, J. Fialho, Parametric rigidness of germs of analytic unfoldings with a Hopf bifurcation. Port. Math. 73, 153–170 (2016)
M. Berthier, D. Cerveau, A. Lins Neto, Sur les feuilletages analytiques réels et le problème du centre. Bull. J. Differ. Equ. 131(2), 244–2661 (1996)
F. Calogero, Isochronous Systems, vol. 1 (Oxford University Press, Oxford, 2008)
P. Elizarov, Y. Il’yashenko, A. Shcherbakov, S. Voronin, Finitely generated groups of germs of one-dimensional conformal mappings and invariants for complex singular points of analytic foliations of the complex plane, in Nonlinear Stokes Phenomena. Advances in Soviet Mathematics, vol. 14, ed. by Y. Il’yashenko (American Mathematical Society, Providence, 1993), pp. 57–105
J. Giné, Isochronous foci for analytic differential systems. International Journal of Bifurcation and Chaos 13(6), 1617–1623 (2003)
J. Giné, M. Grau, Characterization of isochronous foci for planar analytic defferential systems. Proc. R. Soc. Edinb. Sect. A 135, 985–998 (2005)
P. Grinevich, P.M. Santini, Newtonian dynamics in the plane corresponding to straight and cyclic motions on the hyperelliptic curve \(\nu ^2= v^n-1,\)\(n\in {\mathbb{Z}}:\) ergodicity, isochrony, periodicity and fractals. Physica D 232, 22–33 (2007)
L. Hormander, An Introduction to Complex Analysis in Several Variables (North-Holland Publishing Company, Amsterdam, 1973)
Y. Il’yashenko, Nonlinear Stokes Phenomena. Advances in Soviet Mathematics, vol. 14 (American Mathematics Society, Providence, 1993)
Y. Il’yashenko, S. Yakovenko, Lectures on Analytic Differential Equations. Graduate Studies in Mathematical, vol. 86 (American Mathematical Society, Providence, 2008)
J. Martinet, J.P. Ramis, Classification analytique des équations différentielles non linéaires résonnantes du premier ordre. Ann. Scient. Éc. Norm. Sup. 4e série 16, 571–621 (1983)
J.F. Mattei, R. Moussu, Holonomie et intégrales premières. Ann. Scient. Éc. Norm. Sup. 4e série 13, 469–523 (1980)
M. Sabatini, Non-periodic isochronous oscillations in plane differential systems. Anali di Matematica 182, 487–501 (2003)
M. Shishikura, Bifurcation of Parabolic Fixed Points. The Mandelbrot Set, Theme and Variations. Lond. Math. Soc. Lect. Notes 274, 325–363 (2000)
L.J. Teyssier, Équation homologique et classification analytique des germes de champs de vecteurs holomorphes de type noeud-col, Mathématiques [math]. Université Rennes 1 (2003)
S. Yakovenko, A geometric proof of the Bautin theorem. Am. Math. Soc. Transl. Ser. 2 165, 203–219 (1995)
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The author is grateful to the unknown referee for providing important remarks and clever observations after a preliminary revision of the original manuscript.
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Arriagada, W. Parametric rigidity of the Hopf bifurcation up to analytic conjugacy. Period Math Hung 84, 1–17 (2022). https://doi.org/10.1007/s10998-021-00385-y
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DOI: https://doi.org/10.1007/s10998-021-00385-y