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A characterization of multiplicity-preserving global bifurcations of complex polynomial vector fields. (English) Zbl 07273483
Summary: For the space of single-variable monic and centered complex polynomial vector fields of arbitrary degree $$d$$, it is proved that any bifurcation which preserves the multiplicity of equilibrium points admits a decomposition into a finite number of elementary bifurcations, and the elementary bifurcations are characterized.
##### MSC:
 37F46 Bifurcations; parameter spaces in holomorphic dynamics; the Mandelbrot and Multibrot sets 37C29 Homoclinic and heteroclinic orbits for dynamical systems 37F75 Dynamical aspects of holomorphic foliations and vector fields 32M25 Complex vector fields, holomorphic foliations, $$\mathbb{C}$$-actions 34C23 Bifurcation theory for ordinary differential equations
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##### References:
 [1] Álvarez, M.; Gasull, A.; Prohens, R., Topological classification of polynomial complex differential equations with all the critical points of center type, J. Differ. Equ. Appl., 16, 5, 411-423 (2010) · Zbl 1201.37022 [2] Andronov, AA; Leontovich, EA; Gordon, II; Maier, AG, Qualitative Theory of Second-Order Dynamic Systems (1973), New York: Wiley, New York [3] Artés, JC; Llibre, J.; Schlomiuk, D., The geometry of quadratic differential systems with a weak focus of second order, Int. J. Bifurc. Chaos, 16, 11, 3127-3194 (2006) · Zbl 1124.34014 [4] Benzinger, HE, Plane autonomous systems with rational vector fields, Trans. Am. Math. Soc., 326, 2, 465-483 (1991) · Zbl 0727.34004 [5] Benzinger, HE, Julia sets and differential equations, Proc. Am. Math. Soc., 117, 4, 939-946 (1993) · Zbl 0772.34038 [6] Branner, B.; Dias, K., Classification of polynomial vector fields in one complex variable, J. Differ. Equ. Appl., 16, 5, 463-517 (2010) · Zbl 1203.37080 [7] Brickman, L.; Thomas, ES, Conformal equivalence of analytic flows, J. Differ. Equ., 25, 310-324 (1977) · Zbl 0348.34034 [8] Buff, X.; Chéritat, A., Ensembles de julia quadratiques de mesure de lebesgue strictement positive, C. R. Acad. Sci. Paris, 341, 11, 669-674 (2005) · Zbl 1082.37049 [9] Buff, X.; Tan, L., Dynamical convergence and polynomial vector fields, J. Differ. Geom., 77, 1, 1-41 (2007) · Zbl 1126.37029 [10] Christopher, C.; Rousseau, C., The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point, Int. Math. Res. Not., 9, 2494-2558 (2014) · Zbl 1351.37192 [11] Dias, K., Enumerating combinatorial classes of complex polynomial vector fields in $$\mathbb{C}$$, Ergod. Theory Dyn. Syst., 33, 416-440 (2013) · Zbl 1285.37015 [12] Dias, K.; Tan, L., On parameter space of complex polynomial vector fields in $$\mathbb{C}$$, J. Differ. Equ., 260, 628-652 (2016) · Zbl 1362.37099 [13] Douady, A., Estrada, F., Sentenac, P.: Champs de vecteurs polynomiaux sur $$\mathbb{C}$$. Unpublished manuscript [14] Fathi, A., Laudenbach, F., Poenaru, V.: Traveaux de thurston sur les surfaces. Asterisque, pp. 66-67 (1979) [15] Gardiner, F.P., Hu, J.: Finite earthquakes and the associahedron. In: Teichmüller theory and moduli problems, pp. 179-194 (2010) · Zbl 1203.52010 [16] Garijo, A.; Gasull, A.; Jarque, X., Normal forms for singularities of one dimentional holomorphic vector fields, Electron. J. Differ. Equ., 2004, 122, 1-7 (2004) · Zbl 1075.34089 [17] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (1983), New York: Springer, New York · Zbl 0515.34001 [18] Hájek, O., Notes on meromorphic dynamical systems, I, Czech. Math. J., 16, 1, 14-27 (1966) · Zbl 0145.32401 [19] Hájek, O., Notes on meromorphic dynamical systems, II, Czech. Math. J., 16, 1, 28-35 (1966) · Zbl 0145.32401 [20] Hájek, O., Notes on meromorphic dynamical systems, III, Czech. Math. J., 16, 1, 36-40 (1966) · Zbl 0145.32401 [21] Jenkins, J., Univalent Functions (1958), Berlin: Springer, Berlin [22] Llibre, J.; Schlomiuk, D., The geometry of quadratic differential systems with a weak focus of third order, Can. J. Math., 56, 2, 310-343 (2004) · Zbl 1058.34034 [23] Mardešić, P.; Roussarie, R.; Rousseau, C., Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms, Mosc. Math. J., 4, 455-498 (2004) · Zbl 1077.37035 [24] Muciño-Raymundo, J.; Valero-Valdés, C., Bifurcations of meromorphic vector fields on the Riemann sphere, Ergod. Theory Dyn. Syst., 15, 6, 1211-1222 (1995) · Zbl 0863.58056 [25] Needham, D.; King, A., On meromorphic complex differential equations, Dyn. Stab. Syst., 9, 99-121 (1994) · Zbl 0813.34005 [26] Neumann, D., Classification of continuous flows on 2-manifolds, Proc. Am. Math. Soc., 48, 1, 73-81 (1975) · Zbl 0307.34044 [27] Oudkerk, R.: The parabolic implosion for $$f\_0(z)=z+z^{\nu +1}+o(z^{\nu +2})$$. Ph.D. thesis, University of Warwick (1999) [28] Rousseau, C., Analytic moduli for unfoldings of germs of generic analytic diffeomorphims with a codimension $$k$$ parabolic point, Ergod. Theory Dyn. Syst., 35, 274-292 (2015) · Zbl 1308.32018 [29] Rousseau, C.; Teyssier, L., Analytical moduli for unfoldings of saddle-node vector fields, Mosc. Math. J., 8, 3, 547-614 (2008) · Zbl 1165.37016 [30] Shishikura, M.; Lei, T., Bifurcation of parabolic fixed points, The Mandelbrot Set, Theme and Variations (2000), Cambridge: Cambridge University Press, Cambridge [31] Sverdlove, R., Vector fields defined by complex functions, J. Differ. Equ., 34, 427-439 (1979) · Zbl 0431.34034
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