Abstract
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.
Similar content being viewed by others
Notes
For \(d=2\), there are technically no separatrices since infinity is not a pole. The results in this paper still hold in this case if we consider the trajectories through infinity as separatrices.
References
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)
Andronov, A.A., Leontovich, E.A., Gordon, I.I., Maier, A.G.: Qualitative Theory of Second-Order Dynamic Systems. Wiley, New York (1973). (Original: Nakua, Moscow 1967)
Artés, J.C., 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)
Benzinger, H.E.: Plane autonomous systems with rational vector fields. Trans. Am. Math. Soc. 326(2), 465–483 (1991)
Benzinger, H.E.: Julia sets and differential equations. Proc. Am. Math. Soc. 117(4), 939–946 (1993)
Branner, B., Dias, K.: Classification of polynomial vector fields in one complex variable. J. Differ. Equ. Appl. 16(5), 463–517 (2010)
Brickman, L., Thomas, E.S.: Conformal equivalence of analytic flows. J. Differ. Equ. 25, 310–324 (1977)
Buff, X., Chéritat, A.: Ensembles de julia quadratiques de mesure de lebesgue strictement positive. C. R. Acad. Sci. Paris 341(11), 669–674 (2005)
Buff, X., Tan, L.: Dynamical convergence and polynomial vector fields. J. Differ. Geom. 77(1), 1–41 (2007)
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)
Dias, K.: Enumerating combinatorial classes of complex polynomial vector fields in \(\mathbb{C}\). Ergod. Theory Dyn. Syst. 33, 416–440 (2013)
Dias, K., Tan, L.: On parameter space of complex polynomial vector fields in \(\mathbb{C}\). J. Differ. Equ. 260, 628–652 (2016)
Douady, A., Estrada, F., Sentenac, P.: Champs de vecteurs polynomiaux sur \(\mathbb{C}\). Unpublished manuscript
Fathi, A., Laudenbach, F., Poenaru, V.: Traveaux de thurston sur les surfaces. Asterisque, pp. 66–67 (1979)
Gardiner, F.P., Hu, J.: Finite earthquakes and the associahedron. In: Teichmüller theory and moduli problems, pp. 179–194 (2010)
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)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)
Hájek, O.: Notes on meromorphic dynamical systems, I. Czech. Math. J. 16(1), 14–27 (1966)
Hájek, O.: Notes on meromorphic dynamical systems, II. Czech. Math. J. 16(1), 28–35 (1966)
Hájek, O.: Notes on meromorphic dynamical systems, III. Czech. Math. J. 16(1), 36–40 (1966)
Jenkins, J.: Univalent Functions. Springer, Berlin (1958)
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)
Mardešić, P., Roussarie, R., Rousseau, C.: Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms. Mosc. Math. J. 4, 455–498 (2004)
Muciño-Raymundo, J., Valero-Valdés, C.: Bifurcations of meromorphic vector fields on the Riemann sphere. Ergod. Theory Dyn. Syst. 15(6), 1211–1222 (1995)
Needham, D., King, A.: On meromorphic complex differential equations. Dyn. Stab. Syst. 9, 99–121 (1994)
Neumann, D.: Classification of continuous flows on 2-manifolds. Proc. Am. Math. Soc. 48(1), 73–81 (1975)
Oudkerk, R.: The parabolic implosion for \(f\_0(z)=z+z^{\nu +1}+o(z^{\nu +2})\). Ph.D. thesis, University of Warwick (1999)
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)
Rousseau, C., Teyssier, L.: Analytical moduli for unfoldings of saddle-node vector fields. Mosc. Math. J. 8(3), 547–614 (2008)
Shishikura, M.: Bifurcation of parabolic fixed points. In: Lei, T. (ed.) The Mandelbrot Set, Theme and Variations. Cambridge University Press, Cambridge (2000)
Sverdlove, R.: Vector fields defined by complex functions. J. Differ. Equ. 34, 427–439 (1979)
Acknowledgements
There are many people to thank for patiently listening to various stages of this work and providing helpful suggestions. In particular, the author is indebted to Bodil Branner, Frederick Gardiner and the other participants in the Extremal Length seminar at CUNY Graduate Center, Christian Henriksen, Poul G. Hjorth, Louis Pedersen, and Tan Lei. The author would also like to express gratitude to the anonymous referee, whose thorough and thoughtful critiques have greatly improved this manuscript.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was supported by Fondation Idella, the Marie Curie European Union Research Training Network Conformal Structures and Dynamics (CODY), the Research Foundation of CUNY PSC-CUNY Cycle 44 (66148-00 44) and Cycle 47 (69510-00 47) Research Awards, the Bronx Community College Foundation Faculty Scholarship Grant 2016, and the Association for Women in Mathematics Travel Grant October 2019 Cycle (NSF 1642548).
Rights and permissions
About this article
Cite this article
Dias, K. A Characterization of Multiplicity-Preserving Global Bifurcations of Complex Polynomial Vector Fields. Qual. Theory Dyn. Syst. 19, 90 (2020). https://doi.org/10.1007/s12346-020-00424-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-020-00424-y
Keywords
- Global bifurcations
- Homoclinic orbits
- Holomorphic foliations and vector fields
- Complex ordinary differential equations