A Characterization of Multiplicity-Preserving Global Bifurcations of Complex Polynomial Vector Fields
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A Characterization of Multiplicity-Preserving Global Bifurcations of Complex Polynomial Vector Fields Kealey Dias1 Received: 27 July 2019 / Accepted: 9 September 2020 © Springer Nature Switzerland AG 2020
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. Keywords Global bifurcations · Homoclinic orbits · Holomorphic foliations and vector fields · Complex ordinary differential equations Mathematics Subject Classification 37C10 · 34C23 · 34M99 · 37C29 · 37F75
1 Introduction Bifurcations, the qualitative change in dynamics produced by varying parameters, are fundamental to the analysis of any family of dynamical systems but are notoriously difficult to describe in any generality. This paper makes a considerable step towards a complete description of the bifurcations of the global topological structure of the integral curves of the complex polynomial vector fields ξ P = P(z)
d , z ∈ C, dz
(1)
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).
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Kealey Dias [email protected] Department of Mathematics and Computer Science, Bronx Community College of the City University of New York, 2155 University Avenue, Bronx, NY 10453, USA 0123456789().: V,-vol
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or equivalently, the maximal solutions γ (t, z) to the associated autonomous ordinary differential equation (ODE) z˙ = P(z), γ (0, z) = z, z ∈ C, t ∈ R,
(2)
where P(z) = z d + ad−2 z d−2 + · · · + a0 is a monic and centered polynomial of degree d ≥ 2. Namely, we characterize the multiplicity-preserving bifurcations, i.e. bifurcations where the multiplicities of the equilibrium points (the zeros of P) are preserved under small perturbation. For significant work on parabolic bifurcations (which do not preserve multiplicities) of complex vector fields, see [10,23,28]. Complex polynomial ODEs of the form (2) are a subset of the R2 systems x˙ = u(x, y) y˙ = v(x, y), x, y, t ∈ R,
(3)
whose global qualitative structure in general remains a fundamental open problem in dynamics. Famously, part of Hilbert’s 16th problem inquires to the number and configurations of limit cycles in the plane for each degree d polynomial system in two real variables. Even though holomorphic vector fields, which lack limit cycles (e.g., [25,31]), may seem distant from Hilbert’s 16th problem, experts in this area have shown that a substantial class of perturbations to study are non-holomorphic perturbations of holomorphic poly
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