Complex Polynomial Vector Fields Research Articles

Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields.

A Characterization of Multiplicity-Preserving Global Bifurcations of Complex Polynomial Vector Fields

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 can be realized as a composition of a finite number of simpler bifurcations, and these bifurcations are characterized.

Attracting and Natural Invariant Varieties for Polynomial Vector Fields and Control Systems

We discuss real and complex polynomial vector fields and polynomially nonlinear, input-affine control systems, with a focus on invariant algebraic varieties. For a given real variety we consider the construction of polynomial ordinary differential equations dot{x}=f(x) such that the variety is invariant and locally attracting, and show that such a construction is possible for any compact connected component of a smooth variety satisfying a weak additional condition. Moreover we introduce and study natural controlled invariant varieties (NCIV) with respect to a given input matrix g, i.e. varieties which are controlled invariant sets of {dot{x}}=f(x)+g(x)u for any choice of the drift vector f. We use basic tools from commutative algebra and algebraic geometry in order to characterize NCIV’s, and we present a constructive method to decide whether a variety is a NCIV with respect to an input matrix. The results and the algorithmic approach are illustrated by examples.

New Nonlinear Active Element Dedicated to Modeling Chaotic Dynamics with Complex Polynomial Vector Fields

This paper describes evolution of new active element that is able to significantly simplify the design process of lumped chaotic oscillator, especially if the concept of analog computer or state space description is adopted. The major advantage of the proposed active device lies in the incorporation of two fundamental mathematical operations into a single five-port voltage-input current-output element: namely, differentiation and multiplication. The developed active device is verified inside three different synthesis scenarios: circuitry realization of a third-order cyclically symmetrical vector field, hyperchaotic system based on the Lorenz equations and fourth- and fifth-order hyperjerk function. Mentioned cases represent complicated vector fields that cannot be implemented without the necessity of utilizing many active elements. The captured oscilloscope screenshots are compared with numerically integrated trajectories to demonstrate good agreement between theory and measurement.

On parameter space of complex polynomial vector fields in [formula omitted

The space Ξd of degree d single-variable monic and centered complex polynomial vector fields can be decomposed into loci in which the vector fields have the same topological structure. This paper analyzes the geometric structure of these loci and describes some bifurcations. In particular, it is proved that new homoclinic separatrices can form under small perturbation. By an example, we show that this decomposition of parameter space by combinatorial data is not a cell decomposition.The appendix to this article, joint work with Tan Lei, shows that landing separatrices are stable under small perturbation of the vector field if the multiplicities of the equilibrium points are preserved.

Topological enumeration of complex polynomial vector fields

AbstractThe enumeration of combinatorial classes of the complex polynomial vector fields in$ \mathbb{C} $presented by K. Dias [Enumerating combinatorial classes of the complex polynomial vector fields in$ \mathbb{C} $.Ergod. Th. & Dynam. Sys. 33(2013), 416–440] is extended here to a closed form enumeration of combinatorial classes for degree$d$polynomial vector fields up to rotations of the$2(d- 1)\mathrm{th} $roots of unity. The main tool in the proof of this result is based on a general method of enumeration developed by V. A. Liskovets [Reductive enumeration under mutually orthogonal group actions.Acta Appl. Math. 52(1998), 91–120].

Topological and analytical classification of vector fields with only isochronous centres

We study vector fields on the plane having only isochronous centres. The most familiar examples are isochronous vector fields, they are the real parts of complex polynomial vector fields on having all their zeroes of centre type. We describe the number of topologically inequivalent isochronous (singular) foliations that can appear for degree s, up to orientation preserving homeomorphisms. For each s, there exists a real analytic variety parametrizing the isochronous vector fields of degree s, the group of complex automorphisms of the plane acts on it. Furthermore, if , then is a non-singular real analytic variety of dimension , and their number of connected components is bounded by . An explicit formula for the residues of the rational 1-form, canonically associated with a complex polynomial vector field with simple zeroes, is given. A collection of residues (i.e. periods) does not characterize an isochronous vector field, even up to complex automorphisms of . An exact bound for the number of isochronous vector fields, up to , having the same collection of residues (periods) is given. We develop several descriptions of the quotient space using residues, weighted s-trees and singular flat Riemannian metrics associated with isochronous vector fields.

Enumerating combinatorial classes of the complex polynomial vector fields in ℂ

AbstractIn order to understand the parameter space Ξd of monic and centered complex polynomial vector fields in ℂ of degree d, decomposed by the combinatorial classes of the vector fields, it is interesting to know the number of loci in parameter space consisting of vector fields with the same combinatorial data (corresponding to topological classification with fixed separatrices at infinity). This paper answers questions posed by Adam L. Epstein and Tan Lei about the total number of combinatorial classes and the number of combinatorial classes corresponding to loci of a specific (real) dimension q in parameter space, for fixed degree d; these numbers are denoted by cd and cd,q, respectively. These results are extensions of a result by Douady, Estrada, and Sentenac, which shows that the number of combinatorial classes of the structurally stable complex polynomial vector fields in ℂ of degree d is the Catalan number Cd−1. We show that enumerating the combinatorial classes is equivalent to a so-called bracketing problem. Then we analyze the generating functions and find closed-form expressions for cd and cd,q, and we furthermore make an asymptotic analysis of these sequences for d tending to ∞. These results are also applicable to special classes of quadratic and Abelian differentials and singular holomorphic foliations of the plane.

Classification of complex polynomial vector fields in one complex variable

This paper classifies the global structure of monic and centred one-variable complex polynomial vector fields. The classification is achieved by means of combinatorial and analytic data. More specifically, given a polynomial vector field, we construct a combinatorial invariant, describing the topology, and a set of analytic invariants, describing the geometry. Conversely, given admissible combinatorial and analytic data sets, we show using surgery the existence of a unique monic and centred polynomial vector field realizing the given invariants. This is the content of the Structure Theorem, the main result of the paper. This result is an extension and refinement of Douady et al. (Champs de vecteurs polynomiaux sur ℂ. Unpublished manuscript) classification of the structurally stable polynomial vector fields. We further review some general concepts for completeness and show that vector fields in the same combinatorial class have flows that are quasi-conformally equivalent.

Complex polynomial vector fields having an orbit with finite total curvature

In this paper we address the following questions: (i) Let \({C \subset \mathbb{C}^2}\) be an orbit of a polynomial vector field which has finite total Gaussian curvature. Is C contained in an algebraic curve? (ii) What can be said of a polynomial vector field which has a finitely curved transcendent orbit? We give a positive answer to (i) under some non-degeneracy conditions on the singularities of the projective foliation induced by the vector field. For vector fields with a slightly more general class of singularities we prove a classification result that captures rational pull-backs of Poincare-Dulac normal forms.

Complex Polynomial Vector Fields Having a Finitely Curved Orbit

The author proves that a non-singular polynomial vector field without invariant lines and having an entire finitely curved transcendent orbit on ℂ2 must be equivalent to a trivial vector field by a holomorphic change of coordinates. Other classification results are obtained for polynomial vector fields having a finitely curved orbit on ℂ2.

Finitely curved orbits of complex polynomial vector fields

This note is about the geometry of holomorphic foliations. Let X be a polynomial vector field with isolated singularities on C2. We announce some results regarding two problems: 1. Given a finitely curved orbit L of X, under which conditions is L algebraic? 2. If X has some non-algebraic finitely curved orbit L what is the classification of X? Problem 1 is related to the following question: Let C [symbol: see text]C2 be a holomorphic curve which has finite total Gaussian curvature. Is C contained in an algebraic curve?