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

Entropy ◽  
2019 ◽  
Vol 21 (9) ◽  
pp. 871 ◽  
Author(s):  
Petrzela ◽  
Sotner
Keyword(s):  
Vector Fields ◽  
Active Element ◽  
Analog Computer ◽  
Complex Polynomial ◽  
Chaotic Oscillator ◽  
Hyperchaotic System ◽  
Lorenz Equations ◽  
Polynomial Vector Fields ◽  
Active Device ◽  
Current Output

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.

scholarly journals Finitely curved orbits of complex polynomial vector fields

2007 ◽  
Vol 79 (1) ◽  
pp. 13-16
Author(s):  
Albetã C. Mafra
Keyword(s):  
Vector Field ◽  
Gaussian Curvature ◽  
Algebraic Curve ◽  
Vector Fields ◽  
Complex Polynomial ◽  
Holomorphic Foliations ◽  
Holomorphic Curve ◽  
Polynomial Vector ◽  
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 <FONT FACE=Symbol>Ì</FONT> C² be a holomorphic curve which has finite total Gaussian curvature. IsC contained in an algebraic curve?

scholarly journals Topological enumeration of complex polynomial vector fields

2014 ◽  
Vol 35 (4) ◽  
pp. 1315-1344 ◽  
Author(s):  
J. TOMASINI
Keyword(s):  
Closed Form ◽  
Orthogonal Group ◽  
Vector Fields ◽  
Main Tool ◽  
Roots Of Unity ◽  
Complex Polynomial ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Appl Math ◽  
General Method

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].

scholarly journals Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

2021 ◽  
Author(s):  
Niclas Kruff ◽  
Jaume Llibre ◽  
Chara Pantazi ◽  
Sebastian Walcher
Keyword(s):  
Vector Fields ◽  
Arbitrary Dimension ◽  
Classical Problem ◽  
Algebraic Surfaces ◽  
Stationary Points ◽  
Complex Polynomial ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Degree Bounds ◽  
Invariant Algebraic Surfaces

AbstractWe 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.

Enumerating combinatorial classes of the complex polynomial vector fields in ℂ

2012 ◽  
Vol 33 (2) ◽  
pp. 416-440 ◽  
Author(s):  
KEALEY DIAS
Keyword(s):  
Parameter Space ◽  
Vector Fields ◽  
Topological Classification ◽  
Stable Complex ◽  
Complex Polynomial ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Fixed Degree ◽  
Combinatorial Data ◽  
Structurally Stable

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.

Limit cycles of three-dimensional polynomial vector fields

Nonlinearity ◽  
2004 ◽  
Vol 18 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Marcin Bobie ski ◽  
Henryk o a dek
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Polynomial Vector Fields
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