scholarly journals Singular perturbations of the unicritical polynomials with two parameters

2016 ◽  
Vol 37 (6) ◽  
pp. 1997-2016 ◽  
Author(s):  
YINGQING XIAO ◽  
FEI YANG
Keyword(s):  
Julia Set ◽  
Dynamical Behavior ◽  
Rational Maps ◽  
Topological Properties ◽  
Fatou Set ◽  
The Family ◽  
Image Position ◽  
Two Parameters ◽  
Unicritical Polynomials

In this paper, we study the dynamics of the family of rational maps with two parameters $$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$ where $n\geq 2$ and $a,b\in \mathbb{C}^{\ast }$. We give a characterization of the topological properties of the Julia set and the Fatou set of $f_{a,b}$ according to the dynamical behavior of the orbits of the free critical points.

ON THE MARGIN OF COMPLETE STABILITY FOR A CLASS OF CELLULAR NEURAL NETWORKS

2008 ◽  
Vol 18 (05) ◽  
pp. 1343-1361 ◽  
Author(s):  
MAURO DI MARCO ◽  
MAURO FORTI ◽  
ALBERTO TESI
Keyword(s):  
Neural Networks ◽  
Dynamical Behavior ◽  
Stability Margin ◽  
Cellular Neural Networks ◽  
Third Order ◽  
Dimensional Parameter ◽  
The Family ◽  
A New Technique ◽  
Two Parameters

In this paper, the dynamical behavior of a class of third-order competitive cellular neural networks (CNNs) depending on two parameters, is studied. The class contains a one-parameter family of symmetric CNNs, which are known to be completely stable. The main result is that it is a generic property within the family of symmetric CNNs that complete stability is robust with respect to (small) nonsymmetric perturbations of the neuron interconnections. The paper also gives an exact evaluation of the complete stability margin of each symmetric CNN via the characterization of the whole region in the two-dimensional parameter space where the CNNs turn out to be completely stable. The results are established by means of a new technique to investigate trajectory convergence of the considered class of CNNs in the nonsymmetric case.

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

2011 ◽  
Vol 21 (11) ◽  
pp. 3323-3339
Author(s):  
RIKA HAGIHARA ◽  
JANE HAWKINS
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Riemann Sphere ◽  
Julia Set ◽  
Critical Orbit ◽  
Rational Maps ◽  
Complex Numbers ◽  
Period 2 ◽  
The Family ◽  
Dynamical Properties

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

Parabolic-like mappings

2014 ◽  
Vol 35 (7) ◽  
pp. 2171-2197 ◽  
Author(s):  
LUNA LOMONACO
Keyword(s):  
Fixed Point ◽  
Julia Set ◽  
The Family ◽  
Dynamical Properties ◽  
Image Position ◽  
Parabolic Fixed Point

In this paper we introduce the notion of parabolic-like mapping. Such an object is similar to a polynomial-like mapping, but it has a parabolic external class, i.e. an external map with a parabolic fixed point. We define the notion of parabolic-like mapping and we study the dynamical properties of parabolic-like mappings. We prove a straightening theorem for parabolic-like mappings which states that any parabolic-like mapping of degree two is hybrid conjugate to a member of the family $$\begin{eqnarray}\mathit{Per}_{1}(1)=\left\{[P_{A}]\,\bigg|\,P_{A}(z)=z+\frac{1}{z}+A,~A\in \mathbb{C}\right\}\!,\end{eqnarray}$$ a unique such member if the filled Julia set is connected.

Variation of Hausdorff dimension of Julia sets

1993 ◽  
Vol 13 (1) ◽  
pp. 167-174 ◽  
Author(s):  
T. J. Ransford
Keyword(s):  
Hausdorff Dimension ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Analytic Family ◽  
Simply Connected Domain ◽  
Image Position ◽  
Simply Connected

AbstractLet (Rλ)λ∈D be an analytic family of rational maps of degree d ≥ 2, where D is a simply connected domain in ℂ, and each Rλ is hyperbolic. Then the Hausdorff dimension δ(λ) of the Julia set of Rλ satisfieswhere ℋ is a collection of harmonic functions u on D. We examine some consequences of this, and show how it can be used to obtain estimates for the Hausdorff dimension of some particular Julia sets.

SINGULAR PERTURBATIONS OF zn WITH MULTIPLE POLES

2008 ◽  
Vol 18 (04) ◽  
pp. 1085-1100 ◽  
Author(s):  
SEBASTIAN M. MAROTTA
Keyword(s):  
Singular Perturbations ◽  
Julia Set ◽  
Complex Parameter ◽  
Fatou Set ◽  
Small Complex ◽  
The Family ◽  
Topological Characteristics ◽  
Multiple Poles

We study the dynamics of the family of complex maps given by fλ(z) = zn + λ/((z - a)da(z - b)db) where n ≥ 2 is an integer and λ is an arbitrarily small complex parameter. We focus on the topological characteristics of the Julia set and the Fatou set of fλ(z). We prove that despite the large amount of possibilities there are only four different cases that correspond to different positions and orders of the poles a and b.

scholarly journals On (non-)local connectivity of some Julia sets

2014 ◽  
Author(s):  
Alexandre Dezotti ◽  
Pascale Roesch
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Complete Characterization ◽  
Rational Maps ◽  
Local Connectivity ◽  
Rational Map ◽  
Critical Set ◽  
Quadratic Polynomials ◽  
Non Local

This chapter deals with the question of local connectivity of the Julia set of polynomials and rational maps. It discusses when the Julia set of a rational map is considered connected but not locally connected. The question of the local connectivity of the Julia set has been studied extensively for quadratic polynomials, but there is still no complete characterization of when a quadratic polynomial has a connected and locally connected Julia set. This chapter thus proposes some conjectures and develops a model of non-locally connected Julia sets in the case of infinitely renormalizable quadratic polynomials. This model presents the structure of what the post-critical set in that setting should be.

A STUDY OF THE DYNAMICS OF λ sin z

2002 ◽  
Vol 12 (12) ◽  
pp. 2869-2883 ◽  
Author(s):  
PATRICIA DOMÍNGUEZ ◽  
GUILLERMO SIENRA
Keyword(s):  
Unit Circle ◽  
Unit Disc ◽  
Julia Set ◽  
Parabolic Type ◽  
Fatou Set ◽  
Hyperbolic Component ◽  
Q Cycle ◽  
The Family ◽  
Fatou Components ◽  
Connectedness Locus

This paper studies the dynamics of the family λ sin z for some values of λ. First we give a description of the Fatou set for values of λ inside the unit disc. Then for values of λ on the unit circle of parabolic type (λ = exp (i2πθ), θ = p/q, (p, q) = 1), we prove that if q is even, there is one q-cycle of Fatou components, if q is odd, there are two q cycles of Fatou components. Moreover the Fatou components of such cycles are bounded. For λ as above there exists a component Dq tangent to the unit disc at λ of a hyperbolic component. There are examples for λ such that the Julia set is the whole complex plane. Finally, we discuss the connectedness locus and the existence of buried components for the Julia set.

Planar Sublattices of a Free Lattice. II

1979 ◽  
Vol 31 (1) ◽  
pp. 17-34 ◽  
Author(s):  
Ivan Rival ◽  
Bill Sands
Keyword(s):  
Finite Lattice ◽  
Free Lattice ◽  
The Family ◽  
Finite Lattices ◽  
Image Position ◽  
Planar Lattices

In Planar sublattices of a free lattice, I [8] we verify Jonsson's conjecture for finite planar lattices; in particular we obtain a characterization of finite planar sublattices of a free lattice among all finite lattices. In the present paper we use arguments of a quite different flavour to obtain another characterization. Letbe the family of lattices illustrated in Figures 1, 2, 3, and 4. Our goal is to prove the following theorem: a finite lattice is a planar sublattice of a free lattice if andonly if it does not have a member of as a sublattice.

scholarly journals NORMAL FAMILIES OF BICOMPLEX HOLOMORPHIC FUNCTIONS

Fractals ◽  
2009 ◽  
Vol 17 (03) ◽  
pp. 257-268 ◽  
Author(s):  
K. S. CHARAK ◽  
D. ROCHON ◽  
N. SHARMA
Keyword(s):  
Complex Variable ◽  
Julia Set ◽  
Julia Sets ◽  
Holomorphic Functions ◽  
Normal Families ◽  
General Definition ◽  
Fatou Set ◽  
Definition Of ◽  
Bicomplex Holomorphic Functions

In this article, we introduce the concept of normal families of bicomplex holomorphic functions to obtain a bicomplex Montel theorem. Moreover, we give a general definition of Fatou and Julia sets for bicomplex polynomials and we obtain a characterization of bicomplex Fatou and Julia sets in terms of Fatou set, Julia set and filled-in Julia set of one complex variable. Some 3D visual examples of bicomplex Julia sets are also given for the specific slice j = 0.

LIMITING JULIA SETS FOR SINGULARLY PERTURBED RATIONAL MAPS

2008 ◽  
Vol 18 (10) ◽  
pp. 3175-3181 ◽  
Author(s):  
MARK MORABITO ◽  
ROBERT L. DEVANEY
Keyword(s):  
Unit Circle ◽  
Unit Disk ◽  
Riemann Sphere ◽  
Julia Set ◽  
Singularly Perturbed ◽  
Parameter Plane ◽  
Rational Maps ◽  
Closed Unit Disk ◽  
Open Set ◽  
The Family

In this paper, we consider the family of rational maps given by [Formula: see text] where n ≥ 2, and λ is a complex parameter. When λ = 0 the Julia set is the unit circle, as is well known. But as soon as λ is nonzero, the Julia set explodes. We show that, as λ tends to the origin along n - 1 special rays in the parameter plane, the Julia set of Fλ converges to the closed unit disk. This is somewhat unexpected, since it is also known that, if a Julia set contains an open set, it must be the entire Riemann sphere.

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