scholarly journals Capture Zones of the Family of Functions λzmexp(z)

2003 ◽  
Vol 13 (09) ◽  
pp. 2623-2640 ◽  
Author(s):  
Núria Fagella ◽  
Antonio Garijo
Keyword(s):  
Fixed Point ◽  
Critical Point ◽  
Basin Of Attraction ◽  
Capture Zone ◽  
Connected Component ◽  
Dynamical Plane ◽  
Capture Zones ◽  
The Family ◽  
Parameter Values ◽  
The Basin Of Attraction

We consider the family of entire transcendental maps given by Fλ,m(z)=λzm exp (z) where m≥2. All functions Fλ,m have a superattracting fixed point at z=0, and a critical point at z = -m. In the dynamical plane we study the topology of the basin of attraction of z=0. In the parameter plane we focus on the capture behavior, i.e. λ values such that the critical point belongs to the basin of attraction of z=0. In particular, we find a capture zone for which this basin has a unique connected component, whose boundary is then nonlocally connected. However, there are parameter values for which the boundary of the immediate basin of z=0 is a quasicircle.

THE BASIN OF ATTRACTION OF LOZI MAPPINGS

2009 ◽  
Vol 19 (03) ◽  
pp. 1043-1049 ◽  
Author(s):  
DIOGO BAPTISTA ◽  
RICARDO SEVERINO ◽  
SANDRA VINAGRE
Keyword(s):  
Strange Attractor ◽  
Basin Of Attraction ◽  
Parameter Values ◽  
The Basin Of Attraction

For parameter values which assure its existence, we characterize the basin of attraction for the Lozi map strange attractor.

Biaccessibility in quadratic Julia sets

2000 ◽  
Vol 20 (6) ◽  
pp. 1859-1883 ◽  
Author(s):  
SAEED ZAKERI
Keyword(s):  
Fixed Point ◽  
Measure Zero ◽  
Julia Set ◽  
Basin Of Attraction ◽  
Countable Union ◽  
Common Theme ◽  
Chebyshev Map ◽  
Straight Line ◽  
The Common ◽  
The Basin Of Attraction

This paper consists of two nearly independent parts, both of which discuss the common theme of biaccessible points in the Julia set $J$ of a quadratic polynomial $f:z\mapsto z^2+c$.In Part I, we assume that $J$ is locally-connected. We prove that the Brolin measure of the set of biaccessible points (through the basin of attraction of infinity) in $J$ is zero except when $f(z)=z^2-2$ is the Chebyshev map for which the corresponding measure is one. As a corollary, we show that a locally-connected quadratic Julia set is not a countable union of embedded arcs unless it is a straight line or a Jordan curve.In Part II, we assume that $f$ has an irrationally indifferent fixed point $\alpha$. If $z$ is a biaccessible point in $J$, we prove that the orbit of $z$ eventually hits the critical point of $f$ in the Siegel case, and the fixed point $\alpha$ in the Cremer case. As a corollary, it follows that the set of biaccessible points in $J$ has Brolin measure zero.

scholarly journals Superattracting fixed points of quasiregular mappings

2014 ◽  
Vol 36 (3) ◽  
pp. 781-793 ◽  
Author(s):  
ALASTAIR FLETCHER ◽  
DANIEL A. NICKS
Keyword(s):  
Fixed Point ◽  
Spherical Shell ◽  
Fixed Points ◽  
Rate Of Convergence ◽  
Basin Of Attraction ◽  
Quasiregular Mapping ◽  
Quasiregular Mappings ◽  
Local Index ◽  
Polynomial Type ◽  
The Basin Of Attraction

We investigate the rate of convergence of the iterates of an $n$-dimensional quasiregular mapping within the basin of attraction of a fixed point of high local index. A key tool is a refinement of a result that gives bounds on the distortion of the image of a small spherical shell. This result also has applications to the rate of growth of quasiregular mappings of polynomial type, and to the rate at which the iterates of such maps can escape to infinity.

scholarly journals One-Point Optimal Family of Multiple Root Solvers of Second-Order

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 655 ◽  
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma ◽  
Clemente Cesarano
Keyword(s):  
Weight Function ◽  
Basin Of Attraction ◽  
Multiple Root ◽  
Second Order ◽  
Weight Functions ◽  
Modified Newton Method ◽  
The Family ◽  
Dynamical Nature ◽  
Iterative Functions ◽  
The Basin Of Attraction

This manuscript contains the development of a one-point family of iterative functions. The family has optimal convergence of a second-order according to the Kung-Traub conjecture. This family is used to approximate the multiple zeros of nonlinear equations, and is based on the procedure of weight functions. The convergence behavior is discussed by showing some essential conditions of the weight function. The well-known modified Newton method is a member of the proposed family for particular choices of the weight function. The dynamical nature of different members is presented by using a technique called the “basin of attraction”. Several practical problems are given to compare different methods of the presented family.

scholarly journals Strategies for an Efficient Official Publicity Campaign

2021 ◽  
Vol 183 (2) ◽  
Author(s):  
Juan Neirotti
Keyword(s):  
Fixed Point ◽  
Public Opinion ◽  
Phase Space ◽  
Fixed Points ◽  
Basin Of Attraction ◽  
Opinion Formation ◽  
Publicity Campaign ◽  
The Basin Of Attraction

AbstractWe consider the process of opinion formation, in a society where there is a set of rules B that indicates whether a social instance is acceptable. Public opinion is formed by the integration of the voters’ attitudes which can be either conservative (mostly in agreement with B) or liberal (mostly in disagreement with B and in agreement with peer voters). These attitudes are represented by stable fixed points in the phase space of the system. In this article we study the properties of a perturbative term, mimicking the effects of a publicity campaign, that pushes the system from the basin of attraction of the liberal fixed point into the basin of the conservative point, when both fixed points are equally likely.

Intermingled basins for the triangle map

1996 ◽  
Vol 16 (4) ◽  
pp. 651-662 ◽  
Author(s):  
James C. Alexander ◽  
Brian R. Hunt ◽  
Ittai Kan ◽  
James A. Yorke
Keyword(s):  
Lebesgue Measure ◽  
Basin Of Attraction ◽  
The Other ◽  
Positive Lebesgue Measure ◽  
Quadratic Maps ◽  
Open Set ◽  
Parameter Values ◽  
The Basin Of Attraction

AbstractA family of quadratic maps of the plane has been found numerically for certain parameter values to have three attractors, in a triangular pattern, with ‘intermingled’ basins. This means that for every open set S, if the basin of attraction of one of the attractors intersects S in a set of positive Lebesgue measure, then so do the other two basins. In this paper we mathematically verify this observation for a particular parameter, and prove that our results hold for a set of parameters with positive Lebesgue measure.

Accessible saddles on fractal basin boundaries

1992 ◽  
Vol 12 (3) ◽  
pp. 377-400 ◽  
Author(s):  
Kathleen T. Alligood ◽  
James A. Yorke
Keyword(s):  
Fixed Point ◽  
Basin Of Attraction ◽  
Open Disk ◽  
Fractal Basin Boundaries ◽  
Basin Boundary ◽  
Fixed Point Attractor ◽  
Point Attractor ◽  
Simply Connected ◽  
Basin Boundaries ◽  
The Basin Of Attraction

AbstractFor a homeomorphism of the plane, the basin of attraction of a fixed point attractor is open, connected, and simply-connected, and hence is homeomorphic to an open disk. The basin boundary, however, need not be homeomorphic to a circle. When it is not, it can contain periodic orbits of infinitely many different periods.

Entropy Production of the Willamowski-Rössler Oscillator

2005 ◽  
Vol 60 (8-9) ◽  
pp. 599-9 ◽  
Author(s):  
Jörg W. Stucki ◽  
Robert Urbanczik
Keyword(s):  
Fixed Point ◽  
Entropy Production ◽  
Dynamic Behaviour ◽  
Basin Of Attraction ◽  
Steady States ◽  
Original Model ◽  
Core Component ◽  
Mean Values ◽  
The Mean ◽  
The Basin Of Attraction

Some properties of the Willamowski-Rössler model are studied by numerical simulations. From the original equations a minimal version of the model is derived which also exhibits the characteristic properties of the original model. This minimal model shows that it contains the Volterra-Lotka oscillator as a core component. It thus belongs to a class of generalized Volterra-Lotka systems. It has two steady states, a saddle point, responsible for chaos, and a fixed point, dictating its dynamic behaviour. The chaotic attractor is located close to the surface of the basin of attraction of the saddle node. The mean values of the variables are equal to the (unstable) steady state values during oscillations even under chaos, and the variables are always non-negative as in other generalized Volterra-Lotka systems. Surprisingly this was also the case with the original reversible Willamowski-Rössler model allowing to compare the entropy production during oscillations with the entropy production of the steady states. During oscillations the entropy production was always lower even under chaos. Since under these circumstances less energy is dissipated to produce the same output, the oscillating system is more efficient than the non-oscillatory one.

Punctured Parabolic Cylinders in Automorphisms of ℂ2

2020 ◽  
Author(s):  
Josias Reppekus
Keyword(s):  
Fixed Point ◽  
Blow Up ◽  
Basin Of Attraction ◽  
Irrational Rotation ◽  
Parabolic Cylinder ◽  
Density Property ◽  
Fatou Component ◽  
Crucial Step ◽  
Blowing Up ◽  
The Basin Of Attraction

Abstract We show the existence of automorphisms $F$ of $\mathbb{C}^{2}$ with a non-recurrent Fatou component $\Omega $ biholomorphic to $\mathbb{C}\times \mathbb{C}^{*}$ that is the basin of attraction to an invariant entire curve on which $F$ acts as an irrational rotation. We further show that the biholomorphism $\Omega \to \mathbb{C}\times \mathbb{C}^{*}$ can be chosen such that it conjugates $F$ to a translation $(z,w)\mapsto (z+1,w)$, making $\Omega $ a parabolic cylinder as recently defined by L. Boc Thaler, F. Bracci, and H. Peters. $F$ and $\Omega $ are obtained by blowing up a fixed point of an automorphism of $\mathbb{C}^{2}$ with a Fatou component of the same biholomorphic type attracted to that fixed point, established by F. Bracci, J. Raissy, and B. Stensønes. A crucial step is the application of the density property of a suitable Lie algebra to show that the automorphism in their work can be chosen such that it fixes a coordinate axis. We can then remove the proper transform of that axis from the blow-up to obtain an $F$-stable subset of the blow-up that is biholomorphic to $\mathbb{C}^{2}$. Thus, we can interpret $F$ as an automorphism of $\mathbb{C}^{2}$.

Basins of attraction near homoclinic tangencies

1994 ◽  
Vol 14 (2) ◽  
pp. 351-390 ◽  
Author(s):  
J.C. Tatjer ◽  
C. Simó
Keyword(s):  
Fixed Point ◽  
Saddle Point ◽  
Basin Of Attraction ◽  
Periodic Points ◽  
Basins Of Attraction ◽  
Homoclinic Point ◽  
Homoclinic Tangencies ◽  
The Basin Of Attraction

AbstractWe describe the behaviour of the basin of attraction of the attracting periodic points which appear near a non-degenerate tangential homoclinic point of a dissipative saddle fixed point for one-parameter families of planar diffeomorphisms. This behaviour depends on certain relations between the eigenvalues of the saddle point and on the geometry of the tangency.

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