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.

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.

APPLICATION OF THE SZNAJD SOCIOPHYSICS MODEL TO SMALL-WORLD NETWORKS

2001 ◽  
Vol 12 (10) ◽  
pp. 1537-1544 ◽  
Author(s):  
A. S. ELGAZZAR
Keyword(s):  
Time Series ◽  
Fixed Point ◽  
Time Series Analysis ◽  
Fixed Points ◽  
Small World ◽  
Small World Networks ◽  
Opinion Formation ◽  
Series Analysis ◽  
Definition Of ◽  
Simple Definition

The Sznajd model for the opinion formation is generalized to small-world networks. This generalization destroyed the stalemate fixed point. Then a simple definition of leaders is included. No fixed points are observed. This model displays some interesting aspects in sociology. The model is investigated using time series analysis.

scholarly journals Basin of Attraction Analysis of New Memristor-Based Fractional-Order Chaotic System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Long Ding ◽  
Li Cui ◽  
Fei Yu ◽  
Jie Jin
Keyword(s):  
Phase Space ◽  
Lyapunov Exponent ◽  
Fractional Order ◽  
Chaotic System ◽  
Exponential Function ◽  
Local Minimum ◽  
System Analysis ◽  
Basin Of Attraction ◽  
Stable Point ◽  
The Basin Of Attraction

Memristor is the fourth basic electronic element discovered in addition to resistor, capacitor, and inductor. It is a nonlinear gadget with memory features which can be used for realizing chaotic, memory, neural network, and other similar circuits and systems. In this paper, a novel memristor-based fractional-order chaotic system is presented, and this chaotic system is taken as an example to analyze its dynamic characteristics. First, we used Adomian algorithm to solve the proposed fractional-order chaotic system and yield a chaotic phase diagram. Then, we examined the Lyapunov exponent spectrum, bifurcation, SE complexity, and basin of attraction of this system. We used the resulting Lyapunov exponent to describe the state of the basin of attraction of this fractional-order chaotic system. As the local minimum point of Lyapunov exponential function is the stable point in phase space, when this stable point in phase space comes into the lowest region of the basin of attraction, the solution of the chaotic system is yielded. In the analysis, we yielded the solution of the system equation with the same method used to solve the local minimum of Lyapunov exponential function. Our system analysis also revealed the multistability of this system.

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.

THE EFFECT OF WEAK DISSIPATION IN TWO-DIMENSIONAL MAPPING

2012 ◽  
Vol 22 (10) ◽  
pp. 1250248 ◽  
Author(s):  
JULIANO A. DE OLIVEIRA ◽  
EDSON D. LEONEL
Keyword(s):  
Fixed Point ◽  
Phase Space ◽  
Basin Of Attraction ◽  
Two Dimensional ◽  
Weak Dissipation ◽  
Attracting Fixed Point ◽  
Dimensional Mapping

The influence of weak dissipation and its consequences in a two-dimensional mapping are studied. The mapping is parametrized by an exponent γ in one of the dynamical variables and by a parameter δ which denotes the amount of the dissipation. It is shown that for different values of γ the structure of the phase space of the nondissipative model is replaced by a large number of attractors. The approach to the attracting fixed point is characterized both analytically and numerically. The attracting fixed point exhibits a very complicated basin of attraction.

The effects of parameter variation on the gaits of passive walking models: simulations and experiments

Robotica ◽  
2009 ◽  
Vol 27 (4) ◽  
pp. 511-528 ◽  
Author(s):  
Liu Ning ◽  
Li Junfeng ◽  
Wang Tianshu
Keyword(s):  
Fixed Points ◽  
Basin Of Attraction ◽  
Step Length ◽  
Moment Of Inertia ◽  
Mass Centre ◽  
Random Slope ◽  
Simulation Results ◽  
Good For ◽  
The Times ◽  
The Basin Of Attraction

SUMMARYWe have made a systematic study of the gait of a straight leg planar passive walking model through simulations and experiments. Three normalised parameters, which represent the foot radius, the position of the mass centre and the moment of inertia, are used to characterise the walking model.In the simulation, we have obtained the fixed points and the basins of attraction of the walking models with different parameter combinations by the aid of the cell mapping method. With the results of fixed points, we investigated the effects of parameter variations on the gait descriptors, including step length, period, average speed and energy inefficiency. A model that has a large basin of attraction has been obtained, and it can start walking far from its fixed point. However, the size of the basin of attraction is not a good measurement of robustness. Thus, we proposed floors with random slope angles or stairs with random heights to test robustness. Five hundred times of simulations with 100 non-dimensional time units were implemented for each parameter combination. The times that the walker failed to arrive at the end were recorded. The simulation results showed that the model with a larger foot radius and higher position of mass centre has a lower possibility of falling on uneven floors. A large moment of inertia is helpful for walking on a random slope angle floor, while low values of moment of inertia are good for navigating random stairs.Prototype experiments have validated the simulation results, which showed that models with larger feet have a longer step length and high speed. However, period differences were difficult to obtain in the experiments since the differences were very small. We have tested the sensitivity with the initial conditions of the models with different foot radii on a flat floor, and have also tested the robustness of the models on a floor with random slope angles. The times that the model reached the end of the floor were recorded. The experimental results showed that a large foot radius is good for improving the basin of attraction and robustness on uneven floors. Finally, the exceptions of the experiment are explained.

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.

A COMPUTER-ASSISTED STUDY OF GLOBAL DYNAMIC TRANSITIONS FOR A NONINVERTIBLE SYSTEM

2007 ◽  
Vol 17 (04) ◽  
pp. 1305-1321 ◽  
Author(s):  
RAYMOND A. ADOMAITIS ◽  
IOANNIS G. KEVREKIDIS ◽  
RAFAEL DE LA LLAVE
Keyword(s):  
Phase Space ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Computer Assisted ◽  
Discrete Time System ◽  
Time System ◽  
Invariant Circle ◽  
Global Dynamic ◽  
Assisted Analysis ◽  
The Basin Of Attraction

We present a computer-assisted analysis of the phase space features and bifurcations of a noninvertible, discrete-time system. Our focus is on the role played by noninvertibility in generating disconnected basins of attraction and the breakup of invariant circle solutions. Transitions between basin of attraction structures are identified and organized according to "levels of complexity," a term we define in this paper. In particular, we present an algorithm that provides a computational approximation to the boundary (in phase space) separating points with different preimage behavior. The interplay between this boundary and other phase space features is shown to be crucial in understanding global bifurcations and transitions in the structure of the basin of attraction.

scholarly journals On embedding of invariant manifolds of the simplest Morse-Smale flows with heteroclinical intersections

2018 ◽  
Vol 20 (4) ◽  
pp. 378-383
Author(s):  
E. Ya. Gurevich ◽  
D. A. Pavlova
Keyword(s):  
Phase Space ◽  
Invariant Manifolds ◽  
Basin Of Attraction ◽  
Classification Problem ◽  
Closed Curve ◽  
Topological Classification ◽  
Dimensional Phase Space ◽  
Smale Flows ◽  
Source Sink ◽  
The Basin Of Attraction

We study a structure of four-dimensional phase space decomposition on trajectories of Morse-Smale flows admitting heteroclinical intersections. More precisely, we consider a class G(S4) of Morse-Smale flows on the sphere S4 such that for any flow f∈G(S4) its non-wandering set consists of exactly four equilibria: source, sink and two saddles. Wandering set of such flows contains finite number of heteroclinical curves that belong to intersection of invariant manifolds of saddle equilibria. We describe a topology of embedding of saddle equilibria’s invariant manifolds; that is the first step in the solution of topological classification problem. In particular, we prove that the closures of invariant manifolds of saddle equlibria that do not contain heteroclinical curves are locally flat 2-sphere and closed curve. These manifolds are attractor and repeller of the flow. In set of orbits that belong to the basin of attraction or repulsion we construct a section that is homeomoprhic to the direct product S2×S1. We study a topology of intersection of saddle equlibria’s invariant manifolds with this section.

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.

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