scholarly journals Control and synchronization of Julia sets of the complex dissipative standard system

2016 ◽  
Vol 21 (4) ◽  
pp. 465-476
Author(s):  
Weihua Sun ◽  
Yongping Zhang
Keyword(s):  
Control Parameter ◽  
Julia Set ◽  
Julia Sets ◽  
The Other ◽  
Standard System ◽  
Control And Synchronization ◽  
Backward Orbits

The fractal behaviors of the complex dissipative standard system are discussed in this paper. By using the boundedness of the forward and backward orbits, Julia set of the system is introduced and visualization of Julia set is also given. Then a controller is designed to achieve Julia set shrinking or expanding with the changing of the control parameter. And synchronization of two different Julia sets is discussed by adding a coupling item, which makes one Julia set change to be the other. The simulations illustrate the efficacy of these methods.

Control and Synchronization of Julia Sets in the Forced Brusselator Model

2015 ◽  
Vol 25 (09) ◽  
pp. 1550113 ◽  
Author(s):  
Weihua Sun ◽  
Yongping Zhang
Keyword(s):  
Feedback Control ◽  
Numerical Simulations ◽  
Julia Set ◽  
Julia Sets ◽  
The Other ◽  
Discrete Version ◽  
Brusselator Model ◽  
Control And Synchronization ◽  
And Control

The forced Brusselator model is investigated from the fractal viewpoint. A Julia set of the discrete version of the Brusselator model is introduced and control of the Julia set is presented by using feedback control. In order to discuss the relations of two different Julia sets, a coupled item is designed to realize the synchronization of two Julia sets with different parameters, which provides a method to discuss the relation and the changing of two different Julia sets, one Julia set can be changed to be the other. Numerical simulations are used to verify the effectiveness of these methods.

CONTROL AND SYNCHRONIZATION OF JULIA SETS OF THE COMPLEX PERTURBED RATIONAL MAPS

2013 ◽  
Vol 23 (05) ◽  
pp. 1350083 ◽  
Author(s):  
YONGPING ZHANG
Keyword(s):  
Optimal Control ◽  
Function Method ◽  
Control Method ◽  
Control Function ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Control And Synchronization ◽  
Function Synchronization ◽  
Optimal Control Function

The dynamical and fractal behaviors of the complex perturbed rational maps [Formula: see text] are discussed in this paper. And the optimal control function method is taken on the Julia set of this system. In this control method, infinity is regarded as a fixed point to be controlled. By substituting the driving item for an item in the optimal control function, synchronization of Julia sets of two such different systems is also studied.

scholarly journals Consensus of Julia Sets

2022 ◽  
Vol 6 (1) ◽  
pp. 43
Author(s):  
Weihua Sun ◽  
Shutang Liu
Keyword(s):  
Fractal Theory ◽  
Julia Set ◽  
Julia Sets ◽  
The Other ◽  
Multi Agent Systems ◽  
Practical Applications ◽  
Agent Systems ◽  
Multi Agent

The Julia set is one of the most important sets in fractal theory. The previous studies on Julia sets mainly focused on the properties and graph of a single Julia set. In this paper, activated by the consensus of multi-agent systems, the consensus of Julia sets is introduced. Moreover, two types of the consensus of Julia sets are proposed: one is with a leader and the other is with no leaders. Then, controllers are designed to achieve the consensus of Julia sets. The consensus of Julia sets allows multiple different Julia sets to be coupled. In practical applications, the consensus of Julia sets provides a tool to study the consensus of group behaviors depicted by a Julia set. The simulations illustrate the efficacy of these methods.

Fractal analysis and control of the competition model

2016 ◽  
Vol 09 (03) ◽  
pp. 1650045 ◽  
Author(s):  
Mianmian Zhang ◽  
Yongping Zhang
Keyword(s):  
Feedback Control ◽  
Mathematical Models ◽  
Fractal Analysis ◽  
Fractal Geometry ◽  
Julia Set ◽  
Julia Sets ◽  
Competition Model ◽  
Simulation Results ◽  
Population Competition ◽  
And Control

Lotka–Volterra population competition model plays an important role in mathematical models. In this paper, Julia set of the competition model is introduced by use of the ideas and methods of Julia set in fractal geometry. Then feedback control is taken on the Julia set of the model. And synchronization of two different Julia sets of the model with different parameters is discussed, which makes one Julia set change to be another. The simulation results show the efficacy of these methods.

scholarly journals Transient fields produced by a cylindrical electron beam flowing through a plasma

2012 ◽  
Vol 30 (3) ◽  
pp. 497-507
Author(s):  
Marie-Christine Firpo
Keyword(s):  
Magnetic Field ◽  
Electron Beam ◽  
Laplace Transform ◽  
Control Parameter ◽  
Bessel Functions ◽  
Skin Depth ◽  
The Other ◽  
Beam Radius ◽  
Equilibrium Situation ◽  
Detailed Computation

AbstractThe out-of-equilibrium situation in which an initially sharp-edged cylindrical electron beam, that could, e.g., model electrons flowing within a wire, is injected into a plasma is considered. A detailed computation of the subsequently produced magnetic field is presented. The control parameter of the problem is shown to be the ratio of the beam radius to the electron skin depth. Two alternative ways to address analytically the problem are considered: one uses the usual Laplace transform approach, the other one involves Riemann's method in which causality conditions manifest through some integrals of triple products of Bessel functions.

scholarly journals Wiman–Valiron Discs and the Dimension of Julia Sets

2020 ◽  
Author(s):  
James Waterman
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Julia Sets ◽  
Singular Values ◽  
Simply Connected ◽  
Set Of Points ◽  
Meromorphic Map

Abstract We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman–Valiron theory.

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions II

2000 ◽  
Vol 20 (3) ◽  
pp. 895-910 ◽  
Author(s):  
GWYNETH M. STALLARD
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Meromorphic Function ◽  
Statistical Mechanics ◽  
Rational Function ◽  
Meromorphic Functions ◽  
Julia Set ◽  
Julia Sets ◽  
Real Analytic ◽  
Analytic Maps

Ruelle (Repellers for real analytic maps. Ergod. Th. & Dynam. Sys.2 (1982), 99–108) used results from statistical mechanics to show that, when a rational function $f$ is hyperbolic, the Hausdorff dimension of the Julia set, $\dim J(f)$, depends real analytically on $f$. We give a proof of the fact that $\dim J(f)$ is a continuous function of $f$ that does not depend on results from statistical mechanics and we show that this result can be extended to a class of transcendental meromorphic functions. This enables us to show that, for each $d \in (0,1)$, there exists a transcendental meromorphic function $f$ with $\dim J(f) = d$.

scholarly journals Any counterexample to Makienko’s conjecture is an indecomposable continuum

2009 ◽  
Vol 29 (3) ◽  
pp. 875-883 ◽  
Author(s):  
CLINTON P. CURRY ◽  
JOHN C. MAYER ◽  
JONATHAN MEDDAUGH ◽  
JAMES T. ROGERS Jr
Keyword(s):  
Rational Function ◽  
Julia Set ◽  
Julia Sets ◽  
Rational Functions ◽  
Fatou Set ◽  
Indecomposable Continuum

AbstractMakienko’s conjecture, a proposed addition to Sullivan’s dictionary, can be stated as follows: the Julia set of a rational function R:ℂ∞→ℂ∞ has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienko’s conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational function whose Julia set is an indecomposable continuum.

Density of hyperbolicity for rational maps with Cantor Julia sets

2011 ◽  
Vol 32 (5) ◽  
pp. 1711-1726 ◽  
Author(s):  
WENJUAN PENG ◽  
YONGCHENG YIN ◽  
YU ZHAI
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Rational Map

AbstractIn this paper, taking advantage of quasi-conformal surgery, we prove that each non-hyperbolic rational map with a Cantor Julia set can be approximated by hyperbolic rational maps with Cantor Julia sets of the same degree.

Optimal Control and Synchronization of Alternated Julia Sets

2015 ◽  
Vol 18 (5) ◽  
pp. 1698-1705 ◽  
Author(s):  
Pei Wang ◽  
Shutang Liu
Keyword(s):  
Optimal Control ◽  
Julia Sets ◽  
Control And Synchronization
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