Julia Sets and Their Control in a Three-Dimensional Discrete Fractional-Order Financial Model

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
Zhongyuan Zhao ◽  
Yongping Zhang
Keyword(s):  
Fixed Point ◽  
Fractional Order ◽  
Julia Set ◽  
Julia Sets ◽  
Three Dimensional ◽  
Financial System ◽  
Order Theory ◽  
System Model ◽  
Financial Model ◽  
Fractal Characteristics

It is of great significance to study the three-dimensional financial system model based on the discrete fractional-order theory. In this paper, the Julia set of the three-dimensional discrete fractional-order financial model is identified to show its fractal characteristics. The sizes of the Julia sets need to be changed in some situations, so it is necessary to achieve control of the Julia sets. In combination with the characteristics of the model, two different controllers based on the fixed point are designed, and the control of the three-dimensional Julia sets is realized by adding the controllers into the model in different ways. Finally, the simulation graphs show that the controllers can effectively control the fractal behaviors.

scholarly journals Julia sets of Zorich maps

2021 ◽  
pp. 1-37
Author(s):  
ATHANASIOS TSANTARIS
Keyword(s):  
Complex Plane ◽  
Pairwise Disjoint ◽  
Exponential Family ◽  
Julia Set ◽  
Julia Sets ◽  
Three Dimensional ◽  
Periodic Points ◽  
Exponential Map ◽  
Simple Curves ◽  
Entire Complex

Abstract The Julia set of the exponential family $E_{\kappa }:z\mapsto \kappa e^z$ , $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<\kappa \leq 1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three-dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the whole of $\mathbb {R}^3$ , generalizing Misiurewicz’s result. Moreover, we show that the periodic points of the Zorich map are dense in $\mathbb {R}^3$ and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.

scholarly journals 0-1 Test for Chaos in a Fractional Order Financial System with Investment Incentive

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Baogui Xin ◽  
Yuting Li
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Three Dimensional ◽  
Financial System ◽  
Investment Incentive ◽  
Integer Order ◽  
System A ◽  
Weighted Integral ◽  
Test Algorithm ◽  
Predictor Corrector

A new integer-order chaotic financial system is extended by introducing a simple investment incentive into a three-dimensional chaotic financial system. A four-dimensional fractional-order chaotic financial system is presented by bringing fractional calculus into the new integer-order financial system. By using weighted integral thought, the fractional order derivative's economics meaning is given. The 0-1 test algorithm and the improved Adams-Bashforth-Moulton predictor-corrector scheme are employed to detect numerically the chaos in the proposed fractional order financial system.

Elliptic Functions with Disconnected Julia Sets

2016 ◽  
Vol 26 (06) ◽  
pp. 1650095 ◽  
Author(s):  
Lorelei Koss
Keyword(s):  
Fixed Point ◽  
Critical Points ◽  
Elliptic Function ◽  
Julia Set ◽  
Julia Sets ◽  
Elliptic Functions ◽  
Equivalence Classes ◽  
Weierstrass Elliptic Function ◽  
Typical Function ◽  
Rhombic Lattice

In this paper, we investigate elliptic functions of the form [Formula: see text], where [Formula: see text] is the Weierstrass elliptic function on a real rhombic lattice. We show that a typical function in this family has a superattracting fixed point at the origin and five other equivalence classes of critical points. We investigate conditions on the lattice which guarantee that [Formula: see text] has a double toral band, and we show that this family contains the first known examples of elliptic functions for which the Julia set is disconnected but not Cantor.

scholarly journals Fractal Control and Synchronization of the Discrete Fractional SIRS Model

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-16
Author(s):  
Miao Ouyang ◽  
Yongping Zhang ◽  
Jian Liu
Keyword(s):  
Infectious Disease ◽  
Fixed Point ◽  
Fractional Order ◽  
Fractal Theory ◽  
Julia Sets ◽  
Order System ◽  
Fractional Order Systems ◽  
Sirs Model ◽  
Simulation Results ◽  
Control And Synchronization

SIRS model is one of the most basic models in the dynamic warehouse model of infectious diseases, which describes the temporary immunity after cure. The discrete SIRS models with the Caputo deltas sense and the theories of fractional calculus and fractal theory provide a reasonable and sensible perspective of studying infectious disease phenomenon. After discussing the fixed point of the fractional order system, controllers of Julia sets are designed by utilizing fixed point, which are introduced as a whole and a part in the models. Then, two totally different coupled controllers are introduced to achieve the synchronization of Julia sets of the discrete fractional order systems with different parameters but with the same structure. And new proofs about the synchronization of Julia sets are given. The complexity and irregularity of Julia sets can be seen from the figures, and the correctness of the theoretical analysis is exhibited by the simulation results.

Existence, uniqueness and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
JinRong Wang ◽  
Chun Zhu ◽  
Michal Fečkan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Integral Equations ◽  
Fractional Order ◽  
Point Theorem ◽  
Measure Of Noncompactness ◽  
Existence Results ◽  
Limit Property ◽  
Type Integral

AbstractIn this paper, we apply certain measure of noncompactness and fixed point theorem of Darbo type to derive the existence and limit property of solutions to quadratic Erdélyi-Kober type integral equations of fractional order with three parameters. Moreover, we also present the uniqueness and another existence results of the solutions to the above equations. Finally, two examples are given to illustrate the obtained results.

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