Control And Synchronization Of Julia Sets In A Three-Dimensional Discrete Fractional-Order Financial Model

In recent decades,research on the financial system has shown a trend of vigorous development.With the globalization of the economy,the global economic market presents diversity and complexity.The linear analysis method can not be used simply to study financial systems.Combined with the progress of computer technology,nonlinear analysis method has been paid more attention to,and gradually become a new discipline,called nonlinear dynamics.Financial systems are complex nonlinear dynamic systems.Huang Dengshi proposed a differential equation model of chaotic financial systems,which has been widely studied due to its good dynamic characteristics.This paper will carry out a series of studies on Julia sets of this three-dimensional financial model.The research contents are listed as follows:1.Based on fractional calculus and fractal theory,a three-dimensional discrete integer-order financial model and a three-dimensional discrete fractionalorder financial model are established,and their Julia sets are given.Fractional differential equations are good tools for describing memory and heredity characteristics.Many financial variables in the financial system have strong memory characteristics.Consider applying fractional calculus theory to financial system.Fractal theory is an important branch of nonlinear science,and Julia set is one of the important fractal sets in fractal theory,which is often used in the theoretical research of nonlinear dynamic systems.Therefore,this paper first establishes a discrete integer-order financial model,and combines it with the Caputo fractional calculus definition to establish a discrete fractional-order financial model.Subsequently,Julia sets of the models with different model parameters and fractional orders are given.2.Design controllers to control Julia sets of the three-dimensional discrete fractional-order financial model.In some cases,people want models to exhibit different behaviors and characteristics,so it is necessary to study the control of Julia sets of the models in fractal field.According to the characteristics of the financial model,two controllers are designed based on the fixed point,and the Julia sets can be controlled by changing the control parameters.Furthermore,different controllers exhibit different effects and efficiencies,and the effectiveness is verified by Julia sets graphics.3.Design synchronous couplers to synchronize Julia sets of different forms of the three-dimensional discrete financial models.When aiming to ensure consistency in fractal characteristics between models,consider the synchronization of the Julia sets of models.According to the model forms,the synchronization of Julia sets of financial models is divided into three situations,and two kinds of synchronous couplers are designed to achieve synchronization of Julia sets in different models by changing synchronization parameters.Theoretical derivations of the proposed synchronization methods are given,and the efficiencies of different synchronous couplers are compared.The effectiveness is verified by Julia sets graphics.4.Add different noises and discuss the impact of noise on Julia sets of financial models,and provide a filtering method to reduce the noises of Julia sets on financial models.In real problems,noise will produce a non-negligible influence on models.By adjusting random parameters to change the noise,the interference of different noises on the Julia sets of financial models is observed.Then,the Julia sets of financial models with noise are filtered,and its denoising effect is verified by Julia sets graphics.In summary,this paper discusses the control,synchronization,and noise interference of the Julia sets of the three-dimensional discrete integer-order and fractional-order financial models,which play a positive role in the development and application of fractal theory.