Fractal Analysis And Control Of Several Kinds Of Nonlinear Systems

Julia set,as an important set in fractal theory,is also a significant subject of scientific research,possessing complex internal structures and interesting properties.The analysis of Julia set can help us better understand system complexity,analyze system stability,and design system control strategies.Additionally,fractional order models can explain and address phenomena that traditional integer order models cannot explain,an increasing number of researches indicate that many systems in nature actually exhibit fractional order characteristics,while integer order models can only approximately describe local properties.The combination of fractal theory and fractional order system provides powerful tool for the modeling of nonlinear system,the simulation of discrete system,and the description of dynamic behavior and properties of fractal phenomena.Introducing fractal theory and fractional order system into the study of mathematical models for infectious diseases,this enables a more comprehensive understanding of the transmission modes and variations of diseases within populations,deepens our knowledge of disease transmission dynamics and provides scientific foundations and decision support for epidemic intervention and control.Against the backdrop of the global COVID-19 pandemic,research on infectious disease models holds great significance for global public health,it not only aids in understanding disease transmission mechanisms and key factors,enhances our ability to respond to and control sudden infectious diseases,but also promotes the sustainable development of global public health security.Currently,the study of complex dynamical system remains a focal point,encompassing its qualitative theory and control of bounded domains of fractal sets,the study of complex dynamical system offers new approaches for studying various complex shapes and structures found in nature,and it has extensive applications in fields such as physics,engineering,theoretical biology,management,psychology,education,chemistry and others.Therefore,the control research on Julia sets of complex dynamical system holds important practical significance.This paper is grounded in theory and practical applications,it utilizes fractal theory to investigate discrete infectious disease models and achieves control and synchronization of system fractal sets;it incorporates fractional order system theory to construct the fractional order infectious disease model,it innovatively designs fractional order control strategies and achieves control and synchronization of fractional order fractal sets;it applies fractal sets to analyze the common properties of complex dynamical planar and spatial systems.The main research contents of this paper include the following four aspects:1.Fractal control and synchronization of discrete system Julia sets based on the infectious disease SIRS modelThe SIRS model is one of the fundamental dynamic compartmental models of infectious diseases,primarily describing the temporary immunity after recovery from an infection.This part employs Julia sets in the study of the infectious diseases SIRS model and analyzes the fractal dynamics of the discrete SIRS model using Julia sets,it introduces an infectious disease model with seasonal factors,designs controllers to modify the Julia sets,selects appropriate control parameters,and visualizes the system control effects using images,demonstrates the model complexity by calculating the box dimension of the system Julia sets,and introduces nonlinear coupling methods to achieve synchronization between Julia sets of systems with the same structure but different parameters.2.Control of discrete fractional order system Julia sets based on the Caputo fractional order SIRS modelBy utilizing the definition of Caputo fractional calculus and fractal theory,this section investigates the fractional order infectious disease SIRS model on the time scale,it analyzes the fractal dynamical behavior of this model by using Julia set of discrete fractional order SIRS model.It designs three different controllers,with each controller acting as a multiplicative factor,a whole,or a part added at different locations within the system to modify the Julia sets.The graphics illustrate the variation of Julia sets of controlled system with changes in the fractional orders and control parameters,effectively showcasing the complexity of the model and intuitively comparing the effectiveness of the control.3.Fractal control and synchronization of fractional order system Julia sets based on the discrete fractional order SIRS modelThis section employs discrete fractional order SIRS model with the definition of Caputo fractional calculus to study infectious diseases,it discusses the fixed point of the fractional order system and designs controllers containing fixed point,where controllers are incorporated into the system as a whole and as a part.It introduces two completely different coupling synchronization strategies to achieve synchronization between Julia sets of discrete fractional order infectious disease systems with different parameters but the same structure,it also innovatively provides rigorous mathematical proofs regarding the synchronization effectiveness of Julia sets.It characterizes the degree of synchronization between Julia sets of systems by combining Hausdorff distance with visual images,and the graphics demonstrate the irregularity and controllability of Julia sets.4.Fractal control of Julia sets based on complex dynamical planar and spatial systemsThis part applies Julia sets to investigate a class of complex dynamical planar and spatial systems with varying coefficients,proposes control methods for the system stability domain,explores the formatting of Julia sets of complex planar system where two circles intersect and three circles intersect,provides proofs and examples for each theorem,visually represents the control effects of stable regions through images,demonstrates novel Julia sets of complex planar systems by exploiting structural symmetry and shows that the position,size,area,and shape of Julia sets can be controlled by selecting different coefficients.Furthermore,it extends the analysis from planar system to spatial system,and explores the commonalities of Julia sets of spatial system,offering potential reference for controlling the stability domain of specific complex dynamical systems in practical applications.In conclusion,this paper innovatively integrates fractional order theory into the investigation of infectious disease phenomenon,and achieves control and synchronization of Julia sets of fractional order infectious disease systems;it innovatively designs control strategies for fractional order models,discusses the influence of system parameters and fractional orders on system control;it innovatively proves the synchronizations between two fractional order systems,enriching the theoretical research of fractional order systems,and this research holds significant reference value for the fractal analysis and control in other fractional order systems;it innovatively investigates the formatting of Julia sets of complex dynamical planar and spatial systems,and analyzes the common characteristics of Julia sets,providing theoretical support for the application of fractal theory.