Fractal Analysis And Control Of Fractional Nonlinear Dynamical Systems

Deepening of people's understanding of nature with the rapid development of science and technology,fractals and fractional systems have become the curren-t theoretical hotspots and technological frontiers.They are powerful tools for modeling,describing,analyzing,and controlling various nonlinear processes and abnormal phenomena in varied fields,especially in interdisciplinary fields,for that reason they have attracted the continuous attention of numerous scholars at home and abroad.On the one hand,fractal sets that are represented by Ju-lia sets intuitively characterize the asymptotic properties of system states.The complexity of a nonlinear dynamical system can be better to understand and grasp thorough analysis and estimation of this kind of set;some properties of the system can be realized by controlling its Julia set.Besides,Julia sets and diffusion-limited aggregation models are crucial fractals with intricate internal structures and unusual interesting properties.On the other hand,fractional sys-tems are commonly used to describe phenomena and behaviors with memory,heredity,or non-locality.Such phenomena or behaviors are inherently nonlinear and highly complex,which generally cannot be explained succinctly and clearly through conventional integer-order models.Withal,more and more studies have confirmed that most systems in nature are essentially fractional,and integer-order models obtained through classical methods can only reflect a few partial prop-erties or get some rough results.Therefore,with combining fractal theory and fractional system theory to study fractional systems from a fractal perspective and to introduce fractional elements into traditional fractals,it is able to develop some new analysis tools and control methods for the nonlinear system theory and also to provide a new way for the modeling and application of nonlinear dynamic problems,which has both important theoretical significance and urgent practical value.Based on theory and applications,this work constructs several kinds of fractals by integrating fractal theory with fractional system theory,discusses the fractal dynamic properties of fractional systems from both qualitative and quantitative levels,realizes the control and synchronization of fractional fractal sets,and pro-vides a new perspective and feasible method for the in-depth understanding of fractional dynamics as well as describing certain nonlinear phenomena in nature.The research content mainly includes the following four aspects:1.Fractal dynamic analysis and control of Julia sets from a contin-uous fractional system based on the fractional Lotka-Volterra model.This part generalizes an existing fractional Lotka-Volterra model,contrives a coupled Jacobian matrix to analyze the stability of the resulting system equilib-rium point,defines the model's Julia set and discuss its fractal characteristics,controls and compares the Julia set through three distinct control strategies,and designs coupling items to achieve synchronization of two Julia sets with different system parameters.Furthermore,the fractional Lotka-Volterra model is extend-ed to the complex domain and dynamic noise disturbance is introduced.Then the structure and properties of its spatial Julia set are studied,and the Julia deviation index is defined to quantitatively analyze the impact of several kinds of dynamic noise on the set.The symmetry of the Julia set and the destructive effect from noise is also discussed.2.Dynamic analysis and synchronization of fractal sets from a dis-crete fractional system based on the fractional difference logistic map.This part considers a logistic map derived from a difference equation based on the discrete fractional calculus framework.Through Julia sets and Poincare plots,fractal and chaotic characteristics of the map are discussed comparing with the defined fractional difference quadratic map,which clarifies the memory effect of the fractional difference map reflected by these dynamic phenomena.A cou-pling controller is designed to realize the synchronization between the fractional difference logistic map and the fractional difference quadratic map.Moreover,a criterion of fractionalization is proposed for fractal sets of traditional maps.Then several specific fractional generalizations of the Julia set and the Mandelbrot set generated by the classic quadratic map are presented,and the differences among these generalizations are compared and discussed.Through visualization tech-nology and dimensionality analysis,the effect of a fractional map's order on its fractal sets is examined.3.Fractal dynamic analysis and synchronization of Julia sets from a fractional function iteration based on the Mittag-Leffler function.This part investigates Julia sets generated from a class of uncertain discrete complex dynamic systems composed of fractional functions involving Mittag-Leffler functions.Julia sets of some classical non-polynomial function iterations are generalized;the influence of the function parameter on the characteristics of these fractal sets is discussed.An adaptive control strategy that can be directly applicable to complex dynamic systems is proposed to synchronize Julia sets of two systems with different parameters and simultaneously identify their unknown parameters.4.Fractal dynamic analysis of a fractional partial differential system based on the fractional diffusion-limited aggregation model.This part,based on a fractional diffusion mechanism,introduces a modified diffusion-limited aggregation that is derived from the classic DLA model as a nov-el approach to modeling fractal growth.The specific memory performance which is inherent in fractional operators can be reflected macroscopically as the direc-tivity of aggregated patterns,eventually.The influence of the fractional model's order on the aggregating behavior and cluster structure is further quantitative-ly described by defining the anisotropy index and combining it with the fractal dimension.To sum up,this paper innovatively studies several kinds of fractals based on typical fractional systems,analyzes the properties and characteristics of fractional fractals,discusses the effect of fractional systems' order on their fractals,realizes the control and synchronization of fractional Julia set as well as the identification of unknown system parameters,and improves the corresponding visualization al-gorithms.It expands the knowledge framework of fractal research,enriches the research methods of fractional systems,provides certain technical support for the application of fractal theory and fractional system theory,and also has refer-ence significance for fractal analysis and fractal control of more general fractional systems.