Study On The Dynamical Behavior Of Fractional Order Chaotic System

Chaos is a focus in the study of nonlinear science. In recent years, chaos has beenstudied deeply, and chaos is widely applicated in many fields, like securecommunications, signal processing, chemistry, biology, sociology. Many systems areknown to display fractional-order dynamics. It is found that the dynamics offractional-order chaotic system is much more complex than integral-order chaoticsystem. Fractional-order chaotic system is more in line with the engineeringapplications in the real world.This PhD thesis studies the dynamic behaviors of fractional-order chaotic systemwith new method; we are the first to achieve the wavelet phase synchronization infractional-order chaotic system; we achieve projective synchronization in a fractionalorder multi-scroll chaotic system and a fractional-order multi-wing hyper-chaoticsystem; we investigate the dynamical behaviors of variable fractional-order chaoticsystem.The main contents and results are organized as follows:1. Multi-scroll chaotic system has complicated topological structures and a richvariety of dynamical behaviors. We propose a novel fractional order multi-scroll chaoticsystem and investigate the dynamical behaviors of this system. Complicated dynamicalbehavior similar to integer-order system can be generated with specific parameters. Thesystem parameter can be an effective controller, with the changing of system parameters,the system can generate periodic orbit, fixed point, chaotic attractors with differentamount of scrolls.It is also found that the fractional-order can be an effective controller, whisch icorrelated with the amount of scrolls. Chaotic behaviors can be found in this systemwhen the fraction-order is lower than3. Until no chaotic behaviors can be found in thisfractional order multi-scroll chaotic system, the number of scrolls is decreasing with thereducing of system order. The lowest fractional-order for chaos to exist in this system is2.1.2. Multi-wing chaotic system also has complicated topological structures; hyper-chaotic system has more complicated dynamic behaviors than chaotic system. Wecombine multi-wing chaotic system and hyper-chaotic system together, a rare4dimensional fractional-order multi-wing hyper-chaotic system is investigated.Hyper-chaotic behaviors can be found in this system when the order is lower than4，andfour-wing hyper-chaotic attractors similar to integer order system can be generated.The fractional-order can also be an effective controller. With the reducing ofsystem order, the number of wings is decreasing. The lowest order for hyper-chaos toexist in this system is3.6and the lowest order for chaos to exist in this system is2.4.3. Synchronization is a focus of interest in the study of chaotic system. We are thefirst to investigate the wavelet phase synchronization of fractional-order chaotic system.This novel method is based on wavelet transform and phase synchronization. Wepropose a fractional-order dynamical system which is composed of two coupled R sslerchaotic attractors and analyze in detail the synchronization behaviors with the changingof coupling strength, the central frequency and the time scale of the wavelet. It isconcluded that the wavelet phase synchronization can be achieved only when thecoupling strength, the central frequency exceeds a certain threshold, and the time scaleis in a specific area.Projective synchronization is also an efficient method in the study ofsynchronization. In this thesis, we briefly introduce the concept of projectivesynchronization and design effective synchronization controllers for a fractional ordermulti-scroll chaotic system and a fractional-order multi-wing hyper-chaotic systemseparately. Through making the output synchronization errors system satisfy thestability conditions of fractional-order chaotic system, we verify that projectivesynchronization can be achieved in this fractional order multi-scroll chaotic system andthis fractional-order multi-wing hyper-chaotic system.4. In this thesis, based on the research of complex network, we investigate thephase synchronization in small world network of fractional-order chaotic oscillator forthe first time. We propose a small world network model composed of fractional-orderR ssler chaotic oscillators and investigate the coupling strength and the topologicalstructure’s impact on the system’s phase synchronization behaviors.5. Most of the fractional-order dynamical system before only consider thefractional-order as a constant, but in the real world, many physical phenomenon, engineering applications appear to exhibit fractional-order behavior that may vary withtime or space. This PhD thesis proposes the concept of variable fractional-order chaoticsystem for the first time and investigates the dynamic behaviors of memoryless variablefractional-order chaotic system. The signal obeys different distribution is adopted asvariable order. In the simulation of variable fractional-order chaotic system, the Croneapproximation method based on fuzzy theory is adopted.We analyze the dynamical behaviors of variable fractional-order R ssler chaoticsystem and variable fractional-order multi-scroll chaotic system. It is presented that thesystem parameters can be effective controllers and the distribution function of variablefractional-order has important impact on the dynamical behaviors of variablefractional-order chaotic system.