Chaos Control Of Rucklidge System & Generalized Synchronization Of A Class Of Systems

Chaos in nature has been extensively demonstrated and is very common. In resentyears, after the pioneer work of Ott, Gtebogi and Yorke (OGY), chaos control has becomemore active in academic research and practical applications. In addition, an important re-search activity has been devoted to the dynamics of coupled and driven chaotic oscillators,and this is still an active field of research. And generalized synchronization is an importantbranch of this field.The Rucklidge system studied in this paper is a ?uid mechanical model. So its equi-librium points and periodic orbits are of great significance. But, since the system is chaotic,the equilibrium points and the periodic orbits are all unstable. In order to guide the chaotictrajectories of the Rucklidge system to unstable equilibrium points or bring forth periodicorbits, a simple but effective scheme is designed by means of single feedback method in thispaper.The other purpose of this paper is to introduce a special kind of chaotic driven system,uniformly submissive system, whose trajectories, for any chaotic drives, are synchronizedto the trajectories of the drives in a generalized sense. The Routh-Hurwitz criteria, theory ofHopf bifurcation and some usual analysis methods are used to complete the theoretic proof,and the auxiliary system approach is used to observe the generalized synchronization, andthe conclusions is verified by means of numerical simulations.This paper is organized as follows.In chapter 1, we introduce the chaos background, summarize the current research ofchaos control and generalized synchronization, and present the research frame of this paper.In chapter 2, we talk about the chaos control of Rucklidge system. This chapter iscomposed of two section. In the first section, we summarize the question and make ananalysis of the properties of the Rucklidge system; In the second section, we present a simple but effective control scheme and analyze its conditions in suppressing chaos to equilibriumpoints and bringing forth periodic orbits, and verify the conclusion by means of numericalsimulation.In chapter 3,we discuss the generalized synchronization and put forward the uniformlysubmissive system. This chapter is composed of four section. In section 1, we summarizethe question and present the definitions and the sufficient and necessary conditions of thegeneralized synchronization, and according to this, we get the definition of the uniformlysubmissive system; In section 2, we put forward some methods on detecting the uniformlysubmissive system and give their proof; In section 3, we introduce some applications ofsuch methods; In section 4, we verify the conclusions by means of numerical simulationaccording to the auxiliary system approach.