Some Problems In Chaotic Systems And Their Applications

This thesis is divided into six chapters. In Chapter 1, the review of the chaosresearch background and developments involving the definitions of chaos, mea-surements of chaos, concept of transversal homoclinic points, concept of strangeattractors, and control and anti-control of chaos is presented. Besides, the math-ematical, particularly, the dynamical methodology in the developments of biolog-ical neurology is discussed, and several mathematical biology models are intro-duced. We show that it is necessary to analysis not only the stable phenomenonbut also the compIex dynamics in concrete models.Chapter 2 involves several theoretical problems in chaos research. We firstc1arify the problems in the proof of Marotto theorem by serious deductions. FOrthe sake of the convenience in realistic application, we deduced some criteria forchaos existence with uniform norm such as Euclidean norm and 1-norm. Further-more, by constructing the relations between the fiflite subshift on the symbolicset and the chaotic iterated map by means of Marotto theorem, we investigatethe structure of the invariant as well as the estimation of topological entropy,Zeta--function, Lyapunov exponent. By the end of the chapter, we present severalconclusions on the perturbed chaotic systems and a theorem about "heteroclinicalrepellers imply chaos".In Chapter 3, we are mathematically modelling two particular classes of impul-sive differential equations having chaos dynamics. The perturbed theory givenin Chapter 2 and a new definition on sensitive dependence on initial values ofimpulse-interval functions are adopted. From the abstract to real, we discussa real model, named integrate-and-fire circuits, simulating the dynamics of in-fOrmation processing in neurons. By the means of Marotto theorem, we proveIde g viithe existence of chaotic dynamics in this model with the parameters in somerestricted regions. The conclusions on the invariant set deduced in last chapterare applied to this model. Although some results are deduced with the aid ofcomputer computing, we stil1 regard it is rigorous because of the nonempty of theregions we obtained. The following numerical simulations further reinforce ourresults.We, in Chapter 4, comprehensively discuss the dynamics of discrete chaoticneural networks, including f the existence of fixed points, the stable, unstabledynamics of the fixed point, the saddle-node and period doubling bifurcations insingIe neuron model, and chaotic dynamics of the networks. The proofS and de-ductions involve Schauder fixed point principle, constructions of Lyapunov func-tions, bifurcation theory, contraction map principle, and anti-integrable limitmethod. It should be mentioned that all the results obtained in this chapter isrelevant to the hypnosis that the weight matrix is symmetric.Chapter 5 is made up of two sections. In the first section, we investigatethe stable dynamics of a class of functional differential equations with the aidof Lasalle invariant principle. Besides, we show the method and result in theresearch of the heart rate variability. It is proved that the nonlinear analysis ismuch more effective in the clinical diagnoses of heart disease.By the end of this dissertation, we list some problems fOr our future worksincluding chaos in discrete time~varying systems, SRB measures of the chaoticmap in the sense of Marotto, complex dynamics of both H.H. model and cou-pled integrate-and-fire models, strange attractors in H6non systems with classicalparameters.