Research On Control, Stochastic Control, Synchronization And Noise-Induced Synchronization Of Nonautonomous Chaotic Systems

Chaos is often analogized as the third great scientific revolution of the 20th century,along with theory of relativity and quantum mechanics. In fact, chaos, as a newconception, is universal in the world, and has a very deep impact on the research anddevelopment of mathematics, classic mechanics, physics, biology, medicine,humanities and social science, and so on. Chaos control and synchronization is thekey tache in chaos research and has found its applications in many fields, such asvibration engineering, electric engineering, chemical engineering, systemsengineering, biological engineering and communication engineering.Since the nonautonomous system is a kind of typical system, which often perturbedby noise in reality, this dissertation is devoted to study of chaos in nonautonomouschaotic systems with harmonic excitations and its deterministic control, randomcontrol, synchronization and "noise induced synchronization".Firstly, in deterministic control of chaos two different kinds of resonant harmonicexcitations, additive and parametric ones are used. With the Melnikov method, wehave obtained the regions of excitation amplitude, where heteroclinic chaos may begenerated or suppressed. Meanwhile, for suppressing heteroclinic chaos, we havedetermined the relationship between parameters of the system excitation and thecontrol excitation must be satisfied. The analytical results show that phase differencebetween the two excitations has important effect. Moreover, numerical methods showthat the phase control method is feasible not. only to controlling heteroclinic chaos,but also to other type of chaos in nonautonomous systems. Comparing the effect of anadditive harmonious excitation with that of a parametric one, we find the former oneis more effective at the small resonant frequencies, while the latter one is moreeffective at the large ones. Secondly,. the corresponding random control is studied with the StochasticMelnikov Process method and numerical methods respectively. Based on theStochastic Melnikov Process method and a mean-square criterion, the critical valuesfor heteroclinic chaos in the Josephson junction soMy driven by bounded noise canbe derived. Meanwhile, even keeping the noise strength invariant, we can increase thenoise bandwidth to suppress chaos. The dissipation control can also suppress chaos.Numerical methods obtain similar results for stable chaos and both two methods leadto fully consistent results. Through further considering the Josephson junction drivenby bounded noise and harmonic excitation, two utterly different kinds of controleffect, i.e. generating chaos or suppressing chaos through adjusting stochastic phase,are discovered and illustrated by numerical examples.Thirdly, synchronization of nonautonomous chaotic systems is considered. The twocontrollers, the Active-Sliding Mode controller and the Adaptive Feedback Controller,are suggested to synchronize two identical nonautonomous chaotic systems withunknown parameters. Furthermore, we have explored how to estimate the unknownparameters in these two kinds of controllers. The former one has the merits of theActive Controller and the Sliding Mode Controller, and its robustness to noise andparameter uncertainty is shown in two illustrative examples, a Duffing system subjectto a harmonic excitation and a 4-D chaotic system. The latter controller is a universalone able to synchronize almost all kinds of chaos in nonautonomous systems.Especially, the latter can lessen the number of control variables for many systems oreven utilize a scalar controller, such as for the Duffing system.Finally, "noise-induced synchronization" and "chaos-induced synchronization" inthe above stated systems are discussed. The "noise-induced synchronization" impliesa kind of generalized synchronization between the noise and the chaotic systemdriven by noise, as the two stochastic systems without direct feedbacks can besynchronized under the induction of the common noise. In the dissertation a directstochastic feedback method, with some noise modulated driving signals straightlyfeedbacking to the responding system, is proposed to synchronize the driving andresponding systems. In fact, the proposed method has less limitation than the linear feedback synchronization method, hence the feedback strength need not be constantor self-adaptive, and the method has great flexibility for just requiring its feedbackstrength varying randomly in a certain range. Numerical results have verified itsfeasibility and effectiveness.