Analysis, Control, Synchronization And Circuit Implementation Of Chaotic And Hyperchaotic Systems

Chaos is a special phenomenon in nonlinear systems and it has rich and complexnonlinear dynamical characteristics. Modeling chaotic systems with novel features, aswell as chaos control and chaos synchronization, has become one of the majorsubjects in mathematics and engineering for a long time. In this dissertation, novelchaotic and hyperchaotic systems will be formulated, and some novel methods ofchaos control and chaos synchronization will be proposed. Furthermore, thecorresponding analog circuit implementations will also be presented.A new three-dimensional autonomous chaotic system is constructed by adding aproduct term with variable coefficient to the first equation of Chen’s system. Based onthis new chaotic system, a novel five-dimensional autonomous hyperchaotic system isformulated via feedback.In the study of chaos control and hyperchaos control, the controller is firstlydesigned via linearization of the controlled system. Then, the specific controlparameters will be selected to ensure the uniqueness of the equilibrium point and theglobal asymptotic stability of the controlled system. This method can make thecontroller less conservative. For the stability control of the high dimensional system, avalid method is proposed based on the passivity of the feedback system. The highdimensional controlled system can be divided into some low order sub-systemsrepresented as a feedback connection, such that the controller can be designed fromthe low order sub-systems to realize the control of the high dimensional system.A novel method, to synchronize Chua’s circuits whose error system can berepresented as a feedback connection of a linear dynamical system and a nonlinearelement, is proposed based on absolute stability. The nonlinear element contains thecontrol parameters and globally satisfies the sector condition. With the aid of theNyquist plot of the transfer function, a one-dimensional univariate unidirectionalcoupling linear synchronization controller is designed from the circle criterion. Forthe synchronization of corresponding novel chaotic and hyperchaotic systems,according to the passivity of the nonlinear system, the control parameters can beobtained just from making the coefficient matrix of the linear part in the error systemHurwitz, because the chaotic and hyperchaotic attractors are bounded and ergodic. With this method, the control parameters are independent on the chaos ranges andcontain no state variables of the drive and the response systems. This method isproposed based on the theoretical assumption and verified by the numericalsimulation.All the chaotic and hyperchaotic systems in this dissertation are implemented byanalog circuit, as well as the designed controllers and the synchronization systems.The validity of the theory can be proven by the running results of the circuits.Moreover, all the circuits are constructed on the basis of a good qualitative agreementbetween the numerical and the circuit simulations. Thus, the parameters of the circuitcomponents can easily be determined, and the consistency between the actual and thesimulation results can be improved. The circuits can run in a safe situation.