Several Types Of Synchronization Problems For Hyperchaotic Systems With Complex Variables

Nonlinear science is a very active subject nowadays,which is widely used and has been paid more and more attention in various fields.Chaos is a special advanced and complex type of nonlinear science.Since the discovery of chaos,more and more scholars have studied chaos phenomenon and studied it more and more deeply.Nowadays,many types of chaos synchronization are emerging,and many types of synchronization control emerge one after another.However,most of these are researches on hyperchaotic systems with real variables,while researches on hyperchaotic systems with complex variables are rare.At the same time,the actual situation should be considered.In the study of synchronization of chaotic systems,there are often uncertainty of the system and external disturbance.The control method based on UDE is a very effective method to deal with chaotic systems with model uncertainty and external disturbance,but this method can only solve the case of bounded external disturbance,and it is still helpless for unbounded case.Therefore,based on comprehensive consideration of the above aspects,this thesis studies several types of synchronization problems of hyperchaotic systems with complex variables subjected to external disturbance,aiming at different types of external disturbance,and obtains some important theoretical results.The full thesis will be expanded from the following seven aspects:(1)Firstly,several types of synchronization problems of chaotic systems are considered,and they are classified according to the synchronization problems of the whole system or part of the subsystems.In addition,the relationship between projective synchronization problems and partial anti-synchronization problems of chaotic systems is also given as an innovation of this thesis.(2)Inspired by the idea of control method based on UDE,it is an original feature of this thesis to select the appropriate function and design the filter of corresponding disturbance estimation by considering different types of external disturbance through selecting the appropriate function.(3)The complete synchronization of a complex variable hyperchaotic system subjected to external disturbance is studied.Firstly,the hyperchaotic system of complex variables is transformed into the equivalent hyperchaotic system of real variables by introducing the semi-tensor product tool.Then,aiming at the synchronization problem of hyperchaotic system with complex variables,the controller is designed step by step.In the first step,the dynamic feedback gain method is used to achieve complete synchronization of the nominal(undisturbed)hyperchaotic system with complex variables.The second step,considering the type of external disturbance,design a reasonable filter.The third step is to design a controller based on disturbance estimation by combining the controller and filter.Finally,a practical example is given to verify the validity and correctness of the proposed method.The method of transforming hyperchaotic system of complex variables into equivalent hyper-system of real variables is another innovation in this thesis.(4)The anti-synchronization problem of a complex variable hyperchaotic system subjected to external disturbance is studied.First,the semi-tensor product complex variable hyperchaotic system is transformed into real variable hyperchaotic system.Then,the controller is designed step by step for the anti-synchronization of the hyperchaotic system with complex variables.Firstly,the dynamic feedback gain method is used to realize the anti-synchronization of the nominal(undisturbed)hyperchaotic system with complex variables.Considering the type of external disturbance,a reasonable filter is designed.Combined with the controller and filter,a controller based on disturbance estimation is designed.Finally,a practical example of Chua system is given to verify the validity and correctness of the proposed method.(5)Simultaneous synchronization and anti-synchronization of a complex variable hyperchaotic system subjected to external disturbance are studied.Firstly,hyperchaotic systems of complex variables are transformed into equivalent hyperchaotic systems of real variables by means of semi-tensor product.Then,a controller is designed step by step for simultaneous synchronization and anti-synchronization of hyper-chaotic system with complex variables.Firstly,a dynamic feedback gain method is proposed to realize simultaneous synchronization and anti-synchronization of the nominal(undisturbed)hyperchaotic system with complex variables.Then a new type of external disturbance is given and a reasonable filter is designed.Combined with the controller and filter,a controller based on disturbance estimation is designed.Finally,a practical example is given to verify the validity and correctness of the proposed method.(6)Partial anti-synchronization of a hyperchaotic system with complex variables subjected to external disturbance is studied.Firstly,a sufficient and necessary condition for the existence of partial anti-synchronization for hyperchaotic systems with complex variables is given.Secondly,hyperchaotic system of complex variables is transformed into hyperchaotic system of real variables by means of semi-tensor product.Then,aiming at the problem of partial desynchronization of complex variable hyperchaotic system,the controller is designed step by step.In the first step,the dynamic feedback gain method is used to realize partial desynchronization of complex variable hyperchaotic nominal(undisturbed)system.The second step,considering the type of external disturbance,design a reasonable filter.The third step is to design a controller based on disturbance estimation by combining the controller and filter.Finally,a practical example is given to verify the validity and correctness of the proposed method.(7)The projective synchronization of a complex variable hyperchaotic system subjected to external disturbance is studied.Firstly,the existence of projective synchronization problem for hyperchaotic systems with complex variables is considered and a sufficient and necessary condition is given.Secondly,the hyperchaotic system of complex variables is transformed into a hyperchaotic system of real variables by using the semi-tensor product.Then,aiming at the problem of projective synchronization of hyperchaotic system with complex variables,the controller is designed step by step.Firstly,the dynamic feedback gain method is used to realize projective synchronization of the nominal(undisturbed)hyperchaotic system with complex variables.Considering the type of external disturbance,a reasonable filter is designed.Combined with the controller and filter,a controller based on disturbance estimation is designed.Finally,a practical example is given to verify the validity and correctness of the proposed method.