| Fractional-order complex chaotic systems are typical complex dynamical systems and a class of complex nonlinear systems.Fractional-order chaotic systems have the characteristics of initial value sensitivity and pseudo-randomness of chaotic systems,but also have the unique properties of fractional-order systems such as complexity,historical memory,and dynamical properties closely related to the order of the system.The study of fractional-order chaotic systems by complex domain analysis,whose state variables are in the complex domain,belongs to the intersection of complex analysis techniques and dynamical systems;the system structure is more complex and the dynamical behavior is richer.It has a broader application prospect in the fields of security communication,signal monitoring and biomedicine.Based on the characteristics of chaotic systems,this thesis develops a series of fundamental studies on fractional-order complex chaotic systems and their synchronization control by stability theorems of fractional-order systems and related properties in combination with the complex domain analysis technique.Unlike previous studies,all the analytical procedures in the thesis are performed in the complex domain,and the synchronization conditions are derived using complex functional analysis,which eliminates the need to divide the real and imaginary parts of the complex chaotic system into two real-valued systems and greatly reduces the complexity of analysis and computation.The above-mentioned research ideas are used in this paper when studying the modified projection synchronization of fractional-order complex chaotic systems with unknown parameters,the adaptive robust synchronization of fractional-order complex chaotic systems with unknown parameters,and the synchronization of fractional-order complex chaotic neural networks.The main research contents are summarized in the following sections.(1)Two numerical simulation methods for fractional-order complex chaotic systems are constructed by combining Caputo and Grünwald-Letnikov derivative definitions,and the feasibility and effectiveness of the proposed numerical simulation methods are verified by implementing system simulations in MATLAB.(2)The stability determination methods of complex fractional-order linear systems and complex fractional-order nonlinear systems are proposed by combining the complex function analysis method and fractional-order stability theory;and the feasibility and effectiveness of the proposed methods are verified by numerical simulations of linear control of fractional-order complex chaotic Chen systems and fractional-order complex chaotic neural networks.(3)The dynamical behaviors of fractional-order complex Lü chaotic systems and fractional-order complex Lorenz systems are investigated,and three control models based on drive-response,active-passive,and linear feedback methods for fractional-order complex chaotic systems with known complex parameters are proposed.The proposed synchronization methods all require only one drive signal to achieve fractional-order complex chaotic synchronization,which greatly reduces the constraints of chaotic synchronization and is especially convenient for the engineering implementation of secure communication.(4)The problem of corrected projection synchronization of fractional-order complex chaotic systems is studied.Based on fractional-order inequalities and complex domain analysis techniques,a synchronization control method is given to gradually synchronize the constructed response system to the driving system.In numerical simulations,the complex corrected projection synchronization of fractional-order Chen complex chaotic systems and the single-state variable single controller hybrid projection synchronization are realized respectively,thus verifying the effectiveness of proposed methods.(5)A new robust adaptive control scheme is proposed based on the stability theory of fractional-order complex chaotic systems by trying to analyze the behavior and characteristics of fractional-order complex chaotic systems in the complex domain,considering the effects of unknown external perturbations and unknown parameters.The paper investigates the synchronous control models of fractional-order complex chaotic systems with known parameters,unknown parameters,unknown parameters and the presence of unknown external perturbations.When the parameters are known,a single variable controller is designed to achieve a simple and effective scheme that is easy to implement;when the parameters are unknown,a new robust adaptive control scheme is designed to achieve synchronization of fractional-order complex chaotic systems.The proposed method can estimate all the unknown parameters and eliminate the influence of external uncertain perturbations on the system.(6)An adaptive synchronization scheme for fractional-order complex chaotic neural networks considering time delays is proposed using fractional-order complex variable differential inequalities,stability theory of fractional-order complex systems,and adaptive control methods.According to the given synchronization scheme,the synchronization problems of neural networks with known parameters and unknown parameters are studied in numerical simulations,and the effectiveness of the above scheme is verified.In conclusion,this paper focuses on the stability theory and synchronization control of fractional-order complex chaotic systems,and realizes the adaptive synchronization and corrective projection synchronization of chaotic systems and chaotic neural networks with unknown and perturbed parameters in the complex domain,which promotes the development of the theory of fractional-order complex chaotic systems and provides the theoretical foundation and basis for their applications in the fields of secure communication and signal monitoring. |
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