| Recently, with the deep study of system science, researchers find that systems contain a variety of complexities in practice. In the further improvement of scientific research, the complexity science has been a hot topic. In many practical complex systems, complexities such as parameter uncertainties, external disturbance and time delays commonly exist and make the systems more complex. From regular coupling to irregular coupling, this paper studies properties and synchronizations of some representative complex systems based on the Lyapunov stability theory, the matrix theory and the fractional differential calculus. The main contents and innovations can be summarized as following:(1) The properties and synchronizations of chaotic system are studied. Firstly, a class of chaotic maps with parameter q is proposed based on the fractional differential calculus and the predictor-corrector method. The system chaotic parameter space is enlarged. With the largest Lyapunov exponent diagrams, bifurcation diagrams and system phase diagrams, the dynamical properties of chaotic maps are studied. Secondly, the lag synchronization of complex permanent magnet synchronous motor system is studied and synchronization controllers are designed with the Backstepping method. At last, the complex hyperchaotic Lorenz system is established and the module-phase synchronization after complex projection is derived.(2) The spatiotemporal chaotic system with regular couplings is studied. At first, multiple chaotic maps and chaotic systems are coupled connected by using the coupled lattice model and spatiotemporal chaotic maps with parameter q are derived. With the largest Lyapunov exponent diagrams, bifurcation diagrams and spatiotemporal evolution diagrams, the interaction of coupling strength and system parameters is considered and chaotic parameter range is derived. Next, mismatched dimensions, complex number variables, fractional order and parameter uncertainties are taken into consideration based on the coupled spatiotemporal chaotic system model. In addition, the spatiotemporal chaotic system synchronization with above complexities is established. At last, a robust secondary chaotic secure communication is proposed based on the spatiotemporal chaotic system synchronization and hyperchaotic system synchronization. The complexity and security are improved.(3) The synchronizations of neural network with irregular network-based couplings are studied. Firstly, with fractional differential calculus, the fractional neural network model is derived and its synchronization is studied with linear matrix inequality. Secondly, the neural network synchronization and module-phase synchronization are considered in the complex number field with the Lyapunov stability theory. In addition, complexities such as parameter uncertainties and time delay are considered. Based on this, the discrete neural network with time-varying delays and its synchronization are considered and synchronization theorems are derived.(4) The synchronizations of Boolean network with irregular network-based couplings and logical variables are studied. At first, with the semi-tensor product and its linear representation, the network inner synchronization concept is introduced into the Boolean network and the inner synchronization and outer synchronization conditions are derived. Based on this, Boolean networks under different updating schemes are investigated. Next, the asynchronous switch Boolean network model with several updating models and asynchronous updating scheme is established.In addition, free Boolean sequence controllers and feedback controllers are designed and synchronization criterions are derived. At last, the cluster synchronization of Boolean network is proposed. Both given asynchronous Boolean network and state-related Boolean network are considered. The controllers for Boolean network and state-related Boolean network are given to make the synchronization established. |
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