Research On Some Problems In Chaos And Asynchronous Boolean Networks

In recent years, complexity science has been one of the front research fileds. Chaotic systems and gene regulatory newtorks as typical complex systems are the important and hot research fields of complexity science. In this dissertation, by using theoretical analysis and numerical simuliations, some problems in chaotic systems, such as the dynamics, synchronization and applications in secure commnucation have been discussed. Furthermore, based on linear representations of logical systems, the dynamics of asynchronous Boolean networks and controllability of asynchronous Boolean control networks have been studied. The main work of this paper is as follows:(1) The fractional-order complex chaotic systems are first proposed, which extend the chaotic systems from real space into complex space. By utilizing phase portraits, the largest Lyapunov exponents and the bifurcation diagrams, dynamical behaviors of fractional-order complex Lorenz system and Chen system are studied, respectively. Chaotic regions and periodic windows are explored as well as different types of motions such as tangent bifurcations, flip bifurcations and chaotic crisis shown along the routes to chaos. By means of the fractional-order complex chaotic systems, two kinds of digital secure communication schemes have been achevied based on synchronous and asynchronous methods. In the synchronous scheme, an improved switch-modulated digital secure communication system is designed, which is based on a hybrid synchronization in coupled fractional-order complex Chen systems. And in the asynchronous scheme, an error-correcting secure communication is discussed, which has the ability of error checking and correcting in real time and high probability against the errors existing in transmitted signals. Numerical simuliations show the feasibility of the proposed schemes.(2) Modified function projective synchronization (MFPS) is a kind of generalized synchronization scheme, and some other chaotic synchronizations, such as complete synchronization, anti-synchronization, projective synchronization and function projective synchronization, can be seen as the special cases of MFPS. Here, MFPS in frational-order chaotic systems, complex chaotic systems and different dimensions chaotic systems have been discussed, respectively. Based on Lyapunov stability theory, the adaptive controllers have been achieved. Theoretical proof and numerical simulations demonstrate the effectiveness of the proposed schemes. A hybrid modulus-phase synchronization in coupled complex systems is first proposed, which means different synchronization schemes can be achieved simultaneously in modulus and phase spaces of complex systems. A general controller is designed to implement modified projective synchronization in modulus space and anti-synchronization in phase space. And the theoretical proof is given based on the Lyapunov stability theory. This proposed scheme is suitable for some real cases that only modulus and phase of complex state variables are measurable.(3) By using the semi-tensor product, the linear representations of generalized asynchronous Boolean networks (GABNs) and generalized asynchronous mutilple-valued networks (GAMVNs) are achevied. The dynamical behaviors of GABNs and GAMVNs are investigated analytically. Some results about the fixed points and loose attractors are provided and algorithms to find all of attractors and basins of attrations are given. Based on the algebraic representation, controllability of asynchronous Boolean control networks (ABCNs) is discussed. Firstly, deterministic controllability in ABCNs is discussed. Secondly, via three different kinds of controls, controllability of ABCNs is analytically discussed by revealing the reachable sets, respectively. Examples are shown to demonstrate the feasibility of the related discussions.