| Because of the further development of physics, biology, medicine, population dynamics and economics, natural sciences and social sciences etal, the researchers have put forward a number of concrete mathematical models describing differ- ential equation, for example, the famous insect Model- Logistic map and the two-dimensional Henon map that can product chaos. For many problems described by continuous dynamical systems, people often simplify them into a simple discrete system so as to easier and more convenient investigate the complicate behavior of the system, so studying discrete chaotic map, whether in theory or in practical applications, has important significance.With the continuous deepening studies for chaos, the researchers found that state variables in a considerable portion of dynamical systems lag time, namely the evolutionary trend of the system is not only relate with the current state of the system but also the last state variables. Different from the dynamic systems Described by ordinary differential equations, a time-delayed system is infinite dimensional due to its own characteristics, which makes the analysis is often very difficult. The time-delayed system is infinite dimensional chaotic dynamic system and can produce multiple positive Lyapunov exponents, so a time-delayed chaotic system with simple structure can generate highly random and unpredictable time series, which can overcome the short- comings of complicated structure of non-delayed high- dimensional hyperchaotic system in chaotic synchronization and control and improve the confidentiality, thus studying time-delayed chaotic systems is of great significance, it is expected to become a new growth point for high-tech.Chaos synchronization has become one of the hot chaotic researchs due to its important role in secure communication. On the other hand, because of chaotic motion with unpredictable long-term trend of movement and pseudo- random noise characteristics, the chaotic output sometimes does not meet people's needs, so in many practical problems, chaos is considered a harmful form of movement, how to effectively suppress or eliminate chaos become the need for the practical application in these systems.Experimental research is an important intermediate and the only way that theoretical results are used in practice. At present, research of discrete chaotic systems and time-delay chaotic systems mainly focus on theoretical analysis and numerical calculation, but experimental study is rarely reported. Research work of this paper is to design and build some discrete chaotic circuits and time-delayed circuits based on the actual application, then respectively realize synchronization and control of these circuits combining theoretical analysis and numerical calculation. This paper's objectives are: on the one hand it can validate the feasibility and correctness of the theoretic design and results; meanwhile it also expects that this work will provide experimental reference for the practical application.This paper is mainly divided into four parts:1. Study of discrete Chaotic Synchronization. A chaotic and hyperchaotic two-dimensional discrete circuit was designed and realized, and synchronization of hyperchaotic circuits was achieved by pulse-driven approach based on active-passive method. Experimental results were agree with Theoretical analysis and numerical calculation results. We further studied the effects of noise on synchronization. Experimental results showed that in low noise conditions, chaotic and hyperchaotc synchronization can still achieve by pulse-driven approach, the system has a certain robustness. Our work has lay a good foundation for application of chaotic synchronization in digital secure communication.2. Study of discrete chaotic control. For a given nonlinear system, its bifurcation diagram is mostly obtained by numerical calculation. The Logistic map and Henon map circuits were designed and implemented in this article, the circuit demonstrating bifurcation diagram is also realized. The output voltage of the circuits as a parameter changes was observed in a oscilloscope; Taking example for Henon map, based on the principle of controlling chaos by OGY method, analysis conditions werw given and chaos control was realized in the circuit; variable pulse perturbation method was used to control the Logistic chaotic circuit and Henon chaotic circuit, more different periodic orbits were achieved by adjusting the pulse width; Control of a two-dimensional chaotic circuit with characteristics of chaos and hyperchaos was realized by time-delayed feedback method, the analytical conditions were given by theoretically analysis, the experimental results agreed with theoretical analysis and numerical calculation results. We used enhanced control method to achieve synchronization of between systems that were not synchronized by single time-delayed feedback variable, this method can both effectively make use of of the output signal without additional complex experimental device and significantly improve the control effect and quality.3. Study of time-delayed chaotic synchronization. Taking a two-dimensional time-delayed system as example, its bifurcation phenomena and power spectrum of chaotic signal with limited bandwidth were analyzed. A two- dimensional time-delayed chaotic circuit was designed and set up using ordinary capacitors and inductors, similarly a N-dimensional time-delayed chaotic circuit was realized either. The precise synchronization of these two chaotic circuits was achieved by using linear feedback method, and The sufficient condition for synchronization was achieved in theory; the precise synchronization of Logistic time-delay chaotic systems was realized by disturbing a parameter using the difference between signals of the drive circuit and respond circuit; Taking a two-dimensional time-delayed system, based on active-passive synchronization method the generalized synchronization of chaotic systems was realized by linear transforming method. The generalized synchronization condition was obtained by mathematical analysis, and generalized synchronization was realized in an electronic circuit. The experimental results were agreed with results of theoretical analysis and numerical calculation. 4. Study of time-delayed chaotic control. Control of time- delayed chaotic systems was investigated in Logistic chaotic circuit and the two-dimensional chaotic circuit. Firstly, the control of Logistic chaotic circuit was realized by means of phase space compression method, the amplitude of chaotic was limited by a diode, different periodic states were achieved by changding the Amplitude. Secondly, control of a two-dimensional time-delayed chaotic circuit was realized by the linear feedback method, the periodic orbits such as 1P, 2P, 4P were observed, the control coefficient was achieved by theoretical analysis. It was also found that when the feedback coefficient was changed the first bifurcation point with delay time as bifurcation parameter was changed. The feedback coefficient was obtained when Hofpf bifurcation took place; Finally, the PP-SV method was used to successfully control logistic time-delayed chaotic system, the experimental results show that: the circuit can be controlled to different periodic states by adjusting control coefficient or pulse width and cycle.In this paper, the method of theoretical analysis and circuit design is applicable not only to low-dimensional discrete and time-delay chaotic systems, but also to higher-dimensional discrete and time-delay systems.
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