| First, we introduce the development of chaos theory generally in this paper. We summarize the concepts of chaos bringing forward from different aspects, then describe and interpret the chaos phenomenon using typical examples. Based on this, we put our attention on generalizing the definition of chaos synchronization and classical synchronization methods, then briefly introduce the research status of chaos synchronization at home and abroad. Because of the importance of the synchronization between two different-structural chaotic systems, we study two groups systems which have different features using two different methods. Taking Xin system, Lorenz system and Ameodo system as an example, applying the non-linear variable feedback method, Lyapunov function is constructed by backstepping, the controller is confirmed and the complete synchronization of two different-structural chaotic systems is studied. Not only is exponential convergent the synchronization error, but also presents a systematic procedure for selecting a proper controller in this method. Thus, systems achieve synchronization quickly and easily. Then, the Sprott-B and Sprott-C systems which have topological equivalence features are taken as our models, based on the stability criterion and the modified active method, we study the lag synchronization of the Sprott-B and Sprott-C systems. By choosing conformable matrix, we obtain the condition of synchronization. Numerical simulations results show the effectiveness of two chaotic principles.
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