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Top1. Introduction
Recently, chaotic systems are widely used in information encryption and secure communication because of their good random characteristics (Chen et al., 2009; El-Dessoky, 2009; Lou Albert, 2009; Wang et al., 2006; Yoshimura et al., 2008). And the synchronization between drive chaotic system and response chaotic system, is a very popular encryption and decryption method that is currently widely adopt. At present, many researches focus on the synchronization of uncertain chaotic systems, and all kinds of adaptive and robust control methods are applied to synchronization of chaotic systems to solve the problem of uncertainties caused in the process of communication (Chen et al., 2008; El-Taha, 2006; Grassi, 2009; Pototsky & Janson, 2009; Qi et al., 2008). However, the rapidity of chaos synchronization and its convergence time is also worth of great concern. Because the rapidity of chaos synchronization will affect the implementation of information decryption and recovery process. If the convergence time of synchronization of the chaos is too long, it certainly will affects the recovery of information.
Terminal sliding mode has attracted much attention in recent years because of its finite time and fast convergence. Especially since Yu Xinghuo's (1998) in 1998, a systematic study has been carried out, and various forms have been proposed, such as nonsingular terminal sliding mode (Feng et al., 2002), chattering free terminal sliding mode (Feng et al., 2014) and so on. In 2016, Chaoxu Mu(2016) used homogeneous theory to explain the finite time convergence of terminal sliding mode. A continuous homogeneous sliding mode control law was designed by Shyam Kamala(2016) in 2016, and its finite time convergence was proved by homogeneous theory.
Haimo (1986) put forward the concept of finite time stability, while Wang (2009) proposed a judgment method of finite time stability of a common system, then many researchers introduced the concept of finite time stability into the chaotic synchronization, also cases of finite time synchronization chaotic systems and many theorems with many implementation methods were presented . But currently there is no helpful method for the construction of a finite time stable system(FTSS), so based on the previous theorem and concept on the finite time stability, we proposed the main principles for constructing a FTSS which satisfies the limit conditions as proposed in this paper , thus sufficient conditions for finite time stability of a system is achieved on the basis of this principle. What is more important is that by using the superposition principle, a large class of new FTSS can be constructed with a ordinary stable system and an old FTSS, so the difficult problem of constructing a new FTSS was solved.
At present, many researches on chaotic secure communication only stay in the realization of chaotic synchronization, but the process of information encryption and decryption has not been studied deeply. It should be said that chaotic synchronization is the basis of chaotic secure communication, but chaos synchronization is not enough, so it is necessary to match the corresponding information hiding and recovery strategy to complete chaos synchronization.
There are two popular methods using chaotic system to realize information hiding. One is the direct method, the useful information is superimposed on the chaotic signal. This method demand a very high requirement for chaos synchronization and recovery of information, because chaotic signal superimposed on the chaotic states sometimes will cause a bad effect and make the chaos synchronization not very accurate, then the information recovery is difficult to achieve a good effect. The second method is the indirect method, the information is hidden in one of the parameters of the chaotic system, which affects all the states of chaotic system, and in the receiver side, parameter identification method is used to recover the original signal. This method is more ingenious and safe. By using the relevant theorems and lemmas on finite time stability and synchronization proposed in this paper , the whole process of information encryption and recovery with the above second method is realized, and also the numerical simulation and analysis were done to verify the correctness of the proposed scheme and theory.