| Chaos is a complex phenomenon in nonlinear dynamical systems.In recent years,more and more research and application have been done in the fields of medicine,biology,circuit and confidential communication and so on.There is no accurate definition on chaos.However,related theoretical research,chaos control,chaos synchronization and many theoretical and technical problems to be improved and perfected,especially in the field of multi-scroll chaotic systems,which are composed of ordinary differential equations(ODEs).In this paper,a new type of chaotic system is proposed for the research and exploration in the first-order chaotic systems,which is described in detail from the aspects of theoretical simulation,mathematical proof,control synchronization and engineering application.The research results extend the theoretical basis and provide the theoretical basis for perfecting chaos theory,chaos synchronization control and chaotic application.Now,the main work is summarized as follows:Firstly,it summarizes the research background of the existing chaos theory and chaos synchronization.The basic definition and research method will lay the theoretical foundation for the research of the following chapters.Secondly,this paper introduces several classical chaotic systems,such as Lorenz system,Chen system and Lü system.It summarizes and analyzes the characteristics of these three kinds of chaotic systems.In this paper,we devoted to seeking three new chaotic systems,which can generation of N scroll attractor,and the unification of the three systems called the translation-type chaotic systems.Therefore,the three systems are subsystems,named the translation chaotic system,the translation-2 chaotic system and the translation-3 chaotic system,respectively.In this paper,the dynamic characteristics of them have been analyzed in detail,such as the equilibrium parameters,dissipative condition,the bifurcation diagram,Lyapunov exponent,Pioncaré mapping,and the circuit design principle.Thus,the theoretical system of the first order chaotic system is improved.Thirdly,in order to ensure the tightness of the theoretical results,the paper gives the mathematical proof of two subsystems in the translation-type chaotic system from the following two aspects.They are:(1)Based on the undetermined coefficients,the translation chaotic system has been proved;(2)Based on Si'lnikov bifurcations theorem the translation-2 chaotic system has been proved.The theoretical analysis shows that the chaotic characteristics of the two subsystems,and the homoclinic orbits and heteroclinic orbits together determine the geometric structure of the attractor.Fourthly,the chaotic synchronization methods have been analyzed and selected,which have already been realized,and the self synchronization and the different synchronization of the three subsystems in the translation-type chaotic system have been realized.The results of numerical simulation show that the adaptive controller with unknown parameters has good synchronization effect and shows the potential application value in the field of engineering technology,such as secure communication and information processing.Finally,the translation-type chaotic system is applied to the field of image encryption and decryption.In this chapter proposes two different attacks on a recently proposed image based on the translation chaotic system,which are plaintext attack and ciphertext attack.In the first of all,the cryptosystem under study proceed by shuffling the image rows and columns to disturb the high correlation among pixels by iterating the logistic map.Secondly,a key stream is generated to mix it with the pixels of the shuffled image using the translation chaotic system.These two processes in the encryption stage present weakness,and a chosen plaintext attack and a chosen ciphertext attack can be done to recover the ciphered-image without any knowledge of the key value.It just demands three couples of plaintext/ciphertext to break totally the cryptosystem. |
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