Analysis And Control Of Chaotic Systems And Their Application In Image Encryption

Nonlinear phenomenon is one of the essential states of the nature and the social.Chaos,one of the branches of nonlinear dynamics,it reveals the nature and society's widespread unity of certainty and randomness and unpredictability.Study on nonlinear dynamics can help us more profoundly understand its essence and complexity.In recent years,the chaotic systems are widely used in secure communications,signal processing,physics,biology discipline,and so on.In the same time,the fractional order chaotic systems are more in line with the engineering practice than integer order ones.This PhD dissertation studies nonlinear dynamics,and probes the mechanism of several new chaotic models,and investigates robust synchronization of chaotic system,and analyzes preliminarily study on fractional order chaotic system used in digital image encryption.The contents and main results are organized as follows:(1)By utilizing the definition of Caputo fractional calculus,a new form of discrete fractional map is introduced.The new discrete fractional map holds memory effect.The Caputo-like discrete fractional of the discrete Duffing map is given,and its phase,bifurcation diagrams and the largest Lyapunov exponent are studied.Moreover,one gives the bifurcation diagrams and the largest Lyapunov exponent with varied fractional order.This is a new discipline,and is also a new hot topic on the investigation for discrete map.By the application of the topological horseshoe theorem,a horseshoe can be found in this map,and so this map is a chaotic system.(2)It is an attractive business for us to find a chaotic system with specific dynamical behaviors.A new quadratic chaotic system with three stable equilibrium points is presented.The dissipativeness of this chaotic system,and equilibrium points and their stability,the Lyapunov exponents and bifurcation diagrams are analyzed.Local bifurcation and singularly degenerate heteroclinic cycles are investigated.Meanwhile,the circuit design of this system is also derived.Its experimental results are agreement with the numerical simulation.Finally,a topological horseshoe is found by using the topological horseshoe theorem,and this implies that the dynamical system is chaotic.(3)A new four dimensional hyper-chaotic system is proposed by modifying a known chaotic system.As the parameters vary,this dynamical system has line equilibrium points,one equilibrium or five equilibrium points,and their stability is analyzed.In general,a chaotic system with line equilibrium has a hidden attractor,thus this one is attractive to scholars.Furthermore,the other dynamical behaviors,such as the phase,the Lyapunov exponent and bifurcation diagrams are analyzed.The circuit diagram is also described,and its experimental results are consistent with the numerical simulation.From the numerical analysis,this system has hyper-chaotic,chaotic,periodic and quasi-periodic behaviors,and so on.When this system is chaotic,it is a four-wing chaotic attractor.To a hyper-chaotic system,it has two positive Lyapunov exponents,which means that this hyper-chaotic system can extend to two directs.A topological horseshoe is found in three-dimensional space,therefore,this system is hyper-chaotic.(4)A chaotic model for generating multi-directional multi-scroll attractors via hyperbolic tangent function series is proposed.The dynamical mechanisms of this chaotic model are further investigated,including one direct,two directs and three directs chaotic attractors.Moreover,the dynamical behaviors of this system are theoretically analyzed and numerically simulated,such as equilibrium points and their stability,Lyapunov exponents and bifurcation diagrams.Observing the phase,bifurcation chart and Poincaré section,one can find the number of the scroll is consistent.The circuit design is given,but the nonlinear function,i.e.hyperbolic tangent function series is difficult to be realized,it can be replaced with other similar nonlinear function.This system can be widely used in data encryption and secure communication.(5)A new method for generating M×N-grid double-scroll Rucklidge chaotic attractors is presented.By designing arctangent function series to construct a nonlinear function,which replace the state variables of the Rucklidge system,a new M×N-grid double-scroll chaotic system is created.The formation mechanism of this M×N-grid double-scroll chaotic model is further investigated.Moreover,some basic properties are theoretically analyzed and numerically simulated,including equilibrium points and their stability,Lyapunov exponents and bifurcation diagram.From the observation of the phase,bifurcation diagrams and Poincaré section,it is easy to find the number of the double-scroll is consistent.The simple circuit diagram is described,but the arctangent function series is difficult to be realized today.This system can be widely utilized in secure communication and data encryption.(6)On the aspect of control,firstly we study the chaos control by using sliding mode strategy,and discuss finite-time synchronization of fractional order chaotic system with uncertainty and disturbance.Furthermore,we also investigate parameter identification and robust synchronization of chaotic system with uncertainty and disturbance.Secondly,The robust synchronization problem of a class of uncertain chaotic system with external disturbance is studied,and identifications of the parameters are analyzed.On the basis of the stability theory of the finite time synchronization,the presented control scheme can be successfully used to this typical chaotic system.Thirdly,the robust H synchronization problem of a class of uncertain chaotic system with external disturbance is investigated.By using Lyapunov stability theory,one designs the controller and adaptive laws,and obtains the sufficient conditions for chaotic robust H synchronization.Finally,we verify the effectiveness of these schemes,and convergence of parameter estimation is consistent with the theory.The experimental results imply the feasibility of control methods,and denote practicability of these schemes.(7)A chaos encryption scheme based on Arnold transformation is presented.We transform fractional order chaotic sequences,and make it become pseudo random,unpredictable.The pseudo random sequences serve as key-streams to encrypt images.After the image is encrypted,key space become larger,sensitivity is increased,and image information entropy increases.The image has a strong resistance to attack,and the results prove the superior security and high efficiency of the scheme.