| In recent years,piecewise smooth mappings were researched in depth,and the mapping was widely applied in various scientific fields,such as power and electronics,biology,medicine,economics,etc.The theory of piecewise smooth mappings is still imperfect,and the unsolved problems should be further studied.Based on the analysis and summary of existing research results of piecewise smooth mapping,periodic orbit,chaotic boundary,chaotic control and its application for a piecewise linear discontinuous mapping with bilateral increase were studied.Our specific results are as follows:(1)Study the dynamic behavior of a piecewise linear discontinuous mapping with left and right sides increase.This model may be applied to the physical sciences and engineering,and also helps to research some economic models.By adjusting the important parameters of system,period-adding phenomenon,that is,the number of period grows in an arithmetic progression,and chaos as well as divergence are found with approach of qualitative analysis and numerical simulation.Through the derivation of the parameter curves of border-collision bifurcation,the region where stable periodic orbits exist is determined.And according to border-collision bifurcation curve of the periodic orbit with high level of complexity,combined with the two-parameter bifurcation diagram,the phenomenon of period adding and period superposition is explained.(2)For the piecewise linear mapping dynamic system with the left and right sides increase,the bifurcation diagram of system was obtained using an important parameter as bifurcation parameter.It finds that there is certain existing region for every periodic point of periodic orbit in the attractive,invariant interval of the system so that the iterative restricted phenomenon occurs in a bifurcation structure.The existing range of the periodic point of periodic orbit and boundary of the iterative restricted area are determined through theoretical derivation.Furthermore,the analytic expressions of boundaries of chaotic area and period-n orbit are gotten from the exploration of the penalty area boundary.The analysis result is verified by applying Lyapunov exponent.(3)Mainly based on chaos produced by the piecewise discontinuity mapping to achieve two goals,the first is the chaos control.First of all,according to the characteristics of the piecewise mapping possessing several branches,linear controllers are added on both sides of piecewise mapping.Then,parameters of the controllers are determined by the stability criterion of periodic orbit of piecewise discontinuous mapping.Finally,simulation results show that the chaos can be controlled to any period-2 orbit by adding linear controllers onto the double sides of the mapping.Adding Gaussian white noise to the control system,it is found that the above control method has a certain ability to resist noise.The second is to use the chaos for image encryption.Based on chaotic sequence generated by the linear discontinuity mapping with bilateral increase,an image encryption algorithm with double chaotic sequences scrambling and dual chaotic XOR is designed.Safety analysis of the experimental result shows that the encryption algorithm has large key space,high sensitivity,and the very high security. |
