| As a complex dynamic phenomenon in deterministic nonlinear systems,chaos has characteristics of high sensitivity to initial conditions,pseudo-randomness and long-term unpredictability.These characteristics make chaotic systems widely used,for example,the pseudo-randomness of chaotic sequences can be applied to encryption.In order to make the confidentiality and security of chaotic secure communication more ideal,scholars are dedicated to constructing chaotic systems to produce more complex dynamic behaviors,which has become a very important research hotspot in nonlinear science in recent years.Multi-stability has been used in many fields such as image encryption and regarded as an additional source of randomness in many information engineering applications.It has caused in-depth discussions by scholars with its complex dynamic characteristics.Based on this,the research on multistable chaotic systems is carried out,and the main work is as follows:1.A four-dimensional chaotic system with complex multi-stability is proposed and analyzed.First,phase diagrams,Lyapunov exponential spectra,bifurcation diagrams and other tools are uesed to study the dynamic behavior of the proposed system,it is found that:(1)The system has coexisting attractors related to ten different combinations of initial values.The number of attractors for each combination is at least two and at most five.The types of attractors involved are point attractors,periodic attractors,quasi-periodic attractors and chaotic attractors indicating that the system has complex and rich multistable phenomena;(2)The system’s topology has strong sensitivity to parameter a,i.e.when a is [1.2,1.4],the shape of the attractor varies from single scroll to double scroll and further to four scroll;(3)The system’s characteristics have strong sensitivity to the parameters b and c,i.e.when b is [-5,3]and c is [3,8],the system motion state is alternatively periodic and chaotic states.Secondly,An circuit of the proposed system is designed,and a FPGA(Field Programmable Gate Array,FPGA)circuit of the system is made.The existence of chaotic behaviors and the coexistence of attractors of the system is verified,and its feasibility is also demonstrated.Then,based on the stability theory of Lyapunov,an adaptive sliding mode synchronization controller is designed.This controller successfully tracks the sine and cosine signals in 0.35 s and identifies unknown parameters in 0.14 s.Finally,a pseudo-random sequence generator of the chaotic system is designed,and the generated sequence successfully passes the frequency distribution test and randomness test,which shows good randomness of the sequence.The successfullytested sequence is applied to digital image encryption experiments.2.A four-dimensional hyperchaotic system with extremely multi-stability is proposed and analyzed.Through the study of the dynamic behavior of the system,it is found that:(1)the system has two non-hyperbolic equilibrium points or infinite non-hyperbolic equilibrium points under different values of system parameters;(2)The system has complex multi-stability and extreme multi-stability phenomena in non-hyperbolic equilibrium.The involved multistability attractors are point attractors,different periodic attractors,different chaotic attractors,different hyperchaotic attractors.Secondly,the analog circuit of the proposed system is designed,simulated by Multisim,and realized the circuit by FPGA,which proves the system’s hyperchaotic behavior and practicality.Then,based on the stability theory of Lyapunov,an adaptive sliding mode synchronization controller is designed.This controller successfully trackes the sine and cosine signals in 0.3s and identified unknown parameters in0.13 s.Finally,a pseudo-random sequence generator for a hyperchaotic system is designed.The generator generates hyperchaotic sequences and chaotic sequences.This sequence successfully passes all statistical tests,and can be used for image encryption. |
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