Research On New Chaotic Systems With Multiple Equilibrium Point Types

As an interdisciplinary subject,nonlinear science has gradually become a hotspot and frontier of scientific research.Many nonlinear systems are accompanied by chaotic phenomena,and the stronger of system nonlinearity,the more abundant the dynamic characteristics of chaotic phenomena.Therefore,chaos theory and applied research have gradually become one of the main frontiers of nonlinear science.Equilibrium plays an important role in chaotic system analysis.Chaotic systems with multiple equilibrium points have multiple types of attractors and more complex dynamic behaviors,which are of great application value in chaotic cryptography and secure communication.At present,most chaotic systems with multiple equilibrium points have been proposed to be more complex.Usually,it is necessary to change a number of nonlinear terms or parameter values in state equation to obtain multiple equilibrium points.Therefore,it is necessary to study chaotic systems with simple structure and more equilibrium points.Therefore,chaotic and hyper-chaotic systems with multi-class equilibrium points are studied in this thesis.The main work is as follows:1.Based on the Lu system,a new three-dimensional continuous chaotic system with multiple types of equilibrium points is proposed and analyzed in detail in this thesis.The system's important property is that there are a stable equilibrium point,an unstable equilibrium point and a line equilibrium for a given value of a system parameter while not changing any nonlinear items or linear items in the system's state equation.Meanwhile,there are the coexistence of chaotic attractor,periodic attractor and stable point,the coexistence of quasi-periodic and periodic attractor and the coexistence of two different periodic attractors.An analog circuit based on Multisim and a digital circuit based on FPGA are designed and simulated,and the practicability of the system is proved.In addition,a discrete algorithm is used to generate a pseudo-random sequence for image encryption application.2.A chaotic system with various equilibrium types has rich dynamic behaviors.Its state can switch flexibly among different families of attractors,which is beneficial to practical applications,so it has been widely concerned in recent years.In this thesis,a new 5D hyper-chaotic system is proposed.The important characteristic of the system is that it may have multiple types of equilibrium points by changing system parameters,namely,linear equilibrium point,no equilibrium point,non-hyperbolic unstable equilibrium point and stable hyperbolic-type equilibrium point.Furthermore,there are hyper-chaotic phenomena and multi-stability about the coexistence of multiple chaotic attractors and the coexistence of hyper-chaotic attractors and chaotic attractors in the system.In addition,the complexity of the system is analyzed,and it is found that the complexity of the system is close to 1 in the hyperchaotic state.The bit-sequence generated by the system has passed all statistical tests,and can be used for image encryption.Finally,an analog circuit of the system is designed and simulated.