Research On Complex Dynamics Of A Controlled Four-dimensional Conjugate Lorenz Hyperchaotic System

Since the middle of the 20 th century,nonlinear science has developed rapidly.Chaos,as a unique complex phenomenon in nonlinear science,plays an important role in the field of nonlinear science.At present,chaos has become the focus and hotspot of nonlinear scientific research.As the first mathematical model that abstracted chaos,Lorenz system has laid an important foundation for the development of chaos theory.In the fields of biological science,electronic engineering and computer science,Chaos has been widely studied and applied.Based on the three-dimensional conjugate Lorenz system,a four-dimensional autonomous simple hyperchaotic system with no equilibria or unique equilibria is obtained by introducing control feedback terms.The attractors of the system are hidden when the new hyperchaoic system without equilibrium.The two types of hyperchaotic systems are physically implemented by the inverse integration circuit.Further,the system's local dynamics is analyzed by using the central manifold theorem and Hopf bifurcation theory,including the stability of the system equilibrium point,the existence of Hopf bifurcation and the stability of the Hopf bifurcation periodic orbit,and give the approximate expression of the bifurcation periodic orbit.Secondly,by using numerical analysis methods such as Lyapunov exponent diagram,phase diagram,bifurcation diagram and Poincar?e map,the global dynamics of the four-dimensional hyperchaotic system are discussed,and attractor coexistence phenomenon are found.Finally,the chaotic self-synchronization problem of the new four-dimensional hyperchaotic system is explored.Chaotic synchronization is realized by designing simple controllers and vertified by Matlab program simulation.The first chapter introduces the historical background,development process and research status of chaos science.The definitions of chaos and central manifold theorem are summarized.The conjugate Lorenz system and typical hyperchaotic system are introduced.Finally,the theoretical knowledge of chaotic synchronization is given.In the second chapter,based on the three-dimensional conjugate Lorenz system,a new four-dimensional autonomous simple hyperchaotic system with three quadratic terms is proposed.Under different parameters,the system has two different typess:without equilibria or with a unique equilibria.And both have two positive Lyapunov exponential hyperchaotic attractors.Finally the inverse integration circuit is used to physically implement the two types of hyperchaotic systems.The third chapter analyzes the system characteristics of the controlled conjugate Lorenz system.Through mathematical theories such as central manifold theorem and bifurcation theory,the local dynamics of the new four-dimensional hyperchaotic system are investigated including Stability,Hopf bifurcation existence,Hopf bifurcation period orbit stability.And the approximate expression of the bifurcation periodic orbit is obtained.The fourth chapter numerical analysis of the global dynamic behavior of the new four-dimensional hyperchaotic system by using numerical analysis methods including:Lyapunov exponent diagram,phase diagram,bifurcation diagram and Poincar?e map,showing the system's complex dynamic behaviors such as hyperchaos,chaos,period,etc.Coexistence of attractors has been found.In the fifth chapter,the chaotic self-synchronization problem of the new four-dimensional hyperchaotic system is analyzed by using linear feedback synchronization method,nonlinear feedback synchronization method and adaptive synchronization method.The correctness of the theoretical analysis results in the synchronization of the drive system and the response system are verified by numerical simulation experiments.