Controlled Dynamics Analysis Of A Class Of Chaotic System

Since the 3D chaotic model i.e.,Lorenz system was presented,the modeling of chaotic systems has been developed rapidly.The following chaos model x(5)(28)a(y-x),y(5)(28)-c(10)xz,z(5)(28)b-y~2has attracted the attention of many scholars because it has a pair of symmetry equilibria with the opposite stability.In order to deeply understand the chaotic complexity of the system,this academic dissertation mainly studies the complex dynamic behaviors of the above-mentioned system when it is under control.Including the singular points,Hopf bifurcation,the singular point at infinity,singular degenerate heteroclinic cycles,hidden attractor and circuit implementation of chaotic information.The contents include:In chapter 2,a 3D controlled chaotic systems is obtained by adding a linear control item to the 3D chaotic system,the control item keep the dimension of the system and the equilibria'position and number are not changed.The normal form theory,Hopf bifurcation theory are used to analysis the Hopf bifurcation of the 3Dcontrolled system in this chapter.At the same time,the dynamics at infinity of the 3D controlled system are studied with the help of Poincarécompactification technology.Through numerical simulation,singular degenerate heteroclinic orbits are found in the specific parameters,when specific parameters were disturbed,heteroclinic loops of the controlled system break,and emerge a new chaotic attractor.Finally,the actual circuit of chaotic attractor is designed by means of the theory of chaotic circuit.In chapter 3,a controlled 4D systemwith noequilibria is constructed based on a 3D chaotic system.The basic dynamic behaviors of the 4D system are analyzed,including Lyapunove exponent,phase portrait,Poincarésection,sensitivity to initial values,power spectrum and so on.It is found that there is a special form of attractor,namely hidden attractor,whose basin does not intersect with any small neighborhood of the unstable fixed point.With the knowledge of chaotic circuit,the chapteralso designs the practical circuit of the attractor.