Research On Strange Nonchaotic Attractors And Their Generation Mechanisms In Nonsmooth System

Strange nonchaotic attractors(SNAs)can be regarded as a special class of attractors between quasiperiodic attractors and chaotic attractors.SNAs usually exist in the transition region where the attractors change from regular to chaotic in quasiperiodic excitation systems.The word strange means that SNAs show the geometric characteristics of chaotic attractors,that is,they have geometric fractal structure.The word nonchaotic means that the system is nonchaotic in the dynamical sense,namely,the maximum Lyapunov exponent is nonpositive.In recent years,the research of SNAs is mainly focused on smooth systems,and there are few studies on the evolution routes and generation mechanisms of SNAs in nonsmooth systems.Therefore,SNAs in some classes of nonsmooth systems are studied in this paper.In Chapter 2,a class of piecewise smooth skew product systems are considered,and the dynamic properties of the system are analyzed by combining theoretical and numerical methods.The existence of invariant curve is proved theoretically,and the Lyapunov exponent of the system is proved to be less than zero by means of Birkhoff’s ergodic theorem and Oseledec’s multiplicative ergodic theorem.In addition,the nonchaotic property of the invariant curve is verified via the maximum Lyapunov exponent,and the strange property is characterized with the help of the phase sensitive function and rational approximation method.In Chapter 3,the relationship between the interior crisis phenomenon and the generation mechanisms of SNAs in the nonsmooth discrete system is studied.It is found that the collision between the SNA and the boundary of the basin of attraction lead to the appearance of interior crisis and new SNA.The multistability in the system is further explored,namely,the coexistence of quasiperiodic attractors and SNAs.In addition,the basin of attraction of the coexisting attractors is given.In Chapter 4,the evolution routes and generation mechanisms of SNAs in the continuous system are discussed.Considering a class of nonsmooth vibro-impact systems,it is found that quasiperiodic attractors can evolve into SNAs through the torus-doubling,fractal,bubbling,and I-type intermittency routes.The generation mechanisms of SNAs in these four routes are described in detail.Moreover,the dynamical property and geometric structure of SNAs are described with the help of numerical methods such as the bifurcation diagrams,maximum Lyapunov exponent,singular continuous spectrum and phase sensitive function.