Study Of Strange Nonchaotic Attractors And Multistability In Dynamical Systems

In dynamical systems,there are four types of attractors: point attractor,periodic attractors,quasiperiodic attractors,and chaotic attractors.Since Grebogi et al.discovered strange nonchaotic attractors in 1984,it has become one of the significant problems in dynamical systems.A strange nonchaotic attractor has a fractal structure(namely the attractor is nonsmooth),but is nonchaotic in the dynamical sense(namely the largest Lyapunov exponent is nonpositive).Recently,strange nonchaotic attractors have been studied by many scholars.In this article,the main work is as follows.We investigate strange nonchaotic attractors in the Ricker family with quasiperiodic excitation and determine the parameter region in which strange nonchaotic attractors exist by combining mathematical proofs with numerical calculations.Theoretical methods are used to prove the existence of strange nonchaotic attractors in some parameter region.The results show that the strange nonchaotic attractors are discontinuous almost everywhere in such a parameter region.Besides,we also use precise numerical methods to explain the existence of strange nonchaotic attractors in a larger parameter region.It also finds that there is a transition region in which strange nonchaotic attractors alternate with chaotic attractors.The main numerical methods include the evaluation of Lyapunov exponents,phase sensitivity,and rational approximations.The generation mechanisms of strange nonchaotic dynamics in discrete dynamical systems are studied.The generation mechanisms of strange nonchaotic attractors are obviously different in smooth and non-smooth systems,respectively.A smooth economic model with quasiperiodic excitation is considered.The routes to strange nonchaotic attractors,including torus-doubling route,Heagy-Hammel route,and fractalization route are found.The global dynamics of a class of quasiperiodically forced piecewise smooth systems are studied by analysising the parameter space.In addition to torus-doubling route,Heagy-Hammel route,and fractalization route,quasiperiodic attractors can also generate strange nonchaotic attractors through intermittency route and nonsmooth route.Nonsmooth route is obviously different from other routes,the number of quasiperiodic attractor decreases rather than increases.As the number of nonsmooth discontinuous points becomes more and more,smooth quasiperiodic attractor becomes nonsmooth quasiperiodic attractor.Then the number of nonsmooth points increases with the variation of parameters,which eventually leads to the generation of singular non-chaotic attractors.Three nonsmooth dynamical systems are selected from the continuous time dynamical systems to analyze strange nonchaotic dynamics.Firstly,we consider a class of single-degreeof-freedom gear dynamical system with quasiperiodic extension.Strange nonchaotic attractors in the quasiperiodic non-smooth dynamical systems are found,and strange nonchaotic attractors are the transition between quasiperiodic motion and chaotic motion.The dynamics is analyzed through Lyapunov exponents,bifurcation diagrams,phase diagrams,singular continuous power spectrum,phase sensitivity of time series and rational approximations.Then,we study a two-degree-of-freedom quasiperiodically forced vibro-impact system.It is shown that strange nonchaotic attractors occur between two chaotic regions,but not between the quasiperiodic region and the chaotic one.Finally,we consider a periodically forced nonsmooth system and find that strange nonchaotic attractors are created by a small amount of noise.Strange nonchaotic attractors can be generated in different periodic windows with weak noise perturbation.If the parameter is varied further from the chaotic range,a larger noise intensity is required to induce strange nonchaotic attractors.Besides,noise-induced strange nonchaotic attractors can be generated by the periodic attractors near the boundary crisis.In addition,the intermittency between strange nonchaotic attractors and periodic attractors can be induced by transient chaos with the increasing noise intensity.The stange nonchaotic multistability in quasiperiodically forced nonsmooth dynamical systems is studied.Firstly,the coexistence of attractors in quasiperiodically forced nonsmooth map is studied,it is shown that strange nonchaotic attractors and quasiperiodic attractors coexist.Then,considering a quasiperiodically forced cantilever beam vibro-impact system with two-sided elastic constraints.It is not only found that strange nonchaotic attractors and quasiperiodic attractors coexist,but also strange nonchaotic attractors and chaotic attractors coexist.Finally,we study the strange nonchaotic dynamics of a two-degree-of-freedom quasiperiodically forced vibro-impact system,it is found three kinds of multistability: the coexistence of quasiperiodic attractors of different frequencies,the coexistence of strange nonchaotic attractors and quasiperiodic attractors,the coexistence of quasiperiodic attractors and chaotic attractors.Moreover,we obtain the domains of attraction of these coexisting attractors.In addition,the quasiperiodic attractor can transform into a chaotic attractor directly through torus break-up without passing through stange nonchaotic attractors.Besides,on the basis of studying the multistability,we control the multistability of stange nonchaotic attractors in nonsmooth systems.Firstly,the Lyapunov function is used to prove that the system can converge to the desired orbit by feedback control.Then,the control signal is optimized by linear augmentation and intermittent control.The results show that the strange nonchaotic attractor can be successfully driven to a stable quasiperiodic orbit by several control methods.