Strange Nonchaotic Dynamics In Nonsmooth Systems

Strange nonchaotic attractors(SNAs)are attractors that are geometrically fractal(strange),but its largest Lyapunov exponents are negative after calculation(nonchaotic).Usually,SNAs can exist in an extremely narrow region between periodic attractors and chaotic attractors.Since Grobegi et al.first proposed this concept in 1984,strange nonchaotic attractors have become a research hotspot in nonlinear dynamics and have been widely studied in smooth systems.It is not very clear to understand the genesis and mechanisms for the creation of SNAs and coexisting SNAs in the nonsmooth systems.In this paper,we take several kinds of nonsmooth systems as examples to study the strange nonchaotic dynamics(the genesis and mechanisms for the creation of SNAs and coexisting SNAs).In this paper,we explore the new mechanisms for the creation of strange nonchaotic attractors in different types of nonsmooth systems,and use the largest Lyapunov exponents,phase sensitivity exponents,power spectrum,recurrence analysis,the distribution of finite time Lyapunov exponents,spectrum distribution function and its scaling law to analyze the strange nonchaotic attractors and find many unique statistical characteristics.The innovation of this paper is as follows:Firstly,SNAs are explored in two kinds of nonsmooth monostable difference equations with special bifurcations(Grazing bifurcaton and bifurcation with Farey tree).An unusual route to the creation of SNA is found in a quasiperiodically forced interval map.The smooth quasiperiodic torus becomes nonsmooth due to the grazing bifurcation of the torus.The nonsmooth points on the torus increase more and more with the change of control parameter.Finally,the torus gets extremely fractal and becomes a SNA which is termed the grazing bifurcation route to the SNA.A significant feature of the finite-time Lyapunov exponential distribution of the route is that the positive tails in the distribution decay linearly and the negative tails exhibit recurrent fluctuations.In addition,the existence of SNAs is verified in quasiperiodically driven piecewise smooth systems with Farey tree.It can be seen that more and more jumping discontinuities appear on the smooth torus and the torus becomes extremely fragmented with the change of control parameter.Finally,the torus becomes an SNA with fractal property.At this time,the largest Lyapunov exponents diagram represents adevil's staircase as a function of the control parameter.The distribution of finite time Lyapunov exponents exhibits a remarkable feature with several peaks and zero distribution.Secondly,a new mechanism for the creation of SNAs is found in a class of nonsmooth multistable difference equations with special bifurcations(boundary collision torus doubling bifurcations).In addition,there are other different kinds of SNAs birth routes after the truncation of the border-collision torus-doubling bifurcation,i.e.Heagy-Hammel route,fractalization route and intermittent routes.It has been shown that there exist two critical tongue-type regions in the parameter space,where the different mechanisms for the birth of SNAs are investigated.These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents.Different types of SNAs are also characterized by the singular continuous spectrum,Fourier transform,rational approximations,distribution of finite-time Lyapunov exponents and recurrence analysis.Finally,the strange nonchaotic dynamics of a class of nonsmooth multistable differential equations(coexistence of SNAs in compliant marine structure)is analyzed.It is found that the transformation from different quasiperiodic torus to coexisting SNAs.For example,coexisting period-1 tori,coexisting period-2 tori and coexisting period-4 tori transform to coexisting SNAs,respectively.If the initial conditions are changed slightly,there will be a large number of strange non chaotic attractors coexisting in the system.It is worth noting that such coexisting attractors have peculiar rotation properties.