Yongxiang Zhang

The multi-torus intermittent paths of strange nonchaotic attractors in quasi-periodic forced piecewise smooth systems are investigated. Due to Farey tree bifurcations, different tori are converted to intermittent strange nonchaotic attractors through a series of non-smooth saddle-node bifurcations. First, the singularity is observed by the phase diagrams, and then the non-chaos is determined by calculating the maximum Lyapunov exponent. Then some characteristics of SNAs are described by analyzing the change of phase sensitive function with the number of tori, the structure of recursive plot and the finite-time Lyapunov exponential distribution. What is different from previous studies about SNAs is that the distribution of the finite-time Lyapunov exponents peaks at extremely negative values, while the positive tail of the distribution decreases in a linear manner.

Previous results show that some oscillators possess finite number of Wada basins. Here we find that a nonlinear oscillator can possess a countable infinity of Wada basins and these Wada basins are connected. Infinite number of coexisting attractors and their Wada basins are investigated by the basin cell theorem and generalized basin cell theorem. Infinite number of Wada basins are systematic, which identical basins structure can be identified in each periodic X-axis coordinate interval. This type of Wada basin boundary can lead to a high level of indeterminacy and an extreme sensitive dependence on initial condition.

A new route to strange nonchaotic attractors (SNAs) is investigated in a quasiperiodically driven nonsmooth map. It is shown that the smooth quasiperiodic torus becomes nonsmooth (continuous and non-differentiable) due to the border-collision bifurcation of the torus. The nonsmooth torus gets extremely fractal and becomes a strange nonchaotic attractor and it is termed the border-collision bifurcation route to SNAs. A novel feature of this route is that SNAs are abundant and the size of SNAs makes up about 40% of the given regions. These SNAs are identified by the Lyapunov exponents and the phase sensitivity exponents and they are also characterized by the singular-continuous spectrum and distribution of finite-time Lyapunov exponents. The distribution of local Lyapunov exponents has its maximum at a relatively small negative value, which contributes largely to lead to the abundance of SNAs.

In this paper, we investigate coexisting strange nonchaotic attractors (SNAs) in a quasiperiodically forced system. We also describe the basins of attraction for coexisting attractors and identify the mechanism for the creation of coexisting attractors. We find three types of routes to coexisting SNAs, including intermittent route, Heagy–Hammel route and fractalization route. The mechanisms for the creation of coexisting SNAs are investigated by the interruption of coexisting torus-doubling bifurcations. We characterize SNAs by the largest Lyapunov exponents, phase sensitivity exponents and power spectrum. Besides, the SNAs with extremely fractal basins exhibit sensitive dependence on the initial condition for some particular parameters.

A fractal basin boundary is a Wada fractal basin boundary if it contains at least three different basins. The corresponding basin is called a Wada fractal basin. Previous results show that some oscillators possess Wada fractal basins with common basin boundaries. Here we find that a nonlinear vibration isolation system can possess abundant coexisting basins and every basin is a Wada fractal basin. These Wada fractal basin boundaries separate different basins in the different regions. A proper classification of these Wada fractal basins is provided according to the order of saddles and Wada numbers. Basin organization is systematic and all basins spiral outward toward the infinity. The entangled basin boundaries are described by the manifolds of saddles and basins (tongues) accumulation.

We identify an unusual route to the creation of a strange nonchaotic attractor (SNA) in a quasiperiodically forced interval map. We find that 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. We characterize the SNA by maximal Lyapunov exponents and their variance, phase sensitivity exponents and power spectra. We also describe the transition between a torus and a SNA by the recurrence analysis. A remarkable feature of the route to SNAs is that the positive tails decay linearly and the negative tails exhibit recurrent fluctuations in the distribution of the finite-time Lyapunov exponents.