Different routes to chaos via strange nonchaotic attractor in a quasiperiodically forced system

摘要

This paper focusses attention on the strange nonchaotic attractors (SNA) of aquasiperiodically forced dynamical system. Several routes, including thestandard ones by which the appearance of strange nonchaotic attractors takesplace, are shown to be realizable in the same model over a two parameters($f-\epsilon$) domain of the system. In particular, the transition throughtorus doubling to chaos via SNA, torus breaking to chaos via SNA and perioddoubling bifurcations of fractal torus are demonstrated with the aid of the twoparameter ($f-\epsilon$) phase diagram. More interestingly, in order toapproach the strange nonchaotic attractor, the existence of several newbifurcations on the torus corresponding to the novel phenomenon of torusbubbling are described. Particularly, we point out the new routes to chaos,namely, (1) two frequency quasiperiodicity $\to$ torus doubling $\to$ torusmerging followed by the gradual fractalization of torus to chaos, (2) twofrequency quasiperiodicity $\to$ torus doubling $\to$ wrinkling $\to$ SNA $\to$chaos $\to$ SNA $\to$ wrinkling $\to$ inverse torus doubling $\to$ torus $\to$torus bubbles followed by the onset of torus breaking to chaos via SNA orfollowed by the onset of torus doubling route to chaos via SNA. The existenceof the strange nonchaotic attractor is confirmed by calculating severalcharacterizing quantities such as Lyapunov exponents, winding numbers, powerspectral measures and dimensions. The mechanism behind the various bifurcationsare also briefly discussed.