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

Abstract

This paper focuses attention on the strange nonchaotic attractors (SNAs) of a quasiperiodically forced dynamical system. Several routes, including the standard ones by which the strange nonchaotic attractors appear, are shown to be realizable in the same model over a two-parameter $f\ensuremath{-}\ensuremath{\epsilon}$ domain of the system. In particular, the transition through torus doubling to chaos via SNAs, torus breaking to chaos via SNAs and period doubling bifurcations of the fractal torus are demonstrated with the aid of the two-parameter $f\ensuremath{-}\ensuremath{\epsilon}$ phase diagram. More interestingly, in order to approach the strange nonchaotic attractor, the existence of several bifurcations on the torus corresponding to the hitherto unreported phenomenon of torus bubbling are described. Particularly, we point out the new routes to chaos, namely, (i) two-frequency quasiperiodicit$\stackrel{\ensuremath{\rightarrow}}{\mathrm{y}}$torus doublin$\stackrel{\ensuremath{\rightarrow}}{\mathrm{g}}$torus merging followed by the gradual fractalization of torus to chaos, and (ii) twofrequency quasiperiodicit$\stackrel{\ensuremath{\rightarrow}}{\mathrm{y}}$torus doublin$\stackrel{\ensuremath{\rightarrow}}{\mathrm{g}}$wrinklin$\stackrel{\ensuremath{\rightarrow}}{\mathrm{g}}$SN$\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}$chao$\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}$SN$\stackrel{\ensuremath{\rightarrow}}{\mathrm{A}}$wrinklin$\stackrel{\ensuremath{\rightarrow}}{\mathrm{g}}$inverse torus doublin$\stackrel{\ensuremath{\rightarrow}}{\mathrm{g}}$toru$\stackrel{\ensuremath{\rightarrow}}{\mathrm{s}}$torus bubbles followed by the onset of torus breaking to chaos via SNA or followed by the onset of torus doubling route to chaos via SNAs. The existence of the strange nonchaotic attractor is confirmed by calculating several characterizing quantities such as Lyapunov exponents, winding numbers, power spectral measures, and dimensions. The mechanism behind the various bifurcations are also briefly discussed.