Fractalization route to strange nonchaotic dynamics

In the fractalization route for the formation of strange nonchaotic attractors (SNA's) in quasiperiodically driven nonlinear dynamical systems, a smooth torus gradually, becomes a fractal as the forcing amplitude is increased, while the Lyapunov exponent remains nonpositive. Using techniques introduced by Kim et al. to identify unstable sets in SNA's, we study torus fractalization in a sequence of approximations wherein the quasiperiodic drive is replaced by periodic forcing of increasing period. This allows us to identify an unstable set embedded in the attractor: In the periodically forced system, we show that there is a cascade of attractor merging crises, and argue that the quasiperiodic analogue of such crises causes fractalization of tori to create, SNA's.