Dynamical mechanism for band-merging transitions in quasiperiodically forced systems

Abstract

As a representative model for quasiperiodically forced period-doubling systems, we consider the quasiperiodically forced logistic map, and investigate the mechanism for the band-merging transition. When the smooth unstable torus loses its accessibility from the interior of the basin of an attractor, it cannot induce the “standard” band-merging transition. For this case, we use the rational approximation to the quasiperiodic forcing and show that a new type of band-merging transition occurs for a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor through a collision with an invariant ring-shaped unstable set which has no counterpart in the unforced case. Particularly, a two-band smooth torus is found to transform into a single-band intermittent strange nonchaotic attractor via a new band-merging transition, which corresponds to a new mechanism for the appearance of strange nonchaotic attractors. Characterization of the intermittent strange nonchaotic attractor is made in terms of the average time between bursts and the local Lyapunov exponents.