Now increase a to 1.2:
The smooth curve has "crinkled". Enlarging various regions of the plot (outline an interesting region with with the mouse on a stopped plot), e.g. the fold around x=0.8, shows a multi-layered structure, so that the curve no longer has the topology of a circle. The torus corresponding to quasiperiodic motion has broken down, and the dynamics is now chaotic. The largest Lyapunov exponent is now positive.