A nonlinear system can make a transition from quasiperiodic motion directly to chaos. This is known as the quasiperiodic route to chaos . It is of great practical and historical importance since it was one of the first proposed mechanisms leading to the formation of a strange attractor [23].
There are, in fact, many routes to chaos even from a humble 
torus
attractor. For instance, when the attractor loses stability, a
stable higher-dimensional torus attractor sometimes forms. Another
possibility in the string system is the formation of a
doubled torus, illustrated schematically in Figure 3.21.
Figure 3.21: Schematic of a torus doubling bifurcation.
In the torus doubling route to chaos, our original torus (which is a closed curve in cross section) appears to split into two circles at the torus doubling bifurcation point [24]. The torus doubling route to chaos is reminiscent of the period doubling route to chaos. However, it differs in at least two significant ways. First, in most experimental systems, there are only a finite number of torus doublings before the onset of chaotic motion. In fact, no more than two torus doublings have ever been observed in the string experiment. Second, the torus doubling route to chaos is a higher-dimensional phenomenon, requiring at least a four-dimensional flow, or a three-dimensional map. It is not observed in one-dimensional maps, unlike the period doubling route to chaos.
Now that we have reviewed some of the more salient dynamical features of a string's motions, let's turn our attention to assembling the tools required to view these motions in a real string experiment.