Transition from strange nonchaotic to strange chaotic attractors

We investigate the transition from strange nonchaotic to strange chaotic attractors in quasiperiodically driven dynamical systems. It is found that whether the asymptotic attractor of the system is strange nonchaotic or strange chaotic is determined by the relative weight of the contraction and expansion for infinitesimal vectors along a typical trajectory on the attractor. When the average contraction dominates the average expansion, the attractor is strange nonchaotic. Strange chaotic attractors arise when the average expansion dominates the average contraction. The transition from strange nonchaotic to strange chaotic attractors occurs when the average contraction and expansion are balanced. A characteristic signature of this route to chaos is that the Lyapunov exponent passes through zero linearly. We provide numerical confirmation using both a quasiperiodically driven map and a quasiperiodic flow.