The Global Bifurcations That Lead to Transient Tumbling Chaos in a Parametrically Driven Pendulum

Criteria for occurrence of transient tumbling chaos in the parametrically driven pendulum are examined by computer aided methods. It is shown that stable manifolds of the saddles associated with the rotating attractors play a crucial role in separating the oscillating and rotating attractors in the phase-plane. A sequence of global bifurcations related to the saddles generates the fractal structure of the basins of attraction of the coexisting attractors. The fractal structure of the basin boundaries implies occurrence of the transient tumbling motion, i.e. the transients that involve rotations with changing directions and oscillatory motion.