Physics & Computer Science
Well-behaved dynamical properties have been found in a parametrically damped pendulum. For various dampings, all minimum forcing amplitudes E for chaos to occur are at the forcing frequency Ω=1.66, and all minimum E for a stationary solution to be unstable are identical (E=2) and at Ω=2 in the Ω-E state space. Between these two frequencies, the variation to chaos along stability boundaries (where the stationary solution becomes unstable) is solely Ω dependent and insensitive to dampings. These two frequencies separate parameter regions with distinct dynamical behaviors. For Ω < 1.66, the route to chaos is due to an inverse boundary crisis while for Ω > 1.66, it is associated with period doubling. For Ω < 2 the transition from stationary solution to periodic solution is a jump, while for Ω > 2 it is a Hopf bifurcation.
Wu, Binrou and Blackburn, James A., "Well-Behaved Dynamics in a Parametrically Damped Pendulum" (1992). Physics and Computer Science Faculty Publications. 58.