Period-3 motions to chaos in an inverted pendulum with a periodic base movement

Abstract

A bifurcation tree of period-3 motions to chaos in an inverted pendulum with a periodic base movement is presented through a discrete implicit mapping method. The stable and unstable periodic motions on the bifurcation tree are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are also carried out. Frequency-amplitude characteristics of the bifurcation tree are presented through the finite Fourier series analysis. Numerical simulations of periodic motions on the bifurcation are completed. The numerical and analytical results are presented for comparison. Except for period-1 motion to chaos studied before, this study focuses on other periodic motions to chaos existing in the inverted pendulum with periodic base movement. In the earthquake testing, one tests structures from 1hz to 33 hz. However, during such a frequency range, one not only can observe the period-1 motion to chaos, but one can observe period-3 motion to chaos in such an inverted pendulum with a periodic base movement. Thus, in the building design, period-3 motions to chaos should be considered, which have different dynamical behaviors from the period-1 motions to chaos.