Bifurcation analysis of double pendulum with a follower force

Abstract

The stability and bifurcations of a double pendulum with a follower force are considered. The focus is placed on a doubly degenerate system which possesses a zero eigenvalue and a pair of pure imaginary eigenvalues: that is, coupled flutter and divergence bifurcation. The local qualitative behavior of the system is examined in the neighborhood of the degenerate system. The four-dimensional equations of motion are reduced to the two-dimensional ones by using some qualitative reduction theory for dynamical systems, and it is shown that two distinct bifurcations can occur in the system according to the ratio of two damping coefficients of the double pendulum. Numerical simulations have been performed using the original four-dimensional equations to confirm the analytical results. Some global forms of behavior of the solutions have also been investigated numerically.