Bifurcation analysis of the motion of an asymmetric double pendulum subjected to a follower force: Codimension three problem

Local and global bifurcations in the motion of a double pendulum subjected to a follower force have been studied when the follower force and the springs at the joints have structural asymmetries. The bifurcations of the system are examined in the neighborhood of double zero eigenvalues. Applying the center manifold and the normal form theorem to a four-dimensional governing equation, we finally obtain a two-dimensional equation with three unfolding parameters. The local bifurcation boundaries can be obtained for the criteria for the pitchfork and the Hopf bifurcation. The Melnikov theorem is used to find the global bifurcation boundaries for appearance of a homoclinic orbit and coalescence of two limit cycles. Numerical simulation was performed using the original four-dimensional equation to confirm the analytical prediction.