Bifurcations in a Horizontally Driven Pendulum

Bifurcations in a Horizontally Driven Pendulum

We consider a forced pendulum with a horizontally oscillating suspension point. Bifurcations associated with stability at the symmetric period-l orbit (SPO), arising from the "unforced" stationary point, are investigated in details by varying the two parameters A (the normalized driving amplitude) and Ω (the normalized natural frequency). We thus obtain the phase diagram showing the bifurcation curves or the SPO in the Ω-A plane through numerical calculations or the Floquet (stability) multipliers and winding numbers. We note that a specific substructure in the bifurcation set of the SPO recurs in the parameter plane.