Dynamic Stabilization in The Double-Well Duffing Oscillator

Bifurcations associated with stability of the saddle fixed point of the Poincar\'{e} map, arising from the unstable equilibrium point of the potential, are investigated in a forced Duffing oscillator with a double-well potential. One interesting behavior is the dynamic stabilization of the saddle fixed point. When the driving amplitude is increased through a threshold value, the saddle fixed point becomes stabilized via a pitchfork bifurcation. We note that this dynamic stabilization is similar to that of the inverted pendulum with a vertically oscillating suspension point. After the dynamic stabilization, the double-well Duffing oscillator behaves as the single-well Duffing oscillator, because the effect of the central potential barrier on the dynamics of the system becomes negligible.