Bifurcation structure of the double-well Duffing oscillator

We consider a forced Duffing oscillator with a double-well potential, which behaves as an asymmetric soft oscillator in each potential well. Bifurcations associated with resonances of the asymmetric attracting periodic orbits, arising from the two stable equilibrium points of the potential, are investigated in details by varying the two parameters A (the driving amplitude) and omega (the driving frequency). We thus obtain the phase diagram showing rich bifurcation structure in the omega-A plane. For the subharmonic resonances, the corresponding period-doubling bifurcation curves become folded back, within which diverse bifurcation phenomena such as "period bubblings" are observed. For the primary and superharmonic resonances, the corresponding saddle-node bifurcation curves form "horns", leaning to the lower frequencies. With decreasing w, resonance horns with successively increasing torsion numbers recur in a similar shape. We note that recurrence of self-similar resonance horns is a "universal" feature in the bifurcation structure of many driven nonlinear oscillators.