Vibration of the Duffing Oscillator: Effect of Fractional Damping
  We have applied the Melnikov criterion to examine a global homoclinic bifurcation and transition to chaos in a case of the Duffing system with nonlinear fractional damping and external excitation.   Using perturbation methods we have found a critical forcing amplitude above which the system may behave chaotically.   The results have been verified by numerical simulations using standard nonlinear tools as   Poincare maps and a Lyapunov exponent. Above the critical Melnikov amplitude $\mu_c$, which is the sufficient condition of a global homoclinic bifurcation, we have observed the region with a transient chaotic motion. 
Marek Borowiec, Grzegorz Litak, Arkadiusz Syta