  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.
