The resonant band is a chaotic motion region in the vicinity of the resonant separatrix instead of the generic separatrix.
Such a chaotic motion region is very difficult to determine. To understand the chaotic motion in nonlinear
Hamiltonian system, it is very necessary to investigate the dynamics of such a resonant band. Since the Duffing oscillator
can be used to investigate the single-mode, nonlinear vibration in structural dynamics, a better understanding of
the dynamical behaviors of the Duffing oscillator will be very significant to understand, predict and observe nonlinear
phenomena in structural vibrations. In 1918, Duffing [1] for the first time presented the 1-DOF oscillator with a cubic
nonlinear term. In 1995, Luo [2] gave the analytical investigation of chaotic motion in resonant layer (also see [3]). In
such an investigation, even though the numerical simulations therein showed the existence of resonant layer indeed and
the resonant layers were also demonstrated, no any efficient numerical approach provides a prediction of the onset and
disappearance of resonant layer to verify the analytic predictions. With the wide use of computer, the numerical verification
is much cheaper than the experimental one. In 1999, Luo et al. [4] developed the energy spectrum method for
investigation of the chaotic motion in the stochastic layer and this method is able to give the efficient numerical prediction
of the onset of the resonant webs in the stochastic layer. Such an investigation made us very easily detect the
range of the resonant webs in the stochastic layer and the corresponding stochastic layer widths were computed.
However, through numerical investigations of chaotic motion in resonant layers (or bands), it is found that this approach
cannot be used for the numerical investigation of the onset, forming and destruction of resonant bands. In 2002,
Luo [5] modified the energy spectrum method and developed an incremental energy spectrum approach to investigate
the resonant layers in a parametrically excited pendulum. The incremental energy spectrum approach gives a very good
prediction of the onset, forming and destruction of resonant bands. Therefore, in this paper, suc