The first step in analyzing any nonlinear system is the identification of
its equilibrium states. The equilibrium states are the stationary points
of the system, that is, where the system comes to rest. For a system of
differential equations, the equilibrium states are calculated by setting all
the time derivatives equal to zero in the unforced system . Setting 
, 
, and A = 0 in equation
(3.14), we immediately find that the location
of the equilibrium solutions is given by
which, in general, has three solutions:
Clearly, there is only one real solution if K ;SPMgt; 0, 
,
since the other two solutions, 
, are imaginary in this case. If
K ;SPMlt; 0, then there are three real solutions.
To understand the stability of the stationary points
it is useful to recall the
physical model that goes with equation (3.14). If
, then K ;SPMlt; 0 (see eqs. (3.4 and 3.5))
and the Duffing equation (3.14) is a simple model
for a beam under a
compressive load. As
illustrated in Figure
3.4, the solutions correspond to the two asymmetric stable beam
configurations.
Figure 3.4: Equilibrium states of a beam under a compressive load.
The position corresponds to the symmetric unstable beam
configuration--a small tap on the beam would immediately send it to one of
the configurations. If 
, then K ;SPMgt; 0 and
the Duffing equation is
a simple model of a string or wire under tension,
so there is only one symmetric
stable configuration, (Fig. 3.5).
Figure 3.5: Equilibrium state of a wire under tension.
