A nonlinear resonance curve is produced when the frequency is scanned with a moderate forcing amplitude, F. Figure 3.10 shows the results of both a backward and a forward scan, which can be constructed from a numerical solution of the Duffing oscillator, equation (3.28) (see Appendix C on Ode [14] for a description of the numerical methods).
Figure 3.10: Schematic of the response curve for a cubic oscillator.
The two scans are
identical except in the region marked by 
. Here, the forward scan produces the upper branch
of the response curve. This upper branch makes a sudden
jump to the lower branch at the frequency 
. Similarly, the
backward (decreasing) scan makes a sudden jump to the
upper branch at 
. In the region ,
at least two stable periodic orbits coexist. The sudden jump between these
two orbits is indicated by the upward and downward arrows
at and . This phenomenon is known
as hysteresis .
The nonlinear response curve also reveals several other intriguing features.
For instance, the maximum response amplitude no longer occurs at 
,
but is shifted forward to the value . This is expected in the string
because, as the string's vibration amplitude increases, its length increases, and
this increase in length (and tension) is accompanied by a shift in the
natural frequency of free oscillations.
Several secondary resonances are evident in Figure 3.11.
Figure 3.11: Nonlinear resonance curve showing secondary resonances in addition
to the main resonance.
These secondary resonances are the bumps in the amplitude resonance curve that occur away from the main resonance.
The main resonance and the secondary resonances are associated with periodic
orbits in the system. The main resonance occurs near
when the forcing amplitude is small and
corresponds to the period one orbits in the system, those orbits whose period
equals the forcing period. The secondary resonances are located near some
rational fraction of the main resonance and are associated with periodic
motions whose period is a rational fraction of 
.
These periodic orbits
(denoted by 
) can often be
approximated to first order by a sinusoidal function of the form
where A is the amplitude of the periodic orbit, (m/n) is its
frequency, and 
is the phase shift.
These periodic
motions are classified by the integers m and n as follows
( 
):
Equation (3.35) is used as the starting point for the method of harmonic balance , a pragmatic technique that takes a trigonometric series as the basis for an approximate solution to the periodic orbits of a nonlinear system (see Prob. 3.12) [11]. It is also possible to have solutions to differential equations involving frequencies that are not rationally related. Such orbits resemble amplitude-modulated motions and are generally known as quasiperiodic motions (see Figure 3.12).
Figure 3.12: Amplitude-modulated, quasiperiodic motions on a torus.
A more complete account of nonlinear resonance theory is found in Nayfeh and Mook [10]. Parlitz and Lauterborn also provide several details about the nonlinear resonance structure of the Duffing oscillator [15].
