The case of whirling motions subject to a planar excitation is described by the equation
where the phase space is four-dimensional:
.
The extended phase space, when we add the forcing variable,
is five-dimen-sional:
.
Equation (3.56) can be analyzed for periodic motions by a combination of averaging and algebraic techniques not unlike the harmonic balance method. Because our text is an experimental introduction to nonlinear dynamics, we present here a qualitative description of the results of this analysis. For further details see references [19], [20], and [21].
As mentioned in the introduction to this section, an experimental frequency scan that passes through a main resonance can result in the following sequence of motions:
The basic features of these experimental observations agree with those predicted by equation (3.56), and are summarized in the response curve shown in Figure 3.17.
Figure 3.17: Response curve for planar and nonplanar motion.
(Adapted from Johnson and Bajaj [19].)
In the parameter range 
, the
response curve indicates the coexistence of three planar
periodic motions and one nonplanar periodic orbit.
In this parameter regime, the planar periodic orbit becomes
unstable; the string ``pops out of the plane'' and begins to
execute a whirling motion.
At some parameter value 
, the nonplanar periodic motion
itself becomes unstable and the system may do any number of things depending
on the exact system parameters and initial conditions. For instance,
it may hop to the small-amplitude planar periodic orbit. Or the
ballooning orbit itself may become unstable and begin to precess (quasiperiodic
motion). In addition, chaotic motions can sometimes be observed in this parameter
range. These various dynamical possibilities are illustrated schematically
in Figure 3.17.
We repeat, the motion observed depends on the exact system parameters and
the initial conditions, because there can be many coexisting attractors with
complicated basins of attraction in this region.
In particular, the chaotic motions are difficult to isolate
(and observe experimentally) without a thorough understanding of the system.
