The spectral signatures for periodic, quasiperiodic, and chaotic motion are illustrated in Figure 3.26.
Figure 3.26: Time series and power spectra: (a) periodic (period one), (b) periodic
(period two), (c) quasiperiodic, (d) chaotic.
The power spectrum of
a period one orbit is dominated by one central peak, call it 
.
The power spectrum of a period two orbit also has a sharp peak at ,
and additional peaks at the subharmonic 
and the ultrasubharmonic
. These new spectral peaks are the
``sidebands'' about the primary frequency that come into
existence through, say, a period doubling bifurcation of the period one orbit.
More generally, the power spectrum of a period n orbit consists of a collection
of discrete peaks showing the primary frequency and its overtones.
In periodic motion, all the peaks are rationally related to the primary
peak (resonance).
Quasiperiodic motion is characterized by the
coexistence of two incommensurate frequencies.
Thus, the power spectrum for a quasiperiodic motion is made up from
at least two primary peaks, and 
,
which are not rationally related. Additionally, each of the primary peaks
can have a complicated overtone spectrum. The mixing of the overtone
spectra from and usually allows one to
distinguish periodic motion from quasiperiodic motion.
A quick examination of the time series, or the Poincaré section, can also help
to distinguish periodic motion from quasiperiodic motion.
The power spectrum of a chaotic motion is easy to distinguish from periodic or quasiperiodic motion. Chaotic motion has a broad-band power spectrum with a rich spectral structure. The broad-band nature of the chaotic power spectrum indicates the existence of a continuum of frequencies. A purely random or noisy process also has a broad-band power spectrum, so we need to develop methods to distinguish noise from chaos. In addition to its broad-band feature, a chaotic power spectrum can also have many broad peaks at the nonlinear resonances of the system. These nonlinear resonances are directly related to the unstable periodic orbits embedded within the chaotic attractor. So the power spectrum of a chaotic attractor does provide some limited information concerning the dynamics of the system, namely, the existence of unstable periodic orbits (nonlinear resonances) that strongly influence the recurrence properties of the chaotic orbit.
Additionally, in the periodic and quasiperiodic regimes, new humps and peaks can appear in a power spectrum whenever the system is near a bifurcation point. These spectral features are called transient precursors of a bifurcation. A detailed theory of these precursors with many practical applications has been developed by Wiesenfeld and co-workers [29].