Oscillations, Waves, and Interactions - GWDG

Complex dynamics of nonlinear systems 409
driven by a gas turbine. If the driving turbines did not run smoothly due to extra
sparking the generators lost their synchrony and did not return to their previous
oscillations when the extra sparking of the turbines was over. 4
If the driving amplitude is further increased the cubic nonlinearity becomes more
dominant and very different types of motion occur. Figure 3 shows five examples
as time series, phase space projections and power spectra. The first time series in
Fig. 3(a) is periodic with the same period as the driving signal. Therefore, this stable
oscillation is also called a period-1 attractor. Figure 3(b) shows the corresponding
trajectory in the (x, ˙x)–plane (i. e., a projection of the attractor) where a red marker is
plotted whenever a period T = 2π
ω of the drive elapsed. This results in a stroboscopic
phase portrait which is a convenient way for plotting Poincaré sections of periodically
driven systems. Figure 3(c) shows a Fourier spectrum of the time series where
only odd harmonics (multiples) of the fundamental frequency occur that coincides
here with the driving frequency ν0 = ω/2π. This feature is closely connected to the
symmetry of the orbits in Fig. 3(b) that can be broken by a symmetry-breaking bifurcation
as shown in Figs. 3(d–f). Symmetry breaking is a precursor of period-doubling
because only asymmetric orbits can undergo a period-doubling bifurcation. An example
for such a period-2 attractor is shown in Figs. 3(g–i) where now subharmonics
occur in the spectrum (Fig. 3(i)) and two markers appear on the orbit (Fig. 3(h)).
When the driving amplitude is increased furthermore a full period-doubling cascade
takes place leading to chaotic dynamics as shown in Figs. 3(j–l). The oscillation is
aperiodic (Fig. 3(j)) with a broadband spectrum (Fig. 3(l)) and a stroboscopic phase
portrait (Fig. 3(k)) that constitutes a fractal set. Finally, Figs. 3(m–o) show a period-
3 oscillation occuring for f = 56 which is an example for general period-m attractors
that can be found for any m in some specific parts of the parameter space of Duffing’s
equation. Again, this period-3 attractor is symmetric (with odd harmonics of
the fundamental frequency in the spectrum) and will undergo a symmetry-breaking
bifurcation before entering a period-doubling cascade to chaos.
Figure 4 shows the Poincaré section of the chaotic attractor from Fig. 3(k) in
more detail which possesses a self-similar structure as can be seen in the enlargement
Fig. 4(b).
Poincaré cross sections are also an elegant way to visualise the parameter dependence
of the dynamics. For this purpose a projection of the points in the Poincaré
section (i. e., z1(n) = x(nT )) is plotted vs. a control parameter that is varied in small
steps. As initial conditions for the solutions of the equations of motions at a new
parameter value the last computed state from the previous parameter is used. In this
way, transients are kept short and one follows an attractor (and its metamorphoses
4 In Ref. [4] on page 1 and 2 Duffing wrote: “Jene synchronen Drehstrommaschinen waren
durch Gasmaschinen angetrieben. Die Frequenz der Antriebsimpulse und die sogenannte
Eigenfrequenz waren genügend auseinander, so daß nur mäßige Pendelungen auftraten, wenn
die Antriebsmaschinen im Beharrungszustande waren. Wurde dieser Beharrungszustand
jedoch nur durch einige wenige heftigere Zündungen gestört, so wurden, auch nachdem die
Verbrennungen wieder regelmäßig geworden war, die Pendelung immer noch größer und
größer, so daß die Maschinen schließlich außer Tritt kamen. Nach den Ergebnissen der
Theorie hätten, nach Eintreten der regelmäßigen Verbrennung, infolge der Dämpfung die
Schwingungen im Laufe der Zeit wieder ihre normale Größe erhalten müssen.”