Oscillations, Waves, and Interactions - GWDG

408 U. Parlitz
Figure 2. Amplitude resonance
curve of the Duffing oscillator
Eq. (7). Plotted is a =
max(x(t)) vs. ω for d = 0.2
and f = 0.15 (black curve),
f = 0.3 (blue curve), and f = 1
(red curve). At some critical
frequencies (bifurcation points)
the oscillations loose their stability
and the system undergoes
a transient to another stable periodic
solution as indicated by
the arrows.
a
2.5
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3
resulting in two branches. When slowly increasing the driving frequency ω starting
from small values the oscillation amplitude grows until the end of the upper branch
is reached. At that point the periodic oscillation looses its stability due to a saddlenode
bifurcation and the driven oscillator converges to a stable periodic oscillation
with much smaller amplitude corresponding to the lower branch. This transient is
indicated by the grey arrow pointing downward and it typically takes several periods
of the driving signal (i. e., it is not abrupt as may be suggested by the arrow). In the
opposite direction, for decreasing excitation frequencies ω the system first follows the
lower branch until this periodic oscillation becomes unstable and a transient to the
upper branch occurs. In the frequency interval between both bifurcation points two
stable periodic solutions exist for the oscillator and it depends on initial conditions
whether the system exhibits the small or the large amplitude oscillations. Each stable
periodic oscillation is associated with an attracting closed curve in state space called
an attractor and we observe here a parameter interval with coexisting attractors. If
the driving amplitude is increased to f = 1 this interval becomes larger and it is
shifted further towards high driving frequencies (red curve in Fig. 2). Additionally,
small peaks at low driving frequencies occur corresponding to nonlinear resonances.
They bend to the left and they also overhang if the driving amplitude is sufficiently
large. However, what is shown in Fig. 2 is just the tip of the iceberg and many
additional coexisting attractors undergoing different types of bifurcations occur if
the oscillator is forced into its full nonlinear regime. Before some of these features of
the Duffing oscillator will briefly be discussed we want to have a look back again at
Georg Duffing’s pioneering work. He wrote that he first hoped to solve Eq. (7) using
elliptic functions but then he realized soon that this is not possible. Then he applied
perturbation theory and tested the resulting approximate solutions with mechanical
experiments (also briefly presented in his book). His main interest was the shift of
the resonance frequency and the occurrence of coexisting stable solutions including
the resulting hysteresis phenomena (as shown in Fig. 2). Duffing’s motivation for this
study was mainly due to his interest in technical systems. In the introduction of his
book [4] he describes some observations with synchronous electrical generators each
ω