Oscillations, Waves, and Interactions - GWDG

412 U. Parlitz
two-parameter studies are often desired and there are many bifurcation phenomena
requiring (at least) two control parameters (called codimension-two bifurcations).
Two-dimensional charts of the parameter space showing parameter combinations
where bifurcations occur are called phase diagrams. In these diagrams (codimensionone)
bifurcations are associated with curves as can be seen in Fig. 6 for the Duffing
oscillator. The damping constant d = 0.2 is kept fixed and the driving frequency ω
and the driving amplitude f are plotted along logarithmic axes, to emphasise the repeated
structure of bifurcation curves. This superstructure of the bifurcation set [5]
is closely connected to the nonlinear resonances of the oscillator (partly visible in
Fig. 2) and the torsion of the flow along the periodic orbits [6]. In the parameter
regions coloured in orange asymmetric period-1 oscillations exist (similar to the example
shown in Fig. 3(d–f)) and in the yellow regions period-doubling cascades and
chaos occur. There is some very characteristic pattern of bifurcation curves (not
shown here) that occurs not only in all resonances of Duffing’s equation but also
with many other driven nonlinear oscillators [7,8]. Furthermore, the phase diagram
in Fig. 3 doesn’t show all bifurcations in the chosen section of the parameter space.
There are, for example, coexisting attractors in phase space which independently
undergo their own bifurcation scenarios when ω and f are varied. A complete study
and smart ways for visualising the plethora of attractors and bifurcations of Duffing’s
equation (and other nonlinear oscillators) remains a challenge.
f
200
100
30
20
10
5
1
0.33
1 2
ω
Figure 6. Phase diagram of the Duffing oscillator Eq. (7) for d = 0.2 [9].
2.4