Oscillations, Waves, and Interactions - GWDG

410 U. Parlitz
4
10
5
5
(a) (b) (c)
x
2
0
−2
dx/dt
0
−4
0 10 20 30
−5
−4 −2 0 2 4
10
0 1 2 3 4 5 6
t
x
2
5
20
10
ν / ν
0
5
(d) (e) (f)
x
0
dx/dt
10
0
−10
−5
0 10 20 30
−20
−5 0 5
10
0 1 2 3 4 5 6
t
x
2
20
ν / ν
0
5
10
10 4
10 5
(g) (h) (i)
x
0
dx/dt
0
−10
−5
0 10 20 30
−20
−5 0 5
10
0 1 2 3 4 5 6
t
x
2
10 3
5
20
ν / ν
0
10
10 4
10 5
(j) (k) (l)
x
0
dx/dt
0
−10
−5
0 10 20 30
−20
−5 0 5
10
0 1 2 3 4 5 6
t
x
2
10 3
5
20
ν / ν
0
10
10 4
10 5
(m) (n) (o)
x
0
−5
0 10 20 30
t
dx/dt
0
−10
−20
−5 0
x
5
S(ν)
S(ν)
S(ν)
S(ν)
S(ν)
10 4
10 3
10 4
10 3
10 3
10
0 1 2 3 4 5 6
2
ν / ν
0
Figure 3. Time series, (stroboscopic) phase portraits, and power spectra of the Duffing
oscillator Eq. (7) for d = 0.2, ω = 2 and (a-c) f = 10 (symmetric period-1), (d-f) f = 42
(asymmetric period-1), (g-i) f = 48 (period-2), (j-l) f = 54 (chaos), and (m-o) f = 56
(symmetric period-3). Red markers are plotted whenever a period T = 2π/ω = 1/ν0 of the
drive elapsed.
and bifurcations) in parameter space. Figure 5(a) shows such a bifurcation diagram
where the driving frequency ω is varied and in Fig. 5(b) the driving force f is used
as a control parameter.
Although providing already a very detailed view of the bifurcation structure, bifurcation
diagrams are limited to a single parameter axis. To get an even better overview