Oscillations, Waves, and Interactions - GWDG

430 U. Parlitz
(a)
y
(c)
y
40
20
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(b)
τ1,
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−40
−40 −20 0 20 40
0
−20
40
20
0
−20
−40
−40 −20 0
x
20 40
x
(d)
τ1
k1a
3
−
τ2
k2a
− τ1 −
k1b
k1a
k1b, k2b τ2, k2a
k3b − τ3, k3a
τ1, k1a
−
k1b, k2b
τ2, k2a k3b
τ3, k3a
k1a
2
k1b
1
1
− −
Figure 19. Phase (colour coded) of the complex solution f of the controlled GLE (29)
with parameters (a, b) = (−1.45, 0.34). White rectangles denote control cells where signals
are measured and control is applied. In the region between the control cells chaotic spiral
waves are turned into (a) slanted traveling waves if the control scheme (b) is applied with
parameters τ1 = 31, τ2 = 59, τ3 = 84, k1a = 0.22, k1b = 0.3, k2a = 0.2, k2b = 0.5, k3a = 0.3,
and k3b = 0. Using control scheme (d) with k1a = 0.22, k2a = 0.1, k3a = 0.35, k1b = 0.3,
k2b = 0.5, k3b = 0, τ1 = 41, τ2 = 27, and τ3 = 49 individual spiral waves can be trapped (c).
From Ref. [94].
References
[1] W. Lauterborn, ‘Numerical Investigation of Nonlinear Oscillations of Gas Bubbles in
Liquids’, J. Acoust. Soc. Am. 59, 283 (1976).
[2] W. Lauterborn and E. Cramer, ‘Subharmonic Route to Chaos Observed in Acoustics’,
Phys. Rev. Lett. 47, 1445 (1981).
[3] W. Lauterborn and U. Parlitz, ‘Methods of chaos physics and their application to
acoustics’, J. Acoust. Soc. Am. 84, 1975 (1988).
[4] G. Duffing, Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische
Bedeutung, (Verlag Friedr. Vieweg & Sohn, Braunschweig, 1918).
[5] U. Parlitz and W. Lauterborn, ‘Superstructure in the bifurcation set of the Duffing
equation ¨x + d ˙x + x + x 3 = fcos(wt)’, Phys. Lett. A 107, 351 (1985).
[6] U. Parlitz and W. Lauterborn, ‘Resonances and torsion numbers of driven dissipative
nonlinear oscillators’, Z. Naturforsch. 41a, 605 (1986).
[7] C. Scheffczyk, U. Parlitz, T. Kurz, W. Knop and W. Lauterborn, ‘Comparison of
bifurcation structures of driven dissipative nonlinear oscillators’, Phys. Rev. A 43,
k2b
τ2
k2a
2
k2b
τ3
k3a
−
3
k3b
τ3
k3a
k3b