Oscillations, Waves, and Interactions - GWDG

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406 U. Parlitz here chaos control methods are of interest. As an example, in Sect. 6.1 multipledelay feedback control is applied to a frequency-doubled Nd:YAG laser to suppress its chaotic fluctuations. This control method was developed at the DPI and has some promising features for stabilising and manipulating chaotic systems including complex spatio-temporal dynamics (Sect. 6.2). In Sect. 2.3 a combination of dynamical system governed by ordinary differential equations and an automaton switching between a finite number of states is presented. Such hybrid systems often occur in a technical context and may also exhibit chaotic dynamics. In Sect. 3 we briefly revisit the most important concepts for characterising chaotic dynamics: fractal dimensions and Lyapunov exponents. To investigate experimental systems specific methods for time series analysis are required. Following the long tradition of signal processing at the DPI, new methods for data analysis have been devised, implemented and applied. This topic is addressed in Sect. 4 where state space reconstruction (Sect. 4.1) and forecasting methods (Sect. 4.2) are discussed. 2 Nonlinear systems exhibiting complex dynamics In this section three types of dynamical systems are presented which possess different dynamical features. 2.1 Nonlinear oscillators A cornerstone of physics and engineering is the damped harmonic oscillator ¨x + d ˙x + ω 2 0x = f cos(ωt) (1) driven by some external periodic force f cos(ωt). 2 It is well known that the solution of this ordinary differential equation (ODE) x(t) = xhom(t) + xinhom(t) (2) consists of a general solution xhom(t) of the homogeneous equation (f = 0) converging to zero due to the damping and a special solution of the inhomogeneous system (1) with amplitude xinhom(t) = a cos(ωt − ϕ) (3) a = f � (ω 2 0 − ω 2 ) 2 + d 2 ω 2 and phase � dω ϕ = arctan ω2 � . (5) 0 − ω2 The amplitude a of the (asymptotic) oscillation of the driven linear oscillator is proportional to the driving force f and the shape and location of the resonance curve a(ω) which depends on the damping d and the eigenfrequency ω0 as shown in Fig. 1. 2 By rescaling time the eigenfrequency ω0 can be eliminated (i. e., ω0 → 1) but we shall keep it here for a more transparent interpretation of the results. (4)
Figure 1. Resonance curves of the damped and driven harmonic oscillator Eq. (1) for high (d = 0.5) and low (d = 0.2) damping with f = 1 and ω0 = 1. The maximal amplitude amax occurs at the resonance frequency ωr = p ω 2 0 − d2 /2. Complex dynamics of nonlinear systems 407 a 6 5 4 3 2 1 d = 0.5 0 0 0.5 1 1.5 2 ω d = 0.2 These relations completely describe the response of a (damped) harmonic oscillator to a sinusoidal excitation and can be extended to the case of a general periodic force using the superposition principle. Such a comprehensive treatment is possible only, because this system is linear. Many natural oscillators, however, are nonlinear and the question arises: how do nonlinearities change the dynamical behaviour? With the harmonic oscillator nonlinearity can enter in terms of a nonlinear restoring force and/or due to a nonlinear damping mechanism where the latter may render the oscillator a self-sustained system oscillating without any external driving. A prototypical example of an oscillator with a nonlinear restoring force is the Duffing oscillator ¨x + d ˙x + ω 2 0x + αx 3 = f cos(ωt) , (6) where the restoring force ω 2 0x+αx 3 may be interpreted as a nonlinear approximation of a more general nonlinear force (Taylor expansion). By rescaling time t and x both coefficients ω 2 0 and α can be set to one and the resulting normalised Duffing equation reads ¨x + d ˙x + x + x 3 = f cos(ωt) . (7) The class of nonlinear oscillators with cubic restoring force is named after Georg Duffing engineer in Berlin in the beginning of the 20th century and published in 1918 a detailed study on ‘Erzwungene Schwingungen bei veränderlicher Eigenfrequenz und ihre technische Bedeutung’ 3 [4]. As pointed out in the title of Duffing’s book the “eigenfrequency” of nonlinear oscillators depends in general very much on the amplitude of the oscillation and actually provides a useful notion for relatively small amplitudes only, as we shall show in the following. This phenomenon is illustrated in Fig. 2 where amplitude resonance curves of the Duffing oscillator Eq. (7) are shown for d = 0.2 and three different driving amplitudes. For weak forcing with f = 0.15 the resulting (black) resonance curve still resembles the linear resonance curve (Fig. 1) but it is shifted towards higher frequencies and it bends to the right. If the driving amplitude is increased to f = 0.3 the resonance moves further to the right and the curve overhangs 3 In English: Driven oscillations with variable eigenfrequencies and their technical im- portance.
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Thomas Kurz, Ulrich Parlitz, and Ud
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erschienen im Universitätsverlag G
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Bibliographische Information der De
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iv Contents Laser speckle metrology
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Oscillations, Waves and Interaction
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Applied physics at the “Dritte”
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noise component [GNE] 5 4 3 2 1 can
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3.4 Transfer to running speech Spee
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3.4.2 Analysis of running speech Sp
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Speech research 33 Area [Pixels] 60
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Speech research 35 [13] D. Michaeli
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Specific signal types in hearing re
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80 D. Ronneberger et al. Figure 6.
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82 D. Ronneberger et al. Figure 8.
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102 D. Ronneberger et al. Neverthel
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104 D. Ronneberger et al. [4] J. Br
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106 D. Ronneberger et al. strömung
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108 D. Guicking synchronised tuning
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110 D. Guicking primary noise micro
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112 D. Guicking primary sensor desi
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114 D. Guicking by an antiphase sou
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116 D. Guicking R L C C + + 1 1−C
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118 D. Guicking More involved than
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120 D. Guicking In the 1980s, longi
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122 D. Guicking with electrodynamic
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124 D. Guicking 3.6 Noise reduction
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126 D. Guicking the turbulence of a
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128 D. Guicking References [1] Lord
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130 D. Guicking [44] Falcke, H.,
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132 D. Guicking [84] J. Melcher,
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134 D. Guicking [125] D. Heyland et
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136 D. Guicking [166] S. Zommer et
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138 D. Guicking [212] M. L. Post an
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140 W. Lauterborn et al. liquid κ
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142 W. Lauterborn et al. Figure 2.
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144 W. Lauterborn et al. nator, and
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146 W. Lauterborn et al. by types a
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148 W. Lauterborn et al. bubble rad
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158 W. Lauterborn et al. laser puls
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168 W. Lauterborn et al. R [µ m] 1
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170 W. Lauterborn et al. [17] M. P.
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172 R. Mettin (a) (b) Figure 1. Bub
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174 R. Mettin 0 ms 2 mm 1 ms 2 ms 3
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176 R. Mettin important quantity ch
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178 R. Mettin 100 kPa 200 kPa | | F
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180 R. Mettin Figure 7. Trapped sin
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182 R. Mettin concentration of spec
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184 R. Mettin which is called the p
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186 R. Mettin R 02 [µm] R 02 [µm]
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188 R. Mettin constant to M a = 2π
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190 R. Mettin (a) p a [Pa] 140 120
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192 R. Mettin z [mm] 5 4 3 2 1 0 -1
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196 R. Mettin - a hot microlaborato
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198 R. Mettin [52] R. Mettin, C.-D.
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200 W. Eisenmenger and U. Kaatze Th
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218 A. Vogel, I. Apitz, V. Venugopa
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260 K. D. Hinsch Generally, any of
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262 K. D. Hinsch Figure 1. Monitori
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264 K. D. Hinsch Figure 3. ESPI stu
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266 K. D. Hinsch Figure 4. Deterior
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268 K. D. Hinsch Often, in-plane mo
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270 K. D. Hinsch Figure 7. Optical
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272 K. D. Hinsch Figure 9. Humidity
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276 K. D. Hinsch Figure 13. Map of
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278 K. D. Hinsch References [1] D.
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280 Schreiber not moving along with
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282 Schreiber These properties made
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284 Schreiber Figure 3. The G ring
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286 Schreiber Rotation Rate [rad/s]
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288 Schreiber and it is currently b
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290 Schreiber n1 D A B n Figure 8.
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292 Schreiber Beamwalk [µm] 80.0 7
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294 Schreiber ∆ Perimeter [*10e12
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296 Schreiber and the last term acc
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298 Schreiber Δf [µHz] 100 50 0 -
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300 Schreiber formation of a new wo
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302 Schreiber PSD [*10 16 (rad/s) 2
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308 Schreiber Est. Phase Vel. (m/s)
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310 Schreiber [18] V. Frede and V.
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312 Martin Dressel tice is reduced
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314 Martin Dressel Metallic whisker
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316 Martin Dressel (a) (b) (c) CH 3
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318 Martin Dressel 3.1 Charge densi
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320 Martin Dressel Absorptivity σ
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322 Martin Dressel brought a confir
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324 Martin Dressel a charge disprop
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326 Martin Dressel Conductivity 70
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328 Martin Dressel Reflectivity Con
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330 Martin Dressel References [1] M
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332 Martin Dressel and L. Montgomer
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334 R. Pottel, J. Haller and U. Kaa
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456 S. Lakämper and C. F. Schmidt
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462 Index basin of attraction, 144
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464 Index cone bubble structure, 19
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466 Index turbulent, 111, 126 fluct
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468 Index LPC, 29 luminescence of b
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470 Index peak factor, 38, 43 Peier
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472 Index laser-induced bubble, 152
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474 Index ultraharmonic resonance,
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