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Noisy **oscillators**... Fourier coefficients of h22(φ), Eq. (11), for a van der Pol–Duffing **oscillator** (14) (b=0 corresponds to a van der Pol **oscillator**).
... The Maple 6 program below operates with two-dimensional limit-cycle **oscillators**. It calculates the phase resetting curve (or susceptibility of the phase to noise) and the susceptibility of the phase to linear delayed feedback, and employs these data for calculation of the mean **frequency** and the phase diffusion constant of a limit-cycle **oscillator** subject to recursive delay feedback and white Gaussian noise. Differential equation systems for three sample dynamical systems are already included in the program text (two of them should be commented, the one to be used – uncommented); any required equations can be typed-in instead. The program outputs comments on all calculation steps; these comments can be as well found in the program text immediately before the commands for the commented calculation steps.
... Van der Pol–Duffing **oscillator** (14) with μ=0.7 and b=0.5 subject to the recursive delay feedback control. (a): The phase portrait of the control-free noiseless system. (b): h22(ψ). (c): Analytical dependence D(τ) (T0≈2π/1.14). For detail see caption to Fig. 1.
... Van der Pol–Duffing **oscillator** (14) with μ=1.0 and b=0.5 subject to the recursive delay feedback control. (a): The phase portrait of the control-free noiseless system. (b): h22(ψ). (c): Analytical dependence D(τ) (T0≈2π/1.12). For detail see caption to Fig. 1.
... Van der Pol **oscillator** (7) with μ=1.0 subject to the recursive delay feedback control. (a): The phase portrait of the control-free noiseless system. (b): h22(ψ). (c): Analytical dependence D(τ) (T0≈2π/0.94). (d): Numerical simulation. For detail see caption to Fig. 1.

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High-**frequency** coupling: delta power lags.
... NREM slow **oscillation**... Delta power: high-**frequency** coupling correlations and age.
... Cross-correlation between normalized delta power and the logarithm of the high-**frequency** to low-**frequency** cardiopulmonary coupling (CPC) ratio for 3150 subjects of the Sleep Heart Health Study with an apnea-hypopnea index <5 (gray shading shows the standard deviation of the mean). Note that the maximal average cross-correlation of 0.401 occurs at a lag of −4.267min, indicating that fluctuations in the CPC coupling ratio precede fluctuations in delta power by a median of approximately 4min. Also note that the minimal correlations that occur when delta power and the CPC ratio are maximally out of phase at an interval of 91.7min, consistent with the well-known 90-min period of rapid eye movement and nonrapid eye movement sleep cycling.
... Correlations of delta (electroencephalogram [EEG] 0–4Hz) power and high-**frequency** coupling (HFC) power. Delta power 0–4Hz and cardiopulmonary coupling (CPC) and their cross-correlations for a representative subject in the Sleep Heart Health Study database. From the top: absolute delta power 0–4Hz (μV2) (A). Note the higher absolute delta power in the first half of the night compared to the second half. The 0- to 4-Hz delta power normalized to total EEG power (B). Note that the relative delta power in the first and second halves of the night are of relatively equal maximal magnitude. The logarithm of the ratio of high-**frequency** to low-**frequency** CPC (C). Note the correspondence between delta power fluctuations and the CPC ratios. The CPC sleep spectrogram (D). The cross-correlation between absolute and normalized delta power and HFC in this subject was r=0.61 and r=0.75, respectively.
... Relationship between the correlation of all-night delta power high-**frequency** coupling to polysomnographic and electrocardiogram-spectrographic variables.
... High **frequency** coupling

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Natural vibration **frequency**... The analysis of the quality of fluidization of the potato starch powder appears somewhat different from the FCC powder. Two examples of picture of the bed surface and the corresponding time series of the bed surface position and of the column displacement are reported in Figs. 6 and 7, for a fluidized bed of 1.155kg and vibration acceleration a/g=2.0. Also in this case air rate was kept constant close to the minimum for fluidization. At 20Hz (Video 3, Fig. 6a and b) the bed surface motion appears regular and smooth, and also in this case the amplitude of the bed surface **oscillation** appears always larger than the amplitude imparted to the bed. Time series reported in Fig. 7a with a significant phase lag are well described by a sinusoidal curve of the same **frequency** of that representing the column displacement. In this case the amplitude ratio is about 2.25. The phase lag between the bed surface and the column is significant and close to be in perfect opposition. At 18.5Hz the bed surface presents very large bed surface **oscillations** and visible corrugations as shown in Video 4, Fig. 6c and d. Differently from FCC, for potato starch these corrugations appear to move with the column and probably are the result of material sticking on the column wall. The presence of such corrugations makes the surface tracking more difficult. Local evaluation of surface motion produces bed height time series as those reported in Fig. 7b which show a non sinusoidal oscillatory motion. The average amplitude ratio increases up to about 5.5. The phase lag is close to half a cycle also at this **frequency**.... The quality of the effect of vibration on the surface **oscillation** of fluidized bed of FCC aeratable powder appears to be strongly affected by the **frequency** of the applied vibration. For example at 15Hz the surface of a 0.800kg FCC bed is always well defined and qualitatively appears as in Video 1, Figs. 3 and 4a, with changing surface height both with respect to the image (a fixed reference) and with respect to the column (displayed by the mark). The amplitude of the bed surface appears comparable with the column **oscillation** but with a different phase. Instead, the appearance of the same bed at 20Hz (Video 2, Fig. 4b, c and d) is completely different. At this **frequency** the amplitude of the surface is much larger than the column **oscillation** and the bed, close to the surface, does not appear any more a compact continuum. The material instead is thrown up at the highest surface level concentrated in sort of solid jets and then rains down apparently at a smaller velocity than that of the column (Fig. 4b). At this point a new rising surface within the bed starts to be visible. As it appears in the sequence of Fig. 4c and d, this surface becomes more and more evident as it meets the falling solids. It corresponds to the formation of a solid concentration front probably determined by a difference between the faster rising velocity of the particulate phase, caused by the column movement, and the falling velocity of the particulate phase, limited by the raining velocity of the particles. The corresponding time series of the bed height and of the column displacement are reported in Fig. 5, for a fluidized bed of 0.800kg recorded at conditions close to the minimum for fluidization and vibration acceleration a/g=1.0. At 15Hz (Fig. 5a) the time series confirm the direct observation of the bed surface. In fact, the bed surface motion appears regular and well described by a sinusoidal curve of the same **frequency** of that representing the column displacement. In this case the amplitude ratio is about 2.3. At 20Hz (Fig. 5b) instead the maximum amplitude ratio increases up to about 3.5 and, furthermore, the curve representing the bed surface time series is not a sinusoidal curve. As it appears in Fig. 4c and d, during the descending phase a secondary edge appears below the bed surface describing the formation of the second solid front described above. Correspondingly the curve of Fig. 5b representing the bed height shows multiple branching corresponding to the coexistence of two concentration fronts: an ascending front starting from inside the bed and a descending front stopping when it meets the other. In this and similar situations two values of the amplitude ratio were derived. One was evaluated as the difference between the maximum peak value and the minimum value of the descending branch curve and will be referred to as AR′. The other was evaluated as the difference between the maximum peak value and the minimum value of the ascending branch curve and will be referred to as AR″. The first of these amplitude ratios does not account for the overall surface **oscillation** but it can be evaluated more precisely than the other which, instead, is subject to higher uncertainty in the visualization of the initial phase of the rising concentration front. At all **frequencies** tested the bed and column **oscillations** do not appear to be in phase. Time series confirm this observation and indicate a larger phase difference at 20Hz (Fig. 5b) than at 15Hz (Fig. 5a).

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Time–**frequency** representations of gamma activity over the contralateral sensorimotor ROI in the predictable condition. (A) Controls. (B) Patients. The analyzed gamma band (60–90Hz) is indicated by horizontal black lines. (C) and (D) show the time courses of mean contralateral power changes, in the predictable condition, for the beta (13–30Hz) band in red, and the gamma band (60–90Hz), in blue (the traces are represented±1 SEM, indicated by the shaded areas). The traces are aligned relative to the mean across the epoch.
... Based on the concept of hierarchical coupling of different **oscillation** **frequencies**, we hypothesized that the amount of predictive beta modulation could be related to the delta ITC. Both measures showed a strong contralateral predominance in the predictable condition. We therefore evaluated the correlation between predictive beta modulation with delta phase ITC for all participants, for both hemispheres and conditions. In the predictable condition, entrainment of delta phase in the motor cortex was, across groups, significantly correlated with the cortical predictive beta modulation in the contralateral (r=0.42, P=0.042), but not in the ipsilateral hemisphere (r=0.23, P>0.20). In the random condition this correlation was, across groups, present both in the contralateral (r=0.41, P=0.045) and in the ipsilateral hemisphere (r=0.45, P=0.026) (see Supplementary Fig. 2). This pattern conforms to the Predictability by Hemisphere interaction that was found both for delta ITC and for predictive beta modulation, and suggests a possible joint role in preparation and interlinked entrainment of delta and beta **oscillations** by the regular task structure.... Neural **oscillations**... Group mean time–**frequency** representations of oscillatory power changes, relative to the mean power over the epoch. Data of control subjects are on the left (predictable (A) and random (B) conditions). Data of PD patients are in the right column (predictable (C) and random (D) conditions). Time–**frequency** data are mean spectral power values over ROIs represented in (E). T=0 indicates onset of the stimulus requiring a contralateral hand response.

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A simple harmonic constituent model based on the three **frequencies** of the Schwabe sunspot cycle: (A) Eq. (6); (B) Eq. (8). The figures also depict two reconstructions of solar activity based on 10Be and 14C cosmogenic isotopes (Bard et al., 2000; Steinhilber et al., 2009). The millennial harmonic Eq. (9) is also depicted in (B). The solar/planetary model show clear quasi-periodic multi-decadal beat cycles of low and high activity with a period of about 110–120 years and a quasi-millennial beat cycle, which are well synchronized to the cycles observed in the solar reconstructions. The units are generic, but would correspond to W/m2 at 1AU for Steinhilber's solar reconstruction.
... Proposed solar harmonic reconstructions based on four beat **frequencies**. (Top) Average beat envelope function of the model (Eq. (18)) and (Bottom) the version modulated with a millennial cycle (Eq. (21)). The curves may approximately represent an estimate average harmonic component function of solar activity both in luminosity and magnetic activity. The warm and cold periods of the Earth history are indicated as in Fig. 7. Note that the amplitudes of the constituent harmonics are not optimized and can be adjusted for alternative scenarios. However, the bottom curve approximately reproduces the patterns observed in the proxy solar models depicted in Fig. 5. The latter record may be considered as a realistic, although schematic, representation of solar dynamics.
... Power spectrum analysis of the monthly average sunspot number record. (A) We use the multi-taper method (MTM) (solid) and the maximum entropy method (MEM) (dash) (Ghil et al., 2002). Note that the three **frequency** peaks that make the Schwabe sunspot cycle at about 9.98, 10.90 and 11.86 years are the three highest peaks of the spectrum. (B) The Lomb-periodogram too reveals the existence of the same three spectral peaks.
... Comparison between carbon-14 (14C) and beryllium-10 (10Be) nucleotide records, which are used as proxies for the solar activity, and a composite (HSG MC52-VM29-191) of a set of drift ice-index records, which is used as a proxy for the global surface temperature throughout the Holocene (Bond et al., 2001). The three black harmonic components represent Eq. (19) of the solar/planetary 983-year beat modulation whose phase, P123 = 2059.7 A.D. is calculated using Eq. (7). Note the relatively good correlation between the three solar/climate records and the modeled harmonic function. The correlation coefficients with the 983-year harmonic function are: 14C record, r0=0.43 for 164 data (P(|r|≥r0)**frequencies** at periods >1800 years are removed with a Gaussian filter.
... Modulated three-**frequency** harmonic model, Eq. (8) (which represents an ideal solar activity variation) versus the Northern Hemisphere proxy temperature reconstruction by Ljungqvist (2010). Note the good timing matching of the millenarian cycle and the 17 115-year cycles between the two records. The Roman Warm Period (RWP), Dark Age Cold Period (DACP), Medieval Warm Period (MWP), Little Ice Age (LIA) and Current Warm Period (CWP) are indicated in the figure. At the bottom: the model harmonic (blue) with period P12=114.783 and phase T12=1980.528 calculated using Eq. (7); the 165-year smooth residual of the temperature signal. The correlation coefficient is r0=0.3 for 200 points, which indicates that the 115-year cycles in the two curves are well correlated (P(|r|≥r0)<0.1%). The 115-year cycle reached a maximum in 1980.5 and will reach a new minimum in 2037.9 A.D. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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Direct numerical simulation (DNS) of hydrogen–air turbulent swirling premixed flame in a cuboid combustor is conducted to investigate thermoacoustic instability and dynamic modes of turbulent swirling premixed flame in combustors. A detailed kinetic mechanism and temperature dependency of physical properties are considered in DNS. Two swirl number cases of 0.6 and 1.2 are investigated. Large-scale helical vortical structures are generated near the inlet of combustion chamber, and a lot of fine-scale vortices emerge downstream. Flame structure is engulfed by the vortical structures and depends largely on the swirl number. Spectral analysis of pressure **oscillation** on walls shows that quarter-wave mode of longitudinal acoustics has the largest energy, and that characteristic **oscillations** are found at higher **frequencies**. To investigate **oscillation** modes of pressure and heat release rate fields, dynamic mode decomposition (DMD) is applied to DNS results. It is shown that there are differences between dominant **frequencies** in spectral analysis of time-series pressure data at one point of walls and DMD of time-series pressure field data. DMD of pressure field reveals that the quarter-wave mode in longitudinal acoustics has the largest energy for the case of S=0.6. Furthermore, it is clarified that the transverse acoustic plane waves and pressure **oscillations** induced by large-scale vortical motions play important roles for the pressure **oscillations** in combustors. DMD of heat release rate field reveals that the DMD modes of pressure with high amplitude do not necessarily have coupling with fluctuations of heat release rate. Interactions between dynamic modes of pressure and heat release rate are also discussed.

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(a) X and Y displacements of particle and the magnitude of the 3f current against the **oscillation** amplitude, with db=5.6μm and β=37.5°. (b) X and Y displacements of the particle and the phase of 3f current against **oscillation** amplitude, with db=5.6μm and β=37.5°.
... (a) 3f real and imaginary components of current with streptavidin-coated polystyrene microbeads show ‘wiggles’ resembling the **oscillations** in these components observed from modelling. (b) Magnitude of 3f current due to surface–particle interaction from model and experiment plotted as a functional of inferred QCM displacement.
... (a) Positions of minima of the potential before and after rotation by an angle β=37°. (b) Sliding of the particle on the **oscillating** surface (db=5μm and β=37°).
... (a) X and Y displacements for a single particle and the magnitude of 3f current against **oscillation** amplitude for 3000 particles with db=5.6μm (all averaged over various crystal orientations). (b) X and Y displacements for a single particle and the phase of 3f current against **oscillation** amplitude for 3000 particles with db=5.6μm (all averaged over various crystal orientations.
... (a) X and Y displacements of particle and the real component of the 3f current against the **oscillation** amplitude, with db=5.6μm and β=37.5°. (b) X and Y displacements of the particle and the imaginary component of 3f current against **oscillation** amplitude, with db=5.6μm and β=37.5°.

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Mean power per condition (full/reduced) per experiment (1/2/3) in low-**frequency** range (A) and high-**frequency** range (B). Small dots linked by lines represent individual participants.
... Results of TF Analysis over significant electrodes per experiment. The left panels show the grand-average TF representations for the low-**frequency** range for the full conditions and the middle panels show the reduced conditions, all relative to a 500ms baseline. The right panels present difference topography plots.
... Results of TF Analysis over significant electrodes per experiment. The left panels show the grand-average TF representations for the high-**frequency** range for the full conditions and the middle panels show the reduced conditions, all relative to a 500ms baseline. The right panels present difference topography plots.
... Neuronal **oscillations**

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Inertial **oscillations**... Differential rotation rate of the sphere versus the amplitude of **oscillations** b (a) and the same dependency in the dimensionless form (b). The dashed line shows the dependence ΔΩ/Ωrot∼b2/R1δ obtained theoretically in the case of a cylindrical body in [39].
... The amplitude of the sphere **oscillations** with respect to the cavity shown on a photograph of the radially displaced sphere (a) and plotted versus rotation rate of the cavity for fluids of different viscosity (b).

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