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Driven **qubit**

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Trajectories on the phase torus. (a) resonant two-**frequency** regime with winding number w=1:2, (b) three-**frequency** regime.
... (Color online). Chart of the Lyapunov’s exponents for the system of three coupled phase **oscillators** (5) for Δ2=1. The color palette is given and decrypted under the picture. The numbers indicate the tongues of the main resonant two-**frequency** regimes. These regimes are explained in the description of Fig. 5.
... Rotation numbers ν1–2 and ν2–3 versus the **frequency** detuning Δ1 for the system (11). Values of the parameters are λ1=1.3,λ2=1.9,λ3=1.8,Δ2=1.5 and μ=0.32.
... Subdivision of the chain of four **oscillators** by clusters for the three types of the phase locked pair of **oscillators**. Parameters are chosen in such a way that the system of **oscillators** is near the point of the saddle–node bifurcation.
... Schematic representation of a system of three coupled self-**oscillators**.
... Phase **oscillators**

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Spontaneously occurring fast ‘ripple’ **oscillations** (400–500Hz) in the neocortex of the rat during high-voltage spindles. (A) Averaged high-voltage spindles and associated unit firing histograms from layers IV–VI. (B) Wide-band (a and a′; 1Hz–5kHz), filtered field (b and b′; 200–800Hz), and filtered unit (c and c′; 0.5–5kHz) traces from layers IV and V, respectively. (C) Averaged fast waves and corresponding unit histograms. The field ripples are filtered (200–800Hz) derivatives of the wide-band signals recorded from 16 sites. Note the sudden phase-reversal of the **oscillating** waves (arrows) but locking of unit discharges (dashed lines). These phase reversed dipoles likely reflect synchronous discharge of layer 5 neurons in the vicinity of the recording electrode.
... Self-organized burst of activity in the CA3 region of the hippocampus produces a sharp wave sink in the apical dendrites of CA1 pyramidal neurons and also discharge interneurons. The interactions between the discharging pyramidal cells and interneurons give rise to a short-lived fast **oscillation** (‘ripple’; 140–200Hz), which can be detected as a field potential in the somatic layer. The strong CA1 population burst brings about strongly synchronized activity in the target populations of parahippocampal structures as well. These parahippocampal ripples are slower and less synchronous, compared to CA1 ripples.

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(A) Comparison of the **frequency** of **oscillations** during oblique, pure horizontal and pure vertical saccades. Number of observations is plotted on y-axis, while x-axis represents bins of **oscillation** **frequency**. Each data point represents the number of observations in a given **frequency** bin. Black trace suggests oblique saccade, Gray traces with circular symbols are horizontal saccades and triangular symbols represent vertical saccade. Dashed lines depict median **oscillation** **frequency**. (B) Comparison of **frequency** oblique saccade **oscillations** with the **frequency** of orthogonal saccadic **oscillations** during pure horizontal and vertical saccades. Each data point depicts one subject. Black data points are comparison with pure horizontal saccade, gray data points are comparison with vertical saccade. Dashed gray line is an equality line. (C) Comparison of the amplitude of the sinusoidal modulation of oblique, horizontal, and vertical saccade trajectories. Number of samples is plotted on y-axis, while x-axis represents the amplitude bins. Each data point depicts number of observations in a given bin of the histogram. Black trace shows oblique saccade, Gray trace with circuit symbol is a horizontal saccade and the triangular symbol is a vertical saccade. Dashed lines represent median values.
... An example of horizontal, vertical, and oblique saccade from one healthy subject. The left column depicts horizontal saccade; central column vertical, and right column is oblique saccade. Panels A, B and C illustrate eye position vector plotted along y-axis. Panels D, E and F represent eye velocity vector plotted along y-axis while ordinate in panels G, H and I illustrate eye acceleration. In each panel, x-axis represents corresponding time. Arrows in panels C, F, I show **oscillations** in oblique saccade trajectory.

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Free **oscillation** response of pendulum mechanism.
... Free **oscillation** response... Low **frequency**

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High-**frequency** **oscillations**... Time–**frequency** distributions. On the left side, the full 20–1000 Hz range is displayed for three exemplary subjects. The two graphs per subject show the ERG and VEP activity, respectively. The high-**frequency** **oscillations** appear as a distinct area which in most cases is around or above 100 Hz. The flash was given at t=0. Those parts of the time–**frequency** diagram which would be contaminated by edge effects are displayed in white. Their spread is due to the inevitable **frequency**-dependent finite time resolution, which also causes the spurious pre-stimulus activity at low **frequencies**. The white rectangles in the diagrams mark the regions of interest, which are shown enlarged on the right side for all 7 subjects. The arrows link the high-**frequency** maxima of ERG and VEP. Most subjects produced activity around or above 100 Hz in both VEP and ERG. However, only in one subject (S1) the **frequencies** matched. Asterisks indicate the significance levels of **frequency** differences in standard notation, based on a sequential Bonferroni adjustment. No significance value could be obtained for subject S3.

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Charts of the Lyapunov exponents for the four dissipatively coupled phase **oscillators** on the **frequency** detunings parameter plane (Δ1,Δ3). Values of the parameters are μ=0.4, (а) Δ2=0.4, (b) Δ2=2.4. Resonance conditions in the chain of **oscillators** are shown by arrows.
... Examples of phase portraits for the system (2). (a) Two-**frequency** resonance regime of the type 1:3 for Δ1=−1.5, Δ2=1, μ=0.6; (b) three-**frequency** regime for Δ1=−1, Δ2=1, μ=0.25.
... Chart of the Lyapunov exponents for three coupled van der Pol **oscillators** on the **frequency** detunings parameter plane. Numbers correspond to cycle periods in the Poincaré section. Values of the parameters are λ=0.1,μ=0.04.
... Chart of the Lyapunov exponents for three coupled van der Pol **oscillators** on the **frequency** detunings parameter plane. Numbers correspond to cycle periods in the Poincaré section. Values of the parameters are λ=1,μ=0.4.
... Chain of van der Pol **oscillators**... Full synchronization area for the four phase **oscillators** on the **frequency** detunings parameter space (Δ1,Δ2,Δ3).

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We consider the interaction between a single cavity mode and N≫1 identical **qubits**, assuming that any system parameter can be rapidly modulated in situ by external bias. It is shown that, for the **qubits** initially in the ground states, three photons can be coherently annihilated in the dispersive regime for harmonic modulation with **frequency** 3ω0−Ω0, where ω0 (Ω0) is the bare cavity (**qubit**) **frequency**. This phenomenon can be called “Anti-dynamical Casimir effect”, since a pair of excitations is destroyed without dissipation due to the external modulation. For the initial vacuum cavity state, three **qubit** excitations can also be annihilated for the modulation **frequency** 3Ω0−ω0.

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High-**frequency** **oscillations**... An example of the implantation schedule (patient #1) demonstrating areas with conventional **frequency** ictal patterns, ictal high-**frequency** **oscillations**, hyperexcitability, and radiological lesions.
... An example of the implantation schedule (patient #7) demonstrating areas with conventional **frequency** ictal patterns, ictal high-**frequency** **oscillations**, hyperexcitability, and radiological lesions.
... Summary table for statistical analysis. HFO=high **frequency** **oscillations**, CFIP=conventional **frequency** ictal patterns.

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