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We note that sometimes definition (1) is understood in a very narrow sense (Formula presented) that is only valid for quasiharmonic oscillations and excludes, e.g., the obvious case of synchronization of two relaxational oscillators and other nontrivial phenomena
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We note that sometimes definition (1) is understood in a very narrow sense (Formula presented) that is only valid for quasiharmonic oscillations and excludes, e.g., the obvious case of synchronization of two relaxational oscillators and other nontrivial phenomena.
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The phase of a chaotic oscillator can be determined by means of three different approaches, see 16. Suppose we can define a Poincaré map for the autonomous continuous-time system. Then, we define the phase proportional to time between two cross sections with the Poincaré surface so that the phase increment is 2π at each rotation, (Formula presented) and (Formula presented) where (Formula presented) is the time of the (Formula presented) crossing of the secant surface. Quite often it is possible to find such a projection of a strange attractor that the phase portrait looks like rotations around some origin. In this case the phase can be determined as the angle between the projection of the phase point on the plane and a given direction on the plane. Finally, the phase of a properly chosen oscillatory observable computed via Hilbert transform, can be taken as the phase of the chaotic oscillator. Note that although the phases determined by different techniques often do not coincide microscopically, i.e., on a time scale less than the average period of oscillation, they have equal average growth rates. In other words, the mean frequencies defined as the average of (Formula presented) over a large period of time, coincide
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The phase of a chaotic oscillator can be determined by means of three different approaches, see 16. Suppose we can define a Poincaré map for the autonomous continuous-time system. Then, we define the phase proportional to time between two cross sections with the Poincaré surface so that the phase increment is 2π at each rotation, (Formula presented) and (Formula presented) where (Formula presented) is the time of the (Formula presented) crossing of the secant surface. Quite often it is possible to find such a projection of a strange attractor that the phase portrait looks like rotations around some origin. In this case the phase can be determined as the angle between the projection of the phase point on the plane and a given direction on the plane. Finally, the phase of a properly chosen oscillatory observable computed via Hilbert transform, can be taken as the phase of the chaotic oscillator. Note that although the phases determined by different techniques often do not coincide microscopically, i.e., on a time scale less than the average period of oscillation, they have equal average growth rates. In other words, the mean frequencies defined as the average of (Formula presented) over a large period of time, coincide.
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46
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85036288537
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Indeed, the heart rate is always several times larger than the frequency of respiration; therefore, the choice of the frequency ratio (Formula presented) for the purposes of illustration is reasonable; this is confirmed by experimental results presented in Sec. V
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Indeed, the heart rate is always several times larger than the frequency of respiration; therefore, the choice of the frequency ratio (Formula presented) for the purposes of illustration is reasonable; this is confirmed by experimental results presented in Sec. V.
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C. Schäfer, Ph.D. thesis, Potsdam University, 1998
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This observation was made by T. G. Anishchenko (private communication). This fact might be explained by age and gender differences in heart rate variability. Indeed, for young persons the latter is more pronounced in women, see K. Jensen-Urstad , Acta Physiol. Scand. 160, 235 (1997)
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due to the reduced sympathetic influence that is known to decrease the heart rate variability, see
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