Entrainment, with forced oscillations

For sufficiently large forcing amplitudes the oscillation becomes completely entrained, with a period exactly equal to one forcing period, whatever that value of a>/a>0. The entrainment may arise from a phase-locked response—as seen previously in Fig. 13.9—or from a quasi-periodic pattern. The boundary for full entrainment appears as an almost straight line with positive slope of oj/oj0 > 1 and negative slope for oj/oj0 < 1. 

Fig. 18. Forced oscillations with Pt(llO) harmonic entrainment. Modulation of the O, pressure by 1.2% with a frequency of 0.20 s which is close to that of the autonomous oscillations (0.16 s ). (From Ref. 93.) ptyi = 4 x I0"s torr, pm = 2 x 0 s torr, T = 530 K.

Fig. 21. Theoretical dynamic phase diagram for forced oscillations on Pt(l 10), evaluated from Eqs. (4), (5a), and (6). The calculations have been performed up to larger forcing amplitudes than were experimentally accessible (compare with Fig. 17). For small A, the bands characterizing entrainment and quasiperiodicity are discernible, whereas for larger A more complex bifurcations result. Types of bifurcations ns, Neimark-Sacker pd, period doubling snp, saddle-node of periodic orbits. (From Ref. 69.)...

Function q is 27r-periodic, and in the simplest case q -) = sin( ) Eq. (13.3) is called the Adler equation. One can easily see that on the plane of the parameters of the external forcing (cu, e) there exist a region eqmin < oo loq < eqmax, where Eq. (13.3) has a stable stationary solution. This solution corresponds to the conditions of phase locking (the phase 0 just follows the phase of the force, i.e. (f> = uit + constant) and frequency entrainment (the observed frequency of the oscillator Cl = (0) exactly coincides with the forcing frequency tu brackets () denote time averaging). 

We discuss now how the synchronization transition occurs, taking the applause in an audience as an example (experimental study of synchronous clapping is reported in [35]). Initially, each person claps with an individual frequency, and the sound they all produce is noisy.As long as this sound is weak, and contains no characteristic frequency, it does not essentially affect the ensemble. Each oscillator has its own frequency oJk, each person applauds and each firefly flashes with its individual rate, but there always exists some value of it that is preferred by the majority. Definitely, some elements behave in a very individualistic manner, but the main part of the population tends to be like the neighbor . So, the frequencies u>k are distributed over some range, and this distribution has a maximum around the most probable frequency. Therefore, there are always at least two oscillators that have very close frequencies and, hence, easily synchronize. As a result, the contribution to the mean field at the frequency of these synchronous oscillations increases. This increased component of the driving force naturally entrains other elements that have close frequencies, this leads to the growth of the synchronized cluster and to a further increase of the component of the mean field at a certain frequency. This process develops (quickly for relaxation oscillators, relatively slow for quasilinear ones), and eventually almost all elements join the majority and oscillate in synchrony, and their common output - the mean field - is not noisy any more, but rhythmic. 

The response to external forcing with frequency and amplitude A may be classified as follows [31-33] If the resulting period Tj. of the system exhibit a fixed phase relation to that of the modulation Tex, the system is entrained. The ratio Tr/Tex may be expressed as that between two small numbers, that is, Tr/Tex = k/l. For k/l =, the entrainment is called harmonic, for k/l> super harmonic, and for k/lphase difference between response and modulation varies continuously, the oscillations are called quasi-periodic. 

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