Harmonic entrainment

Harmonic entrainment, Pt, 37 237-238 Hartmann-Hahn condition, 33 210 Hartree-Fock... 

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.

In Fig. 16.9 we plot the dissipation in an oscillatory state, averaged over one oscillation, minus the dissipation in the unstable stationary state from which the oscillations arise, divided by that same dissipation. To and T are the periods of the self-sustained oscillation and the period of the external perturbation, respectively. The two circles on the curve near Tq/T = 1 indicate the extent of this fundamental entrainment band, and the two circles near Tq/T = 2 indicate the extent of this sub-harmonic entrainment band. The dissipation varies substantially in the fundamental entrainment band, and varies in the sub-harmonic entrainment band, but less so. Fundamental entrainment occurs in the range 0.85 < Tq/T < 1.25, which agrees well with the experimental findings 0.83 < To/T < 1.42, [11]. 

In a series of experiments we have tested the type and range of entrainment of glycolytic oscillations by a periodic source of substrate realizing domains of entrainment by the fundamental frequency, one-half harmonic and one-third harmonic of a sinusoidal source of substrate. Furthermore, random variation of the substrate input was found to yield sustained oscillations of stable period. The demonstration of the subharmonic entrainment adds to the proof of the nonlinear nature of the glycolytic oscillator, since this behavior is not observed in linear systems. A comparison between the experimental results and computer simulations furthermore showed that the oscillatory dynamics of the glycolytic system can be described by the phosphofructokinase model. 

Lattice modes, or external modes, as well as internal modes produce harmonic displacements of the atoms. The internal modes result in local molecular deformation, whilst external modes are supposed to entrain the atoms when a molecule is rigidly displaced from its equilibrium position in the lattice. The time-dependent atomic position vector r can be expressed in terms of the internal displacement vector u nx, taken with respect to the molecular centre of mass, and the external displacement vector, Hext the displacement vectors have units of length. A, Eq. (A2.52). 

The breakthrough experiment was carried out by Whitham et al. [39,40] in France. They used a Smalley-type laser vaporization source (Fig. 4) to provide a molecular beam of Ca atoms entrained in He or Ar gas. The second harmonic (532 nm) from a pulsed Nd YAG laser was focused (Fig. 4) on a rotating calcium rod. About 500 jus prior to this, a pulsed valve (left side of Fig. 4) is opened and the plume of vaporized metal is entrained in Ar or He gas. The carrier gas is seeded with a few percent of the oxidant such as H20. The plume of excited- and ground-state metal atoms are carried down a short channel and react with the oxidant. At the end of the channel, the product molecules such as CaOH expand into the vacuum chamber and cool. After a short expansion, the pressure has dropped so low that the molecules are effectively in a collisionless, ultracold (<10K) environment. 

Fig. 2.29. Entrainment of the oscillations by a periodic source of substrate in the allosteric model for glycolytic oscillations. The period of the sinusoidal source and the resulting period of the oscillatory enzyme are denoted by T and T, respectively. Domains C, B and A correspond to entrainment by the fundamental frequency, and by the 1/2 and 1/3 harmonics of the forcing input (Boiteux et al, 1975).

Numerical integration of eqns (2.7) in which source v is given by relation (2.32) permits the construction of a diagram (fig. 2.29) where the main domains of entrainment of the oscillatory enzyme by the external periodicity are indicated. For a constant source v = 0.5 s" equal to the mean value chosen for the periodic input, the autonomous period of the system is equal to 406 s. Domains C, B and A indicate, respectively, entrainment of the oscillatory enzyme to the fundamental frequency of the external source T=T), and entrainment by the harmonics i (T=2T ) and J (T=3T ). In ill these cases, the period and the phase of the oscillations with respect to the source remain constant after entrainment. 

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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