Subharmonic entrainment

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. 

As indicated in fig. 2.29, subharmonic entrainment at a fraction of the fundamental frequency, which is characteristic of nonlinear systems (Hayashi, 1964), occurs in smaller and smaller domains of the external period as the value of that fraction diminishes. A similar phenomenon is observed in the entrainment of circadian rhythms by light-dark cycles (Pittendrigh, 1960,1965) the latter property imderlies the adaptation of most living organisms to periodic variations in their environment. 

FIGURE 7.11. Time series for periodically forced oscillations in the CO oxidation on a Pt(l 10) surface exhibiting sustained oscillations [34]. (a) 1 2 Subharmonic entrainment, (b) 2 1 Superharmonic entrainment, (c) 7 2 Superharmonic entrainment, (d) Quasi-periodic response. 

If at is an integer, say a> = n we have rather complicated manifestations of the subharmonic resonance. If it is at n + e, where e is a certain small number, one has still the subharmonic resonance, but it is accompanied by another phenomenon of synchronization, which con-, gists in the entrainment of the frequency of the avtoperiodic oscillation (if A = 0), by that of the heteroperiodic oscillation (the externally applied one). 

We now consider the phenomenon of entrainment (the development of resonances) on the torus (Meyer, 1983). When (and if) the off-diagonal band in Fig. 6 crosses the diagonal [Figs. 6(c) and 6(f)], there exist points whose images fall on themselves they are fixed points of the map we study. These points lie on periodic trajectories that are locked on the torus. Such trajectories appear in pairs in saddle-node bifurcations and are usually termed subharmonics . When this occurs there is no quasi-periodic attractor winding around the torus surface, but the basic structure of the torus persists the invariant circle is patched up from the unstable manifolds of the periodic saddle-points with the addition of the node-periodic point (Arnol d, 1973, 1982). As we continue changing some system parameter the periodic points may come to die in another saddle-node bifurcation (see Fig. 5). Periodic trajectories thus... 

It is interesting to consider the shapes of the subharmonic trajectories that lock on the torus in the various entrainment regions of order p/q. The subharmonic period 4 at the 4/3 resonance horn is, for example, a three-peaked oscillation in time [Fig. 7(a)] and has three closed loops in its phase-plane projection [Fig. 7(b)], while the subharmonic period 4 at the 4/ 1 resonance is a single-peaked, single-loop oscillation [Figs. 7(d) and 7(e)]. A subharmonic period 2 at the 2/3 resonance is also included in Figs. 7(g) and 7(h). Multipeaked oscillations observed in chemical systems (Scheintuch and Schmitz, 1977 Flytzani-Stephanopoulos et al., 1980) may thus result from the interaction of frequencies of local oscillators. Such trajectories are the nonlin-... 

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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