Subharmonic

Lounis B, Jelezko F and Orrit M 1997 Single molecules driven by strong resonant fields hyper-Raman and subharmonic resonances Phys. Rev. Lett. 78 3673-6... 

The frequencies of a spectrum can be divided into two parts subharmonic and harmonic (i.e., frequencies below and above the running speed). The subharmonic part of the spectrum may contain oil whirl in the journal bearings. Oil whirl is identifiable at about one-half the running speed (as are several components) due to structural resonances of the machine with the rest of the system in which it is operating and hydrodynamic instabilities in its journal bearings. Almost all subharmonic components are independent of the running speed. 

R. Baumann, K. Kassner, C. Misbah, D. E. Temkin. Spatial subharmonics, irrational patterns, and disorder in eutectic growth. Phys Rev Lett 74 1597, 1995. 

This route should already be familiar to us from our discussion of the logistic map in chapter 4, Prom that chapter, we recall that the Feigenbaum route calls for a sequence of period-doubling bifurcations pitchfork bifurcations versus the Hopf bifurcations of the Landau-Hopf route) such that if subharmonic bifurcations are observed at Reynolds numbers TZi and 7 2, another can be expected at TZ determined by... 

This method was applied for the first time by L. Mandelstam and N. Papalexi in connection with the theory of the subharmonic resonance (Reference 6, p. 464). 

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). 

Subharmonic Resonance.—Another important nonlinear phenomenon is the so-called subharmonic resonance. In the linear theory the concept of harmonics is sufficiently well known so that it requires no further explanation other than the statement that these harmonics have frequencies higher than the fundamental wave. 

In the nonlinear systems, one often encounters subharmonics that have frequencies lower than that of the fundamental wave. As an example, consider a nonlinear conductor of electricity such as an electron tube circuit in which there exists between the anode current ia and the grid voltage v, a relation of the form... 

The first four are the ordinary harmonics, the remaining ones are the so-called combination tones and those of them whose frequencies are lower than the smaller of the frequencies a>i and o>2 are called subharmonics. By a proper choice of frequencies and nonlinearities one can obtain subharmonics of a very low order. 

Once the existence of subharmonics is ascertained, one can easily conclude that if one of them is near the period of the system, a corresponding subharmonic resonance must appear. Unfortunately while the physical nature of this phenomenon is simple, its mathematical expression is not. In fact, one is generally given, not the subharmonic, but the differential equation, and the establishment of the existence of a stable subharmonic is usually not a simple matter. 

The necessary condition for the subharmonic resonance occurs when the following condition is approximately fulfilled. 

L. Mandelstam and N. Papalexi were first to establish the theory of the subharmonic resonance based directly on the theory of Poinear6 (Section 6.18). The derivation of this theory, together with the details of the electronic circuits, is given in25, or in an abridged version in6 (pages 464-473). The difficulty of the problem is due to the fact that this case is nonautonomous so that conditions of stability are determined in terms of the characteristic exponents, which always leads to rather long calculations. 

This means that we are seeking a subharmonic resonance of order n. If (i — 0 the corresponding solution is... 

There are two distinct problems (1) the determination of the exact subharmonic resonance, and (2) the determination of the zone of that resonance. As problem (2) is more complicated than problem (1) we consider only the exact resonance (1). 

In order to simplify the problem still more, we shall consider the subharmonic resonance of the order (i.e., where n = 2) in connection with the differential equation... 

Hence the subharmonic resonance exists as long as the amplitude e of the external periodio excitation is not too large, namely ... 

It is necessary to show that the subharmonic resonance whose existence we have just ascertained is stable. Here the condition of stability is very simple, since in the stroboscopic method we deal with the stability of the singular point (and not of the stationary motion). 

Assume that we have a pendulum (Fig. 6-14) provided with a piece of soft iron P placed coaxially with a coil C carrying an alternating current that is, the axis of the coil coincides with the longitudinal axis OP of the pendulum at rest. If the coil is excited, one finds that the pendulum in due course begins to oscillate, and th oscillations finally reach a stationary amplitude. It is important to note that between the period of oscillation of the pendulum and the period of the alternating current there exists no rational ratio, so that the question of the subharmonic effect is ruled out. 

Resonance, nonlinear, 354 Resonance, subharmonic, 372,376 Rhombus rules, 85 Risk matrix, 315 Risk rule, 315... 

Subharmonic resonance, 372,376 exact values, 378 zone of, 378 Subharmonics, 376... 

The first experimental measurements of the time dependence of the hydrated electron yield were due to Wolff et al. (1973) and Hunt et al. (1973). They used the stroboscopic pulse radiolysis (SPR) technique, which allowed them to interpret the yield during the interval (30-350 ps) between fine structures of the microwave pulse envelope (1-10 ns). These observations were quickly supported by the work of Jonah et al. (1973), who used the subharmonic pre-buncher technique to generate very short pulses of 50-ps duration. Allowing... 

Figure 4 Back-to-back point contact diodes in a subharmonic mixer configuration.

As a direct consequence of the nonlinearity of the equations and of the boundary conditions there appears in (29) an infinite series introducing subharmonic terms that are a source of spatial asymmetry (see Fig. 12). The dominant contribution to this distortion is given by the term whose n is the odd closest to the value n — V3ju.. 

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. 

We get a very similar time behavior in subharmonic generation where/1 = 0 and /2 / 0. Self-pulsation and multiperiodic evolution of intensities have been found. However, these findings are not investigated here. 

Let us draw a plain H, tangent to dw at some point P with an outwards normal np. By the maximum principle for subharmonic functions... 

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