Piecewise-linear oscillator

These phenomena are being actively studied at the present time, and constitute a new chapter in the theory of oscillations that is known as piecewise linear oscillations. There exists already a considerable literature on this subject in the theory of automatic control systems11-34 but the situation is far from being definitely settled. One can expect that these studies will eventually add another body of knowledge to the theory of oscillations, that will be concerned with nonanalytic oscillatory phenomena. 

Optimal flow method, 261 Optimization non-constrained, 286 of functionals, 305 Ordinary value, 338 Orthogonalization, Schmidt," 65 Osaki, S., 664 Oscillation hysteresis, 342 Oscillations autoperiodic, 372 discontinuous theory, 385 heteroperiodic, 372 piecewise linear, 390 relaxation asymptotic theory, 388 relaxation, 383 Oscillatory circuit, 380 "Out field, 648 existence of, 723... 

The main effect of this conditioning function is to prevent the rapid pH shift between 3 and 6 pH, giving rise to a sharp kick in the control response which could otherwise cause oscillations. The input conditioning is assumed to be implemented as a piecewise linear function, as this is the method generally available in control software. 

Fig. 4.25. Piecewise linear map constructed on the basis of the results of fig. 4.23 to account for complex periodic oscillations of the bursting type. The onedimensional return map x +j=/(x ) is defined by eqns (4.5) which contain the three parameters M, a and b. The arrowed trajectory corresponds to the simple pattern of bursting with three peaks per period Tr(3). Parameter values are a = 6,b = 5,M=ll (Decroly Goldbeter, 1987).

The piecewise linear map also allows us to comprehend how the variation of another parameter, such as b, can elicit the transition from complex periodic oscillations to simple periodic behaviour. For such a transition to occur, it suffices that the segment/2(x ) acquire a less negative slope, so that the fixed point x of the return map becomes stable. 

The second approach, successfully followed in the analysis of complex oscillations observed in the model of the multiply regulated biochemical system, relies on a further reduction that permits the description of the dynamics of the three-variable system in terms of a single variable only, by means of a Poincare section of the original system. Based on the one-dimensional map thus obtained from the differential system, a piecewise linear map can be constructed for bursting. The fit between the predictions of this map and the numerical observations on the three-variable differential system is quite remarkable. This approach allows us to understand the mechanism by which a pattern of bursting with n peaks per period transforms into a pattern with (n + 1) peaks. 

The piecewise linear map does not account, however, for the appearance of chaotic behaviour. A slight modification of the unidimensional map, taking into account some previously neglected details of the Poincare section of the differential system, shows how chaos may appear besides complex periodic oscillations of the bursting type. 

In arteries, pressure is usually positive with small oscillations about a nominal operating point (point A in Figure 10.2). Hence the piecewise linear approximation can be reduced to a single line with constant slope 1/C ... 

Fig. 7. (a) The amplitudes and (b) the period T of the oscillations found from integrating Eq. (27), N= 3, for varying values of n. Except for the value n = 4, global limit cycle attractors were found. The arrows on the right-hand side of the diagrams represent the theoretical values for the piecewise linear equation in the limit The arrow on the left-hand ride of (b) is the period predicted by the Hopf bifurcation theorem. 

Since 5>0, d[fip),f(q)]contraction mapping (see Appendix) and there must be a unique stable limit cycle in the four regions. In Fig. 14, we give this construction for the parameters used to compute Fig. 12. Although there is good agreement between the dynamics in the piecewise linear and the continuous equations, no proof of stable limit cycle oscillations has been found for Eq. (48) or (50). However, there has been a recent proof for the existence of nonlocal periodic solutions of Eq. (45) using fixed-point methods. ... 

The EGA in human small intestine was modeled using a chain of 100 bidirectionally coupled relaxation oscillators [Robertson-Dunn and Linkens, 1976]. Goupling was nonuniform and asymmetrical. The model featured (1) a piecewise linear decline in intrinsic frequency along the chain, (2) a piecewise linear decline in coupling similar to that of the intrinsic frequency, (3) forward coupling which is stronger than... 

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