Period piecewise-linear oscillator

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. 

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. ... 

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