Forced Oscillations and the Stroboscopic Phase Plane

When the time dependence takes the form of a periodic perturbation of some parameter, we speak of this as periodic forcing.19 The response will obviously not be a steady state, but can be periodic, quasi-periodic, or chaotic. If the response is periodic, it may be with a period that is a multiple of the period of forcing. It is quasi-periodic if the response winds itself onto the cylinder in a helix whose pitch is an irrational multiple of the forcing period, so that it is never quite truly periodic. An example20 of a forced system is the Gray-Scott autocatalator with the feed concentration sinusoidally perturbed  

the average value of the inlet concentration has been retained as a characteristic quantity and A is the ratio of the amplitude of the forcing to this mean. (The frequency w = 2ttIt will also be used.) 

19 For a good discussion of the efficacy of concentration forcing, see a paper of that title by R. G. Rinker and R. Yadav. Chem. Eng. Sci. 44,2191-2195 (1989), and references there given. 

20 A comparative study was done by Kevrekidis and published as I. G. Kevrekidis, L. D. Schmidt, and R. Aris. Some common features of periodically forced reacting systems. Chem. Eng. Sci. 41,1263-1276 (1986). See also two papers by the same authors Resonance in periodically forced processes Chem. Eng. Sci. 41, 905-911 (1986) The stirred tank forced. Chem. Eng. Sci. 41,1549-1560 (1986). A full study of the Schmidt-Takoudis vacant site mechanism is to be found in M. A. McKamin, L. D. Schmidt, and R. Aris. Autonomous bifurcations of a simple bimolecular surface-reaction model. Proc. R. Soc. Lond. A 415,363-387 (1988) Forced oscillations of a self-oscillating bimolecular surface reaction model. Proc. R. Soc. Lond. A 415,363-388 (1988). 

FIGURE 29 The development of the stroboscopic phase-plane. Segments (a) and (d) show the trajectory settling down to a limit-cycle through a sequence of points at times that are multiples of t. This is drawn out in the time dimension in (b) and shown in its regularity in (c). If there is a unique periodic solution, the stroboscopic plane will show a sequence of states converging on (e). 

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