Multiple periodic attractors birhythmicity and trirhythmicity

It was before establishing the bifurcation diagram as a function of k that the first indication of a coexistence between two simultaneously 

This phenomenon of birhythmicity is observed in two domains of values of parameter k, at a fixed value of parameter v, as indicated by the bifurcation diagram of fig. 4.2 and by table 4.1. The coexistence occurs for the stable cycles LCl and LC2, or for LC2 and LC3. For larger values of constant k, the limit cycle LC2 can also coexist with solutions of period 2 (n = 2, 4, 8,16.) issued from LC3, or with chaos, to which this sequence of period-doubling leads. An example of such a coexis- 

The closeness of the two distinct regions of birhythmicity that involve the coexistence between the limit cycle LC2 with either LCl or LC3 suggests the possibility that the two domains may overlap for some values of the other parameters. This conjecture is indeed verified by varying the rate of substrate injection, the bifurcation diagram of fig. 4.2 can transform into that represented schematically in fig. 4.6. The region of coexistence between LC2 Md LC3 is here displaced tow ds lower values of parameter for which the limit cycle LCl subsists. Thus, in a 

Trirhythmicity is illustrated in fig. 4.7 where the three stable limit cycles are reached from distinct initial conditions, for the same set of parameter values (Decroly Goldbeter, 1984a). Initial conditions in parts a, b and c only differ, in the first decimal place, by variable a. Each of the stable cycles is attained after a brief excursion along an unstable limit cycle. 

The transition between the three stable periodic regimes can also occur, as in the case of birhythmidty, in response to a perturbation such as the addition of an appropriate quantity of substrate, provided that the perturbation occurs at the adequate phase of each oscillation (fig. 4.8). 

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