Birhythmicity coexistence and transition between two simultaneously stable periodic regimes

2 Birhythmicity coexistence and transition between two simultaneously stable periodic regimes 

Let us now see how the nullcline deformation due to recycling gives rise to birhythmicity. Shown in fig. 3.6 are the bifurcation diagrams obtained as a function of parameter v for eight increasing values of the maximum rate of recycling, a , for a fixed value of constant K. In each part, the steady-state value of the substrate ao is indicated, as well as the maximum value qim reached by the substrate during oscillations. Solid lines denote stable steady-state or periodic solutions both types of solution are indicated by dashed lines when unstable. The stability properties of the steady state were determined by linear stability analysis of eqns 

Further increase in the rate of product recycling gives rise to a supercritical Hopf bifurcation corresponding to the creation of a stable, small-amplitude limit cycle. This phenomenon occurs at the border of the stability domain induced by recycling, inside the domain of existence of the large-amplitude limit cycle (fig. 3.6d). Birhythmicity arises from the coexistence of these two stable limit cycles. 

A typical example of birhythmicity in the phase plane is illustrated in fig. 3.7. The large-amplitude limit cycle encloses a limit cycle of more reduced amplitude these two stable cycles are separated by an unstable limit cycle. Birhythmicity is of interest because it allows the existence of two distinct oscillations in the same conditions, i.e. for a given set of parameter values. Moreover, the passage from one stable rhythm to the other can be effected by means of the same type of perturbation. Thus, 

The addition of an appropriate amount of substrate at the right phase thus allows the system to switch reversibly from one periodic regime to the other. The same transitions can of course occur in response to an addition of product, as suggested by the phase plane diagram of fig. 3.7. 

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