Rhythms as limit cycles and temporal dissipative structures

2 Rhythms as limit cycles and temporal dissipative structures 

In the phase space formed by the concentrations of the chemical variables involved in the reaction, sustained oscillations correspond to the evolution towards a closed curve called a limit cycle by Poincare, 

By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in the following chapters. Similarly, the phase space analysis of two-variable models indicates that the oscillatory dynamics of neurons corresponds to the evolution towards a limit cycle (Fitzhugh, 1961 Kokoz Krinskii, 1973 Krinskii Kokoz, 1973 Hassard, 1978 Rinzel, 1985 Av-Ron, Parnas Segel, 1991). A similar evolution is predicted (May, 1972) by models for predator-prey interactions in ecology. 

The seventies saw a true explosion of theoretical and experimental studies devoted to oscillating reactions. Nowadays, this domain continues to expand as more and more complex phenomena are observed in the experiments or predicted theoretically. The initial impetus for the study of oscillations owes much to the concomitance of several factors. The discovery of temporal and spatiotemporal organization in the 

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