Chaos Feigenbaum cascade

This type of transition to a chaotic state is called the Feigenbaum cascade. Such a scenario of generation of chaos involves consecutive losses of stability by successive orbits, see Fig. 103. 

The appearance of aperiodic oscillations beyond a point of accumulation of a cascade of period-doubling bifurcations is one of the best-known scenarios for the emergence of chaos (Feigenbaum, 1978 Berge et al, 1984). It is also along this way that chaos arrives in the multiply regulated enzyme model. Another example of this type of irregular... 

Moreover, the values of parameter for which period-doubling bifurcations occur before the onset of aperiodic oscillations define a sequence (Decroly, 1987a,b) characterized by a value close to that obtained by Feigenbaum (1978) for one of the universal constants associated with the cascade of period-doubling bifurcations leading to chaos. 

Another indication of the occurrence of chaos is given by the route leading to this irregular oscillatory behaviour in parameter space. Aperiodic oscillations indeed arise in the model after a cascade of period-doubling bifurcations as a function of parameter v, which measures the net rate of ATP supply to the adenylate cyclase reaction site. The successive values of parameter v corresponding to these bifurcations obey the universal scheme described by Feigenbaum (1978) for the onset of chaos. Thus, the values of v (in s" ) associated with the first three period doublings, from period 1 to period 8, are as follows ... 

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