Feigenbaum Route

This route should already be familiar to us from our discussion of the logistic map in chapter 4, Prom that chapter, we recall that the Feigenbaum route calls for a sequence of period-doubling bifurcations pitchfork bifurcations versus the Hopf bifurcations of the Landau-Hopf route) such that if subharmonic bifurcations are observed at Reynolds numbers TZi and 7 2, another can be expected at TZ determined by 

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Feigenbaum route Infinite cascade of periodic Chemical reaction... 

The bifurcation diagram of Figure 2.15 is often referred to as the Feigenbaum route to chaos. It applies also to reactions such as CO + O2 and NO + H2 on platinum, as Figure 2.16 shows. The single period oscillation between two states corresponds to point Xi in Figure 2.15, the double period oscillation between four states at A = A2, a four period oscillation between eight states (A3) and the aperiodic, unpredictable oscillation of an essentially chaotic system. 

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 ... 

The route followed by the system to aperiodic oscillations is through a cascade of period-doubling bifurcations as described by Feigenbaum (1978) in a totally different context. This illustrates again the universality of nonequilibrium dynamic behaviour in nonlinear systems, regardless of the evolution laws governing these systems. 

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