Reverse period-doubling bifurcation

Figure 37 shows a bifurcation diagram formed by plotting the value of the variable A in this Poincare section as a function of the bifurcation parameter kx- The 51151 state can be seen to go through a series of period-doubling bifurcations before the chaotic state appears. At the other end of the chaotic region, the state abruptly appears, with no evidence of a reverse... 

Fig. 2. Bifurcation diagram as a function of H2 showing period-doubling and reverse period-doubling cascades to chaos.

The moment dynamics method allows us to quickly discover the bifurcation diagram. It is given in Fig. 1.22. We see that the transition from stable fixed point of the mean to spiking and back to a stable fixed point is exceptionally complicated. We see a period-doubling cascade followed by a Canard explosion of the chaotic attractor, then a reverse period adding sequence towards a state of uninterrupted spiking. At higher noise inten-... 

The Willamowski-RSssler rate equations yield both periodic and chaotic attractors as the system parameters are tuned [10,21]. One common scenario for the appearance of a chaotic attractor is a cascade of ppriod-doubling bifurcations. As can be seen in the bifurcation diagram in Figure 2, the Willamowski-Rossler reaction possesses such a bifurcation sequence the chaotic attractor is followed by a reverse cascade of period doublings leading, ultimately, to a steady state. 

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