Hopf bifurcation first experience

We discussed some aspects of the responses of chemical systems, linear or nonlinear, to perturbations on several earlier occasions. The first was the responses of the chemical species in a reaction mechanism (a network) in a nonequilibrium stable stationary state to a pulse in concentration of one species. We referred to this approach as the pulse method (see chapter 5 for theory and chapter 6 for experiments). Second, we studied the time series of the responses of concentrations to repeated random perturbations, the formulation of correlation functions from such measurements, and the construction of the correlation metric (see chapter 7 for theory and chapter 8 for experiments). Third, in the investigation of oscillatory chemical reactions we showed that the responses of a chemical system in a stable stationary state close to a Hopf bifurcation are related to the category of the oscillatory reaction and to the role of the essential species in the system (see chapter 11 for theory and experiments). In each of these cases the responses yield important information about the reaction pathway and the reaction mechanism. 

The variety of dynamical behavior observed in this apparently simple system is truly remarkable. As r is increased, there are Hopf bifurcations to simple periodic behavior, then period-doubling sequences to chaos, as well as several types of multistability. One of the most interesting findings, shown in Figure 10.7, is the first experimental observation (in a chemical system) of crisis (Grebogi et al., 1982), the sudden expansion of the size of a chaotic attractor as a parameter (in this case, r) is varied. Simulations with the delayed feedback of eq. (10.48) taken into account give qualitative agreement with the bifurcation sequences observed in the experiments. 

We have presented the first clear evidence of quasiperiodicity in a chemical dynamical system. The shape of the torus as well ais its evolution is consistent with the nature of some generic instabilities of this chemical system. Some preliminary experiments in our laboratory indicate that similair tori co ild be obtained after a Hopf bifurcation of a limit cycle in perfect agreement with the theoretical predictions. 

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