The effects of a nonlinear delayed feedback on a chemical reaction

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Abstract
With delay feedback experiments on the minimal bromate oscillator, we show that chemical systems with delay display a variety of dynamical behavior. Using a nonlinear delayed feedback, we induce Hopf bifurcations, period doubling, bifurcations into chaos, and crisis (observed for the first time in a chemical system) into the system, which does not display this behavior without the delay. We test a conjecture [M. Le Berre, E. Ressayre, A. Tallet, H. M. Gibbs, D. L. Kaplan, and M. H. Rose, Phys. Rev. A 3 5, 4020 (1987)] that the dimension of a chaotic attractor is equal to τ/δ f , where δ f is the correlation time of the delayed feedback. Using the Grassberger–Procaccia algorithm [P. Grassberger and I. Procaccia, Phys. Status Solidi D 9, 189 (1983)] to calculate the dimensions of the chaotic attractors from the experimental system, we show that the calculated dimensions are less than those calculated by τ/δ f . We compare numerical integrations of the proposed mechanism for the minimal bromate oscillator with the experimental results and find agreement of the predicted bifurcation sequence with the experimental observations. The results of this study indicate that with appropriate delay feedback functions, and a sufficiently nonlinear dynamical system, it is possible to ‘‘push’’ a dynamical system into further bifurcation regimes, of interest in themselves, which also yield information on the system without delay.

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