We apply the continuous delayed feedback method of Pyragas to the experimental control of chaos in the peroxidase-oxidase (PO) reaction. Unstable periodic 1' and l2 orbits embedded in the chaotic PO attractor were stabilized in CSTR experiments. The stabilization is demonstrated by a minimum in the experimental dispersion function and by the equality of the delay time z and the period of the stabilized attractor. In model calculations we compare the Pyragas method with the discontinuous control method by Ott, Grebogi, and Yorke (OGY). Experimental noise reduces the control efficiency of the OGY method much more severely than that of the Pyragas method precluding the experimental stabilization of unstable periodic orbits by the OGY method in the present PO system.

We present experiments and simulations of two mutually mass-coupled biochemical oscillators represented by the nonlinear peroxidase-oxidase reaction. The uncoupled oscillators show simple period-1 (P1) oscillations of different frequencies for different values of the oxygen concentration in the gas stream. A phase diagram is established where the ratio of the natural frequencies is plotted versus the mass exchange rate. For each frequency ratio four regimes of behavior have been observed for an increasing mass exchange rate:  (I) two independent and uncorrelated P1 oscillations; (II) quasiperiodicity in one and P1 oscillations in the other reactor; (III) quasiperiodicity in both reactors; (IV) periodic in-phase synchronization of both reactors. We use the correlation coefficient, the frequencies of the system, and the phase difference between the two cells to characterize the four regimes. A new feature of this work is the experimental observation of an “infinite” modulation period at the transition from the quasiperiodic region to the limit cycle that can be explained by a collision of a torus with a saddle-node bifurcation of a limit cycle (SNIPER bifurcation of codimension two). Simulations using two coupled four-variable DOP models give very good agreement with the experimental results and support our interpretation of the dynamical behavior and the observed bifurcations.

We apply the continuous delayed feedback method of Pyragas to control chaos in the enzymatic Peroxidase-Oxidase (PO) reaction, using the electric current as the control parameter. At each data point in the time series, a time delayed feedback function applies a small amplitude perturbation to inert platinum electrodes, which causes redox processes on the surface of the electrodes. These perturbations are calculated as the difference between the previous (time delayed) signal and the actual signal. Unstable periodic P1, 11, and 12 orbits (UPOs) were stabilized in the CSTR (continuous stirred tank reactor) experiments. The stabilization is demonstrated by at least three conditions: A minimum in the experimental dispersion function, the equality of the delay time with the period of the stabilized attractor and the embedment of the stabilized periodic attractor in the chaotic attractor.