Keywords: cellular automaton, CO oxidation, oscillatory behaviour, kinetic phase-transitions, control of chaos, periodic forcing.

The chaotic behavior in a chemical reaction can be controlled by means of the method of external forcing. This technique, which is based on the resonance phenomenon, converts chaotic behavior into a periodical one through application of a sinusoidal modulation. This paper analyzes the influence of a periodical perturbation on the environment temperature in a model of a cellular automaton (CA) for the catalytic oxidation of CO. The model includes the variation of the catalyst temperature in order to analyze the oscillatory behavior of the reaction. The results of the simulations show chaotic and quasiperiodical behaviors. We point out that the periodical forcing can remove the chaotic dynamics through the stabilization of periodical solutions.

Oxidation of CO by O on metallic surfaces is one of the best studied heterogeneous catalytic reactions. This reaction, which is a good example of system far from thermodynamic equilibrium, exhibits a rich variety of oscillatory behavior, ranging from a periodical form to a quasiperiodical and chaotic ones. The models proposed to describe this oscillatory behavior involves the nonlinearity of different kinds. So, there are models where the surface temperature and the environment temperature may be different, because the reaction heat may spread on the surface more quickly than it transfers to the surroundings, causing fluctuactions in the surface temperature. These oscillations are due to the strong nonlinear dependence of the reaction rate on temperature.

The technique of cellular automata (CAs) appears as an efficient tool for simulating the behavior of heterogeneous catalytic systems, because the CAs approximation makes easy parallel processing and diminishes significantly the computation time.

The periodical forcing of the kinetics of a reaction through the variation of some external control parameter is one of the most used tools for controlling chaos in the study of heterogeneous catalytic processes [1]. So, in experiments focused on the study of oscillatory catalytic reactions, periodical perturbations are applied to stabilize oscillations of period 1 (P1). This stabilization is easily reached if both external and internal frequencies coincide.

The model assumes that the reaction takes place according to three elementary mechanisms: adsorption of gaseous molecules CO and O₂ on a surface and reaction of the species adsorbed CO and O. The rates for each process are chosen in the Arrhenius form. In order to study the oscillatory behavior of the reaction, the model is completed adding an equation which describes the variation of the surface temperature due to the adsorption and reaction processes. The CA uses a square lattice of 256x256 sites with periodical boundary conditions. The lattice is updated in synchronous way according to probabilistic rules. The simulations of the CA show quasiperiodical and chaotic behaviors, in which the surface temperature fluctuates around a temperature greater than the environment one [2].

For the recognition of chaos we have proceeded in the usual way, studying the time series obtained from the simulations of the CA, through the building of recurrence graphics, Poincaré maps, Fourier transforms and other typical quantities of the nonlinear analysis such as the space-time entropy and the maximum Lyapunov exponent.

In the range of parameters in which the autonomous chaotic behavior exists, we perturb the environment temperature through a sinusoidal function with a single frequency, obtaining a kinetic phase diagram of the model in terms of the normalized amplitude (A) and the angular frequency () of the external perturbation. In the phase diagram it can be observed that when the perturbation applied to the chaotic autonomous region is weak, for low values of A, a periodical response does not appear. However, as perturbing amplitude increases, wider is the coupling band, that is to say, the range of external frequency resonant with the dynamical system for a fixed value of A increases as well. Thus, for intermediate values of A, oscillatory states P1 with frequency equal to the external one can appear: the Poincaré maps show cyclic orbits and characteristic peaks are observed in the Fourier power spectra. Finally, if A is great, the response of the system is always synchronized for any value of .

Thus, the strategy of control through periodical forcing is able to remove the chaotic dynamics through stabilization of periodical solutions, a perturbing harmonic function with a single frequency being enough to transform chaotic states in periodical ones P1. These results are in good agreement with studies performed of the oxidation of CO.

1

V.P. Zhdanov, "Periodic perturbation of the kinetics of heterogeneous catalytic reactions", Surface Science Reports, 55, 1-48, 2004. doi:10.1016/j.surfrep.2004.06.001

2

M.C. Lemos and F. Jiménez-Morales, "A cellular automaton for the modeling of oscillations in a surface reaction", Journal of Chemical Physics, 121, 3206-3211, 2004. doi:10.1063/1.1770455

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