Candidate of science (physics and mathematics) dissertation.

Specialty 01.04.18 – Surface Physics and Chemistry.

O.O. Chuiko Institute of Surface Chemistry of the National Academy of Sciences of Ukraine, Kyiv, 2008.

The Hopf and Turing instabilities as main symmetry-breaking instabilities of nonequilibrium systems are studied. The two model - the FitzHugh-Nagumo system as a canonic reaction-diffusion model of activator-inhibitor type and the model electrochemical system with electrocatalytic reaction on spherical microelectrode are chosen. These are so-called N-systems. In the FitzHugh-Nagumo model the activator nullcline has the N-form which allows the system to reproduce such an important property of nonequilibrium systems as excitability and bistability. In the second model the voltamperometric curve has the N-form with the region of negative differential resistance where discussed instabilities arise. In this system the electrode potential plays the role of activator and the concentration of electroactive particles plays the role of inhibitor. For the point-like FHN system as an empirical model of excitable biological membranes on the base of the Hopf bifurcation theory the conditions of stable periodic oscillations under constant stimulation current were obtained. The dependence of the oscillation period on this constraint was calculated. Using computational simulation the response pattern of the FHN system on the periodic train of current pulses with different shape, amplitude, duration and period of stimulation is investigated. For the reduced one-dimensional FHN model in a certain range of constant external current the exact solution is found. The solution is the kink (hyperbolic tangent) that propagates with constant velocity which depends on the values of constant external current, activator diffusivity and excitation level of the system. The question of appearance and behavior of one-dimensional and two-dimensional spatiotemporal patterns caused by the Hopf and Turing instabilities was examined. Using impedance spectroscopy method the conditions for the Hopf instability in the model electrochemical system with one sort of electroactive species are derived. Mass transfer of the particles was considered in the frame of the Nernst diffusion model, implied that the diffusion layer has the same thickness around spherical electrode. It was shown that under potentiostatic control the oscillations emerge when faradaic impedance approaches its zero value from the side of the negative values of Re(Z_f(w)) when w®¥. The bifurcation frequency values w depend on the electrode size and the thickness of the layer from which electroactive species diffuse. There are two bifurcation point (two frequencies), where real and imaginary parts of faradaic impedance are equal zero. The potential region, where the instability appears, decreases with decreasing electrode size. Such decrease of the electrode radius at the Hopf bifurcation point causes the negative faradaic impedance to drop while the capacity loop undergoes a slight increase. The dependence on the thickness of the Nernst diffusion layer is observed at low frequencies in the negative faradaic impedance region. In the case of semi-infinite diffusion the region of the negative values of faradaic impedance real part increases in comparison with the case of diffusion from a thin layer for the same radius value.

Key words: nonequilibrium patterns, reaction-diffusion system, FitzHugh-Nagumo model, electrocatalytic surface reactions, Hopf and Turing instabilities, faradaic impedance, Nernst diffusion layer, spherical microelectrode.