Antonis Karantonis

GENERALIZED HOPF, SADDLE-NODE INFINITE PERIOD BIFURCATIONS AND EXCITABILITY DURING THE ELECTRODISSOLUTION AND PASSIVATION OF IRON IN A SULFURIC ACID SOLUTION

Periodic current oscillations of relaxation type are observed in a certain potential range during iron electrodissolution in sulfuric acid solutions when the iron electrode turns from the active to the passive state and vice versa. The system displays in the current-potential (I-E) plane a hysteresis loop consisting of the stable active and passive steady states and an unstable steady state which corresponds to an intermediate situation between active and passive states. In the present study, the dynamical characteristics of the I-E polarization curve were studied. It was shown that transition from the oscillatory to the active steady state goes via a generalized Hopf bifurcation, whereas transition from the oscillatory to the passive steady state goes via a saddle-node infinite-period (SNIPER) bifurcation. Single pulse and pulse series stimulation of the active and passive steady states proved that both steady states are excitable. The refractory periods and the thresholds of excitability were measured. Further perturbation experiments were carried out at the two steady states away from the bifurcation points, and it was shown that the active steady state is a stable focus and the passive one is a stable node.

Dynamical response of the sinusoidally perturbed electrodissolution/passivation of iron in sulfuric acid solutions: Entrainment, spike generation, and quasiperiodicity

The iron/sulfuric acid (Fe/2 M H(2)SO(4)) system exhibits periodic current oscillations of relaxation type within the potential transition region formed between the active and passive states of the iron electrode when it is polarized in the 2 M sulfuric acid solution. In the present work the dynamical response of the Fe/2 M H(2)SO(4) electrochemical oscillator is investigated when the applied potential at the iron electrode is sinusoidally perturbed. The behavior of the periodically perturbed Fe/2 M H(2)SO(4) oscillator differs significantly from the response of other forced oscillators, as the potential amplitude E(p) and the frequency ratio omega(p)/omega(0) vary. The omega(p) and omega(0) are the angular frequencies of the perturbed applied potential and the unperturbed oscillator, respectively. A special feature of its response is the appearance of a number of spikes, generated within the passive section of a periodic oscillatory cycle for omega(p)/omega(0)<2.9, for periods of the autonomous oscillator T(0) greater, similar 3 s. The number of the generated spikes depends on the amplitude and frequency of the perturbed applied potential as well as on the period of the autonomous oscillator. Spikes are not generated for omega(p)/omega(0)=1 and the system is harmonically entrained by the forcing frequency. However, when the system is subharmonically entrained for omega(p)/omega(0) close to 2, spike generation does occur. By increasing the perturbation frequency for omega(p)/omega(0) greater, similar 2.9 and T(0) greater, similar 3 s, or by decreasing the autonomous period for T(0)<3 s and all the omega(p)/omega(0)<2.9 ratios, the spike generation pattern, is replaced by a quasiperiodic pattern. The dynamical response of the perturbed Fe/2 M H(2)SO(4) electrochemical oscillator is characterized by using time-delay reconstructions of the attractors, Poincare maps, and Fourier power spectra.

Coherence and coupling during oscillatory metal electrodissolution

The spatio‐temporal behavior during periodic current oscillations in the Fe H2SO4 system is studied for a one-dimensional electrode surface. It is found that during the oscillatory response the electrode surface is never in the same state at the same time but the surface potential forms an accelerating front. The mode of communication between the individual regions of the reacting surface is explored by dividing the system into two identical coupled oscillators. The response is studied by either dividing the electrode surface with an insulating film of infinitesimal thickness or by a physical partition. The behavior of the coupled oscillators can be simple in-phase synchronization, k:n phase locking, out-of-phase locking or complex, depending on the length of the insulating film or the use of the partition and the applied potential. The results are compared with a model considering a simple coupling through the electrode potentials of the single oscillators. It is proposed that the coupling patterns can be attributed to the electrical coupling and the distribution of the frequency of the individual oscillators.

Application of periodic forcing on the simplified Franck-Fitzhugh model for the electrochemical oscillations observed during the electrodissolution of iron in sulphuric acid solutions

We consider that the electrode potential in the simplified Franck-FitzHugh (F-F) model is externally forced by a periodic modulation. The F-F model is a two-dimensional model and describes the potential or current oscillations which are observed during the electrodissolution of iron in sulphuric acid solutions, under galvanostatic or potentiostatic conditions, respectively. The potential and the fraction of the surface coverage are the dynamical variables in the F-F model which reproduces the electrochemical oscillations of the system Fe/0.5 M H2SO4. By applying an external sinusoidal forcing potential, the two-dimensional system of F-F model becomes a three-dimensional one of non-linear equations which was integrated numerically for a wide range of forcing amplitude and frequency. The results of these integrations were interpreted by using various methods of the non-linear dynamics like phase plane, stroboscopic maps, time-delay reconstructions and Liapunov characteristic exponents. The forced F-F model shows different types of behaviour which depends on the frequency and amplitude of the forcing potential. For small forcing amplitude we observed a wide zone of quasi-periodic response interrupted by harmonic entrainment bands. For higher amplitudes the behaviour of the system consists of multi-periodic (subharmonic, harmonic) and quasi-periodic oscillations. The response of the forced F-F model is compared with the response of other forced systems.

Response of relaxation oscillatory electrochemical networks to external input

We have studied the response of simple networks of relaxation oscillatory electrode pairs to inputs through external cells. Such networks are shown to respond to constant inputs by decreasing or increasing the period depending whether the connection with the external cell is excitatory or inhibitory. The response to pulse inputs is determined by the phase of the relaxation oscillations; within the refractory region the network remains unaffected whereas within the excitable region an increase in the period is observed accompanied by the induction of new peaks.

Constructing Normal Forms from Experimental Observations and Time Series Analysis

A method is proposed for the construction of normal forms from experimental observations which describe the dynamics of a system close to the bifurcation points. The method is applied for the bifurcation of the Fe/2 M H2SO4 electrochemical system from a steady state to a chaotic attractor by considering the applied potential and the external ohmic resistance as bifurcation parameters. Steady state and time evolution curves of the response function are recorded. Perturbation experiments, time delay reconstruction of the attractors and calculation of power and Lyapunov spectra are performed. From the above experimental procedure the linear part of the normal form is constructed. The nonlinear part of the normal form is derived only from the knowledge of the linear part. Perturbations of the derived normal form on the bifurcation point are considered through the versal deformation of the normal form and the construction of the versal family of the normal form. The resulting normal form equations reproduce the dynamic characteristics and the bifurcation diagram of the electrochemical system.

Forced electrochemical oscillations of iron in sulphuric acid solutions at the transition between active and passive states

The effects of a periodically perturbed applied potential on the potentiostatic current oscillations observed during the transition of an iron electrode between the active and passive states in 2 M sulphuric acid solutions is studied experimentally in this work. Autonomous current oscillations appear during the electrodissolution of iron in sulphuric acid solutions when the iron electrode turns from the active to the passive state and vice versa. After the application of a potential perturbation the forced oscillations are conducted in a region with such a parameter space that the autonomous limit cycle is observed. The overall study was limited to the hysteresis region. The response of the oscillatory system Fe/2 M H2SO4 to an external perturbation of the potential was studied as a function of two parameters, the frequency, ωp, and the amplitude, Ep, of the forcing potential and it was characterized with the aid of time-delay reconstructions. The various types of response are presented in the phase diagram Ep-ωp/ω0. For low amplitudes of the potential perturbation and for both ωp/ω0 1 the response of the perturbed system exhibits wide zones of quasi-periodicity interrupted by entrainment bands which are harmonic or subharmonic. For higher amplitudes the entrainment bands become broad and as the forcing amplitude increases the system tends to be driven by the forcing frequency. From this experimental study on forced electrochemical oscillations of the system Fe/2 M H2SO4 it is seen that at very positive potentials for sustained periodic current oscillations to occur under stationary conditions, periodic or quasi-periodic current oscillations could be triggered by the local dissolution of the passive film with a potential pulse of a sufficient amplitude in order for the transition to the active state to be feasible. The resulting small uncovered part of the electrode surface allows metal electrodissolution and its propagation along the surface.

Phase flow deformations and coupled electrochemical oscillators

A simple two-dimensional model of coupled electrochemical oscillators is studied, where linear diffusive coupling of the potential causes dephasing of the oscillators. It is shown that coherence breaks down due to a strong deformation of the phase flow near the limit cycle. When a large number of such oscillators is coupled, the temporal response is bursting oscillations.

The combined unidirectional and local coupling in a spatially one-dimensional model of oscillatory metal electrodissolution. Patch-adaptive simulation study

Experimentally observed effects of a spatially localised external laser forcing, on the oscillatory current of the electrodissolution of an iron ring electrode in sulfuric acid, can be reproduced qualitatively by means of a simple, spatially one-dimensional model, expressed by partial differential equations for the double layer potential drop and hydrogen ions concentration, coupled with an ordinary differential equation for the coverage fraction by the passivating hydroxide. These model equations can be solved efficiently and economically by means of the patch-adaptive finite-difference strategy, that automatically concentrates the spatial and temporal grids in the critical regions. The analysis of the model solutions reveals that activation caused by the forcing occurs via accelerating moving fronts, resulting in spatially non-homogeneous distributions of the dynamical variables. When the oscillator is predominantly in the passive state, then a one-dimensional discrete map can be constructed, based on single perturbation simulations, which predicts, with a good quantitative agreement, the temporal behaviour of the electric current simulated under conditions of periodic forcing. However, when the oscillator is not predominantly in the passive state, a more complex spatio-temporal behaviour is revealed by the simulations. The spatial non-homogeneity of the dynamic variables is observed in the most profound way for the concentration which lacks any relaxation characteristics. This suggests that it is not only the double layer potential drop, but also the ionic concentrations, which may play a role in the development of spatio-temporal patterns in electrochemical systems.

Spatio-temporal laser forcing on an oscillatory/excitable electrochemical system

The effect of spatially localized, time-periodic forcing on the Fe∣1 M H2SO4 system is studied experimentally within the oscillatory and excitable region. The electrode surface is perturbed locally or globally by a laser beam. The resulting response depends strongly on the period and phase of the external forcing. When periodic laser perturbations are applied to the excitable state, the resulting response is large excitations for a large forcing period and pairs of small and large excitations for a small forcing period. During autonomous oscillations the response is entrained either completely or by combinations of autonomous and excitable peaks. A systematic procedure is applied, based on single perturbation experiments, for the construction of one-dimensional maps for the periodically forced oscillator. A mathematical model is proposed for the description of the dynamical phenomena induced by the laser light. The model equations predict most of the experimental findings.