54. Nagy, T.; Verner, E.; Gáspár, V.; Kori, H.; Kiss, I.::

 Delayed feedback induced multirhythmicity in the oscillatory electrodissolution of copper
Chaos: An Interdisciplinary Journal of Nonlinear Science, 2015, Art. No. 064608, doi:/10.1063/1.4921694f


Occurrence of bi- and trirhythmicities (coexistence of two or three stable limit cycles, respectively, with distinctly different periods) has been studied experimentally by applying delayed feedback control to the copper-phosphoric acid electrochemical system oscillating close to a Hopf bifurcation point under potentiostatic condition. The oscillating electrode potential is delayed by τ and the difference between the present and delayed values is fed back to the circuit potential with a feedback gain K. The experiments were performed by determining the period of current oscillations T as a function of (both increasing and decreasing) τ at several fixed values of K. With small delay times, the period exhibits a sinusoidal type dependence on τ. However, with relatively large delays (typically τ >> T) for each feedback gain K, there exists a critical delay τcrit above which birhythmicity emerges. The experiments show that for weak feedback, Kτcrit is approximately constant. At very large delays, the dynamics becomes even more complex, and trirhythmicity could be observed. Results of numerical simulations based on a general kinetic model for metal electrodissolution were consistent with the experimental observations. The experimental and numerical results are also interpreted by using a phase model; the model parameters can be obtained from experimental data measured at small delay times. Analytical solutions to the phase model quantitatively predict the parameter regions for the appearance of birhythmicity in the experiments, and explain the almost constant value of Kτcrit for weak feedback.

53. Bîrzu, A.; Gáspár, V.:

Complex spatio-temporal dynamics in metal electrodissolution: three dimensional cell geometry models
J. Comput. Interd. Sci., 2012, 3, 87-98,
doi:10.6062/jcis.2012.03.02.0054, pdf


Complex dynamics of current distribution on ring or disk working electrodes is studied numerically by modelling metal electrodissolution under potentiostatic control in a three dimensional electrochemical cell. Such spatially distributed systems exhibit extremely rich dynamics in the oscillatory regime, ranging from rotating pulses to spatio-temporal chaos. Several dynamical behaviors were selected and characterized using the Karhunen-Ločve (KL) decomposition and the analytical signal approach.

52. Úrvölgyi, M.; Gáspár, V.; Nagy, T.; Kiss, I. Z.:

Quantitative dynamical relationships for the effect of rotation rate on frequency and waveform of electrochemical oscillations
Chem. Eng. Sci.
, 2012, 83, 56-65, doi:10.1016/j.ces.2011.10.073



We investigate the effect of changing mass transfer conditions through variation of rotation rate of a rotating disk electrode on features of oscillatory dynamics of negative differential resistance electrochemical systems. The theoretical analysis and numerical simulation of a prototype two-variable electrochemical model show that for oscillations close to a Hopf bifurcation the frequency (w) increases with increase in rotation rate (d) following an approximate square root formula  w µd1/2. For relaxation oscillations, the oscillations maxima, minima, and transition points between the high- and low-current states do not depend on rotation rate; the oscillation waveform invariance is explained using nullcline analysis by showing that the rotation does not affect the nullcline of the fast variable (electrode potential) along which the oscillations occur. The numerical and theoretical predictions are confirmed in experiments with copper electrodissolution in phosphoric acid electrolyte using a rotating electrode setup. The results thus indicate that simplifying concepts related to invariant manifolds and parameter dependence of bifurcation points (principle of critical simplification) are efficient approaches to obtaining quantitative dynamical relationships for decoding complexity in electrochemical reaction systems.


E1. Gáspár, V.:

PhD Program in Chemistry at the University of Debrecen, Hungary

Magyar Kémiai Folyóirat, 2011, 117, 147-152. (in Hungarian), pdf



In this article, Vilmos Gáspár, Director of the PhD Program in Chemistry at University of Debrecen, Hungary, gives a detailed account on the educational and research program of the internationally recognized graduate school. The PhD Program is based on a credit system: the students have to collect a total of 180 credit points in 3 years of graduate education. These credit points should be earned by research (156 points), completing graduate courses (12 points), and teaching under-graduate courses (12 points). The PhD degree is issued after a successful oral exam generally in the special topics of the graduate courses taken by the student and a successful open defence of the PhD dissertation (thesis). It is a general requirement that the results presented in a dissertation should be based on 2-3 research papers published previously in peer-reviewed, well respected international journals of high impact. In a given semester, the average number of PhD students varies between 30 and 40. The majority of students are supported by governmental fellowships; however, there is also a possibility to earn the degree based on a personal program sponsored by outside sources, e.g. major companies.

In the school, we offer graduate education in six different research areas:
1. Reaction kinetics and catalysis
2. Coordination chemistry
3. Environmental and instrumental analytical chemistry
4. Macromolecular and surface chemistry
5. Chemistry and biochemistry of carbohydrates
6. Synthesis and structure examination of heterocycles of natural origin and their analogues

The high level of research and education in the PhD Program is the result of the joint effort of 58 professors of the Chemistry Department. Three professors are full members of the Hungarian Academy of Sciences (HAS), while 18 professors hold DSc. degree from the HAS. Since the beginning of the organized graduate education at University of Debrecen (1993), 231 students participated in the graduate school; 168 students completed the research and educational program, and 138 PhD degrees have been issued.

51. Gáspár, V.; Field, R. J.:
Comment on „
Simple KBrO3, H2SO4 Batch Oscillator”
J. Phys. Chem. A
, 2009, 113, 7979-7980,



Rachwalska reported oscillations in the redox potential at a Pt-electrode in the well-stirred, aqueous, batch system KBrO3 and H2SO4. The KBrO3 contained ~ 0.02 % Br impurity. We are concerned that the experimentally observed oscillations in redox potential result from some form of physical/chemical artifact and should be treated with care. The major cause of our concern is that we see no basis for their occurrence in homogeneous FKN oxybromine chemistry.


50. Kiss, I. Z.; Kazsu, Z.; Gáspár, V.:

Scaling Relationship for Oscillating Electrochemical Systems: Dependence of Phase Diagram on Electrode Size and Rotation Rate

Phys. Chem. Chem. Phys., 2009, 11, 7669–7677, pdf



Dynamics of oscillations in electrochemical systems are affected by both chemical and physical properties of the systems. Chemical properties include the type of electrochemical reaction, the electrode material, the composition of the electrolyte, etc., while physical properties include the solution resistance, the cell constant, the electrode size, the rotation rate, the external resistance, etc. Earlier, we proposed the application of cell-geometry-independent-phase-diagrams to characterize the oscillatory regions in the electrode potential vs. external resistance parameter plane. In this report, we investigate how this type of phase diagram changes with the surface area (electrode radius) and the rotation rate of an electrode. Based on linear stability analysis of a general, two-variable model for negative-differential resistance (NDR) type electrochemical oscillators we propose a scaling relationship. It predicts that all scaled data points derived from the critical values of parameters (resistance and potential) characterizing the onset of oscillations should fall – independently of the size of the electrode and the rotation rate – on a single plot. The analytical predictions are tested in both numerical simulations and experiments with copper electrodissolution in phosphoric acid.


49. Bîrzu, A.; Gáspár, V.:
Synchronization of electrochemical oscillators of S-NDR type

Electrochimica Acta,
2009, 55, 383-394, pdf

Abstract. Electrochemical oscillators are classified as truly potentiostatic or Negative Differential Resistance (NDR) type considering, respectively, the nonessential or essential role of double-layer potential on the dynamics. NDR type oscillators are further grouped into N-NDR and S-NDR type systems where N and S refer to the characteristic shape of the polarization curve. Here, we report on the effect of global coupling on the dynamics of interacting electrochemical oscillators of S-NDR type where the electrode potential acts as an essential negative feedback variable. The Lee-Jorné scheme has been applied to model Zn electrodeposition on two or an array of 128 globally coupled nonidentical electrodes. The strength of global coupling between the electrodes can be systematically varied by changing collective and individual resistors in the circuit. Several different scenarios for synchronization and partial synchronization have been observed and analyzed. The collective dynamics is characterized by an order parameter, the average phase shift between the oscillators, and phase portraits based on Hilbert transform. The effect of changing the double-layer capacitance on the dynamics is also explored.


48. Kiss, I. Z.; Kazsu, Z.; Gáspár, V.:
Tracking Unstable Steady States and Periodic Orbits of Oscillatory and Chaotic Electrochemical Systems Using Delayed Feedback Control
CHAOS, 2006, 16, 033109, pdf

Abstract. Experimental results are presented on successful application of delayed-feedback control algorithms for tracking unstable steady states and periodic orbits of electrochemical dissolution systems. Time-delay autosynchronization and delay-optimization with a descent gradient method were applied for stationary states and periodic orbits, respectively. These tracking algorithms are utilized in constructing experimental bifurcation diagrams of the studied electrochemical systems in which Hopf, saddle-node, saddle-loop, and period-doubling bifurcations take place.


47. Kiss, I. Z.; Kazsu, Z.; Gáspár, V.:
Experimental Strategy for Characterization of Essential Dynamical Variables in Oscillatory Systems: Effect of Double-layer Capacitance on the Stability of Electrochemical Oscillators

J. Phys. Chem. A
, 2005, 109, 9521-9527. pdf

Abstract.An experimentally accessible algorithm for changing the time scale associated with a dynamical variable is proposed. In general, a differential controller can be applied to a) identify the essential species in oscillatory systems and b) explore their role in the feedback loops. Here, we report on classifying electrochemical oscillators by changing the time-scale over which the electrode potential varies; the type of different electrochemical oscillators is identified based on whether the controlled modification of pseudo-capacitance induces or suppresses current oscillations.


46. Armstrong, G. R.; Taylor, A.; Scott, S.K.; Gáspár, V.:
Modelling wave propagation across a series of gaps
Phys. Chem. Chem. Phys., 2004, 6, 4677-4681. pdf


Abstract. Wave propagation across a series of gaps in a one-dimensional excitable medium is simulated using the Oregonator model of the Belousov-Zhabotinsky reaction. In agreement with recent experiments, we observe features such as the critical gap width Wcr, critical spacing between gaps Scr and frequency transformation of the passage of a train of waves across a gap with width W Ł Wcr. The role of activator kinetics in the gap is studied and the effect of excitability (through variation of parameters f and ε) on the fraction of waves which successfully cross the domain is determined. We also find that the probability of a wave successfully propagating through the entire domain decreases with increasing number of gaps, and the profile of the activator species is examined for evidence of a “weakening” effect in a multiple gap system.



45. Kheowan, O.-U.; Kantrasiri, S.; Uthaisar, C.; Gáspár, V.; Müller, S.C.
Spiral Wave Dynamics Controlled by a Square–Shape Sensory Domain
Chem. Phys. Letters
, 2004, 389, 140-144. pdf


Abstract. Spiral waves rotating rigidly in a thin layer of the light-sensitive Belousov-Zhabotinsky reaction are subjected to a time-dependent uniform illumination. A non-local feedback algorithm computes the illumination intensity to be proportional to the average wave activity within a square-shape sensory domain. The investigations show a broad spectrum of dynamical responses which results in square- and cross-shaped trajectories of the spiral tip, including reflections at the virtual walls. The geometry of the sensory domain is crucial in determining size and shape of the tip trajectories. A theoretical approach is proposed to explain the observed phenomena.



44. Bamforth, Tóth, R.; Gáspár, V.; Scott, S.K.
Scaling and Dynamics of “Flow Distributed Oscillation Patterns” in the Belousov-Zhabotinsky reaction
Phys. Chem. Chem. Phys., 2002, 4, 1299-1306. pdf


Abstract. The formation of “flow distributed oscillation” (FDO) patterns in the Belousov-Zhabotinsky (BZ) reaction is studied experimentally. We confirm the dependence of the pattern wavelength on the flow velocity and the concentration of the reactant species BrO3- and H+ as predicted in a previous study. We also report on the initial development of the FDO patterns. In contrast to simple interpretations, the patterns arise through a “wave splitting” mechanism in which a pair of counter-propagating wave pulses are created from a pacemaker site at some distance ahead of the pattern: one of these waves propagates with the flow and leaves the reactor, the other propagates against the flow and eventually settle to form the next band of the FDO pattern. This initiation mechanism is confirmed in numerical studies based on the Oregonator model. These computations also indicate the possibility of complex dynamics similar to the “resonance patterns” reported in studies of the propagation of BZ waves through capillary tubes.



43. Kheowan, O.-U.; Gáspár, V.; Zykov, V.S.; Müller, S.C.
Measurements of kinematical parameters of spiral waves in media of low excitability
Phys. Chem. Chem. Phys., 2001, 3, 4747-4752. pdf


Abstract. The dynamics of spiral waves rotating in a thin layer of the light sensitive Belousov-Zhabotinsky reaction mixture are studied under a homogeneous and steady illumination. At given composition of the excitable medium, the spiral waves are meandering, when no or low intensity light is applied, or rigidly rotating, when the light intensity is sufficiently increased. There exists, however,  a critical value of light intensity above which no wave activity is supported by the medium, since its excitability is too strongly reduced by the illumination. In the vicinity of this critical value the basic kinematical parameters of rigidly rotating spirals (such as the rotation period, wavelength, propagation velocity, and the diameter of the spiral core) are measured as a function of the illumination intensity. The experimental observations are in a good agreement with the predictions based on an earlier proposed kinematical theory of spiral waves in media of low excitability



42. Bamforth, J.R.; Merkin, J.H.; Scott, S.K.; Tóth, R.; Gáspár, V.
Flow-distributed oscillation patterns in the Oregonator model
Phys. Chem. Chem. Phys., 2001, 3, 1435-1438. pdf


Abstract. The conditions under which chemical patterns corresponding to “flow-distributed oscillations” are formed are determined analytically for the Oregonator model of the Belousov-Zhabotinsky reaction. These analytical results are confirmed by numerical computation and are also used to predict typical values for the critical flow velocity and how the wavelength varies with the concentrations of the major reactants.



41. Tóth, R.; Papp, A.; Gáspár, V.; Merkin, J.; Scott, S.K.; Taylor, A.
Flow-driven instabilities in the Belousov–Zhabotinsky reaction: Modelling and experiments
Phys. Chem. Chem. Phys., 2001, 3, 957-964. pdf


Abstract. The development of propagating patterns arising from the differential flow of reactants through a tubular reactor is investigated. The results from a series of experimental runs, using the BZ reaction, are presented to show how the wavelength and propagation speed of the patterns depends on the imposed flow velocity and the concentration of BrO3- in the inflow. A model for this system, based on a two-variable Oregonator model for the BZ reaction, is considered. A stability analysis of the model indicates that the mechanism for pattern formation is through a convective instability. Numerical simulations confirm the existence of propagating patterns and are in reasonable agreement with the experimental observations.



40. Rivera, M.; Madrigal, R.; Parmananda, P.; Eiswirth, M.; Nyikos, L.; Kiss, I. Z.; Gáspár, V.
Numerical analysis of spatio-temporal chaos in spatially extended systems and research on chaotic electrodissolution of metals
Rev. Mex. Fis., 2000, 46, 11-15. pdf


Abstract. We report results (numerical and experimental) indicating control of the observed oscillatory dynamics in an electrochemical system using external forcing and variable feedback. Under the influence of external forcing it was observed that the chaotic dynamics could be converted to periodic states. Moreover, regular states with different periodicities could be stabilized. Using a continuous delayed feedback control strategy chaotic oscillations are suppressed via stabilization of fixed point steady states and/or periodic dynamics. Finally we report numerical results indicating control of spatiotemporal chaos observed in a spatially extended chemical system. In one spatial dimension the model dynamics under appropriate parameter conditions exhibit chemical turbulence which is suppressed using feedback and forcing techniques.



39. Parmananda, P.; Madrigal, R.; Rivera, M.; Kiss, I.Z. and Gáspár, V.
Resonant Control of Electrochemical Oscillations
J. Phys. Chem. B , 2000, 104, 11748-11751. pdf


Abstract. We report experimental control of complex (periodic and chaotic) oscillatory dynamics in an electrochemical system by applying sinusoidal forcing. By choosing an appropriate frequency for the periodic modulation of an accessible control parameter (e.g., circuit potential) not only can the chaotic dynamics  be easily converted to regular periodic behavior (controlling chaos), but also the character of the oscillatory dynamics could be altered (for example, 11 -->10). This is different from the previously reported work involving entrainment of oscillatory dynamics, since in our experiments the frequency of sinusoidal modulation is chosen such that the existing unstable dynamics are targeted and subsequently stabilized. Consequently the control signal is less then 5 % of its base value. Since resonant control strategy can be easily implemented without a complicated pre-control procedure, it seems relevant for applications to real systems.



38. Davies, M.L.; Halford-Maw, P.A.; Hill, J.; Tinsley, M.R.; Johnson, B.R.; Scott, S.K.; Kiss, I.Z. and Gáspár, V.
Control of Chaos in Combustion Reactions
J. Phys. Chem. A, 2000, 104, 9944-9952. pdf


Abstract. The chaotic evolution in the combustion of CO in a well-stirred flow reactor is controlled experimentally using a modified form of the simple proportional feedback (SPF) algorithm. An unstable period-1 oscillation is stabilized through the imposition of small, appropriate perturbation which are calculated from the observed experimental response of the system and do not require any information concerning the reaction mechanism. It is observed that the algorithm is significantly more efficient if these perturbations are applied for only a fraction of the oscillatory period. A similar observation is made from a numerical study of a model for the H2 + O2 reaction and it is shown that this arises because the perturbations shift the system significantly from the attractor of the unperturbed system. The duration of the perturbation in each cycle then becomes a second control parameter and effects a higher-dimensional control algorithm in a simple manner appropriate to experimental implementation for such demanding systems. The control strategy is seen to be sufficiently robust to operate even though the system shows a marked drift over the course of the experiment. Some comments concerning strategies for the optimal implementation of SPF methods are then made.



37. Kiss, I.Z. and Gáspár, V.
Controlling Chaos with Artificial Neural Network: Numerical Studies and Experiments
J. Phys. Chem. A, 2000, 104, 8033-8037. pdf


Abstract. Although there are a number of theoretically suggested chaos control methods using Artificial Neural Networks (ANN), experimental tests are still lacking. In this paper, we report on experimental chaos control during the electrochemical dissolution of copper in phosphoric acid. The neural network implementation of simple proportional and recursive feedback algorithms are presented.



36. Kiss, I.Z.; Gáspár, V. and Hudson, J.L.
Experiments on Synchronization and Control of Chaos on Coupled Electrochemical Oscillators
J. Phys. Chem. B, 2000, 104, 7554-7560. pdf


Abstract. Experiments were carried out on synchronization and control of complex chaotic dynamics observed during the dissolution of two and four coupled nickel electrodes in sulfuric acid under potentiostatic conditions. In a given potential range the individually measured currents exhibit asynchronous chaotic oscillations. The complexity (as measured by the correlation dimension) of the chaotic oscillations depends on the extent of coupling among the electrodes.  Thus the effectiveness of a combined synchronization–delayed-feedback procedure can be tested on systems with increasing complexity. We show that the asynchronous chaotic oscillations can be converted to synchronized and simple, periodic current oscillations by a two-step procedure. The chaotic current oscillations of coupled electrodes are first synchronized by perturbations of external resistors that are connected individually to each electrode. Then, the desired periodic orbit is stabilized by perturbations of the potential. We also observed that certain non-vanishing perturbations could lead to only partially synchronized, so called ‘clustered’ chaotic states.



35. Tóth, R.; Gáspár,V.; Belmonte, A.; O'Connell, M.C.; Taylor, A.F. and Scott, S.K.
Wave Initiation in the Ferroin-Catalysed Belousov-Zhabotinsky Reaction with Visible Light
Phys. Chem. Chem. Phys., 2000, 2, 413-416. pdf


Abstract. The initiation of chemical reaction-diffusions waves by visible light of wavelength l = 632.8 nm from a 20 mW He-Ne laser in the ferroin-catalysed BZ reaction on a polysulfone membrane is reported. With low loading of the catalyst on the membrane, oxidation waves can be initiated from the resting steady state and in the recovering tail of a wave. With high loading, waves can only be initiated in the ‘vulnerable’ region behind an existing wavefront. The mechanism of this initiation is discussed in terms of the photoreduction of the metal-ligand catalyst and expressed in terms of a modified Oregonator model. These new observations are in contrast to the inhibitory effect of visible light in the light-sensitive Ru-catalysed BZ system.



34. Taylor, A.F.; Gáspár, V.; Johnson, B.R. and Scott, S.K.
Analysis of Reaction-Diffusion Waves in the Ferroin-Catalysed Belousov-Zhabotinsky Reaction
Phys. Chem. Chem. Phys., 1999, 1, 4595-4599. pdf


Abstract. The Oregonator model is used to analyse reaction-diffusion waves in the ferroin-catalysed Belousov-Zhabotinsky reaction. The value of the effective rate constant of ferriin reduction kJ is determined from experimental wave profiles of the oxidised catalyst obtained from solutions open to air, nitrogen, or sealed with a perspex lid. The value of kJ is found to be lower in systems affected by O2. The dependence of the wave speed c on the concentration of ferriin in the wave profiles is analysed. An alternative method for the determination of kJ from a relationship between the period of target initiation T and [Fe(III)] is presented. The effect of these results on the ‘universal dispersion relation’ is also discussed.



33. Parmananda, P.; Madrigal, R.; Rivera, M.; Nyikos, L.; Kiss, I.Z. and Gáspár, V.
Stabilization of unstable steady states and periodic orbits in an electrochemical system using delayed-feedback control
Phys. Rev. E, 1999, 59, 5266-5271. pdf


Abstract. We report numerical and experimental results indicating successful stabilization of unstable steady states and periodic orbits in an electrochemical system. Applying a continuous delayed-feedback technique not only periodic and chaotic oscillations are suppressed via stabilization of steady-state solutions but also the chaotic dynamics can be converted to periodic behavior. In all cases the feedback perturbation vanishes as a target state is attained.



32. Flesselles, J.-M.; Belmonte, A. and Gáspár, V.
Dispersion Relation for Waves in the Belousov-Zhabotinsky Reaction
J. Chem. Soc. Faraday Trans., 1998, 94, 851-855. pdf


Abstract. Analysis of a chemical model for the Belousov-Zhabotinsky reaction leads to an analytic form for the dispersion relation of waves travelling in such a medium. It is found that the velocity varies like the hyperbolic tangent of the normalized period. Data analysis suggests that the normalization time is the selected spiral period for the medium. This result agrees with previously published data, one-dimensional as well as two-dimensional, all of which can be rescaled onto a single dimensionless curve. It thus provides a unifying approach to all waves in this reaction.



31. Kiss, I.Z.; Gáspár, V. and Nyikos, L.
Stability Analyis of the Oscillatory Electrodissolution of Copper with Impedance Spectroscopy
J. Phys. Chem. A, 1998,102, 909-914.  pdf


Abstract. Impedance spectroscopy is applied to quantitatively characterize the bifurcations leading to current oscillations during anodic dissolution of a copper rotating-disk electrode in sodium acetate–glacial acetic acid and o-phosphoric acid electrolytes under potentiostatic control. The line of Hopf bifurcations in a diagram spanned by the uncompensated series resistance and the true electrode potential has been constructed by measuring the linear frequency response of the electrochemical systems. On the basis of impedance data we have also determined the critical frequency by which oscillations emerge at the bifurcation points. Qualitative differences in the origin of current oscillations in the studied systems are explained by the results of experimental linear stability analysis with impedance spectroscopy.



30.  Kiss, I.Z.; Gáspár, V.; Nyikos, L. and Parmananda, P.
Controlling Electrochemical Chaos in the Copper-Phosphoric Acid System
J. Phys. Chem. A, 1997, 101, 8668-8674. pdf


Abstract. Model calculations indicate that chaotic current oscillations during anodic electrodissolution of copper into phosphoric acid may be controlled by applying a simple map-based algorithm. In the experiments, the unstable period-one and period-two orbits embedded in the chaotic attractor have been stabilized by small perturbations of the anodic potential. We present the results of an experimental test for a power law relating the average chaotic transient time to the size of maximum perturbation allowed during control. The reported experimental results are in good agreement with the theoretical predictions by Ott, Grebogi, and Yorke.



29. Volford, A.; Wittmann, M.; Marlovits, G.; Noszticzius, Z. and Gáspár, V.
Platinum as a Chlorine Dioxide/Chlorite Redox Electrode in ClO2 Based Oscillating Reactions and a New Semibatch Oscillator: the ClO2-Acetone System with I- Inflow
J. Phys. Chem. B, 1997,101, 3720-3726. pdf


Abstract. A systematic study of the potential response of platinum electrode to various redox species in ClO2 based chemical oscillators proves that it is the Cl(+4)/Cl(+3) redox pair that determines the potential in these systems. According to the results reported here, the electrode potential can be predicted from known concentrations of the Cl(+4) and Cl(+3) species or, inversely, these concentrations can be estimated from the measured potential. Thus, at constant ClO2 levels the platinum behaves as a „chlorite selective” electrode. To demonstrate the feasibility of this approach, potentiometric oscillations were recorded in two different ClO2 based oscillating systems, and the experimental results were compared with potentiometric traces calculated by two alternative models.



28. Dajka, J.; Károly, T.; Nagy, I.; Gáspár, V. and Noszticzius, Z.
Transition between Circular Fronts and Spiral Waves in Marginally Excitable Media
J. Chem. Soc. Faraday Trans., 1996, 92, 2897-2901. pdf


Abstract. Belousov-Zhabotinsky (BZ)-type waves are studied on a polysulfone membrane loaded with bathoferroin catalyst. The membrane is placed on a glass-fiber filter-disc soaked with a BZ medium containing no catalyst. Waves are periodically initiated on the membrane surface by using silver- and platinum-wire electrodes. When the area around the silver electrode is made marginally excitable by sufficiently shortening the period of perturbations, an unusual transition between circular fronts and spiral waves is observed.



27. Kiss, I.Z. and Gáspár, V.
Predicting the Dynamics of an Oligo-oscillatory Reaction by an Artificial Neural Network
ACH–Models in Chemistry, 1995, 132, 887-901. pdf


Abstract. Artificial neural networks (ANNs) are model-free computational tools that can "learn" the linear or nonlinear rules embedded in a dataset.  We report the results of an attempt to utilize ANNs in the field of reaction kinetics.  A feedforward network is trained to predict the main dynamical features of oligo-oscillations in the acidic bromate—ascorbic acid—malonic acid reacting mixture, in which the concentration of bromide ion (an intermediate) shows three extrema as a function of time.  Inputs to the network are the initial concentrations of reactants, while outputs are the predicted values of bromide-ion concentration and reaction time at the extrema.  The network is first tested on a numerically generated dataset and then applied to experiments.  The results provide evidence that ANNs can be efficiently employed for the prediction of the dynamics of complex chemical systems, especially, when the mechanism of a reaction is not fully understood. 



26. Marlovits, G.; Wittmann, M.; Gáspár, V. and Noszticzius, Z.
A new chemical oscillator in a novel open reactor: ClO2-I2-acetone system in a membrane fed stirred tank reactor
J. Phys. Chem., 1995, 99, 5359-5364. pdf


Abstract. The title reaction was carried out in a new type of semibatch reactor, where iodine or both iodine and chlorine-dioxide are fed to the stirred bulk of the reactor through thin and selective silicon rubber membranes. For model calculations the iodine inflow rate and the pseudo-first-order rate constant of the iodide production from iodine were determined experimentally. All other reactions and rate constants were taken from the literature. The period and the duration of the experimentally found and theoretically predicted oscillations agree rather well. An exact electrochemical interpretation of the amplitude of the electrode potential oscillations requires further research.



25. Tóth, Á.; Gáspár, V. and Showalter, K.
Signal transmission in chemical systems: propagation of chemical waves through capillary tubes
J. Phys. Chem., 1994, 98, 522-531. pdf


Abstract. The propagation of chemical waves through narrow channels has been investigated. Thin layers of excitable Belousov-Zhabotinsky mixtures are connected by precision-bore capillary tubes of different internal diameters. A wave initiated on one side of an otherwise impenetrable barrier enters and travels trough the capillary tube, forming a hemisphere of excited solution at the exit. When the tube diameter is greater than a critical value, the excitation serves to initiate a circular wave in the second compartment; otherwise, the hemisphere collapses and no wave is initiated. Electrochemically generated periodic wave trains give rise to resonance patterns characterized by firing numbers 1/n, where n=1,2, etc. is the number of waves entering the tube for every wave exiting. The firing numbers correspond to one branch of a Farey tree; higher periodic resonances in modeling calculations indicate that more fully developed Farey sequences may also occur. A one-dimensional mapping procedure is proposed to describe the appearance and ordering of the resonance patterns.



24. Petrov, V.; Gáspár, V.; Masere, J. and Showalter, K.
Controlling chaos in the Belousov-Zhabotinsky reaction
Nature, 1993, 361, 240-243. pdf


Abstract. Deterministic chaos is characterized by long-term unpredictability arising from an extreme sensitivity to initial conditions. Such behaviour may be undesirable, particularly for processes dependent on temporal regulation. On the other hand, a chaotic system can be viewed as a virtually unlimited reservoir of periodic behaviour which may be accessed when appropriate feedback is applied to one of the system parameters. Feedback algorithms have now been successfully applied to stabilize periodic oscillations in chaotic laser, diode, hydrodynamic and magnetoelastic systems, and more recently in myocardial tissue. Here we apply a map based, proportional-feedback algorithm to stabilize periodic behaviour in the chaotic regime of an oscillatory chemical system: the Belousov-Zhabotinsky reaction.



23. Kéki, S.; Magyar, I.; Beck, M.T. and Gáspár, V.
Modeling the oscillatory bromate oxidation of ferroin in open systems
J. Phys. Chem., 1992, 96, 1725-1729. pdf


Abstract. A model for the ferroin-bromate-bromide-sulfuric acid system in a continuously stirred (flow-through) tank reactor has been constructed by extending the Noyes-Field-Thompson mechanism with the following composite processes: (a) ferroin-bromate, (b) ferroin-bromous acid, (c) ferroin-hypobromous acid, (d) ferroin-bromine, (e) feriin-bromide, and (f) ferriin-bromine. The calculated high amplitude oscillations and kinetic phase diagram are in good accordance with the experiments reported earlier. By completing the scheme with a reaction step accounting for the precipitation and dissolution of a ferroin-tribromide salt, the batch oscillations found at high concentrations of reactants can also be simulated.



22. Gáspár, V.; Maselko, J. and Showalter, K.
Transverse coupling of chemical waves
Chaos, 1991, 1, 435-444. pdf


Abstract. The transverse coupling of chemical waves is investigated using a model scheme for excitable media. Chemical waves supported on the surfaces of a semipermeable membrane couple via diffusion through the membrane, resulting in new types of spatiotemporal behavior. The model studies show that spontaneous wave sources may develop from interacting planar waves, giving rise to a complex sequence of patterns accessible only by perturbation. Coupled circular waves result in the spontaneous formation of spiral waves, which subsequently develop patterns in distinct domains with characteristic features. The long time entrainment behavior of coupled spiral waves reveals regions of 1:2 phase locking.



21. Peng, B.; Gáspár, V. and Showalter, K.
False Bifurcations in Chemical Systems: Canards
Phil. Trans. R. Soc. Lond. A, 1991, 337, 275-289. pdf


Abstract. A canard is a false bifurcation in which the amplitude of an oscillatory system may change by orders of magnitude while the qualitative dynamical features remain unchanged. Recent theoretical considerations suggest that canards are characteristic of fast-slow dynamical systems and are associated with the stable and unstable manifolds of the phase plane. An alternative characterization of canard behaviour is proposed involving the crossing of an inflection line by a limit cycle growing out from an unstable stationary state. The inflection line comprises the locus of the points at which the curvature of any phase plane trajectory is zero. The role of the inflection line in the onset of canard behaviour as well as in the continuity of the transaction is examined in a two-variable model for the oscillatory EOE reaction., the Autocatalator, and in the two-variable Oregonator. The approach is also applied to the van der Pol oscillator, the system in which canard behaviour was first examined.



20. Winston, D.; Arora, M.; Maselko, J.; Gáspár, V. and Showalter, K.
Cross-membrane coupling of chemical spatiotemporal patterns
Nature, 1991, 351, 132-135. pdf


Abstract. Chemical systems may communicate by exchange of common species through mass transport, and such coupling may give rise to dynamical complexity beyond that possible in the independent systems. We report here on dynamical spatiotemporal patterns across a membrane. Chemical waves appear on Nafion membranes that are loaded with ferroin catalyst and bathed in a mixture of the reagents of the Belousov-Zhabotinsky oscillatory reaction. The waves on each side of the membrane couple by diffusive transport through the membrane. The coupling initially gives rise to the spontaneous appearance of spiral waves, and subsequent behaviour reveals several distinct phases of evolution, ultimately leading to complete spatiotemporal entrainment.



19. Gáspár, V. and Showalter, K.
A simple model for the oscillatory iodate oxidation of sulfite and ferrocianide
J. Phys. Chem., 1990, 94, 4973-4979. pdf


Abstract. A four-variable reduced model is developed from an empirical rate law model for the oscillatory iodate-sulfite-ferrocianide reaction. The dynamical behavior of the reduced model is compared to that of the original 10-variable model and to the behavior found in experimental studies. The four-variable model is further reduced to a minimal two-variable model which retains most of the dynamical features of the original system. The transitions from oscillatory to steady-state behavior are examined in both reduced models, revealing an apparent infinite period bifurcation and a canard associated with a supercritical Hopf bifurcation.



18. Gáspár, V. and Showalter, K.
Period lengthening and associated bifurcations in a two-variable, flow Oregonator
J. Chem. Phys., 1988, 88, 778-791. pdf


Abstract. The dynamic behavior of a two-variable Oregonator with CSTR flow terms is examined as a function of the stoichiometric factor f . Three steady states are exhibited over a range of f and seven bifurcation points are identified and characterized. Slowing down of the limit cycle oscillations on approaching the upper limit of oscillatory behavior is explained by a saddle-loop bifurcation.



17. Lengyel, I.; Gáspár, V. and Beck, M.T.
Kinetics and mechanism of oxidation of nitrous acid by chlorite ion
J. Phys. Chem., 1988, 92, 137-140. pdf


Abstract. Although the oxidation of nitrite by chlorite both stoichiometrically and kinetically is a fairly simple reaction when nitrite is in excess and the pH is above 4, it becomes exceedingly complex in acidic solution containing chlorite in excess. Cl2O2 appears to be a key intermediate, the transitory formation of which explains both the formation of ClO2 and peculiar three-extrema E versus time curve. Assuming that the electrode process on the bright Pt electrode is HOCl + e- + H+ = Cl2 + H2O, the reaction is an oligooscillatory one in which the concentration of HOCl exhibits three extrema at most.



16. Gáspár, V. and Showalter, K.
The oscillatory Landolt reaction. Empirical rate law model and detailed mechanism
J. Am. Chem. Soc., 1987, 109, 4869-4876. pdf


Abstract. The iodate oxidation of sulfite and ferrocyanide when carried out in a continuous flow stirred tank reactor exhibits large-amplitude oscillations in pH accompanied by an almost constant concentration of iodide. A description of the reaction in terms of component processes and associated empirical rate laws is used to model the dynamical behavior. Limitations and potential refinements of the empirical rate law model are discussed. A detailed mechanism consistent with the component process description is presented .



15. Gáspár, V. and Beck, M.T.
Kinetics of the reversible photoaquation of the octacyanomolybdate(IV) ion
Polyhedron, 1987, 6, 269-273. pdf


Abstract. The kinetics of the photoaquation of the octacyanomolibdate(IV) ion in aqueous solution were studied by potentiometric and spectrophotometric methods. In an alkaline medium a simple scheme analogous to the photoaquation of the hexacyanoferrate(II) ion describes the process. The values of the constants of the kinetic equation are: f = 1.0, k8 = (6.55 ± 0.8) ´ 10–9 s –1 and k-8 = (7.88 ± 0.5) ´ 10-2 mol-1 dm3 s-1 (pH = 10.5). The reversibility of the photoaquation is also explained by the scheme. A simultaneous measurement of free cyanide ion concentration and the absorbance at 512 nm shows that the red coloured transition product is a heptacyano complex.



14. Gáspár,V.; Noszticzius, Z. and Farkas, H.
Numerical Simulation of the BZ Reaction of Oxalic Acid with a Simple Four-Variable Model
React. Kinet. Catal. Letters, 1987, 33, 81-86. pdf


Abstract. A simple model of the bromate-cerium-oxalic acid batch oscillatory system has been constructed and examined by calculations. The model predicts a parameter range and a frequency which agree rather well with the observed values, but it does not describe the peculiar saddlenode infinite period bifurcations found experimentally in the system.



13. Gáspár, V. and Beck, M.T.
Depressing the bistable behavior of the iodate-arsenous acid reaction in a continuous flow stirred tank reactor by the effect of chloride or bromide Ions. A method for determination of rate constants
J. Phys. Chem., 1986, 90, 6303-6305. pdf


Abstract. Chloride or bromide ion catalysis of the iodate-iodide reaction causes changes in the location of the bifurcation point of the iodate-arsenous acid bistable system under continuous flow stirred tank reactor conditions. By the critical values of the halide ion concentrations, the rate constants of the catalyzed steps were determined: 3.2 ´ 108 and 3 ´ 1010 M-5s-1 in the case chloride and bromide ions, respectively.



12. Gáspár, V. and Galambosi, P.
Bifurcation diagram of the oscillatory Belousov-Zhabotinskii system of oxalic acid in a continuous flow stirred tank reactor. Further possible evidence of saddle-node infinite period bifurcation behavior of the system
J. Phys. Chem., 1986, 90, 2222-2226. pdf


Abstract. This paper presents the bifurcation diagram of one of the simplest Belousov-Zhabotinskii (BZ) type oscillators (the bromate-cerium-oxalic acid system) in a continuous flow stirred tank reactor (CSTR). The results support that the transitions from steady states to periodic orbits, and vice versa, proceed via saddle node infinite period (SNIPER) bifurcations. This finding agrees with the results in batch when the produced bromine was removed by an inter gas stream (Noszticzius et al.). Computations demonstrate that a simple revised Oregonator type and bromine-hydrolysis-controlled model cannot describe the SNIPER bifurcation behaviour of the system in a CSTR.



11. Gáspár, V.; Bazsa, Gy. and Beck, M.T.
Bistability and bromide-controlled oscillation during bromate oxidation of ferroin in a continuous flow stirred tank reactor
J. Phys. Chem., 1985, 89, 5495-5499. pdf


Abstract. Autocatalytic oxidation of Fe(phen)32+ (ferroin) by bromate has been investigated in a continuous flow stirred tank reactor (CSTR). A closed region of bistability exitst in the plot of input stream concentrations of ferroin and bromate. The bromate-ferroin-bromide systems exhibits bistability and high amplitude oscillations over a broad range of concentrations. As a first approximation a revised Oregonator type model extended by reactions of ferroin and ferriin describes the system. However, further reactions have to be taken into consideration for a quantitative description.



10. Noszticzius, Z.; Gáspár, V. and Försterling, H.-D.
Experimental test for the control intermediate in the Belousov-Zhabotinsky (BZ) Reaction
J. Am. Chem. Soc., 1985, 107, 2314-2315. pdf


Abstract. The role of the control intermediate in the BZ reaction is to remove the autocatalytic intermediate HBrO2. As from a theoretical point of view both elementary bromine and bromide ions may act as control intermediates, an experimental test was performed. The experiments show that only the Br- ions can react directly with the bromous acid and the elementary Br2 reacts indirectly via its hydrolysis.



9. Gáspár, V.; Bazsa, Gy. and Beck, M.T.
The influence of visible light on the Belousov-Zhabotinskii oscillating reactions applying different catalysts
Z. Phys. Chem. (Leipzig), 1983, 264, 43-48. pdf


Abstract. The influence of visible light on the Belousov-Zhabotinskii (1, 2) (BZ) oscillating systems using different catalysts (Ce4+, Fe(phen)32+ and Ru(dipy)32+) were investigated. Light has no effect on the BrO3--malonic acid-H2SO4-Ce4+system. However, using the complexes as catalysts the amplitude and frequency changed and even initiation and inhibition of oscillation was observed by the effect of illumination. In a modified (viz. containing an excess of Fe2+, oxalic acid and that of phenantroline) BZ system applying ferroin catalyst, at a certain composition, oscillation occurred only on illumination.



8. Gáspár, V. and Beck, M.T.
Kinetics of the photoaquation of hexacyanoferrate(II) ion
Polyhedron, 1983, 2, 387-391. pdf


Abstract. Kinetics of the photoaquation of hexacyanoferrate(II) ion in aqueous solution were studied potentiometrically and spectrophotometrically. Supposing the simplest mechanism (see Fig. 3. in text), the photoaquation in alkaline medium can be well described. The value of the constants at pH = 11.0 are: f = 0.8-1.0, k6 = (3.0 ± 0.5) ´ 10-8 s-1 and k-6 = 1.5 ± 0.3 mol-1 dm3 s-1. To describe the photoaquation in neutral medium the scheme was extended (k’ = 3.33 ´ 102 mol-1 dm3 s-1). The quantum yield in acidic medium can be calculated by combinationof f values of different protonated complexes. The reversibility of photoaquation in alkaline medium is also explained by the scheme.



7. Gáspár, V. és Beck, M.T.
A hexaciano-ferrát(II) fotokémiai akválásának kinetikája
Magy. Kém. Foly., 1982, 88, 433-437. pdf


Összefoglalás. A hexaciano-ferrát(II) vizes oldatban (lúgos, semleges és savas közegben), 365 nm-es besugárzás hatására bekövetkező fotoakválásának kinetikáját tanulményoztuk a cianidion és a pH potenciometriás meghatározásával, valamint spektrofotometriás módszerrel. A lúgos oldatban lejátszódó fotoakválást jó egyezéssel írtuk le a legegyszerűbb séma segítségével. pH = 11,0 értéken (T = 298 ± 0,1 K hőmérsékleten) az állandók értékei a következők: f = 0.8-1.0, k6 = (3.0 ± 0.5) ´ 10-8 s-1 and k-6 = 1.5 ± 0.3 mol-1 dm3 s-1. A semleges oldatbeli fotoakválás leírásához a modellt a [Fe(CN)5H2O]3- + HCN reakcióval bővíteni kellett (k' = 3,33 ´ 102 mol-1 dm3 s-1. A savas oldatban történő fotoakválás kvantumhasznosítási tényezője a különbözőképpen protonált komplexek f értékeinek kombinációjaként számítható. Értelmeztük a fotoakválás reverzibilitását lúgos és semleges oldatokban.



6. Gáspár, V. and Beck, M.T.
The influence of the ionic strength on the dissociation constant of hidrogen cyanide
Acta Chimica Hung., 1982, 110, 425-427. pdf


Abstract. For the study of the photoaquation of hexacyanoferrate(II) and octacyano-molybdate(IV) the dissociation constant of hydrogen cyanide (Kd) was determined at several ionic strengths in the range from 0.1 to 5.0 mol dm-3 NaClO4 (T=298 K).



5. Gáspár, V. és Beck, M.T.
A hidrogén-cianid disszociációs állandójának ionerősségfüggése
Magy. Kém. Foly., 1980, 86, 177-179. pdf


Összefoglalás. Meghatároztuk a hidrogén-cianid disszociációs állandóját különböző nagy (0,1- 0,5) ionerősségek esetén, cianid-szelektív- és üvegelektród együttes alkalmazásával t = 25 ± 0,1 oC-on. A számításokat a titrálási görbe ekvivalencia pontjától távoli szakaszokban nyert adatok alapján, linearizált forma alkalmazásával végeztük. Az ionerősség növelésének hatására a disszociációs állandó jelentősen növekedik.



4. Beck, M.T.; Gáspár, V. and Ling, J.
Formation of glycine in the hydrolysis of coordinated cyanogen
Inorg. Chim. Acta, 1979, 33, L177-178. pdf


Abstract. A new cyanogen-containing complex of the composition CuC3N3H3O has been prepared and characterized. Its probable sctructure is H2 [(NC)2CuNCCN(CN)2] 2H2O. When it is boiled with hydrochloric acid, a considerable part of cyanogen is hydrolysed to glycine.This result indicates the potential role of cyanogen complexes in prebiotic syntheses.



3. Beck, M.T.; Gáspár, V. és Ling, J.
Glicin képződése a koordinált dicián hidrolízisekor
Magy. Kém. Foly., 1979, 85, 147-148. pdf


Összefoglalás. Egy új dicián tartalmú rézkomplexet állítottunk elő, melynek összetétele CuC3N3H3O, valószínű szerkezete pedig H2 [(NC)2CuNCCN(CN)2] 2H2O. A vegyület sósavas hidrolízisekor a dicián jelentős mennyiságe glicinné alakul. Ez az eredmény a diciánkomplex esetleges szerepére hívja fel a figyelmet a prebiotikus szintézsisekben.



2. Beck, M.T.; Gáspár, V. and Goel, D.P.
Formation and stability of tricyano Ion
Inorg. Chim. Acta, 1979, 33, L147-148. pdf


Abstract. The first step in the reaction between cyanide ion and cyanogen is the formation of the rather unstable tricyanide. The value of the stability constant is too low to be determined spectrophotometrically, only the Keproduct can be obtained by this method. By liquid-liquid distribution 0.31 ± 0.031 mol-1dm3 was obtained for the stability constant at 25 oC, I = 3 mol dm-3.



1. Beck, M.T.; Gáspár, V. és Goel, D.P.
A dicián és a cianidion közötti kölcsönhatás
Magy. Kém. Foly., 1979, 85, 133-135. pdf


Összefoglalás. A dicián és a cianid közötti reakciók első lépéseként a kevéssé stabilis (CN)3-ion képződik. A stabilitási állandó értéke olyan kicsiny, hogy spektrofotometriás módszerrel csak a Ke értéke határozható meg. Extrakciós módszerrel Kértékére 25 oC-on 3,0 ionerősségű oldatban0.31 ± 0.031 mol-1dm3 érték adódik.