Vilmos Gáspár

Controlling chaos in the Belousov—Zhabotinsky reaction

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 parameters1. Feedback algorithms have now been successfully applied to stabilize periodic oscillations in chaotic laser2, diode3, hydrodynamic4 and magnetoelastic5 systems, and more recently in myocardial tissue6. Here we apply a map-based, proportional-feedback algorithm7,8 to stabilize periodic behaviour in the chaotic regime of an oscillatory chemical system: the Belousov–Zhabotinsky reaction.

False bifurcations in chemical systems: canards

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 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 transition is examined in a two-variable model for the oscillatory EOE reaction, the Auto-catalator, and the two-variable Oregonator. The approach is also applied to the van der Pol oscillator, the system in which canard behaviour was first examined.

Dispersion relation for waves in the Belousov–Zhabotinsky reaction

Analysis of a chemical model for the Belousov–Zhabotinsky reaction leads to an analytic form for the dispersion relation for waves travelling in such a medium. It is found that the velocity varies as 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.

Wave initiation in the ferroin-catalysed Belousov-Zhabotinsky reaction with visible light

The initiation of chemical reaction–diffusion waves by visible light of wavelength λ=633 nm from a 20 mW He–Ne laser in the ferroin-catalysed BZ reaction on a polysulfone membrane is repor ted. 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.

Kinetics of the photoaquation of Hexacyanoferrate(II) ion

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 = ll.0 are: o = 0.8-1.0, k 6 = (3.0 ± 0.5) × 10 −8 s −1 and k −6 = 1.5 ± 0.2 mol −1 dm 3 s −1 . To describe the photoaquation in neutral medium t was extended ( k ′ = 3.33 x 10 2 mol −1 dm 3 s −1 ). The quantum yield in acidic medium can be calculated by combination of o values of different protonated complexes. The reversibility of photoaquation in alkaline medium is also explained by the scheme.

Tracking unstable steady states and periodic orbits of oscillatory and chaotic electrochemical systems using delayed feedback control

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.

Measurements of kinematical parameters of spiral waves in media of low excitability

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 a given composition of the excitable medium, the spiral waves meand, when no or low intensity light is applied, or rigidly rotate, when the light intensity is increased sufficiently. 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 good agreement with the predictions based on an earlier proposed kinematical theory of spiral waves in media of low excitability.

Period lengthening and associated bifurcations in a two‐variable, flow Oregonator

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.

Transverse coupling of chemical waves

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.

Kinetics of the reversible photoaquation of the octacyanomolybdate(IV) ion

Abstract The kinetics of the photoaquation of the octacyanomolybdate(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: (Φ = 1.0, k 8 = (6.55 ± 0.8) x 10 −9 s −1 , and k −8 = (7.88 ± 0.5) x 10 −2 mol −1 dm 3 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 512nm shows that the red coloured transition product is a heptacyano complex.