A subcritical Hopf bifurcation in the methylene blue oscillator: effects of imposed fluctuations

Abstract

plexed to either the solvent or the electrode surface (or both). There is no evidence from the PMV data to suggest that surface effects are responsible for the long lifetime of the intermediate. It is, however, possible that the estimated lifetime is that of the nitrilium ion formed by reaction of acetonitrile with the diphenylmethyl cation provided that the nitrilium ion is electroactive (presumably by dissociative reduction to form the radical and acetonitrile).It is possible to estimate the potential for the dissociative reduction of the nitrilium ion. Steenken and McClelland showed that the formation of this species is reversible.* Under conditions where it is formed almost exclusively, the major product is actually diphenylmethanol with the Ritter product always a minor component of the mixture. The equilibrium constant for this process must be >1000,35 giving a free energy change of <-4 kcal mol-' (<0.18 eV). Saveant and co-workers have suggested that the overpotential for a dissociative electron-transfer reaction will be essentially equivalent to the enthalpy change for the bond dissociation of the bond in the oxidized form of the electroactive specie^.'^ Using this reasoning, the reduction of the nitrilium ion is expected near 0.17 V vs SCE. A mechanism consistent with all of the experimental data is shown in Scheme I. (35) Steenken, S., private communication. (36) Saveant, J. M. J . Am. Chem. Soc. 1987, 109, 6788. ConclusionsImplicit finitedifference simulation of photomodulation voltammograms can be used to obtain lifetimes of radicals and product ions as well as values for the heterogeneous electron-transfer rate and Eo for redox reactions of radicals. Our experimental setup restricts lifetime measurement to an upper limit of ca. 300 ps (due to the limiting time response of the electrochemical cell). The effects of changes in the kinetic parameters can be summarized as follows:(1) At high potentials (above the diffusion limit) the phase of the electrochemical signal depends only on the lifetime of the radical.(2) Changes in the lifetime of the ion affect the quadrature signals more than those signals that are in phase with the excitation lamp. The effect of changes in ion lifetime, however, is to shift the position of E,/* but not to change the shape of the wave (Le., the breadth of the wave does not change).(3) Decreases in the heterogeneous electron-transfer rate (k,) increase the breadth and ElFor the oxidation of the dipienylmethyl radical the data suggest that nitrilium ion, which is formed in a reaction between the carbocation and acetonitrile, will either undergo a dissociative electron transfer to form the radical or slowly react to form products with a rate that depends on the concentration of water in the solvent system.of the voltammetric waves.The dynamical behavior of the methylene blue-oxygen-sulfide oscillator in a continuous flow stirred tank reactor is investigated. At low flow rates oscillations of period one emerge at a supercritical Hopf bifurcation, whereas a subcritical Hopf bifur...

Bistable emergence of oscillations in growing Bacillus subtilis biofilms

“…However, as stated previously, our biofilm oscillations start with nonzero amplitude. This is suggestive of a subcritical Hopf bifurcation, also commonly observed in natural systems (32)(33)(34). In this case, the limit cycle born at the bifurcation is unstable, overlapping in parameter space with the stable steady state.…”

Control of Mosaic and Turing Patterns by Light and Electric Field in the Methylene Blue−Sulfide−Oxygen System

“…Effects of Visible Light on Oscillations and Turing Structures. The behavior of the Methylene Blue−sulfide−oxygen oscillator under CSTR conditions is well-known from experimental and numerical work. − , No work, however, has been done to study the effect of light on the oscillations in the MBO. During our examinations of Turing patterns we found that light affected the pattern formation process (described below), and it thus was obvious to study the light effect on oscillations, too.…”

Stochastic Resonance under Two-Parameter Modulation in a Chemical Model System

“…In chemical reaction systems, both theoretical and experimental studies have revealed the effects of external fluctuations on the nonlinear dynamics. Periodic modulations by light, − flow rates, − or electrical current have been found to induce entrainment, ,,,,, resonance, ,, and chaos. − Stochastic perturbations applied to flow rates have lead to bursting oscillations in the methylene blue oscillator, sustained oscillations in the glycolysis of yeast, and convergence of chaos in a model of the Belousov−Zhabotinsky (BZ) reaction. , …”