Delay effects and differential delay equations in chemical kinetics

Abstract The effects of time lags on the kinetic behaviour of chemical systems are considered. Such lags can result from spatial separation of components of a system, from the introduction of a delayed feedback, or as a way of representing the effects of intermediate steps in a series of reactions. The mathematical apparatus for treating systems with delay is introduced. Then, a number of examples, both experimental and theoretical, are discussed. It is seen that time lags can produce major, qualitative changes in the behaviour of a system, and that it is sometimes possible to reduce significantly the number of variables needed to describe a system if one utilizes differential delay rather than ordinary differential equations.

[1]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[2]  F. W. Schneider,et al.  Computer simulation of T3/T7 phage infection using lag times. , 1987, Biophysical chemistry.

[3]  Tamás Turányi,et al.  Mechanistic details of the oscillatory Belousov-Zhabotinskii reaction , 1990 .

[4]  Ott,et al.  Anomalous diffusion in "living polymers": A genuine Levy flight? , 1990, Physical review letters.

[5]  Mark Schell,et al.  Effects of time delay in rate processes , 1986 .

[6]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[7]  J. Boissonade,et al.  Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system , 1980 .

[8]  A. M. Zhabotinskii.  [PERIODIC COURSE OF THE OXIDATION OF MALONIC ACID IN A SOLUTION (STUDIES ON THE KINETICS OF BEOLUSOV'S REACTION)]. , 1964, Biofizika.

[9]  L. Glass,et al.  Oscillation and chaos in physiological control systems. , 1977, Science.

[10]  L. Glass,et al.  PATHOLOGICAL CONDITIONS RESULTING FROM INSTABILITIES IN PHYSIOLOGICAL CONTROL SYSTEMS * , 1979, Annals of the New York Academy of Sciences.

[11]  F. W. Schneider,et al.  Delayed-output-controlled chemical oscillations , 1989 .

[12]  Mark Schell,et al.  Stabilization of unstable states and oscillatory phenomena in an illuminated thermochemical system: Theory and experiment , 1984 .

[13]  J. Yorke,et al.  CHAOTIC ATTRACTORS IN CRISIS , 1982 .

[14]  N. Macdonald.  Biological Delay Systems: Linear Stability Theory , 1989 .

[15]  Kapral,et al.  Analysis of a delay-differential equation in optical bistability. , 1986, Physical review. A, General physics.

[16]  A Katchalsky,et al.  Network thermodynamics: dynamic modelling of biophysical systems , 1973, Quarterly Reviews of Biophysics.

[17]  R. M. Noyes,et al.  Oscillations in chemical systems. IV. Limit cycle behavior in a model of a real chemical reaction , 1974 .

[18]  H. Atlan,et al.  Resistance and Inductance-Like Effects in Chemical Reactions: Influence of Time Delays , 1973 .

[19]  K. Ikeda,et al.  Optical Turbulence: Chaotic Behavior of Transmitted Light from a Ring Cavity , 1980 .

[20]  P. Jacobs,et al.  Theory of nucleation in solid state reactions , 1968 .

[21]  Richard M. Noyes,et al.  Mechanism of reaction of bromine(V) with weak one-electron reducing agents , 1971 .

[22]  Irving R. Epstein,et al.  OSCILLATING CHEMICAL REACTIONS. , 1983 .

[23]  P. Grassberger,et al.  Measuring the Strangeness of Strange Attractors , 1983 .

[24]  Rose,et al.  Conjecture on the dimensions of chaotic attractors of delayed-feedback dynamical systems. , 1987, Physical review. A, General physics.

[25]  C. W. Gear,et al.  Numerical initial value problem~ in ordinary differential eqttations , 1971 .

[26]  Irving R. Epstein,et al.  Differential delay equations in chemical kinetics. Nonlinear models: The cross‐shaped phase diagram and the Oregonator , 1991 .

[27]  A. Hindmarsh,et al.  GEAR: ORDINARY DIFFERENTIAL EQUATION SYSTEM SOLVER. , 1971 .

[28]  J. Ross,et al.  The effects of a nonlinear delayed feedback on a chemical reaction , 1991 .

[29]  Irving R. Epstein,et al.  Minimal bromate oscillator: Bromate-bromide-catalyst , 1982 .

[30]  A. Hindmarsh,et al.  Stiff ode slovers: a review of current and coming attractions , 1987 .

[31]  R. J. Field,et al.  Oscillations and Traveling Waves in Chemical Systems , 1985 .

[32]  R. M. Noyes,et al.  Oscillations in chemical systems. II. Thorough analysis of temporal oscillation in the bromate-cerium-malonic acid system , 1972 .

[33]  N. D. Hayes.  Roots of the Transcendental Equation Associated with a Certain Difference‐Differential Equation , 1950 .

[34]  J. Ross,et al.  Light induced bistability in S2O6F2⇄2 SO3F: Theory and experiment , 1984 .

[35]  Irving R. Epstein,et al.  Differential delay equations in chemical kinetics: Some simple linear model systems , 1990 .

[36]  Kenneth W. Smith,et al.  Gas evolution oscillators. 3. A computational model of the Morgan reaction , 1983 .

[37]  K. Bar-Eli,et al.  Bistability of the oxidation of cerous ions by bromate in a stirred flow reactor , 1981 .