1980 Preface * 1999 Preface * 1999 Acknowledgements * Introduction * 1 Circular Logic * 2 Phase Singularities (Screwy Results of Circular Logic) * 3 The Rules of the Ring * 4 Ring Populations * 5 Getting Off the Ring * 6 Attracting Cycles and Isochrons * 7 Measuring the Trajectories of a Circadian Clock * 8 Populations of Attractor Cycle Oscillators * 9 Excitable Kinetics and Excitable Media * 10 The Varieties of Phaseless Experience: In Which the Geometrical Orderliness of Rhythmic Organization Breaks Down in Diverse Ways * 11 The Firefly Machine 12 Energy Metabolism in Cells * 13 The Malonic Acid Reagent ('Sodium Geometrate') * 14 Electrical Rhythmicity and Excitability in Cell Membranes * 15 The Aggregation of Slime Mold Amoebae * 16 Numerical Organizing Centers * 17 Electrical Singular Filaments in the Heart Wall * 18 Pattern Formation in the Fungi * 19 Circadian Rhythms in General * 20 The Circadian Clocks of Insect Eclosion * 21 The Flower of Kalanchoe * 22 The Cell Mitotic Cycle * 23 The Female Cycle * References * Index of Names * Index of Subjects

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/596B270197FE27DE5FABB598B6210622/S0025557200150892a.pdf/div-class-title-span-class-italic-the-geometry-of-biological-time-span-by-a-t-winfree-pp-544-dm68-corrected-second-printing-1990-isbn-3-540-52528-9-springer-div.pdf

The geometry of biological time , by A. T. Winfree. Pp 544. DM68. Corrected Second Printing 1990. ISBN 3-540-52528-9 (Springer)

Transition fronts and localized structures in bistable reaction-diffusion equations

Understanding how biological homochirality emerged remains a challenge for the researchers interested in the origin of life. During the last decades, stable non-racemic steady states of nonequilibrium chemical systems have been discussed as a possible response to this problem. In line with this framework, a description of recycled systems was provided in which stable products can be activated back to reactive compounds. The dynamical behaviour of such systems relies on the presence of a source of energy, leading to the continuous maintaining of unidirectional reaction loops. A full thermodynamic study of recycled systems, composed of microreversible reactions only, is presented here, showing how the energy is transferred and distributed through the system, leading to cycle competitions and the stabilization of asymmetric states.

Energetic and Entropic Analysis of Mirror Symmetry Breaking Processes in a Recycled Microreversible Chemical System

For a model glycolytic diffusion-reaction system, an amplitude equation has been derived in the framework of a weakly nonlinear theory. The linear stability analysis of this amplitude equation interprets the structural transitions and stability of various forms of Turing structures. This amplitude equation also conforms to the expectation that time-invariant amplitudes in Turing structures are independent of complexing reaction with the activator species, whereas complexing reaction strongly influences Hopf-wave bifurcation.

Amplitude equation for a diffusion-reaction system: The reversible Sel'kov model

Understanding how biological homochirality emerged remains a challenge for the researchers interested in the origin of life. During the last decades, stable nonracemic steady states of nonequilibrium chemical systems have been discussed as a possible response to this problem. In line with this framework, a description of recycled systems was provided in which stable products can be activated back to reactive compounds. The dynamical behavior of such systems relies on the presence of a source of energy, leading to the continuous maintaining of unidirectional reaction loops. A full thermodynamic study of recycled systems, composed of microreversible reactions only, is presented here, showing how the energy is transferred and distributed through the system, leading to cycle competitions and the stabilization of asymmetric states.

Energetic and entropic analysis of mirror symmetry breaking processes in a recycled microreversible chemical system.

Uncatalyzed gallic acid oscillating system has been investigated in a continuously fed stirred tank reactor (CSTR). In the [Bromate]0–[Bromide]0 concentration space, a region has been located where a bistability is observed between an oscillatory branch and a flow branch. To our knowledge this is the first evidence of bistability in an uncatalyzed bromate oscillator. Some observations have been explained in terms of the skeleton mechanism proposed in the past.

Bistability in an uncatalyzed bromate oscillator in a continuously fed stirred tank reactor

We show that the chemical entropy production in the reversible Oregonator changes discontinuously at critical values of the concentration parameter at which points a limit cycle bifurcates out of an unstable steady state. The discontinuities in the entropy production are reminiscent of the behavior of the entropy change accompanying a first order phase transition in thermodynamics. They appear to be an example of dynamic phase transitions. For this model, by using Poore’s algorithm, we show that in the case of f=1 the limit cycle is orbitally asymptotically unstable and the bifurcation is subcritical, but in the case of f=0.5 the limit cycle bifurcation at the higher critical concentration of P is subcritical whereas the one at the lower critical concentration of P is supercritical. Therefore, a discontinuous change in chemical entropy production accompanies subcritical Hopf bifurcations and possibly a supercritical Hopf bifurcation. It is conjectured that if a bifurcation is subcritical, attractors of different topological dimensions have different characteristic entropy productions in the same manner as two different states of aggregation of matter have different entropies associated with them.

Limit cycles and discontinuous entropy production changes in the reversible Oregonator

Thermodynamic analysis of the reversible Selkov model (a simple kinetic model describing glycolytic oscillations) has been done by an entropy production technique of Prigogine and it is shown that only the autocatalytic step can destabilize the steady state in this model. It is derived that at thermodynamic equilibrium, the product δS δP is always a positive quantity which appears to be a Chatelier‐like principle for local concentration fluctuation applicable to the autocatalytic step S ⇄ P of this model.

Linear thermodynamic analysis of the reversible Selkov model: An interpretation of the Chatelier-like principle for local concentration fluctuations near thermodynamic equilibrium

The symmetry breaking instability in the irreversible Oregonator model for sufficiently low and unequal diffusion coefficients of its three intermediate species has been investigated using the rate constants of Field and Forsterling. The effects of diffusion and the control parameters on the shift of Hopf bifurcation points associated with the emergence of heterogeneous waves of short wavelengths are reported. The results are found to be in good agreement with those obtained by Prigogine et al. for Brusselator and Turing models.

Arun K. Dutt

Papers

Bistability in an uncatalyzed bromate oscillator in a continuously fed stirred tank reactor

Limit cycles and discontinuous entropy production changes in the reversible Oregonator

Linear thermodynamic analysis of the reversible Selkov model: An interpretation of the Chatelier-like principle for local concentration fluctuations near thermodynamic equilibrium

The effect of diffusion on the Hopf bifurcation in a model chemical reaction exhibiting oscillatory behavior