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)

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.

Dynamics and stability of thin liquid films

Part I. Stationary waves: Scalar equation Leray-Schauder degree Existence of waves Structure of the spectrum Stability and approach to a wave Part II. Bifurcation of waves: Bifurcation of nonstationary modes of wave propagation Mathematical proofs Part III. Waves in chemical kinetics and combustion: Waves in chemical kinetics Combustion waves with complex kinetics Estimates and asymptotics of the speed of combustion waves Asymptotic and approximate analytical methods in combustion problems (supplement) Bibliography.

/pdf/traveling-wave-solutions-of-parabolic-systems-21vts2b7ns.pdf

Traveling wave solutions of parabolic systems

Numerical simulations of a simple reaction-diffusion model reveal a surprising variety of irregular spatiotemporal patterns. These patterns arise in response to finite-amplitude perturbations. Some of them resemble the steady irregular patterns recently observed in thin gel reactor experiments. Others consist of spots that grow until they reach a critical size, at which time they divide in two. If in some region the spots become overcrowded, all of the spots in that region decay into the uniform background.

http://www.sci.wsu.edu/math/faculty/schumaker/Math415/Pearson1993.pdf

Complex patterns in a simple system.

Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C

Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability

1. Introduction Part 1: The Techniques 2. Oscillations in a closed isothermal system 3. Oscillations in a closed isothermal system: mathematical analysis 4. Thermokinetic oscillations in a closed system 5. Hopf bifurcations, the growth of small oscillations, relaxation oscillations, and excitability 6. Autocatalysis in well-stirred open systems: the isothermal CSTR 7. Reaction in a non-isothermal CSTR: Stationary states and singularity theory 8. Oscillatory behaviour in the isothermal CSTR: autocatalytic systems 9. Autocatalytic reactions in plug-flow and diffusion reactors 10. Chemical diffusion pattern formation (Turing structures) 11. Travelling waves 12. Heterogeneous reactions 13. Complex oscillations and chemical chaos Part 2: Experiments 14. Experimental systems 1: solution-phase reactions 15. Experimental systems 2: gas-phase reactions References Index

Chemical Oscillations and Instabilities: Non-Linear Chemical Kinetics

A prototype model is exploited to reveal the origin of mixed‐mode oscillations The initial oscillatory solution is born at a supercritical Hopf bifurcation and exhibits subsequent period doubling as some parameter is varied This period‐2 solution subsequently loses stability, but continues to exist−regaining stability to form the 11 mixed‐mode state (one large plus one small excursion) Other mixed‐mode states lie on isolated branches or ‘‘isolas’’ of limit cycles in the one‐parameter bifurcation diagram and are separated by regions of chaos As a second parameter is varied, the number of isola solutions increases and the ‘‘gaps’’ between them become narrower, leading to correspondingly more complete Devil’s staircases An exactly comparable scenario is shown to arise in the three variable model of the Belousov–Zhabotinsky reaction proposed recently by Gyorgyi and Field [Nature 335, 808 (1992)]

http://heracles.chem.wvu.edu/Garbage/current_papers/MixedMode.JCP.pdf

Mixed‐mode oscillations in chemical systems

Simple reaction‐diffusion fronts are examined in one and two dimensions. In one‐dimensional configurations, fronts arising from either quadratic or cubic autocatalysis typically choose the minimum allowable velocity from an infinite spectrum of possible wave speeds. These speeds depend on both the diffusion coefficient of the autocatalytic species and the pseudo‐first‐order rate constant for the autocatalytic reaction. In the mixed‐order case, where both quadratic and cubic channels contribute, the wave speed depends on the rate constants for both channels, provided the cubic channel dominates. Wave propagation is completely determined by the quadratic contribution when it is more heavily weighted. In two‐dimensional configurations, with unequal diffusion coefficients, the corresponding two‐variable planar fronts may become unstable to perturbations. The instability occurs when the ratio of the diffusion coefficient for the reactant to that for the autocatalyst exceeds some critical value. This critical value, in turn, depends on the relative weights of the quadratic and cubic contributions to the overall kinetics. The spatiotemporal form of the nonplanar wave in such systems depends on the width of the reaction zone, and a sequence showing Hopf, symmetry‐breaking, and period‐doubling bifurcations leading to chaotic behavior is observed as the width is increased.

http://heracles.chem.wvu.edu/Papers/FrontInstabilities.JCP.pdf

Stephen K. Scott

Papers

Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A + 2B → 3B; B → C

Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability

Chemical Oscillations and Instabilities: Non-Linear Chemical Kinetics

Mixed‐mode oscillations in chemical systems

Instabilities in propagating reaction-diffusion fronts