Skip to main content
main-content

Über dieses Buch

Ever since the seminal works on traveling waves and morphogenesis by Fisher, by Kolmogorov, Petrovski and Piscunov, and by Turing, scientists from many disciplines have been fascinated by questions concerning the formation of steady or dynamic patterns in reactive media. Contributions to this volume have been made by chemists, chemical engineers, mathematicians (both pure and applied), and physicists. The topics covered range from reports of experimental studies, through descriptions of numerical experiments, to rather abstract theoretical investigations, each exhibiting different aspects of a very diverse field.

Inhaltsverzeichnis

Frontmatter

Simple Resonance Regions of Torus Diffeomorphisms

Abstract
This paper discusses resonance regions for two parameter families of diffeomorphisms and vector fields on the two dimensional torus. Resonance regions with at most two resonant periodic orbits (in the discrete case) or two equilibrium points (for flows) are studied. We establish global geometric properties of these regions with topological arguments.
Claude Baesens, John Guckenheimer, Seunghwan Kim, Robert Mackay

A Minimal Model for Spatio-Temporal Patterns in Thin Film Flow

Abstract
We consider the development of spatio-temporal oscillations in the Kuramoto-Sivashinsky amplitude model of thin film flow. These develop from Hopf bifurcations off of steady state solutions and are observed to undergo symmetry breaking and period doubling bifurcations. Oscillatory branches in the parameter regime studied apparently terminate in Silnikov type homo-clinic connections. A minimal, three mode nonlinear Galerkin discretization, capable of capturing this bifurcation behavior is constructed. A simple shooting algorithm which exploits this sharp reduction in dimensionality is used to accurately locate the homoclinic connections.
H. S. Brown, I. G. Kevrekidis, M. S. Jolly

Localized and Extended Patterns in Reactive Media

Abstract
We study the dynamics of interacting localized structures in homogeneous reactive media. Equations of motion for solitary waves in excitable media and for vortices in oscillatory media are derived under the assumption of weak interactions. We show that excitable media with oscillatory recovery can support a multitude of stable, nonuniform spatial patterns and that phase field effects in oscillatory media may lead to the formation of bound vortex pairs. The implications of the latter result on the transition to turbulence in oscillatory media are discussed.
Christian Elphick, Ehud Meron

Some Recent Results in Chemical Reaction Network Theory

Abstract
The aim of chemical reaction network theory is to draw connections between reaction network structure and qualitative properties of the corresponding differential equations. Some recent results are discussed, in particular those relating to the possibility of multiple steady states in very complex continuous flow stirred tank reactors, to mechanism discrimination in heterogeneous catalysis, and to the possibility of traveling composition waves on isothermal catalyst surfaces
Martin Feinberg

Genericity, Bifurcation and Symmetry

Abstract
In these lectures I would like to discuss how the existence of symmetries alters the type of bifurcation behavior that one expects to observe. In the first lecture I will concentrate on the structure and dynamics of steady-state bifurcation from equilibria. It is here that the influence of symmetries on linearized equations will be discussed and some facts from elementary representation theory introduced. The second lecture will be devoted to effects of symmetry on period-doubling in maps with a short description of an application to large arrays of Josephson junctions. In the final lecture I will describe how certain standard choices of boundary conditions (particularly Neumann) can be thought of as symmetry constraints and how this fact alters notions of genericity. It accord with the style that has developed in the lectures at this workshop, the lectures are of different length.
Martin Golubitsky

Dynamics of Some Electrochemical Reactions

Abstract
Experiments on a few electrochemical reactions are discussed. Time series of either current or voltage, obtained under potentiostatic or galvanostatic conditions respectively, are presented and characterized. We first some some examples of dynamic behavior such as chaos, quasiperiodicity, and period doubling of tori obtained during the electrodissolution of copper. Some apparent higher order chaos during electrodissolution of iron is then discussed. Finally, we treat briefly coupled electrochemical oscillators.
J. L. Hudson

Construction of the Fitzhugh-Nagumo Pulse Using Differential Forms

Abstract
Systems of singularly perturbed ordinary differential equations can often be solved approximately by singular solutions. These singular solutions are pieced together from solutions to simpler sets of equations obtained as limits from the original equations. There is a large body of literature on the question of when the existence of a singular solution implies the existence of an actual solution to the original equations. Techniques that have been used include fixed point arguments (Conley [1], Carpenter [2], Hastings [3] and Gardner and Smoller [4]), implicit function theorem and related functional-analytic techniques (Fife [5], Fujii et al. [6], Hale and Sakamoto [7,8]) differential inequalities (see for instance Chang and Howes [9]) and nonstandard analysis (Diener and Reeb [10]).
C. Jones, N. Kopell, R. Langer

Kinetic Polynomial: A New Concept of Chemical Kinetics

Abstract
A system of quasi-steady-state equations for a single pathway mechanism of a catalytic reaction can always be reduced to a polynomial in terms of the steady state reaction rate, a kinetic polynomial. The coefficients of this polynomial are polynomials in the parameters of the elementary reaction rates. The form of the lowest coefficient of the polynomial ensures the thermodynamic validity of this form of representation of quasi-steady-state equations. The properties of the kinetic polynomial are discussed in connection with such concepts of chemical kinetics as “molecularity”, “stoichiometric number”.
Possible applications of this form are: asymptotic analysis of steady-state kinetic models as well as analysis of steady-state multiplicity; description of the steady-state dependences of the reaction rate, determining relations between kinetic constants when solving the inverse problem.
On the basis of kinetic polynomial explicit equations for the steady-state rate in case when one of the steps is rate-limiting, and in the neighbourhood of equilibrium have been derived.
Algorithm of computation of the kinetic polynomial and its realisation on the basis of computer algebra are described.
Mark Z. Lazman, Gregory S. Yablonskii

Convergence of Travelling Waves for Phase Field Equations to Sharp Interface Models in the Singular Limit

Abstract
We show the convergence of travelling waves for the phase field equations to that of a modified Stefan model in an appropriate singular limit. Finite surface tension effect is crucial to prove this convergence.
Yasumasa Nishiura, Gunduz Caginalp

Standing and Propagating Temperature Waves on Electrically Heated Catalytic Surfaces

Abstract
Reaction rates are often measured using a catalytic wire or ribbon, the resistance (average temperature) of which is kept at a preset value via electrical heating. Previous investigators assumed that the wire temperature was uniform. An IR-thermal imager shows that some temperature profiles have the shape of a stationary standing wave. A bifurcation map describes the organization of the regions with these standing wave temperature profiles. In some cases, the high temperature wave moves back and forth on the ribbon, with continuous changes in its shape. This leads to both oscillatory and chaotic changes in the overall rate of heat generated by the reaction. The dynamic behavior of the overall reaction rate is different and less regular than that of local temperatures on the ribbon. The power spectrum of the overall rate of heat generation decays exponentially, while that of the local temperatures decays as a power law. Ignoring the nonuniform nature of the temperature of the ribbon may lead to severe pitfalls in the determination of the kinetic rate expression and/or its parameters
Georgios Philippou, Dan Luss

Mixed-Mode Oscillations in the Nonisothermal Autocatalator

Abstract
The internal coupling of chemical and thermal feedback processes is investigated through a non-isothermal autocatalator model. The isothermal oscillatory scheme is augmented to include an exothermic “termination” step and a temperature-dependent initiation: P → A (rate = k 0(T)p); A + 2B → 3B (rate = k 1 ab 2); A → B (rate = k 3 a); B → C+ heat (rate = k 2 b). The pool chemical approximation is applied to reactant P. Multiple stationary states exist for all parameter values and complex dynamic behaviour is also found over wide ranges of these parameters. Path following techniques are used to follow the periodic solutions and to find bifurcations such as period doubling cascades leading to chaos. Other types of complex dynamics such as mixed mode oscillations are categorised, although we find no quasiperiodic behaviour in the model. This gives rise to the belief that the mixed mode oscillations found, do not stem from phase locking on a torus in this instance.
S. K. Scott, A. S. Tomlin

Bifurcations and Global Stability in Surface Catalyzed Reactions Using the Monte Carlo Method

Abstract
The vector of state variables characterizing the dynamics of heterogeneous systems is often spatially nonuniform. Ordinary differential equations (ODEs) or partial differential equations (PDEs) can describe spatial and temporal evolution of nonuniform field systems only under severe assumptions. The Monte Carlo method (MCM) is applied to model such heterogeneous systems. The conditions under which ODEs or PDEs fail are examined by studying generic model systems using the stochastic method. In the presence of nonlinearities in the governing deterministic equations, spatial inhomogeneities are observed. Multiplicities, cusp points, and periodic solutions are calculated by systematic investigation in parameter space and the global stability of solutions is presented. It is demonstrated that the presence of imperfections on surfaces introduces local nonuniformities in the adlayer (nucleation centers) and their interaction on the synchronization of the surface is discussed.
D. G. Vlachos, L. D. Schmidt, R. Aris
Weitere Informationen