scholarly journals Degenerate bifurcations near a double eigenvalue in the Brusselator

1987 ◽  
Vol 28 (4) ◽  
pp. 486-535 ◽  
Author(s):  
Gerhard Dangelmayr
Keyword(s):  
Steady State ◽  
Normal Forms ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Double Zero ◽  
Zero Eigenvalue ◽  
Double Eigenvalue ◽  
Recent Classification ◽  
Imperfect Bifurcations

AbstractThe steady state bifurcations near a double zero eigenvalue of the reaction diffusion equation associated with a tri-molecular chemical reaction (the Brusselator) are analysed. Special emphasis is put on three degeneracies where previous results of Schaeffer and Golubitsky do not apply. For these degeneracies it is shown by means of a LiapunovSchmidt reduction that the steady state bifurcations are determined by codimension-three normal forms. They are of types (9)31, (8)221 and (6a)ρ,κ in a recent classification of Z(2)-equivariant imperfect bifurcations with corank two. Each normal form couples an ordinary corank-1 bifurcation in the sense of Golubitsky and Schaeffer to a degenerate Z(2)-equivariant corank-1 bifurcation of Golubitsky and Langford in a specific way.

Molecular Motion through a Fluctuating Bottleneck

1994 ◽  
Vol 366 ◽  
Author(s):  
N. Eizenberg ◽  
J. Klafter
Keyword(s):  
Steady State ◽  
Diffusion Equation ◽  
Molecular Motion ◽  
Reaction Diffusion ◽  
Time Dependent ◽  
Reaction Diffusion Equation ◽  
Langevin Equations ◽  
Molecular Pathways

ABSTRACTMolecular motion in a series of cavities dominated by time dependent bottlenecks is studied as a model for molecular pathways in biomolecules. The problem is formulated by coupled rate and Langevin equations and is shown to be equivalent to n-dimensional reaction-diffusion equation where n is the number of cavities visited by the molecules. Results are presented for two cavities and a comparison is made between steady state and non steady state results.

COMPUTATION OF SIMPLEST NORMAL FORMS OF DIFFERENTIAL EQUATIONS ASSOCIATED WITH A DOUBLE-ZERO EIGENVALUE

2001 ◽  
Vol 11 (05) ◽  
pp. 1307-1330 ◽  
Author(s):  
Y. YUAN ◽  
P. YU
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Computational Efficiency ◽  
Normal Forms ◽  
Computer Systems ◽  
Computer Programs ◽  
Double Zero ◽  
Zero Eigenvalue ◽  
Explicit Formulae ◽  
User Friendly

In this paper a method is presented for computing the simplest normal form of differential equations associated with the singularity of a double zero eigenvalue. Based on a conventional normal form of the system, explicit formulae for both generic and nongeneric cases are derived, which can be used to compute the coefficients of the simplest normal form and the associated nonlinear transformation. The recursive algebraic formulae have been implemented on computer systems using Maple. The user-friendly programs can be executed without any interaction. Examples are given to demonstrate the computational efficiency of the method and computer programs.

The classification of reversible-equivariant steady-state bifurcations on self-dual spaces

2008 ◽  
Vol 145 (2) ◽  
pp. 379-401 ◽  
Author(s):  
P. H. BAPTISTELLI ◽  
M. MANOEL
Keyword(s):  
Steady State ◽  
Lie Group ◽  
Normal Forms ◽  
Singularity Theory ◽  
Real Parameter ◽  
Compact Lie Group ◽  
Dual Spaces ◽  
Dual Representations ◽  
Group Of Symmetries

AbstractIn this paper we apply singularity theory methods to the classification of reversible-equivariant steady-state bifurcations depending on one real parameter. We assume that the group of symmetries and reversing symmetries is a compact Lie group Γ, and the equivalence is defined in order to preserve these symmetries and reversing symmetries in the normal forms and their unfoldings. When the representation of Γ is self-dual, we show that the classification can be reduced to the standard equivariant context. In this case, we establish a one-to-one association between the classification of bifurcations in the reversible-equivariant context and the classification of purely equivariant bifurcations related to them. As an application of the results, we obtain the classification of self-dual representations ofZ2⊕Z2andD4on the plane.

scholarly journals Solution of nonlinear equations and computation of multiple solutions of a simple reaction-diffusion equation

2000 ◽  
Vol 42 (1) ◽  
pp. 55-64
Author(s):  
Adrian Swift ◽  
Easwaran Balakrishnan
Keyword(s):  
Steady State ◽  
Diffusion Equation ◽  
Nonlinear Equations ◽  
Multiple Solutions ◽  
Path Following ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Research Involvement ◽  
Path Following Methods ◽  
Nonlinear Equation Systems

AbstractThe first part of this paper starts with a brief discussion of some methods for solution of nonlinear equations which have interested the first author over the last twenty years or so. In the second part we discuss a recent research involvement, the success of which relies heavily on the numerical solution of nonlinear equation systems. We briefly describe path-following methods and then present an application to a simple steady-state reaction-diffusion equation arising in combustion theory. Results for some regular geometric shapes are shown and compared with those from an approximate method.

scholarly journals Bifurcation of steady-state solutions of a scalar reaction-diffusion equation in one space variable

1992 ◽  
Vol 52 (3) ◽  
pp. 343-355 ◽  
Author(s):  
Shin-Hwa Wang ◽  
Nicholas D. Kazarinoff
Keyword(s):  
Steady State ◽  
Diffusion Equation ◽  
Space Variable ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Order Interval ◽  
Exact Number ◽  
Time Map ◽  
Steady State Solutions

AbstractWe study the bifurcation of steady-state solutions of a scalar reaction-diffusion equation in one space variable by modifying a “time map” technique introduced by J. Smoller and A. Wasserman. We count the exact number of steady-state solutions which are totally ordered in an order interval. We are then able to find their Conley indices and thus determine their stabilities.

scholarly journals Numerical Analysis of Sensitivity of SAW Structure to the Effect of Toxic Gases

2016 ◽  
Vol 41 (4) ◽  
pp. 747-755
Author(s):  
Tomasz Hejczyk ◽  
Tadeusz Pustelny ◽  
Bartłomiej Wszołek ◽  
Wiesław Jakubik ◽  
Erwin Maciak
Keyword(s):  
Steady State ◽  
Numerical Analysis ◽  
Reaction Diffusion ◽  
Gas Diffusion ◽  
Numerical Results ◽  
Reaction Diffusion Equation ◽  
Saw Sensor ◽  
Python Language ◽  
Sensor Configuration ◽  
Velocity Changes

AbstractThe paper presents the results of numerical analysis of the SAW gas sensor in the steady and non-steady states. The effect of SAW velocity changes vs surface electrical conductivity of the sensing layer is predicted. The conductivity of the porous sensing layer above the piezoelectric waveguide depends on the profile of the diffused gas molecule concentration inside the layer. The Knudsen’s model of gas diffusion was used.Numerical results for the effect of gas CH4on layers: WO3, TiO2, NiO, SnO2in the steady state and CH4in the non-steady state in recovery step in the WO3sensing layer have been shown. The main aim of the investigation was to study thin film interaction with target gases in the SAW sensor configuration based on simple reaction-diffusion equation.The results of the numerical analysis allow to select the sensor design conditions, including the morphology of the sensor layer, its thickness, operating temperature, and layer type. The numerical results basing on the code elaborated numerical system (written in Python language), were analysed. The theoretical results were verified and confirmed experimentally.

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