scholarly journals Stability Analysis of Pulses via the Evans Function: Dissipative Systems

2008 ◽  
pp. 407-428 ◽  
Author(s):  
T. Kapitula
Keyword(s):  
Stability Analysis ◽  
Evans Function ◽  
Dissipative Systems

scholarly journals Stability analysis of combustion waves for competitive exothermic reactions using Evans function

2018 ◽  
Vol 54 ◽  
pp. 347-360 ◽  
Author(s):  
Z. Huang ◽  
H.S. Sidhu ◽  
I.N. Towers ◽  
Z. Jovanoski ◽  
V.V. Gubernov
Keyword(s):  
Stability Analysis ◽  
Evans Function ◽  
Combustion Waves ◽  
Exothermic Reactions

Reaction-Diffusion Dissipative Systems—Detailed Stability Analysis-Pattern of Growth and Effect of Inhomogenity

1984 ◽  
Vol 39 (9) ◽  
pp. 899-916
Author(s):  
P. J. Nandapurkar ◽  
V. Hlavacek ◽  
J. Degreve ◽  
R. Janssen ◽  
P. Van Rompay
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Time Scales ◽  
Bifurcation Point ◽  
Selection Process ◽  
Reaction Diffusion ◽  
Dissipative Systems ◽  
Local Concentration ◽  
Fast Variable ◽  
Steady State Mode

A detailed stability analysis of the one dimensional steady state solutions for the Brusselator model under the conditions of diffusion of initial (non-autocatalytic) components has been performed both for zero flux as well as fixed boundary conditions. In addition to subcritical as well as supercritical bifurcations, situations have been observed where all solution branches at a bifurcation point are unstable. A case of degenerate steady state bifurcation (2 solutions emanating from the same bifurcation point) has also been noticed. A transient simulation of the system in growth reveals the importance of growth rate on the pattern selection process and suggests that the selection of branches at a bifurcation point may be influenced by perturbations/ fluctuations. It also indicates that a stability analysis of the bifurcation diagram alone cannot decide the state of the system in a transient process, and under certain situations complex behavior may be observed at limit points. Numerical calculations on coupled cells indicate that a heterogenity in the system can introduce multiple (two) time scales in the system. As the ratio of time scales increases, aperiodic or irregular oscillations are observed for the 'fast' variable. A combination of cells with one cell in a steady-state mode and the other in a periodic motion results in a combined motion of the entire system. For a distributed parameter system, a heterogenity can cause development of sharp local concentration gradients, alter the stability properties of steady state as well as periodic solutions and can cause partitioning of the system.

Stability Analysis in Mean-Field Games via an Evans Function Approach

2018 ◽  
Author(s):  
Piyush Grover
Keyword(s):  
Optimal Control ◽  
Stability Analysis ◽  
Mean Field ◽  
Evans Function ◽  
Function Approach ◽  
Mean Field Games ◽  
Mean Field Game ◽  
Large Populations ◽  
Multi Agent ◽  
Hamilton Jacobi Bellman

This work is concerned with stability analysis of stationary and time-varying equilibria in a class of mean-field games that relate to multi-agent control problems of flocking and swarming. The mean-field game framework is a non-cooperative model of distributed optimal control in large populations, and characterizes the optimal control for a representative agent in Nash-equilibrium with the population. A mean-field game model is described by a coupled PDE system of forward-in-time Fokker-Planck (FP) equation for density of agents, and a backward-in-time Hamilton-Jacobi-Bellman (HJB) equation for control. The linear stability analysis of fixed points of these equations typically proceeds via numerical computation of spectrum of the linearized MFG operator. We explore the Evans function approach that provides a geometric alternative to solving the characteristic equation.

scholarly journals Stability of Traveling Waves Based upon the Evans Function and Legendre Polynomials

2020 ◽  
Vol 10 (3) ◽  
pp. 846 ◽  
Author(s):  
H. M. Srivastava ◽  
H. I. Abdel-Gawad ◽  
Khaled M. Saad
Keyword(s):  
Stability Analysis ◽  
Fiber Optics ◽  
Traveling Waves ◽  
Legendre Polynomials ◽  
Reaction Diffusion ◽  
Evans Function ◽  
Reaction Diffusion Equations ◽  
Engineering Sciences ◽  
The Stability ◽  
Linearized Operator

One of the tools and techniques concerned with the stability of nonlinear waves is the Evans function which is an analytic function whose zeros give the eigenvalues of the linearized operator. Here, in this paper, we propose a direct approach, which is based essentially upon constructing the eigenfunction solution of the perturbed equation based upon the topological invariance in conjunction with usage of the Legendre polynomials, which have presumably not considered in the literature thus far. The associated Legendre eigenvalue problem arising from the stability analysis of traveling waves solutions is systematically studied here. The present work is of considerable interest in the engineering sciences as well as the mathematical and physical sciences. For example, in chemical industry, the objective is to achieve a great yield of a given product. This can be controlled by depicting the initial concentration of the reactant, which is determined by its value at the bifurcation point. This analysis leads to the point separating stable and unstable solutions. As far as chemical reactions are described by reaction-diffusion equations, this specific concentration can be found mathematically. On the other hand, the study of stability analysis of solutions may depict whether or not a soliton pulse is well-propagated in fiber optics. This can, and should, be carried out by finding the solutions of the coupled nonlinear Schrödinger equations and by analyzing the stability of these solutions.

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