Multiscale analysis for diffusion-driven neutrally stable states

The Turing instabilities for reaction–diffusion systems are studied from the Fourier normal modes which appear by searching the solution obtained from linearization of the reaction–diffusion system at the spatially homogeneous steady state. The linear stability analysis is only appropriate when the temporal eigenvalues associated to every given spatial eigenvalue have non-zero real part. If the real part of the temporal eigenvalue in a normal mode is equal to zero there is no enough information coming from the linearized system. Given an arbitrary spatial eigenvalue, by equating to zero the real part of the corresponding temporal eigenvalue will lead to a neutral stability manifold in the parameter space. If for a given spatial eigenvalue the other parameters in the reaction–diffusion process drive the system to the neutral manifold, then neither stability nor instability can be warranted by the usual linear analysis. In order to give a sketch of the nonlinear analysis we use a multiple scales method. As an application, we analyze the behavior of solutions to the Schnakenberg trimolecular reaction kinetics in the presence of diffusion.     

Stability and approximations of eigenvalues and eigenfunctions for the Neumann Laplacian, Part 2

We extend the results obtained earlier in a joint paper with R. Banuelos, on the stability and approximations of the second Neumann eigenvalue and its corresponding eigenfunction, to the case when the second Neumann eigenvalue has multiplicity at least 2. We then show that our stability result can be applied to the Koch snowflake and the usual sequence of polygons approximating it from inside.     

Nonlinear Eigenvalue Problems in the Stability Analysis of Morphogen Gradients

This paper is concerned with several eigenvalue problems in the linear stability analysis of steady state morphogen gradients for several models of Drosophila wing imaginal discs including one not previously considered. These problems share several common difficulties including the following: (a) The steady state solution which appears in the coefficients of the relevant differential equations of the stability analysis is only known qualitatively and numerically. (b) Though the governing differential equations are linear, the eigenvalue parameter appears nonlinearly after reduction to a problem for one unknown. (c) The eigenvalues are determined not only as solutions of a homogeneous boundary value problem with homogeneous Dirichlet boundary conditions, but also by an alternative auxiliary condition to one of the Dirichlet conditions allowed by a boundary condition of the original problem. Regarding the stability of the steady state morphogen gradients, we prove that the eigenvalues must all be positive and hence the steady state morphogen gradients are asymptotically stable. The other principal finding is a novel result pertaining to the smallest (positive) eigenvalue that determines the slowest decay rate of transients and the time needed to reach steady state. Here we prove that the smallest eigenvalue does not come from the nonlinear Dirichlet eigenvalue problem but from the complementary auxiliary condition requiring only to find the smallest zero of a rational function. Keeping in mind that even the steady state solution needed for the stability analysis is only known numerically, not having to solve the nonlinear Dirichlet eigenvalue problem is both an attractive theoretical outcome and a significant computational simplification.     

Guaranteed lower eigenvalue bounds for the biharmonic equation

The computation of lower eigenvalue bounds for the biharmonic operator in the buckling of plates is vital for the safety assessment in structural mechanics and highly on demand for the separation of eigenvalues for the plate’s vibrations. This paper shows that the eigenvalue provided by the nonconforming Morley finite element analysis, which is perhaps a lower eigenvalue bound for the biharmonic eigenvalue in the asymptotic sense, is not always a lower bound. A fully-explicit error analysis of the Morley interpolation operator with all the multiplicative constants enables a computable guaranteed lower eigenvalue bound. This paper provides numerical computations of those lower eigenvalue bounds and studies applications for the vibration and the stability of a biharmonic plate with different lower-order terms.     

Comparison of theories for stability of truss structures. Part 1: Computation of critical load

Two largely different theories, i.e. the geometric nonlinear eigenvalue theory and the geometric nonlinear critical point theory, of the stability analysis for truss structures are reviewed by the authors. In this paper, it is pointed out through numerical examples as well as thoroughly theoretical investigations that the eigenvalue theory leads to mistakenly very large solutions of critical load. Though it is correct in theory, the applicability of the critical point theory was inadequately extended to all shallow trusses. To overcome the shortcomings of the stability theories, the authors present two theories of their own with two new approaches for geometric nonlinear analysis and for finding the critical loads for shallow truss structures. Several conclusions are drawn, including: (1) the geometric nonlinear eigenvalue theory is mistaken and (2) the capabilities of various theories are discussed.     

A NOTE ON THE BACKWARD ERRORS FOR INVERSE EIGENVALUE PROBLEMS

In this note,we consider the backward errors for more general inverse eigenvalus prob-lems by extending Sun‘‘‘‘s approach.The optimal backward errors defined for diagonal-ization matrix inverse eigenvalue problem with respect to an approximate solution,and the upper and lower bounds are derived for the optimal backward errors.The results may be useful for testing the stability of practical algorithms.     

Bifurcation from trivial solution for elliptic systems with nonlinear boundary conditions

In this paper, we investigate semilinear elliptic systems having a parameter with nonlinear Neumann boundary conditions over a smooth bounded domain. The objective of our study is to analyse bifurcation component of positive solutions from trivial solution and their stability. The results are obtained via classical bifurcation theorem from a simple eigenvalue, by studying the eigenvalue problem of elliptic systems.     

Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi–Chebyshev algorithm

An efficient way of solving 2D stability problems in fluid mechanics is to use, after discretization of the equations that cast the problem in the form of a generalized eigenvalue problem, the incomplete Arnoldi–Chebyshev method. This method preserves the banded structure sparsity of matrices of the algebraic eigenvalue problem and thus decreases memory use and CPU time consumption.     

Instability of solitons under flexure and twist of an elastic rod

We study the stability of planar soliton solutions of equations describing the dynamics of an infinite inextensible unshearable rod under three-dimensional spatial perturbations. As a result of linearization about the soliton solution, we obtain an inhomogeneous scalar equation. This equation leads to a generalized eigenvalue problem. To establish the instability, we must verify the existence of an unstable eigenvalue (an eigenvalue with a positive real part). The corresponding proof of the instability is done using a local construction of the Evans function depending only on the spectral parameter. This function is analytic in the right half of the complex plane and has at least one zero on the positive real axis coinciding with an unstable eigenvalue of the generalized spectral problem.     

Duality Theory in Nonlinear Buckling Analysis for von Kármán Equations

The nonlinear eigenvalue problem in buckling analysis is studied for von Kármán plates. By using the general duality theory developed by Gao-Strang [1, 2] it is proved that the stability criterion for the bifurcated state depends on a reduced complementary gap function. The duality theory is established for nonlinear bifurcation problems. This theory shows that the nonlinear eigenvalue problem is eventually equivalent to a coupled quadratic dual optimization problem. A series of equivalent variational principles are constructed and a lower bound theorem for the first eigenvalue of the buckling factor is proved.     

Electrohydrodynamic instability of a rotating layer of a viscoelastic fluid heated from below

The stability of a rotating layer of viscoelastic dielectric liquid (Walters liquid B) heated from below is considered. Linear stability theory is used to derive an eigenvalue system of ten orders and exact eigenvalue equation for a neutral instability is obtained. Under somewhat artificial boundary conditions, this equation can be solved exactly to yield the required eigenvalue relationship from which various critical values are determined in detail. Critical Rayleigh heat numbers and wavenumber for the onset of instability are presented graphically as function of the Taylor number for various values of electric Rayleigh number and the elastic parameters.     

Global exponential synchronization criterion for switched linear coupled dynamic networks

We in this paper develop a global exponential synchronization stability criterion for switched linear coupled network. By introducing a switching symmetric matrix, we prove that the stability of global exponential synchronization is governed by the largest eigenvalue of this switching symmetric matrix and the largest switching coupling strength. Meanwhile, we give the threshold of switching coupling strength which can make the switched linear network reach global exponential synchronization. Because the proposed criterion is on the basis of the original synchronization definition and the largest eigenvalue of the switching symmetric matrix, therefore, it is convenient to use in verifying global exponential synchronization of dynamic network with switching linear couplings.     

A Legendre spectral element method for eigenvalues in hydrodynamic stability

A Legendre polynomial-based spectral technique is developed to be applicable to solving eigenvalue problems which arise in linear and nonlinear stability questions in porous media, and other areas of Continuum Mechanics. The matrices produced in the corresponding generalised eigenvalue problem are sparse, reducing the computational and storage costs, where the superimposition of boundary conditions is not needed due to the structure of the method. Several eigenvalue problems are solved using both the Legendre polynomial-based and Chebyshev tau techniques. In each example, the Legendre polynomial-based spectral technique converges to the required accuracy utilising less polynomials than the Chebyshev tau method, and with much greater computational efficiency.     

Construction of General Linear Methods with Runge–Kutta Stability Properties

We describe the construction of explicit general linear methods of order p and stage order q=p with s=p+1 stages which achieve good balance between accuracy and stability properties. The conditions are imposed on the coefficients of these methods which ensure that the resulting stability matrix has only one nonzero eigenvalue. This eigenvalue depends on one real parameter which is related to the error constant of the method. Examples of methods are derived which illustrate the application of the approach presented in this paper.     

Stability for the Dirichlet Problem under Continuous Steiner Symmetrization

In this paper we prove the existence of a deformation transforming an arbitrary open set into the ball, which has the following properties: it keeps constant the measure, the kth eigenvalue of Laplace–Dirichlet operator is continuous from the left and the first eigenvalue is decreasing. The deformation is given by a sequence of continuous Steiner symmetrizations, and the behavior of the eigenvalues is related to the stability of the Dirichlet problem.     

A new approach for calculating the real stability radius

We present a new fast algorithm to compute the real stability radius with respect to the open left half plane which is an important problem in many engineering applications. The method is based on a well-known formula for the real stability radius and the correspondence of singular values of a transfer function to pure imaginary eigenvalues of a three-parameter Hamiltonian matrix eigenvalue problem. We then apply the implicit determinant method, used previously by the authors to compute the complex stability radius, to find the critical point corresponding to the desired singular value. This corresponds to a two-dimensional Jordan block for a pure imaginary eigenvalue in the parameter dependent Hamiltonian matrix. Numerical results showing quadratic convergence of the algorithm are given.     

Upper Bounds for the First Eigenvalue of the Laplacian of Hypersurfaces in terms of Anisotropic Mean Curvatures

We prove upper bounds for the first eigenvalue of the Laplacian of hypersurfaces of Euclidean space involving anisotropic mean curvatures. Then, we study the equality case and its stability.     

On visco - plasticity with hardening

A stability theory is developed for finite difference approximations on nonuniform grids for m-th order linear integrodifferential equations under linear side conditions. The stability inequalities are obtained in l^ρ-spaces, pε[1,∞]. The eigenvalue problem is included.     

Preface

Numerical Algorithms - This paper looks at the tensor eigenvalue complementarity problem (TEiCP) which arises from the stability analysis of finite dimensional mechanical systems and is closely...     

An Improved Eigenvalue Perturbation Bound on a Nonnormal Matrix and Its Application in Robust Stability Analysis

This paper presents an improved estimation of the eigenvalue perturbation bound developed by the author. The result is useful for robust stability analysis of linear control systems.