Stability of solutions bifurcating from multiple eigenvalues

Let X and Y be real Banach spaces and G:X × $\text{R}$ be a twice continuously differentiate function which is not necessarily linear. Suppose G(u₀, α₀) = 0 and the dimension of the null space of G_u(u₀, α₀) is m, where 1 ? m < ∞. Usually, S = {(u, α):G(u, α) = 0}, in a neighborhood of (u₀, α₀), consists of a finite number of curves emanating from (u₀, α₀). We will determine the stability of points, (u, α), in S (i.e., the maximum of the real parts of the spectrum of G_u(u, α) for each (u, α) ∈ S) using a general perturbation theorem of Kato. Our results contain as a special case the stability theorems of Crandall and Rabinowitz for the case m = 1. We will also tie our stability theorems together with some bifurcation results of Decker and Keller. Finally we apply our results to systems of reaction diffusion equations.     

Stability of solutions bifurcating from a multiple eigenvalue

The asymptotic form of the Titchmarsh-Weyl m-coefficient for the differential equation ?(py′)′+q j = λwy on [a, b). (′≡d/dx), as |λ|→∞, is obtained under general conditions on the real-valued coefficients p, q and w, with p≧0 and w≧0. Examples are given to show that the result is near to but possible, although there are many interesting cases for which the asymptotic form of the m-coefficient is still not known.     

Incomplete comparison principle in problems about surface waves trapped by fixed and freely floating bodies

We consider problems about interaction of surface waves with fixed and freely floating bodies. We propose a new boundary value problem with integro–differential boundary conditions. This problem has physical sense only in the case of a totally submerged body, but, in the case of symmetry in the geometric or physical sense, the nonemptiness of its discrete spectrum provides the existence of eigenvalues of both problems for a fixed body and a freely floating body (in the last case, under some additional requirements which make the comparison principle incomplete). Bibliography: 25 titles. Illustrations: 5 figures.     

Stability of the solutions of an axisymmetric problem regarding the convection of a viscous fluid

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 2, pp. 182–188, February, 1989.     

Analytical solutions of the boundary layer flow of power-law fluid over a power-law stretching surface

This article discusses analytical solutions for a nonlinear problem arising in the boundary layer flow of power-law fluid over a power-law stretching surface. Using perturbation method analytical solution is presented for linear stretching surface. This solution covers large range of shear thinning and shear thickening fluids and matches excellently with the numerical solution. Furthermore, some new exact solutions are found for particular combination of m (power-law stretching index) and n (power-law fluid index). This leads to generalize the case of linear stretching to nonlinear stretching surface. The effects of fluid index n on the boundary layer thickness and the skin friction for nonlinear stretching surface is analyzed and discussed. It is observed that the boundary layer thickness and the skin friction coefficient increase as non-linear parameter n decreases. This study gives a new dimension to obtain analytical solutions asymptotically for highly nonlinear problems which to the best of our knowledge has not been examined so far.     

Stability of solutions in optimal reinsurance problem

A discrete-time insurance model with stop-loss reinsurance is considered. One-period insurance claims form a sequence of independent identically distributed nonnegative random variables with finite mean. The insurer maintains the company surplus above a chosen level a by capital injections. We study the stability of optimal capital injections to the variability of claims distribution. The term “optimal” means the minimal amount of injections that can be found from the corresponding Bellman equation.     

Continuous solutions of some boundary layer problem

The continuous solutions for BVP of third order nonlinear differential equations appears in [1] mathematical model of the melt spinning process. The existence theorem is proved for such BVP. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A model problem of vortical surface waves

The research was conducted having the support of the Russian Foundation for Fundamental Research (Grant 93-013-17621).     

Stability of streaming layer in an unbounded self-gravitational fluid

The stability of a streaming layer of incompressible, inviscid, self-gravitational fluid bounded in an infinite similar medium is investigated. The critical wave numbers are found when the layer has a linear streaming velocity profile.     

Stability of an infinite horizontal layer of fluid with a spatial heat source in the presence of a magnetic field

The stability of an electrically conducting infinite layer of a viscous fluid which loses heat throughout its volume at a constant rate in the presence of a magnetic field is discussed. The value of the critical Rayleigh number R is found to decrease as the rate of heat loss increases which implies that the layer becomes more unstable. It is observed that the destabilizing effect of the heat source term Q is more prominent when the strength of the magnetic field is low. The aspect ratio ‘a’ increases with increasing Q showing that the horizontal dimension of the cells decreases with increasing Q.     

Stability of bifurcating periodic solutions for a single delayed inertial neuron model under periodic excitation

Using the averaged method, we investigate the dynamical characteristics of a single inertial neuron model with time delay under periodic external stimuli. It is shown that the system will lose its stability when the time delay is increased and will give rise to a quasi-periodic motion and chaos under the interaction of the periodic excitation. Numerical simulations show that the results of the analytical method are correct by a comparison with those of direct numerical integration.     

Exact solutions of the linear problem of the steady-state waves created by a dipole in a flow of a stratified fluid

The problem of the steady-state waves which are formed when there is uniform flow of a non-viscous, incompressible, vertically stratified fluid round a dipole is considered in a linear formulation. Using the analytical properties of the solutions, two formulae are obtained for the vertical displacement field in the form of series of single integrals taken over the spectral curves. These formulae are simpler than those which have been previously proposed /1/ since the integrands do not contain special functions with logarithmic singularities and enable one to simplify the numerical analysis of the close domain of the wave field in which the asymptotic forms /2–4/ are applicable /5/.     

Stability and multiplicity of solutions for the thermistor problem

The stability of the stationary solution of the thermistor as a circuit element is studied using a Liapunov functional and the Hale–LaSalle invariance principle. The asymptotic stability of a class of periodic solutions is also considered. Received: November 22, 1999; in final form: May 23, 2001?Published online: May 29, 2002