Hydrodynamic and hydromagnetic stability

The application of centre manifolds to amplitude expansions. I. Ordinary differential equations

This investigation studies the dynamic behaviour of a population in which there is a geographic variation in fitness of a one-locus, two allele trait. The results derived here complement certain results by W. H. Fleming who described the existence of a stable polymorphism when the ratio of selection intensity to dispersion rate exceeds a critical value. Here an approximation to the polymorphic state is obtained, and the evolution of other initial states to it is studied.

Analysis of a stable polymorphism arising in a selection-migration model in population genetics

A method of asymptotically determining the bifurcating solutions of a nonlinear eigenvalue problem is described. The method is based on the smallness of a parameter which is different from the usual parameter used in the LyapunovSchmidt procedure. A discussion of the stability and evolution of bifurcating solutions is included. It is shown how the method may be useful for determining secondary bifurcations and turning points on bifurcation curves.

Asymptotic Methods for Bifurcation and Stability Problems

A diffusive energy-balance climate model, governed by a nonlinear parabolic partial differential equation, was studied. Three positive steady-state solutions of this equation are found; they correspond to three possible climates of our planet: an interglacial (nearly identical to the present climate), a glacial, and a completely ice-covered earth. Models similar to the main one are considered, and the number of their steady states was determined. All the models have albedo continuously varying with latitude and temperature, and entirely diffusive horizontal heat transfer. The stability under small perturbations of the main model's climates was investigated. A stability criterion is derived, and its application shows that the present climate and the deep freeze are stable, whereas the model's glacial is unstable. The dependence was examined of the number of steady states and of their stability on the average solar radiation.

Steady-State Solutions of a Diffusive Energy-Balance Climate Model and Their Stability

Nonlinear stability analysis of static states which arise through bifurcation

A layer of viscous fluid lying between two heated horizontal, parallel plates is considered. When a critical adverse temperature gradient is established across the layer, heat is no longer transferred by conduction alone, and a convective motion is established. We model this physical phenomenon with the Boussinesq approximation to the Navier-Stokes equations, and using this mathematical model, we study the evolution of a single periodic disturbance for values of the adverse temperature gradients near the critical value. A method of matched asymptotic expansions is used to construct the time dependent solutions of the system. In the course of this analysis, the Landau equation is rigorously derived and a domain of stability of the convective state is determined. Another result of this analysis is the rigorous justification of a perturbation method which is quite similar to that introduced by Stuart and Watson and other investigators.

Noam Gordon

Papers

Nonlinear stability analysis of static states which arise through bifurcation

An analysis of transient behavior in the onset of convection