Stability of Steady Solutions of Evolution Equations in Two Dimensions and n Dimensions

Abstract

We noted in the introduction that the solutions of three nonlinear ordinary differential equations can be turbulent-like and outside the scope of elementary analysis. In fact, the most complete results known in bifurcation theory are for problems which can be reduced to one or two dimensions. So we shall start our analysis with two-dimensional autonomous problems, reduced to local form

$$\frac{{du}}{{dt}} = f(\mu ,u),$$

(IV.1)

where

$${{f}_{i}}(\mu ,u) = {{A}_{{ij}}}(\mu ){{u}_{j}} + {{B}_{{ijk}}}(\mu ){{u}_{j}}{{u}_{k}} + {{C}_{{ijkl}}}(\mu ){{u}_{j}}{{u}_{k}}{{u}_{l}} + O(\parallel u{{\parallel }^{4}}).$$

(IV.2)

The same equations (IV.1) and (IV.2) hold in ℝⁿ In general, the subscripts range over (1, 2,…, n); in ℝⁿ, n=2.