Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension

Abstract

We consider an evolution equation in ℝ¹ of the form

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

(II.1)

where F(.,.) has two continuous derivatives with respect to µ and u. It is conventional in the study of stability and bifurcation to arrange things so that

$$ F(\mu ,0) = 0 for all real numbers \mu $$

(II.2)

. But we shall not require (II.2). Instead we require that equilibrium solutions of (II.1) satisfy u =ε, independent oft and

$$F(\mu ,\varepsilon ) = 0.$$

(II.3)

The study of bifurcation of equilibrium solutions of the autonomous problem (II.1)is equivalent to the study of singular points of the curves (II.3) in the (µ, ε) plane.