Bifurcation near Equilibrium

Abstract

In this chapter, we discuss various types of dynamic behavior when a bifurcation arises from the existence of a simple eigenvalue. More specificially, we consider an equation

$$ Cx + N(x,\mu ) = 0 $$

(1.1)

in a Banach space X for μ in a Banach space E, N(0, 0) = 0, ∂N(0, 0)/∂x = 0 under the assumption that the linear operator C has zero as a simple eigen-value. The method of Liapunov—Schmidt gives a scalar bifurcation function G(a, μ) defined for (a, μ) in neighborhood of (0, 0) ∈ ℝ × E. Suppose Cx + N(x, μ)is the vector field for an evolutionary equation

$$ \tfrac{{dx}}{{dt}} = Cx + N(x,\mu ) $$

(1.2)

.