Bifurcation with Higher Dimensional Null Spaces

Abstract

To motivate the problems to be discussed in this section, consider M: Λ × X → Z, M(0, 0) = 0, and suppose D _x M(0, 0) is Fredholm of index zero. The method of Liapunov–Schmidt reduces the study of local bifurcation to the study of a system of nonlinear equations f(λ, u) = 0 for u ∈ ℝ^k, f ∈ ℝ^k, where k = dim N(D _x M(0,0)). If k = 1 and ∂² f(0,0)/∂u² ≠ 0, we have seen in Section 6.2 that the local bifurcations are determined by a scalar function of λ. If ∂²f(0,0)/∂u² = 0,∂³ f(0,0)/∂u³ ≠ 0, we have also seen in Section 6.4 that the local bifurcations are determined by two scalar functions of λ. In Section 6.8, we have also discussed the situation for arbitrary k—the basic result being that the local bifurcations are determined by a polynomial of degree k whose coefficients are functions of λ.