Lyapunov–Schmidt reduction

Summary

The Lyapunov–Schmidt reduction turns a nonlinear equation $F(x, \lambda) = 0$ between infinite-dimensional spaces into an equivalent system of finitely many equations in finitely many unknowns. It works near a solution where the derivative of $F$ is a Fredholm operator, which is close to being invertible. The part of the equation where the derivative is invertible is solved by the implicit function theorem. What remains is a small bifurcation equation, which describes how solutions branch as the parameter $\lambda$ varies. See the definition.