SPECTRAL FLOW IS A COMPLETE INVARIANT FOR DETECTING BIFURCATION OF CRITICAL POINTS

Given a one-parameter path of equations for which there is a trivial branch of solutions, to determine the points on the branch from which there bifurcate nontrivial solutions, there is the heuristic principle of linearization. That is to say, at each point on the branch, linearize the equation, and justify the inference that points on the branch that are bifurcation points for the path of linearized equations are also bifurcation points for the original path of equations. In quite general circumstances, for the bifurcation of critical points, we show that, at isolated singular points of the path of linearizations, a property of the path that is known to be sufficient to force bifurcation of nontrivial critical points is also necessary.