Bifurcation Theory and the Type Numbers of Marston Morse

Abstract

Let H be a real Hilbert space and f(x,λ) be a C ² operator mapping a small neighborhood U of (x ₀,λ₀) ε (H × R¹) into itself. We investigate the solutions of the equation f(x,λ) = 0 near a solution (x ₀,λ₀), assuming that f(x,λ) is a gradient mapping and 0 < dim Ker f_x(x ₀,λ₀) < ∞. In particular, we show that the type numbers of Marston Morse for an isolated critical point can be used to prove the existence of a point of bifurcation at (x ₀,λ₀). An application of this result is given to the discovery of periodic motions near a stationary point for a large class of nonlinear Hamiltonian systems in “resonant” cases.