Berger M S
Belfer Graduate School of Science, Yeshiva University, New York, N.Y. 10033.
Proc Natl Acad Sci U S A. 1972 Jul;69(7):1737-8. doi: 10.1073/pnas.69.7.1737.
Let H be a real Hilbert space and f(x,lambda) be a C(2) operator mapping a small neighborhood U of (x(0),lambda(0)) epsilon (H x R(1)) into itself. We investigate the solutions of the equation f(x,lambda) = 0 near a solution (x(0),lambda(0)), assuming that f(x,lambda) is a gradient mapping and 0 < dim Ker f(x)(x(0),lambda(0)) < infinity. 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),lambda(0)). 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.
设(H)为实希尔伯特空间，(f(x,\lambda))是一个(C^2)算子，它将((x^{(0)},\lambda^{(0)})\in(H\times\mathbb{R}^1))的一个小邻域(U)映射到自身。我们研究方程(f(x,\lambda)=0)在解((x^{(0)},\lambda^{(0)}))附近的解，假设(f(x,\lambda))是一个梯度映射且(0\lt\dim\ker f_{x}(x^{(0)},\lambda^{(0)})\lt\infty)。特别地，我们表明，对于一个孤立临界点，马斯顿·莫尔斯的型数可用于证明在((x^{(0)},\lambda^{(0)}))处存在分岔点。该结果应用于一大类非线性哈密顿系统在“共振”情形下驻点附近周期运动的发现。
