To make progress toward better computability of Morse–Floer homology and thus enhance the applicability of Floer theory, it is essential to have tools to determine the relative index of equilibria. Since even the existence of nontrivial stationary points is often difficult to accomplish, extracting their index information is usually out of reach. In this paper, we establish a computer-assisted proof approach to determining relative indices of stationary states. We introduce the general framework and then focus on three example problems described by partial differential equations to show how these ideas work in practice. Based on a rigorous implementation, with accompanying code made available, we determine the relative indices of many stationary points. Moreover, we show how forcing results can be then used to prove theorems about connecting orbits and traveling waves in partial differential equations.

为了提高莫尔斯-弗洛尔同调的可计算性，从而增强弗洛尔理论的适用性，有必要拥有确定平衡相对指数的工具。由于即使是非平凡驻点的存在也常常难以实现，因此提取它们的索引信息通常是遥不可及的。在本文中，我们建立了一种计算机辅助证明方法来确定静止状态的相对指数。我们介绍总体框架，然后重点关注偏微分方程描述的三个示例问题，以展示这些想法在实践中如何发挥作用。基于严格的实现，并提供可用的随附代码，我们确定了许多驻点的相对索引。此外，我们还展示了如何使用强迫结果来证明偏微分方程中有关连接轨道和行波的定理。