In a previous paper we studied parametrized autonomous systems and gave a computable criterion that an approximate orbit connecting hyperbolic equilibria is shadowed by a true connecting orbit. This criterion was used to give rigorously verified examples of Shilnikov saddle-focus homoclinic orbits in three dimensions. This involved verifying a condition on the eigenvalues of the linearization at the equilibrium. In dimensions greater than three, there are three more conditions which must be established: general position, asymptotic tangency and a transversality condition. In this paper we give computable criteria for verifying these three conditions. An example in four dimensions, in which detailed rigorous computations are carried out, is given.

在先前的论文中，我们研究了参数化的自治系统，并给出了一个可计算的标准，即一个近似的连接双曲均衡的轨道被一个真实的连接轨道所遮盖。该标准用于给出三个维度的希尔尔科夫鞍形聚焦同宿轨道的经过严格验证的示例。这涉及在平衡时验证线性化特征值的条件。在尺寸大于3的情况下，必须建立另外三个条件：一般位置，渐近切线和横向条件。在本文中，我们给出了验证这三个条件的可计算标准。给出了四个维度的示例，其中进行了严格的详细计算。