Existence of Complex Homoclinic Points

A method of proving the existence of a local transversal intersection between two immersed holomorphic curves in $\mathbb{C}^2$ is suggested. It is based on an application of the inverse function theorem, the corresponding inequalities being checked numerically. The method is applied to the problem of interpretation of tips of ferns on the unstable manifold of the semistandard map as complex homoclinic points.