Morse and NonMorse Functions

Here is a contour map of a Morse function. The island theorem works here. This height function has 6 maxima, 5 minima and 10 saddle points. The total index is 1. Here is a contour map of a non-Morse function. The center is a critical point in which more than two lines cross. It can not be a classical saddle. The Morse Lemma prevents this. In this case, there are 5 maxima, 5 minima and no saddle points. The central point has index -9. It is a non-Morse critical point.