Ugh that is a difficult situation.

I'm hesitant to suggest this as I'm not a doctor, however, I present many of the same habits and cycles, and it sounds like she may have a fear of abandonment. You can find a good introductory article here. This is complicated by what seems like a notion that bisexuality is wrong as or at the very least will complicate here life by societies unacceptance.

In the end you only have three options. Break up, accept her for who she is, or talk to her about how you are feeling. Find out what is motivating these behaviors and set boundaries that you can both agree on. You can not change her behavior but perhaps she is unaware as to how disruptive it really is to your relationship.

Anyway I wish you the best of luck.