In the state of Illinois, if a mother drops her children off at the parents of their biological father without prior arrangements or reason, can this be classified at abandonment? If so, what is the time limit ?

No. This issue is so misunderstood, but if you had searched the Family Law forum (where this was moved to) you would have had an answer.

Abandonment is a criminal action that occurs when a child is left without supervision. So that's not the case here.

Abandonment MAY be used as grounds for something else. So what you need to tell us is what you are trying to do. Are you trying to remove custody from the mother or deny her visitation or what?

This can be used as grounds for the father to go back to court and request custody, esp if she left them for weeks at a time