What are the legal implications of a child refusing to have anything to do with an abusive parent who has all the signs of dementia?

Our mother who is 83 has all the signs of dementia or possibly bi-polar which given her behavior over the years is probably the case. She has always been abusive and aggressive and now she is forgetting things, hiding things and her accusations of us stealng from her have become even worse. She is paranoid and sees and hears people that we cannot see.
Having put up with her abuse for so many years and as she is now refusing our help to get her to a doctor and have her properly diagnosed, we are not able to find enough sympathy and compassion for her to find ways of helpng her. As single parents my sisters and I will not and are not able to assist her financially. She is currently with my sister but has no qualms about abusing my sister and her children at every turn, yet she keeps coming back to live with them every time she fights with them and walks out. She has lived in numerous senior apartments in the area and some no longer want her there because of her aggressive nature.
We want to know what our rights are if she walks out of my sisters place again which she appears to want to do as she is walking the streets looking for an apartment. What do we do if her dementia gets so bad that she can no longer help herself. Will we be forced to care for her or help her get care?
My sister is the only one living in the US with my mother. The rest of us live in Australia and South Africa respectively.
We have no idea what her financial status is, nor have we ever wanted to know,because as far back as we can remember she has been accusing us of stealing from her one way or another and often beaten us for no reason at all.
I have come from Australia to try and help my sister get my mother diagnosed, but as mentioned she refuses to get help and has now stopped talking to me and I have been here just three days.
I am worried that when I leave which I must in a week or so, she will still be with my sister and will continue to verbally abuse her at every turn knowing full well that my sister cannot turf her out into the street. My sister has lived with this abuse for 30 years, why does she have to still put up with it from someone who so plainly does not want to admit that she needs help and does not want to get help. The one time after an episode where she became disoriented and could not remember where she had been staying my mother did see a doctor, but refuses to take the medication she was given saying the doctors are trying to make her crazy. The unfortunate thing is that she has some very lucid moments and is clever enough and sly enough to know when to behave properly, but within the last few weeks many people in the neighborhood and the church have witnessed her behavior and aggression. The police have even been called by others a few times. But she is clever enough to know not to cross the line that will make her a danger to others. She has however told me that she would rather kill herself than go into a care facility.
Please please can someone in the USA tell us how best to deal with this situation and help me protect my sister from having to bear unnecessary costs and stress when our mother gets to the point of no return with her illness.