I grew up in a highly abusive home. My mother is 70 anound beginning to experience health issues. I am being pressured to provide some measure of care and stewardship. Do I have a legal obligation to care for her in any measure? How are these concerns handled for those elderly who do not have family? I live in Michigan in Washtenaw County. My mother lives in Oakland County.

Are there more children? Does she have a husband? Has she named a POA?

There is no legal obligation for you to do anything.

There is no law in any state in the US that requires anyone to care for an adult relative of any kind.
Various social programs will cover her needs (and take all her assets, including her home if she owns one, and has to go into a nursing home).
Basically, that is SSI, Medicaid, food stamps, and public housing. All are welfare. If she ever worked, she probably has retirement SS by now. If she was married for more than 10 years, she can collect on any benefits the husband(s) was/were or is/are entitled to.
Every city and town has a social worker who's job it is to handle this. Areas of small towns might share one.