The singer was "appalled and shocked" when Murphy announced their break-up on a Dutch TV show last year and denied fathering her then-unborn child.
Brown received paternity test results in June proving the Dreamgirls actor is the father of four-month-old baby Angel Iris.
Her lawyer Gloria Allred claims Murphy "has not paid one penny of child support" and that a paternity case has been filed against him.
Allred says, "We've had communication with his attorney, and now we're moving forward.
"Mr. Murphy was promising to do the right thing but to date he has not done the right thing. Mr. Murphy can have all the babies he chooses to have, but his responsibility needs to be to his own children. (Brown) is seeking sole physical and legal custody."
Brown adds, "I want (Angel) to know that she has two parents."
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