The general rule is: your report all your income on your home state return, even the income earned out of state. You file a non-resident state return for the state you worked in and pay tax to that state. Your home state will give you a credit, or partial credit, for what you paid the non-resident state.
But, as the other reply indicates, which state is your resident state depends on the details. From what you describe (living & working in New York full time), you are a NY resident. Having a 2nd home in MI doesn't give you the "option" of declaring MI your tax home.
