When your on the LAST level (the one with both hero and dark together)..

Okay - Rouge's goal is to drain the liquid from the Core. The switch to do this is in the center of the Chamber - but you need to unlock the Cage around it.

Climb up the 4 pillars around the Switch (the pillars will force you to go in a certain path) up to the highest one.

There's a Time Switch on it - before activating it - look around for where the yellow liquid is pouring out (look toward the 3rd pillar - where you came from).

Hit the Switch and then jump and glide in that direction - the hall is covered by a waterfall of the liquid at the lowest level.

Inside - defeat the Shield Robot and drop into the acidic Chamber below. Glide to the far wall and hang there for a Block to come out. Hit the Time Switch beneath the opening right after the Block emerges - and then head into the passage.

Glide into the Hovering Monster to defeat it and then jump up and hit the Time Switch before the Monsters can spring back to life - defeat the Slender one to see a switch - hit it and the Cage at the start will open.

Return to the previous room and hit the Time Switch - then drop to the Spring and bounce up to the passage you entered by.

Run forward - and go through the waterfall before the time flow resumes - then hit the switch and do Knuckles' part - then Sonic's then.....!

Hope that helped !