If you play Akshan to steal an Aurora and the opponent plays Akshan to steal it back, what happens when the second Akshan dies?

When the opponent's Akshan (the second one) dies, you regain control of the Aurora.

Here's the sequence:

1. Your Akshan steals the Aurora, creating a control-change effect that lasts as long as your Akshan is on the board.
2. Opponent's Akshan steals it back. Because their effect is newer, it overrides yours in timestamp order — they now control the Aurora.
3. Opponent's Akshan dies. That control-change effect stops applying.
4. Recalculation: With the opponent's effect gone, the game looks for the next most recent active control-change effect — which is still your Akshan's. You regain control of the Aurora.

This is the same interaction as FAQ #8851 and FAQ #9734. If your Akshan were to also leave the board later, the Aurora would revert to its original owner (the player who brought it into the game), not you — since all control-change effects would be gone.