Akshan, Mischievous

No. You gain control of the enemy Dazzling Aurora after its end-of-turn trigger condition has already been checked.

"At the end of your turn" refers to the Ending Step's end-of-turn point.^317.1.a383.1 At that point, the enemy Dazzling Aurora is still controlled by your opponent, so its controller-relative "your turn" condition is not fulfilled.^{191.5.a383.2.c}

Your Dazzling Aurora's end-of-turn ability plays Akshan after that point in the turn has been reached. After Akshan enters the board, his play effect is added to the chain as a pending item.^{383.4.a383.4.a.2} You gain control of the enemy Dazzling Aurora only when that ability later resolves.^407.5

If you take control of an enemy Dazzling Aurora before the Ending Step and still control it at the end-of-turn point, it triggers normally.

After both play effects have resolved, if either Akshan dies, the gear is controlled by the player whose control effect remains. While both effects remain, the player whose effect was generated most recently controls the gear.

Each resolved play effect creates a delayed passive ability, a control effect that remains active until that Akshan leaves the board.^391.4392 While both effects remain, they change the gear's controller in the same trait-altering layer.^479.2.a Neither effect changes the other's existence, scope, or result, so no dependency is established.^{480.1.a480.1.b480.1.c} Timestamp order therefore applies.⁴⁸² Each effect's timestamp is established when it begins applying, and the newer effect applies last and determines who controls the gear.^482.1482.3

When an Akshan dies, it moves from the board to the trash, ending that Akshan's control effect and leaving the other effect to determine control.^429.2