The Asylum is a location on the Dreadnaught. It contains a large open space in which Tombships fly in the far distance, as well as a portal to Oryx's inner sanctum.
The Asylum may be reached in Patrol by rapidly killing the three Wardens of Oryx in The Founts. If all of them die within a few seconds of each other, the door will open. This must be done as part of the questA Sword Reforged. The Wardens will respawn after a few minutes if the attempt fails.
A fully-charged Agonarch Rune can be used here to activate a world event in which waves of Taken will spawn.