Controlling chaotic transients: Yorke’s game of survival

We consider the tent map as the prototype of a chaotic system with escapes. We show analytically that a small, bounded, but carefully chosen perturbation added to the system can trap forever an orbit close to the chaotic saddle, even in presence of noise of larger, although bounded, amplitude. This problem is focused as a two-person, mathematical game between two players called “the protagonist” and “the adversary.” The protagonist’s goal is to survive. He can lose but cannot win; the best he can do is survive to play another round, struggling ad infinitum. In the absence of actions by either player, the dynamics diverge, leaving a relatively safe region, and we say the protagonist loses. What makes survival difficult is that the adversary is allowed stronger “actions” than the protagonist. What makes survival possible is (i) the background dynamics (the tent map here) are chaotic and (ii) the protagonist knows the action of the adversary in choosing his response and is permitted to choose the initial point $x_0$ of the game. We use the “slope 3” tent map in an example of this problem. We show that it is possible for the protagonist to survive.