Studies have been made on the influence of the attractor on the island of stability which co-exist in the phase space of the system under consideration. Derived is an infinite series of bifurcations of the island-of-stability boundary restructuring due to the attractor influence. An asymptotic law has been formulated for such parameters at which a qualitative restructuring of the island-of-stability boundary occurs for a given infinite bifurcation series. Shown is the effectiveness of discrete parametric control for high-periodic orbit stabilizaton in reversible systems.