The Influence of Delayed Feedback Control on Stabilization of Periodic Orbits

In this paper we solve problems of stabilization of unstable cycle by the delay feedback. We study a model equation with qubic nonlinearity. In this case only one multiplicator is located outside a unit circle. Delay time is proportional to the cycle period. The $D$-partition of the parameter plane is obtained. The main result is analytically found conditions for parameters of delay control such that the initial cycle is stable. Also, we have found necessary and sufficient conditions of solvability of the stabilization problem. As a consequence, the problem of stablity of the Stuart–Landau equation periodic solution is completely solved.