Stability analysis of the delayed feedback control method with commutative gain matrices

Abstract

The purpose of our investigation is to guarantee the delayed feedback control method in chaos control proposed by Pyragas analytically. Recently, we gave the conditions for deciding whether the control method is successful under some assumption(we will say ``Commutative Assumption'') by using the Floquet theory to a delay differential equation. In this article, we summarize the above results and the fundamental matter of the Floquet theory to a delay differential equation. As an application, we will also show numerically that some unstable periodic orbits of the R\"ossler system can be stabilized by the method using a different control gain matrix from that which Pyragas used. Especially, the numerical result that the 3 times periodic orbit can be stabilized is newly obtained result.