Stabilization of unstable periodic solutions of delay equations

Abstract

where x(t) and f(.) are scalar functions and T > 0 is a constant delay. Let x*(t,p) be a periodic trajectory (a limit cycle) of Eq. (1) with period T(p). As a rule, there exists a critical value of the parameter p* such that the limit cycle x*(t,p) is asymptotically orbitally stable for p p*. The problem is to localize and stabilize this cycle for p > p*, including the values of p for which Eq. (1) has chaotic behavior. Let us show that the approach suggested in [1] can be used for this problem. Let C[--T; 0] be the space of continuous real functions q~(.) defining initial conditions for Eq. (1) on the interval [--T;0]. We equip this space with the norm IIx( )ll = sup(Ix( )l, - T < ~ < 0). To Eq. (1) with the initial condition x(0) = ~(0), -~- < 0 < 0, we assign the system of equations dxt/dt = S(t)xt(~9)