Stability of periodic solutions of state-dependent delay-differential equations

We consider a class of autonomous delay-differential equations. ż(t)=f(z_t) which includes equations of the form. ż(t)=g(z(t),z(t-r₁),...,z(t-r_n)),r_i=r_i(z(t))for1≤i≤n, with state-dependent delays ri(z(t))≥0. The functions g and ri satisfy appropriate smoothness conditions.We assume there exists a periodic solution z=x(t) which is linearly asymptotically stable, namely with all nontrivial characteristic multipliers μ satisfying |μ|<1. We prove that the appropriate nonlinear stability properties hold for x(t), namely, that this solution is asymptotically orbitally stable with asymptotic phase, and enjoys an exponential rate of attraction given in terms of the leading nontrivial characteristic multiplier.A principal difficulty which distinguishes the analysis of equations such as (*) from ones with constant delays, is that even with g and ri smooth, the associated function f is not smooth in function space. Techniques of Hartung, Krisztin, Walther, and Wu are employed to resolve these issues.