Given an analytic two-dimensional ordinary differential equation $ \frac{dx}{dt}(t) = X_0( x(t)), $ which admits a limit cycle, we consider the singular perturbation of adding a state-dependent delay: \begin{equation}\label{SDDE} \frac{dx}{dt} (t)= X(x(t), \varepsilon x(t-r(x(t)))), \qquad 0\leq \varepsilon \ll 1. \end{equation} We show that for small enough $\varepsilon$, there exist a limit cycle and a two-dimensional family of solutions to the perturbed state-dependent delay equation above, which resemble the solutions of the unperturbed ODE.

More precisely, we find a parameterization of the limit cycle and its stable manifold and we show that, there is a very similar parameterization that gives a 2-dimensional family of solutions of the delay equation.

Our work consists of two parts. In the theoretical part, we analyze the parameterization, and find functional equations to be satisfied (invariance equations). We prove a theorem in ``a posteriori'' format, that is, if there are approximate solutions of the invariance equations, then there are true solutions of the invariance equations nearby (with appropriate choices of norms). In the numerical part, we implement an algorithm which follows from the constructive proof of above theorem.

This is a joint work with Joan Gimeno and Rafael de la Llave.