Parameterization Method for State-Dependent Delay Perturbation of an Ordinary Differential Equation

We consider state-dependent delay equations (SDDEs) obtained by adding delays to a planar ODE with limit cycle. Situations of this type appear in models several physical processes, where small effects are added. Even if the small, they very singular perturbations since natural phase space an SDDE is infinite-dimensional space. show that for SDDE, there initial values which lead solutions similar those ODE. That is, exist periodic solution and two parameter family whose evolution converges (in case, these called isochrons). The method proof bypasses theory existence, uniqueness, dependence on parameters SDDE. class functions time have well defined behavior (e.g., periodic, or asymptotic periodic) derive functional impose These studied using analysis methods. provide result “a posteriori” format: given approximate equation, has some good condition numbers, we prove true close one. Thus, our can be used validate results numerical computations formal expansions. also leads practical algorithms. In companion paper, present implementation details representative results. One feature presented here it allows us obtain smooth their slow stable manifolds without studying smoothness flow (which seems problematic SDDEs, now optimal $C^1$).