Parameterization method for state-dependent delay perturbation of an ordinary differential equation

Abstract

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