Persistence of hyperbolic solutions of ODE's under functional perturbations: Applications to the motion of relativistic charged particles

Abstract

We rigorously construct a variety of orbits for certain delay differential equations, including the electrodynamic equations formulated by Wheeler and Feynman in 1949. These equations involve delays and advances that depend on the trajectory itself, making it unclear how to formulate them as evolution equations in a conventional phase space. Despite their fundamental significance in physics, their mathematical treatment remains limited. Our method applies broadly to various functional differential equations that have appeared in the literature, including advanced/delayed equations, neutral or state-dependent delay equations, and nested delay equations, under appropriate regularity assumptions. Rather than addressing the notoriously difficult problem of proving the existence of solutions for all the initial conditions in a set, we focus on the direct construction of a diverse collection of solutions. This approach is often sufficient to describe physical phenomena. For instance, in certain models, we establish the existence of families of solutions exhibiting symbolic dynamics. Our method is based on the assumption that the system is, in a weak sense, close to an ordinary differential equation (ODE) with "hyperbolic" solutions as defined in dynamical systems. We then derive functional equations to obtain space-time corrections. As a byproduct of the method, we obtain that the solutions constructed depend very smoothly on parameters of the model. Also, we show that many formal approximations currently used in physics are valid with explicit error terms. Several of the relations between different orbits of the ODE persist qualitatively in the full problem.