Persistence of hyperbolic solutions of ODE's under functional perturbations: Applications to the motion of relativistic charged particles
Joan Gimeno, Rafael de la Llave, Jiaqi Yang
TL;DR
The paper develops a constructive fixed-point framework to demonstrate the persistence of uniformly hyperbolic ODE trajectories under broad functional perturbations, including delay/advance and state-dependent delays. By expressing perturbed solutions in the form $x(t)=(x_0+\widehat x)\circ\phi(t)$ and solving a coupled invariance problem, the authors prove existence, local uniqueness, and smooth parameter dependence for small perturbations $\varepsilon$. The approach yields an a-posteriori format, enabling verification and computer-assisted proofs, and applies to physically relevant models such as Wheeler–Feynman electrodynamics with implicit delays. The results encompass various delay types (neutral, nested, state-dependent) and provide explicit regularity and contraction estimates, showing that hyperbolic dynamics persist under a broad class of singular and nonlocal perturbations. This framework advances the rigorous understanding of complex dynamics in functional differential equations with practical implications for relativistic particle motion and related physical systems.
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.