Differential equations modeling the motion of N charged particles are subject to small but complex delays and advances, defined implicitly by the solutions of the system. To understand the dynamics of such systems, we introduce functional perturbations—incorporating delays or advances—into ordinary differential equations. We demonstrate that near certain special solutions of the unperturbed equations, there exist solutions of the perturbed functional equations. Moreover, we establish smooth dependence of these solutions on parameters. We develop methods to quantify the size of allowable perturbations using a computer-assisted approach. Finally, we present some recent advances in the study of a point charge interacting with a small plasma, including results on existence and uniqueness of solutions to the Vlasov-Poisson system and their asymptotic behavior.