Electromagnetic two-body problem: recurrent dynamics in the presence of state-dependent delay

We study the electromagnetic two-body problem of classical electrodynamics as a prototype dynamical system with state-dependent delays. The equations of motion are analysed with reference to motion along a straight line in the presence of an electrostatic field. We consider the general electromagnetic equations of motion for point charges with advanced and retarded interactions and study two limits, (a) retarded-only interactions (Dirac electrodynamics) and (b) half-retarded plus half-advanced interactions (Wheeler-Feynman electrodynamics). A fixed point is created where the electrostatic field balances the Coulombian attraction, and we use local analysis near this fixed point to derive necessary conditions for a Hopf bifurcation. In case (a), we study a Hopf bifurcation about an unphysical fixed point and find that it is subcritical. In case (b), there is a Hopf bifurcation about a physical fixed point and we study several families of periodic orbits near this point. The bifurcating periodic orbits are illustrated and simulated numerically, by introducing a surrogate dynamical system into the numerical analysis which transforms future data into past data by exploiting the periodicity, thus obtaining systems with only delays.