Limit cycles govern wave energy accumulation in space-time-varying materials

Abstract

We study the amplification and concentration of wave energy in smooth, periodic space-time media through the dynamics of the characteristic curves of the associated wave equation. In an impedance-matched setting, we show that stable limit cycles attract neighbouring characteristics, focusing the wave energy density. The exponential attraction rate is estimated analytically from the system's Lyapunov exponent. We further show that the attraction of characteristics and energy accumulation persist under a range of nonperiodic perturbations. Finally, we present a Floquet-Fourier numerical method for computing stable and unstable limit cycles and their attraction rates. Throughout, we exemplify the results using the example of modulation given by a superposition of travelling waves, revealing saddle--node bifurcations where limit cycles may appear depending on the relative wave speeds. Full-wave simulations confirm that the characteristic mechanism continues to describe energy accumulation beyond the impedance-matched case.