Time delay in the one-dimensional swarmalator model

We study the one-dimensional (1D) swarmalator model with time-delayed interactions. Delay preserves the familiar static states-sync, async, and phase wave-but generates two new families of nonstatic states. One bifurcates from the phase wave at τ=τ_{c}(J,K); the other bifurcates from the async state at K_{c}=-π/τ and includes a limit cycle, an irregular unsteady regime, and a partially synchronized state. We derive both bifurcation boundaries analytically. A surprising result is that the stability of the sync state is independent of τ; it depends only on the coupling strength.