Self-propulsion in the one-dimensional swarmalator model

We study the one-dimensional swarmalator model with self-propulsion v_{0}sinθ_{i}, where θ_{i} is the body orientation. The static states of the ordinary model unfold into traveling, breathing, split-wave, and chaotic states, two of which admit analytic reductions: A traveling two-cluster branch with a closed-form stability spectrum, and a split-wave ansatz whose active pair obeys an Adler equation. At large propulsion the chaos is transient rather than a chaotic attractor: the accumulated tangent growth saturates-at every propulsion speed across the chaos window, and at every system size we can integrate far enough to test-and trajectories end on the cluster states. The transient lifetime grows with system size, leaving the continuum limit open. The resulting states may serve as qualitative signatures for confined active oscillator arrays.