Chaotic imprints of dark matter in extreme mass-ratio inspirals

Extreme mass-ratio inspirals (EMRIs) are among the most powerful probes of strong-field gravity and of the environments surrounding supermassive compact objects. Motivated by the expected presence of dark matter near galactic centers, we investigate the emergence and gravitational-wave imprints of chaotic dynamics in EMRIs evolving in non-vacuum spacetimes. Within a unified dynamical framework, we analyze test-particle motion in a broad class ofdark-matter-embedded geometries, including singular black holes, regular black holes, naked singularities, and Einstein-cluster configurations. We show that environmental perturbations generically break integrability in the strong-field regime, giving rise to chaotic motion whose onset, duration, and termination depend sensitively on horizon structure, core regularization, and matter distribution. Using the numerical Kludge approach, we demonstrate that chaotic trajectories produce systematic qualitative modifications of the emitted gravitational radiation, such as irregular amplitude modulation and loss of phase coherence, in contrast to the smooth, quasi-periodic waveforms generated by regular motion. Our results establish the robustness of chaos in environmentally perturbed EMRIs and provide a clear conceptual link between nonlinear orbital dynamics, spacetime structure, and observable gravitational-wave signatures.