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We study the existence of a homoclinic orbit and the onset of chaotic motion for a massive particle moving around a Schwarzschild-like black hole embedded in a Dehnen-(1, 4, 5/2) type dark matter halo, within the extreme-mass-ratio limit q = m/M ≪ 1, where m and M are the masses of the particle and the central black hole, respectively. The presence of the halo modifies the spacetime curvature and consequently deforms the effective potential governing the particle's motion. Using the Hamiltonian formulation, we derive the conditions under which an unstable circular orbit and the associated homoclinic trajectory arise, marking the separatrix between bound and plunging motion. By analyzing the effective potential and the corresponding phase-space structure, we identify the transition from regular to chaotic dynamics in the near-horizon region. Numerical analyses through Poincaré sections and Lyapunov exponents calculations demonstrate that increasing the halo density, scale radius along with energy amplifies nonlinear effects, which leads to chaos eventually. We demonstrate that within a dark matter halo environment, the dynamical stability of particle motion can be significantly altered without violating the universal surface gravity bound on chaos. This work provides a deeper understanding of horizon-induced chaos in astrophysically realistic environments and serves as a theoretical basis for exploring its possible imprints on gravitational wave signals in extreme-mass-ratio inspirals system.

We derive new exact gravitational wave solutions with dynamical torsion and nonmetricity tensors in the framework of cubic metric-affine gravity (MAG). For this purpose, we consider the full algebraic classification of the gravitational field in general metric-affine geometries and impose a set of type N conditions on the field strength tensors that implement the kinetics of torsion and nonmetricity in a particular cubic MAG model, recently considered to eliminate ghostly instabilities from the vector and axial sectors of the theory. The new solutions represent pp waves characterized by a metric function that includes the dynamical contributions of the torsion and nonmetricity tensors provided by the field equations of the model. In particular, these quantities induce a scalar polarization mode in the gravitational-wave spectrum, thus offering a distinctive phenomenological signature beyond the ordinary tensor polarization modes of general relativity.