arXiv:2610.33114v1 Announce Type: new Abstract: In this paper, new uniqueness criteria for Rayleigh--Li\'enard systems with damping of positive odd degree in the velocity are established. Under suitable conditions, any limit cycle is unique and hyperbolic, with its stability determined by the damping signs. Neither symmetry nor monotonicity of the damping and restoring functions is required. Changes of variables and time yield further criteria for generalized Kukles systems. Applications to three polynomial families yield uniqueness and nonexistence results. In particular, we resolve an open uniqueness problem for a hybrid van der Pol--Rayleigh oscillator left open by Chen et al. (Phys. D, 2021). We also give an alternative uniqueness proof for a system with homogeneous components and obtain uniqueness and nonexistence results for a modified hybrid van der Pol--Rayleigh oscillator.
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