Periodic dynamics in monotone regularizations: realization and isolation

In a planar Sotomayor–Teixeira regularization one profile determines both the restoring force and the damping near a regularized equilibrium. We ask which local periodic dynamics a strictly monotone profile permits, and which additional assumptions restrict them. For jointly analytic side fields and a profile that belongs to one quasianalytic Denjoy–Carleman class away from the equilibrium height, the return map near an equilibrium with zero trace and positive determinant is either the identity or has only isolated fixed points of finite order at positive amplitude; the proof crosses the equilibrium height by analytic dependence on the tangential variable and does not use monotonicity. For two structured classes of side fields, an equal-potential comparison and a profile-order condition give explicit tests for noncentrality and for the absence of small periodic orbits. In the other direction, for a fixed affine two-fold every smooth damping germ J with J(0) = 0 is realized exactly by one strictly monotone profile. For flat, periodically modulated damping primitives, an analysis of the exact return map gives an exhaustive trichotomy, a complete branch count under finite-order phase zeros, and isolation under analyticity on a low positive band. One profile has infinitely many exact double cycles at a reference thickness, while at every thickness its sufficiently small cycles have multiplicity at most two. All results are local at a fixed positive thickness, and finiteness on compact amplitude ranges does not exclude accumulation at the equilibrium.