Burst-relax endpoint geometry on attracting slow manifolds under fast-only actuation

Fast-only actuation in slow-fast systems raises a local endpoint question: which slow tangent directions can be produced by a short controlled fast burst followed by autonomous relaxation near an attracting slow manifold? This paper formulates that question through endpoint maps defined for a fixed small time-scale parameter. We define the burst endpoint map, the relaxation map, and the induced slow displacement map; prove a local rank theorem for the one-cycle endpoint image; and derive a singular-limit formula for the rescaled endpoint derivative. The derivative factorization identifies two mechanisms that determine endpoint rank: access to fast directions during the burst and sensitivity of the slow drift to the fast deviation during relaxation. A fast-actuated FitzHugh–Nagumo calculation shows how the mechanism arises in a standard neurodynamic model. A two-fast, two-slow diagnostic example then demonstrates full rank, fast-accessibility rank loss, and slow-sensitivity rank loss within one system. The resulting endpoint-rank test gives a concrete first-order diagnostic for designing and interpreting single burst-relax cycles. Graphical abstract A short controlled fast burst moves the state away from an attracting slow manifold; autonomous relaxation then returns it to the manifold and produces a small, structured displacement in the slow variables. The paper turns this geometric picture into an endpoint map and characterizes the local directions generated by one burst-relax cycle.