**Winding number forces universal synchronisation**

Abstract

We prove that two completely different oscillators synchronise through arbitrarily weak reciprocal impulses. Any two planar oscillators with hyperbolic attracting limit cycles and equal natural periods lock stably one-to-one under translation impulses with generic directions and a suitable range of positive strength ratios, whatever their vector fields, limit cycles and transverse contraction rates. The mechanism is topological: the winding of the asymptotic phase gradient forces phase advances and delays, and the relative winding number of an effective impulse field gives lower bounds on attracting and unstable locked states. Equal periods are not essential: locking persists under small nonzero mismatches, with persistence and loss governed by a shifted phase balance. The analysis covers the full state-dependent impulsive dynamics, including transverse stability. Nearly simultaneous emissions may make the return discontinuous; under an explicit regularity condition the global return has a normally attracting Lipschitz invariant circle with topologically Morse--Smale dynamics.