  Topological invariants play a key role in the characterization of topological
states. Due to the existence of exceptional points, it is a great challenge to
detect topological invariants in non-Hermitian systems. We put forward a
dynamic winding number, the winding of realistic observables in long-time
average, for exploring band topology in both Hermitian and non-Hermitian
two-band models via a unified approach. We build a concrete relation between
dynamic winding numbers and conventional topological invariants. In
one-dimension, the dynamical winding number directly gives the conventional
winding number. In two-dimension, the Chern number relates to the weighted sum
of dynamic winding numbers of all phase singularity points. This work opens a
new avenue to measure topological invariants not requesting any prior knowledge
of system topology via time-averaged spin textures.
