Exotic diffeomorphisms on $4$-manifolds with $b_2^{+}=2$

While the exotic diffeomorphisms turned out to be very rich, we know much less about the $b_2^{+}=2$ case, as parameterized gauge-theoretic invariants are not well defined. In this paper we present a method (that is, comparing the winding number of parameter families) to find exotic diffeomorphisms on simply-connected smooth closed $4$-manifolds with $b_2^{+}=2$, and as a result we obtain that $2\mathbb{C}{\mathbb{P}}^2\mathrm{\#}10(-\mathbb{C}{\mathbb{P}}^2)$ admits exotic diffeomorphisms. This is currently the smallest known example of a closed $4$-manifold that supports exotic diffeomorphisms.