  Single crystals of the Sr$_2$RuO$_4$-Ru eutectic system are known to exhibit
enhanced superconductivity at 3~K, in addition to the bulk superconductivity of
Sr$_2$RuO$_4$ at 1.5~K. The 1.5-K phase is believed to be a spin-triplet,
chiral $p$-wave state with the multi-component order parameter, giving rise to
chiral domain structure. In contrast, the 3-K phase is attributable to enhanced
superconductivity of Sr$_2$RuO$_4$ in the strained interface region between Ru
inclusion of a few to tens of micrometers in size and the surrounding
Sr$_2$RuO$_4$. We investigate the dynamic behavior of a topological junction,
where a superconductor is surrounded by another superconductor. Specifically,
we fabricated Nb/Ru/Sr$_2$RuO$_4$ topological superconducting junctions, in
which the difference in phase winding between the $s$-wave superconductivity in
Ru micro-islands induced from Nb and the superconductivity of Sr$_2$RuO$_4$
mainly governs the junction behavior. Comparative results of the asymmetry,
hysteresis and noise in junctions with different sizes, shapes, and
configurations of Ru inclusions are explained by the chiral domain-wall motion
in these topological junctions. Furthermore, a striking difference between the
1.5-K and 3-K phases is clearly revealed: the large noise in the 1.5-K phase
sharply disappears in the 3-K phase. These results confirm the multi-component
order-parameter superconductivity of the bulk Sr$_2$RuO$_4$, consistent with
the chiral $p$-wave state, and the proposed non-chiral single-component
superconductivity of the 3-K phase.
