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In this paper, we investigative the large time decay and stability to any given global smooth solutions of the $3$D incompressible inhomogeneous MHD systems. We proved that given a solution $(a, u, B)$ of (\ref{mhd_a}), the velocity field and magnetic field decay to $0$ with an explicit rate, for $u$ which coincide with incompressible inhomogeneous Navier-Stokes equations \cite{zhangping}. In particular, we give the decay rate of higher order derivatives of $u$ and $B$ which is useful to prove our main stability result. For a large solutions of (\ref{mhd_a}) denoted by $(a, u, B)$, we proved that a small perturbation to the initial data still generates a unique global smooth solution and the smooth solution keeps close to the reference solution $(a, u, B)$. Due to the coupling between $u$ and $B$, we used elliptic estimates to get $\|(u, B)\|_{L^{1}(\mathbb{R}^{+};\dot{B}_{2,1}^{5/2})} < C$, which is different to Navier-Stokes equations.

On the decay and stability of global solutions to the 3D inhomogeneous MHD system