  In this paper, we consider the asymptotic behavior of global solutions to 3D
anisotropic incompressible MHD systems. For the 3D MHD system with horizontal
dissipation and full magnetic diffusion, it is shown that $\uh(t)$ decays at
the rate of $O(t^{-(1-\f{1}{p})})$, $u_3(t)$ decays at the rate of
$O(t^{-\f{3}{2}(1-\f{1}{p})})$ and $B(t)$ decays at the rate of
$O(t^{-\f{3}{2}(1-\f{1}{p})-\f{1}{2}})$. Furthermore, we give the asymptotic
expansion of solutions. We prove that the leading term of $\uh(t)$ is a
combination of linear solution and two integrals from nonlinear coupling
effects, while for $u_3(t)$ the leading term is given by only the linear
solution without the influence of magnetic field. Though the dissipation of
velocity is weak, we show that the full magnetic diffusion is robust enough to
keep the asymptotic expansion of magnetic field basically expected. However,
the magnetic field turns out to affect the higher order asymptotic expansions
of $u_3(t)$. The obtained results reveal the crucial role played by the
magnetic field in the asymptotic analysis of anisotropic incompressible MHD
equations and are expected to enhance the understanding of incompressible MHD
equations from the viewpoint of mathematics.
