Examining $H$-Closed Ducci Sequences on ${\mathbb{Z}}_m^n$

Let $D$ be an endomorphism on ${\mathbb{Z}}_m^n$ so that $$D(x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_n)=(x_1+x_2\text{mod}m,x_2+x_3\text{mod}m,\mathrm{.}\mathrm{.}\mathrm{.},x_n+x_1\text{mod}m)\mathrm{.}$$ We call the sequence $\{D^{\alpha }(\mathbf{u}){\}}_{\alpha =0}^{\mathrm{\infty }}$ the Ducci sequence of $\mathbf{u}\in {\mathbb{Z}}_m^n$, which always enters a cycle. Now let $H$ be an endomorphism on ${\mathbb{Z}}_m^n$ such that $$H(x_1,x_2,\mathrm{.}\mathrm{.}\mathrm{.},x_n)=(x_2,x_3,\mathrm{.}\mathrm{.}\mathrm{.},x_n,x_1)\mathrm{.}$$ In this paper, we will talk about a few cases when $\mathbf{u}$ and $H^{\beta }(\mathbf{u})$ have the same Ducci cycle for $\beta >0$, as well as prove a few cases of $n,m$ where this is guaranteed for every $\mathbf{u}\in {\mathbb{Z}}_m^n$.