E=W(x)\ast\rho(t,x),\rho(t,x)=\int_{\mathbb{R}^{3}}f(t,x,v)dv,\\
f(0,x,v)=f_{0}(x,v_{1},v_{2},v_{z}),f^{0}(v)=f^{0}(v_{1},v_{2},v_{z}),
\end{array}\right.$$ where the distribution function $f=f(t,x,v):\mathbb{R}^{+}\times\mathbb{T}^{3}\times\mathbb{R}^{3}\rightarrow\mathbb{R},W(x)=(W_{1}(x),W_{2}(x),0):
\mathbb{T}^{3}\rightarrow\mathbb{T}^{3},$ $ B_{0}$ is a constant magnetic field along the $\hat{z}$ direction, $E=E(t,x)$ is the electric field, $ B_{0}+B$ is the magnetic field.

For any $\eta,v\in\mathbb{R}^{3},k\in\mathbb{N}_{0}^{3},$ we assume that the following conditions hold in equations (0.9).

-   $ W(x)$ is an odd function on $x_{3},$ $|\widehat{W}(k)|\leq\frac{1}{1+|k|^{\gamma}},\gamma>1,$ where $ W(x)=(W_{1}(x),W_{2}(x),0);$

-   $|\hat{f}^{0}(\eta)|\leq C^{0}e^{-2\pi\lambda_{0}|\eta|},|\partial_{\eta_{3}}\hat{f}^{0}(\eta)|\leq C^{0}e^{-2\pi\lambda_{0}|\eta|},$ $|f^{0}(\cdot,v_{3})|\leq C^{0}e^{-2\pi\alpha_{0}|v_{3}|},$ for some constants $\lambda_{0},\alpha_{0},C^{0}>0;$

-   $|\hat{f}_{0}(k,\eta)|\leq C_{0} e^{-2\pi\lambda_{0}|\eta|}$ for some constant $C_{0}>0,$ where $\lambda_{0}$ is defined in (ii);

-   In addition, $(\mathbf{PSC})$ holds,

    $(\mathbf{PSC}):$ for any component velocity in the $\hat{z}$ direction $v_{3}\in\mathbb{R},$ there exists some positive constant $v_{Te}$ such that if $v_{3}=\frac{\omega}{k_{3}} \textmd{ when } k_{3}\neq0; \textmd{ or } k_{3}=0$ where $\omega,k$ are frequencies of time and space $t,x,$ respectively, then $|v_{3}|\gg v_{Te}.$

Then for any fixed $\eta_{3},k_{3},$ and for any $\lambda'_{0}<\lambda_{0},$ we have $$\begin{aligned}
         &|\hat{f}(t,k,\eta)-\hat{f}_{0}(k,\eta)|\leq  e^{-2\pi\lambda'_{0}|\eta_{3}+k_{3}t|},\quad|\hat{\rho}(t,k)-\hat{\rho}_{0}|\leq  e^{-2\pi\lambda'_{0}|\eta_{k1}|}e^{-2\pi\lambda'_{0}|\eta_{k2}|} e^{-\lambda'_{0}|k_{3}|t},\notag\\
        & |\hat{E}(t,k)|\leq e^{-2\pi\lambda'_{0}|\eta_{k1}|}e^{-2\pi\lambda'_{0}|\eta_{k2}|}e^{-2\pi\lambda'_{0}|k_{3}|t},|\hat{B}(t,k)|\leq te^{-2\pi\lambda'_{0}|\eta_{k1}|}e^{-2\pi\lambda'_{0}|\eta_{k2}|}e^{-2\pi\lambda'_{0}|k_{3}|t}.
         \end{aligned}$$ where $\rho_{0}=\int_{\mathbb{T}^{3}}\int_{\mathbb{R}^{3}}f_{0}(x,v)dvdx,\eta_{k1}=\frac{1}{\Omega}(-k_{2}\cos\Omega t+k_{2}-k_{1}\sin\Omega t),\eta_{k2}=\frac{1}{\Omega}(-k_{2}\sin\Omega t-k_{1}+k_{1}\cos\Omega t).$

In the linear case, from (1.2) and (1.24)-(1.26), it is easy to observe that cyclotron damping is almost the same with Landau damping when $B_{0}$ tends to zero. However, when $B_{0}$ is fixed, in the horizon direction there is no damping and the motion of particles 