well as any equilibrium plasma currents.

A stability analysis based on Eq.(0.1) and Eqs.(0.2) proceeds in the following manner. The quantities $f(t,x,v),E(t,x),$ and $B(t,x)$ are expressed as the sum of their equilibrium values plus a time-dependent perturbation: $$\begin{aligned}
  &f(t,x,v)=f^{0}(x,v)+\delta f(t,x,v),\quad E(t,x)=E^{0}(x)+\delta E(t,x),\quad B(t,x)=B^{0}(x)+\delta B(t,x).\notag\\
  \end{aligned}$$ The quantities $f^{0}(x,v),E^{0}(x)$ and $B^{0}(x)$ satisfy (0.3). The time development of the perturbations $\delta f(t,x,v),\delta E(t,x),$ and $\delta B(t,x)$ is studied by using Eq.(0.1) and Eqs.(0.2). For small-amplitude perturbations, the Vlasov-Maxwell equations are linearized about the equilibrium $f^{0}(x,v),E^{0}(x)$ and $B^{0}(x).$ This gives $$\begin{aligned}
 &\frac{\partial\delta f}{\partial t} + v\cdot\nabla_{x}\delta f(t,x,v)+(E^{0}+v\times B^{0})\cdot\nabla_{v}\delta f(t,x,v)
 =-(\delta E+v\times \delta B)\cdot\nabla_{v}f^{0}(x,v),\quad\notag\\
 &\nabla\cdot \delta B(t,x)=0,\nabla\times \delta B(t,x)=\int_{\mathbb{R}^{3}}v\delta f(t,x,v)dv+\frac{\partial\delta E}{\partial t},\quad\nabla\cdot
  \delta E(t,x)=\int_{\mathbb{R}^{3}}\delta f(t,x,v)dv.\notag\\
 \end{aligned}$$ If the perturbations $\delta f(t,x,v),\delta E(t,x),$ and $\delta B(t,x)$ grow, then the equilibrium distribution $f^{0}(x,v)$ is unstable. Otherwise, the perturbations damp, so the system returns to equilibrium and is stable. We assume that the equilibrium $f^{0}$ is independent of space, namely, $f^{0}(x,v)=f^{0}(v).$

From the above analysis and the form of Vlasov-Maxwell equations, it is obvious that when $B\equiv0,$ cyclotron damping is reduced to Landau damping. Hence, the method used is similar to that employed by Mouhot and Villani \[23\]. However, compared with the electric field, a static magnetic 