and space $t,x,$ respectively, then $|v_{3}|\gg v_{Te}.$

$(\mathbf{PSC})$ tells us that the number of particles that the wave velocity greatly exceeds their velocity is much larger than the number of particles’ velocity slower than the wave velocity. And we will show that cyclotron damping occurs under the above conditions, and that isn’t only consistent with the physical observation, but also is the same with the “Small Denominators" condition in KAM theory in \[1\] in some sense.

Second, compared with the electric field case, it is easy to find a new term $v\times B$ in the electromagnetic field setting. And this brings many difficulties because of the unboundedness of $ v.$ Based on the physical facts of Faraday Law of Electromagnetic induction and Lenz’s Law, we know that $\delta B$ generates the force that inhibits the change of the electric field. This helps us estimate such term. And the above fact leads us to study the following dynamics of the particles trajectory:

$$\begin{aligned}
   \left\{\begin{array}{l}
   \frac{d}{dt}X_{t,\tau}(x,v)=V_{t,\tau}(x,v),\quad\frac{d}{dt}V_{t,\tau}(x,v) =V_{t,\tau}(x,v)\times B_{0}
   +E[f](t,X_{t,\tau}(x,v)),\\
 X_{\tau,\tau}(x,v)=x,\quad  V_{\tau,\tau}(x,v)=v,
  \end{array}\right.
           \end{aligned}$$

where $B=B_{0}+\delta B.$ In other words, we reduce inhomogeneous dynamical system to homogeneous dynamical system. Hence, based on the above dynamical system (0.7), we call the adopted Newton iteration 