observation. Therefore, the stability’s or unstability’s problem of the Vlasov-Maxwell equations is still open.

In the following we sketch the difficulties and methods in our paper’s setting. The crucial estimates of cyclotron damping include the following two inequalities:

$\bullet$ a control of $\rho=\int_{\mathbb{R}^{3}}fdv$ in $\mathcal{F}^{\lambda\tau+\mu}$ norm, that is, $\sup_{\tau\geq0}\|\rho_{\tau}\|_{\mathcal{F}^{\lambda\tau+\mu}}<\infty,$

$\bullet$ a control of $f_{\tau}\circ\Omega_{t,\tau}$ in $\mathcal{Z}^{\lambda'(1+b),\mu';1}_{\tau-\frac{bt}{1+b}}$ norm, where $\lambda'<\lambda,\mu'<\mu.$

However, during the iteration scheme, for cyclotron damping, from the view of the original Newton iteration, the characteristics are not only determined by the density $\rho^{n},$ but also related with the velocity at stage $n.$ However, $\rho^{n}$ is independent of the velocity and the key difficulty is that the velocity is unbounded. This makes that we obtain the estimates of the associated deflection $\Omega^{n}$ more difficult. But this difficulty doesn’t exist for Landau damping in \[23\]. To overcome this difficulty, on the basis of a new observation from Lenz’s Law, we reduce the classical dynamical system to the improved dynamical system (0.7). And the corresponding equation of the density $\rho[h^{n+1}]$ at stage $n+1$ becomes $$\rho[h^{n+1}](t,x)=\int^{t}_{0}\int_{\mathbb{R}^{3}}-\bigg[(E[h^{n+1}]\circ\Omega^{n}_{s,t}(x,v)\cdot G_{s,t}^{n})-
            (B[h^{n+1}]\circ\Omega^{n}_{s,t}(x,v)
            \cdot G_{s,t}^{n,v})$$ $$-(B[f^{n}]\circ\Omega^{n}_{s,t}(x,v))\cdot(\nabla'_{v}h^{n+1}\times V^{0}_{s,t}(x,v))\circ\Omega^{n}_{s,t}(x,v)\bigg](s, X^{0}_{s,t}(x,v),V^{0}_{s,t}(x,v))dvds
            +(\textmd{terms}\quad \textmd{from}\quad \textmd{stage}\quad n),$$ From the above equation, we see that, comparing with that in \[23\], there is a 