  We consider normal forms in `magnetic bottle' type Hamiltonians of the form
$H=\frac{1}{2}(\rho^2_\rho+\omega^2_1\rho^2) +\frac{1}{2}p^2_z+hot$ (second
frequency $\omega_2$ equal to zero in the lowest order). Our main results are:
i) a novel method to construct the normal form in cases of resonance, and ii) a
study of the asymptotic behavior of both the non-resonant and the resonant
series. We find that, if we truncate the normal form series at order $r$, the
series remainder in both constructions decreases with increasing $r$ down to a
minimum, and then it increases with $r$. The computed minimum remainder turns
to be exponentially small in $\frac{1}{\Delta E}$, where $\Delta E$ is the
mirror oscillation energy, while the optimal order scales as an inverse power
of $\Delta E$. We estimate numerically the exponents associated with the
optimal order and the remainder's exponential asymptotic behavior. In the
resonant case, our novel method allows to compute a `quasi-integral' (i.e.
truncated formal integral) valid both for each particular resonance as well as
away from all resonances. We applied these results to a specific magnetic
bottle Hamiltonian. The non resonant normal form yields theorerical invariant
curves on a surface of section which fit well the empirical curves away from
resonances. On the other hand the resonant normal form fits very well both the
invariant curves inside the islands of a particular resonance as well as the
non-resonant invariant curves. Finally, we discuss how normal forms allow to
compute a critical threshold for the onset of global chaos in the magnetic
bottle.
