  Motivated by the precedent study of Ordenes-Huanca and Velazquez [JSTAT
\textbf{093303} (2016)], we address the study of a simple model of a pure
non-neutral plasma: a system of identical non-relativistic charged particles
confined under an external harmonic field with frequency $\omega$. We perform
the equilibrium thermo-statistical analysis in the framework of continuum
approximation. This study reveals the existence of two asymptotic limits: the
known Brillouin steady state at zero temperature, and the gas of harmonic
oscillators in the limit of high temperatures. The non-extensive character of
this model is evidenced by the associated thermodynamic limit,
$N\rightarrow+\infty: U/N^{7/3}=const$, which coincides with the thermodynamic
limit of a self-gravitating system of non-relativistic point particles in
presence of Newtonian gravitation. Afterwards, the dynamics of this model is
analyzed through numerical simulations. It is verified the agreement of
thermo-statistical estimations and the temporal expectation values of the same
macroscopic observables. The system chaoticity is addressed via numerical
computation of Lyapunov exponents in the framework of the known \emph{tangent
dynamics}. The temperature dependence of Lyapunov exponent $\lambda$ approaches
to zero in the two asymptotic limits of this model, reaching its maximum during
the transit between them. The chaos of the present model is very strong, since
its rate is faster than the characteristic timescale of the microscopic
dynamics $\tau_{dyn}=1/\omega$. A qualitative analysis suggests that such a
strong chaoticity cannot be explained in terms of collision events because of
their respective characteristic timescales are quite different,
$\tau_{ch}\propto \tau_{dyn}/N^{1/4}$ and $\tau_{coll}\propto \tau_{dyn}$.
