The topic of this talk is a test ion motion in a cylindrical magnetic field.
The back ground field consists of non-zero strong static magnetic field
with periodic boundary condition and null electric field.
The main topic of this talk is on the effect of unstable fixed points
for individual particles,
which bring about the chaos [1,2].
This chaos indicates that there is no invariant associated with the
magnetic moment
which is assumed in a gyrokinetic theory [3,4] widely used for numerical
simulation of magnetized plasmas [5,6].
Further we investigate ``which kinds of magnetic fields exhibit
unstable fixed points at low energy?''
and ``when and where they appear?'' and would like to discuss ``how do
they affect
the mesoscopic properties of plasmas?'' [7],
for instance, local pressure or density gradients in the local equilibrium.

[1] A. I. Neishtadt, Change of an adiabatic invariant at a separa-
trix, Sov. J. Plasma Phys. 12, 568 (1986).
[2] J. L. Tennyson, J. R. Cary, and D. F. Escande, Change of the
adiabatic invariant due to separatrix crossing,
Phys. Rev. Lett. 56, 2117 (1986).
[3] A. H. Boozer, Physics of magnetically confined plasmas, Rev. Mod.
Phys. 76, 1071 (2004).
[4] E. Sugama, Gyrokinetic theory, J. Plasma Fusion Res. 79, 107,
(2003) (in Japanese).
[5] B. Cambon, X. Leoncini, M. Vittot, R. Dumont, and X. Garbet,
Chaotic motion of charged particles in toroidal magnetic
configurations, Chaos 24, 033101 (2014).
[6] S. Ogawa, B. Cambon, X. Leoncini, M. Vittot, D.
del-Castillo-Negrete, G. Dif-Pradalier, and X. Garbet,
Full particle orbit effects in regular and stochastic magnetic
fields,Phys. Plasmas 23, 072506 (2016).
[7] S. Ogawa, X. Leoncini, A. Vasiliev, and X. Garbet,
Creating steep density profile with a separatrix, arXiv:1611.00063v1.
(New version will appear soon, and title will be changed.)