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\title{
                   STABILITY OF SOME PERIODIC ORBITS
                IN AXIALLY SYMMETRICAL POTENTIAL FIELD
}
\author{                     V.A.Antonov, E.I.Timoshkova\\
                Institute of Theoretical Astronomy RAS\\
             Kutuzov Quay 10, 191187 St.Petersburg, Russia
}
\maketitle

   It is known that the exact circular solution in  the equatorial
   plane exists for the problem of the motion of unity mass in axially
   symmetrical potential field $V(r,z)$. The perturbed motion in the
   meridional plane is described by hamiltonian system with two--dimensional
   polinomial potential. Hamiltonian systems with
   polinomial potentials are extensively studied up to now. However
   the problem of the change of motion topology is not yet understood
   in detail. Formerly we found the full list of the trajectories which
   are represented through the first and second order algebraic curves
for systems with Henon-Heiles potential


  \[  U = 0.5 (x^{2}  + \lambda z^{2} ) + \alpha x^{3}  + xz^{2} \]

   Here $x, z$ are the small variations  of cylindrical coordinates $r,z$
   from values $r=const, z=o$  for  circular solution, $\alpha, \lambda$
   are arbitrary parameters. In this paper we examine some from algebraic
   trajectories for the purpose to reveal their stability in the linear
   approximation. The problem of orbit stability is solved by means of
   analysis for differential equation with periodic coefficients.
   The regions of stable and unstable motions in space of parameters
   $\lambda, h$  ($h$ is the energy constant) are found for circular orbits
   $(\alpha=1)$ by analytical methods. For $x$--axis orbits the analysis of
   variation equation in analitical form is feasible only for a sequance

    \[    \alpha = \frac{4}{n(n+1)},\quad  n=1,2,3. \]

   In these cases the dependence for critical amplitude of oscillations on
   the parameter $\lambda$ was determined. The instants when the stability
   replaces by the instability correspond to topological rebuilding of whole
   complex of orbits which are near to periodicity.

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