On the investigation of stability of equilibrium in Sitnikov problem in nonlinear formulation

With regard to nonlinear terms in the equations of motion, the stability of the trivial equilibrium in Sitnikov problem is investigated. For Hamilton's equations of the problem, the mapping of phase space into itself in the time $t=2\pi $ was constructed up to terms of third order. With the help of point mapping method, the stability of equilibrium is investigated for eccentricity from the interval $[0,1)$. It is shown that Lyapunov stability takes place for $e\in [0,1)$, if we exclude the discrete sequence of values $\{ e_{j} \} $ for which the trace of the monodromy matrix is equal to $\pm 2$. The first and second values of the eccentricity of the specified sequence are investigated. The equilibrium is stable if $e=e_{1} $. Eccentricity value $e=e_{2} $ corresponds to degeneracy stability theorems, therefore the stability analysis requires the consideration of the terms of order higher than the third. The remaining values of eccentricity from discrete sequence have not been studied.