Asymptotic solution of the Henon-Heiles Hamiltonian system

The invariant normalization method was used for the construction of the general Cauchy problem for the Hamiltonian system of the Henon-Heiles equations. The error in the solution is on the order of the fourth power of the oscillation amplitude. This system has two degrees of freedom and the quadratic part determines linear oscillations with frequency equal to unity. The general solution describes a periodic process of energy pumping from one degree of freedom to another. The period of this process is inversely proportional to the square of the amplitude, while the previously constructed periodic solution follow from the solution. The computational algorithm for canonical normalizing substitutions and normal forms can be classified according to the method for determining a canonical substitution. The advantage of the Hamiltonian normal form over the other forms is that two integrals exist for this form in any approximation and the equations of the normal form can be integrated in quadratures.