On Research of Henon-Heiles-type Systems

Three-parametrical family of systems with two degrees of freedom of a kind
$$\frac{d^2 x_1}{dt^2} + x_1 =-2 \varepsilon x_1 x_2 \qquad (*) \\ \frac{d^2 x_2}{dt^2} + x_2 - x_2^2 = \varepsilon (-x_1^2 + \delta \dot{x}_2 + \gamma x_2 \dot{x}_2),$$
where $\varepsilon > 0$ is considered. Analytical research of trajectories behaviour of the system (*) is carried out when $\varepsilon$ is small.
The given research is connected, first of all, to the analysis of resonant zones. Alongside with the initial system, another system
$$\ddot{x}_2 + x_2 - x_2^2 = \varepsilon (-A^2 \sin^2 t + \delta \dot{x}_2 + \gamma x_2 \dot{x}_2) , \qquad (**).$$
that is "close" to the original, is considered. A good concurrence of results for Poincare mapping, induced by an equation (**) when $ \delta = \gamma = 0$, and for the mapping that was constructed by Henon and Heiles, is established.
In addition, for system (*) a transition to nonregular dynamics is numerically analyzed at increase of parameter $\varepsilon$ and $ \delta = \gamma = 0$. It is established, that the transition to nonregular dynamics is connected, in particular, with the period doubling bifurcation (known as Feigenbaum's script), and $\varepsilon_{\infty} \approx 0.95$ .