On Research of Henon-Heiles-type Systems

Three-parametrical family of systems with two degrees of freedom of a kind $$\begin{array}{rcl} \dfrac{d^2x_1}{dt^2}+x_1&=&-2\varepsilon x_1x_2\\ \dfrac{d^2x_2}{dt^2}+x_2-x_2^2&=&\varepsilon (-x_1^2+\delta\dot{x}_2+\gamma x_2\dot{x}_2), \end{array}\qquad\qquad(*)$$ 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 $$\ddot{x}_2+x_2-x_2^2=\varepsilon(-A^2\sin^2t+\delta\dot{x}_2+\gamma x_2\dot{x}_2)\qquad\qquad(**)$$ 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 $\varepsilon_\infty\approx0.95$.