Vadim Chernov

Hamiltonian dynamical systems are considered in this article. They come from iterations of area-preserving quadratic maps of the plain. Stable and unstable invariant curves of the map $QM(u,v)=(v+u+u^2,v+u^2)$ passing across the origin are presented in the form of the Laplace's integrals from the same function but along the different contours. Also an asymptotic of their difference calculated splitting of the map $HM(X,Y)=(Y+X+\varepsilon X(1-X),Y+\varepsilon X(1-X))$. An asimptotic formula is given for a homoclinic invariant as $\varepsilon \rightarrow 0$, but it did not prove rigorously.