On separatrix splitting of some quadratic area-preserving maps of the plane
The simplest nontrivial examples of conservative dynamical systems with a chaotic trajectory behaviour are area-preserving maps of the plane. In the present paper, 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 Laplace's integrals from the same function but along different contours. An asymptotic of their difference calculated splitting of the map \[ HM (X,Y)= \bigl(Y+X+\varepsilon X(1-X),Y+\varepsilon X(1-X)\bigr). \] An asymptotic formula is given for a homoclinic invariant as \(\varepsilon\to 0\).
scientific article; zbMATH DE number 1281641