Exponentially small oscillation of 2-dimensional stable and unstable manifolds in 4-dimensional symplectic mappings

Homoclinic bifurcation of 4-dimensional symplectic mappings is asymptotically studied. We construct the 2-dimensional stable and unstable manifolds near the submanifolds which experience exponentially small splitting, and successfully obtain exponentially small oscillating terms in the 2-dimensional manifolds. ∗ E-mail address: [email protected] 1 typeset using PTPTEX.sty Homoclinic (or heteroclinic) bifurcation plays an important role to cause chaotic motion near hyperbolic fixed points. Melnikov’s method (or Melnikov’s integral), which is based on the regular perturbation method, is a powerful tool to detect transversal homoclinic points. 1) On the other hand, it is known that in the rapidly forced systems and the standard map, and so on, splitting of separatrices is exponentially small. Consequently it is impossible to capture exponentially small oscillation straightforwardly with Melnikov’s method. In the last decade the difficulty has been overcome by using the method called asymptotic expansions beyond all orders. This method enables one to capture exponentially small oscillation of the stable and unstable manifolds in 2-dimensional symplectic mappings which are perturbed from linear mappings, 2), 3), 4) and to construct functional approximations of the manifolds. 5), 6), 7) For 4-dimensional symplectic mappings, Gelfreigh and Shatarov treated a coupled standard mapping and computed the crossing angle between the 2-dimensional stable and unstable manifolds. 8) We obtained the functional approximations of particular sub-manifolds of the stable and unstable manifolds in a 4-dimensional double-well symplectic mapping with a weak coupling. 9) In this letter, we construct the functional approximations of 2-dimensional stable and unstable manifolds in 4-dimensional double-well symplectic mappings with more general weak couplings. We start with the 4-dimensional symplectic map (qj , pj) 7→ (q ′ j , p ′ j)