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Homoclinic bifurcation of 4-dimensional symplectic mappings is asymptoticallystudied. We construct the 2-dimensional stable and unstable manifolds near thesub-manifolds which experience exponentially small splitting, and successfullyobtain exponentially small oscillating terms in the 2-dimensional manifolds.Comment: 6 pages, LaTe

Exponentially Small Oscillation of 2-Dimensional Stable and Unstable Manifolds in 4-Dimensional Symplectic Mappings