Shadowing Homoclinic Chains to a Symplectic Critical Manifold

We prove the existence of trajectories shadowing chains of heteroclinic or-bits to a symplectic normally hyperbolic critical manifold of a Hamiltonian system.The results are quite different for real and complex eigenvalues.General results are applied to Hamiltonian systems depending on a parameter which slowly changes with rate ε.If the frozen autonomous system has a hyperbolic equilibrium possessing trans-verse homoclinic orbits,we construct trajectories shadowing homoclinic chains with energy having quasirandom jumps of order ε and changing with average rate of order ε | ln ε |.This provides a partial multidimensional extension of the results of A.Neish-tadt on the destruction of adiabatic invariants for systems with one degree of freedom and a figure 8 separatrix.