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6 is satisfied. This concludes the proof of the theorem. , for singlepulse homoclinic orbits). We also note that the multi-pulse homoclinic orbits obtained from the theorem have the same asymptotic behavior as the single-pulse homoclinic orbits, hence the construction of chaotic invariant sets in their vicinities can be directly adapted from Li and Wiggins [32]. 11. Conclusions In this paper we gave a general criterion for the existence of nontrivial homoclinic orbits in a large class of near-integrable, multi-dimensional systems that usually arise as modal truncations or discretizations of partial differential equations.
They are introduced as the family local index for universal family of operators. This research is supported by NSF grant number: DMS-9114456 48 N-C. C. Leung We use them to define higher Chern-Simons forms of E: ch(E; A0 , . . , Al ) = [∗] l ◦ Ll (A0 , . . , Al ), and discuss their properties. When l = 0 and 1, they are just the ordinary Chern character and Chern-Simons forms of E. The usual properties of Chern-Simons forms are now generalized to (1) ch(E) ◦ ∂A = d ◦ ch(E), (2) ch(E; s · α) = (−1)|s| ch(E; α).
Multi-Pulse Homoclinic Orbits 35 9. An Alternative Formulation of the Results It may happen that the unperturbed limit of system (1) admits an invariant which offers a more convenient base for perturbation methods than the Hamiltonian H0 . For this reason, we also present an easy modification of our results that uses some other integral of the unperturbed limit. This alternative formulation will prove very useful in our study of the discretized NLS equation in the next section. , {H0 , K0 } = ω(ω (DH0 ), ω (DK0 )) = 0.
