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By Frank Egleston Robbins

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5, 6]. Later, we will consider the case β > 0, which was studied first in Li and McLaughlin [31]. 1. The β = 0 limit. As described in Li and McLaughlin [31], the unperturbed integrable system admits an invariant denoted F˜1 such that F˜1 |C = 0. (119) The function F1 is defined as a Floquet discriminant computed for a set of fundamental solutions to a discretized Lax pair for system (111). For brevity, we do not introduce here all the notation and terminology for the exact definition of F1 , but refer the reader to Li and McLaughlin [29].

J ) i ˆ Tr [e 2π FA B1 · · · B2i φ1 · · · φj ]sym A(X) = X In the last section, we will briefly explain how these higher [2i,2j] are related to family stability. We need to look at all connections on the universal bundle which induces the canonical relative universal connection Drel . Let D + S be such a connection, then its curvature is given by: F2,0 (x, A) = FA at x, F1,1 (x, A)(v, B) = B(v) + DA,v (S(A)(B)) at x, F0,2 = dA S + S2 . By twisted transgression of [∗] (defined using D + S), we get M ap(M, G)-invariant closed forms on M ap(M, A), namely, [∗] M .

Making use of the calculations of Li and McLaughlin [31], we showed the existence of multi-pulse Silnikov-type homoclinic orbits for a codimension one set of parameter values. This provides a significant extension of the set of parameter values for which the discretized NLS equation admits chaotic invariant sets in its phase space. Acknowledgement. I am grateful to Dave McLaughlin for several useful discussions on the subject of this paper and for making ref. [31] available to me before its publication.

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Arithmetic in Philo Judaeus by Frank Egleston Robbins

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