By Frank Egleston Robbins
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Additional info for Arithmetic in Philo Judaeus
5, 6]. Later, we will consider the case β > 0, which was studied first in Li and McLaughlin . 1. The β = 0 limit. As described in Li and McLaughlin , 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 .
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 , 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.  available to me before its publication.
Arithmetic in Philo Judaeus by Frank Egleston Robbins