The idea is to look for the source of chaos in the system.   From previous simulations we know that for S5 there is no chaotic boundary if Saturn alone, on an eccentric orbit, is present.   But there is chaos if perturbations from Jupiter are taken into account.  We would like to know what terms in the disturbing function are responsible for chaos.  To do so, we use the model of Bretagnon and take only secular terms in his series for a. \lambda, q,p,h, and k.  The two following figures report i) the results of a numerical simulation for Jupiter under the influence of the OSS, and ii) the result of using the Bretagnon model by including the first 3 terms (only secular terms, no short period perturbations).    The difference between the two plots is mainly in the absence of short period perturbations (like the great-inequality) in the secular output.
 
 

The next step was to modify SWIFT-WHM, so that test particles are perturbed by a Jupiter on the hyphotetical orbit given by the secular model, while the other planets are integrated in the traditional way.   The following figures report the results of numerical simuations by using the usual version of SWIFT-WHM and the modified version.  There are differences in short period fluctuations in eccentricity and inclinarions.  The next step is to check if this modified model can still produce the layer of chaos...