The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity

The Hess—Appelrot system and its nonholonomic analogs

“…The basic theorems in [41] and [2] are the same, but in our opinion their proofs contain gaps. We consider in more detail the paper [41]. There are no doubts about the existence of two limit cycles which can merge at certain values of the energy and area integrals.…”

“…We also mention the recent papers by P. Lubowiecki and H. Zoladek [41,42] (2012) and A. V. Belyaev [2] (2015). We note that the paper [2] contains results similar to those of [41], however, in all probability, the author, although he referred to this paper, did not try to fully understand its implications. The paper [2] also contains very strange asymptotic expansions the meaning of which for the dynamics is completely unclear.…”

“…There is some confusion in [41,42] regarding normal hyperbolicity. In fact, the cycle is hyperbolic on the invariant Hess manifold; on the isoenergetic surface there can be no normal hyperbolicity due to the existence of an invariant measure (induced by the standard volume form of the Euler-Poisson equations), and under perturbations (deviations from the Hess conditions) the invariant surface is not preserved and the splitting of separatrices to hyperbolic points leads to the formation of a stochastic layer 2 .…”

“…Considering the name used for this case in search systems in the Web (see, e.g., the recent papers[41,42,21,20]), we have decided to keep to the correct name, i.e., the Hess case, throughout the paper and to call it the Hess-Appelrot case in the title of the paper.…”

“…By the way, all these effects (like the main conclusions from[41,42,2]) were illustrated by a Poincaré map in[15] (as well as in the book[12]), to which no reference is made in the above-mentioned works.…”