Title: Differential equations on a $k$-dimensional torus: Poincaré type results

Abstract: Ordinary differential equations of the first order on the torus have been investigated in detail by H. Poincar\'e, P. Bohl and A. Denjoy. P. Bohl, back in 1916, emphasised the importance of the transfer of the results for the order $k=1$ to the case $k>1$, adding at the same time: "However, any attempt to do so would be hopeless". The following more than hundred years have only confirmed Bohl's forecast. It became clear that a new approach to this problem is needed.
In this paper, we propose a new (non-Hamiltonian) and promising approach. We use Hamiltonians, that is, ordinary differential systems of equations of the first order, only for heuristics. In the main scheme and corresponding proofs we do not use these systems. Instead of differential systems, we study sets of continuous vector functions $\phi(t,\eta)$ satisfying certain important conditions. Limit sets and left and right rotation vectors appear in the case $k>1$. Some of our results are new even in the case $k=1.$ Under simple and natural conditions, the left and right rotation vectors coincide and a precise analog of the well-known H. Poincar\'e's result is derived for $k>1$.