Differential equations on a $k$-dimensional torus: Poincar\'e type results

Ordinary differential equations of the first order on the torus have been investigated in detail by H. Poincar\'e and A. Denjoy. The long-standing problem of generalising these results for the equations of the order $k>1$ (or for the systems of equations) is both important and difficult, requiring an essentially new approach. (See, e.g., an interesting old paper by P. Bohl.) 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 important conditions, which follow from the analogy with the solutions in the case $k=1$. Some of our results are new even in the case $k=1.$