  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 $k=1.$
