(1) if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with [[lambda].sub.1], [[lambda].sub.2] not satisfying (3), by the Poincare's theorem
, system (4) is not formally integrable;
For a discussion of this result we refer the reader to  where Sotomayor proves the following version of Poincare's theorem
: Polynomial differential systems whose compactifications have only hyperbolic equilibria and no graphics, are generic and have at most finitely many periodic orbits (for the notion of graphic see also ).
Then, by Poincare's theorem
, for each solution 1(n), n = 0, 1, 2,..., of (28), either 1(n) = 0 for all sufficiently large n [greater than or equal to] [n.sub.0], or the limit [lim.sub.n[right arrow][infinity]]l(n + 1)/l(n) exists and equals one 0] of the roots of the characteristic polynomial.
If Poincare's theorem
holds, then the entropy of an individual system cannot monotonically increase since it must eventually return to its starting point.