  Nonhamiltonian interaction of hamiltonian systems is considered. Dynamical
equations are constructed by use of symmetric designs on Lie algebras. The
results of analysis of these equations show that some class of symmetric
designs on Lie algebras beyond Jordan ones may be useful for a description of
almost periodic, asymptotically periodic, almost asymptotically periodic, and,
possibly, more chaotic systems. However, the behaviour of systems related to
symmetric designs with additional identities is simpler than for general ones
from different points of view. These facts confirm a general thesis that
various algebraic structures beyond Lie algebras may be regarded as certain
characteristics for a wide class of dynamical systems.
