  The investigation of nonlinear dynamical systems of the type
$\dot{x}=P(x,y,z),\dot{y}=Q(x,y,z),\dot{z}=R(x,y,z)$ by means of reduction to
some ordinary differential equations of the second order in the form
$y''+a_1(x,y)y'^3+3a_2(x,y)y'^2+3a_3(x,y)y'+a_4(x,y)=0$ is done. The main
backbone of this investigation was provided by the theory of invariants
developed by S. Lie, R. Liouville and A. Tresse at the end of the 19th century
and the projective geometry of E. Cartan. In our work two, in some sense
supplementary, systems are considered: the Lorenz system $\dot{x}=\sigma (y-x),
\dot{y}=rx-y-zx,\dot{z}=xy-bz $ and the R\"o\ss ler system
$\dot{x}=-y-z,\dot{y}=x+ay,\dot{z}=b+xz-cz.$. The invarinats for the ordinary
differential equations, which correspond to the systems mentioned abouve, are
evaluated. The connection of values of the invariants with characteristics of
dynamical systems is established.
