From the author’s introduction: “In many problems on the integration of evolution equations for large intervals of time, for example bifurcations, global attractors, inertial manifolds, chaos, the usual numerical methods are irrelevant for large time

$T$, because a large number of existing numerical integration algorithms lead to error estimates of the form

$C\left(h\right)exp\left(T\right)$, where

$C\left(h\right)$ is an appropriate constant that is small for small

$h$ and

$[0,T]$ is the interval of time under consideration. Now, a new method of integrating evolution differential equations – the nonlinear Galerkin method – is presented and studied in [

*M. Marion* and

*R. Temam*, SIAM J. Numer. Anal. 26, No. 5, 1139-1157 (1989;

Zbl 0683.65083);

*F. Jauberteau*,

*C. Rosier* and

*R. Temam*, Appl. Numer. Math. 6, No. 5, 361-370 (1990;

Zbl 0702.76077);

*R. Temam*, Math. Comput. 57, No. 196, 477-505 (1991;

Zbl 0734.65079)]. In order to study the chaos phenomenon and global attractors in the problems of Benard flow, the nonlinear Galerkin method is adpated in this paper. The existence and uniqueness of global generalized solution of the considered equations and the convergence of approximate solutions are obtained”.