__Keywords:__attractor non-autonomous system finite-difference scheme

V.M. Ipatova

We consider the Lorenz system with time-dependent continuously differentiable coefficients. It is assumed that the coefficients and their derivatives are bounded when time tends to plus infinity. The set of all possible coefficients is constructed so that the positive time shift operators translate it into itself. For the considered non-autonomous Lorenz system a priori estimates of solutions are obtained that enable to establish the existence of a compact uniformly attracting set. From the general theory attractors for families semiprocesses follows that in this case the system has a uniform (with respect to the coefficients) compact attractor. The system is approximated by a semiexplicit finite-difference scheme with a constant step grid. All the nonlinear terms in the equations are taken from the previous time layer, that is taken into account explicitly. A characteristic feature of the Lorenz system is that it reduces to the dissipative form by a linear change of variables which includes the time-dependent coefficients of the system. Such a substitution is made for the finite-difference scheme. For the discrete problem written in the new variables are introduced the closed balls such that: 1) for sufficiently small grid steps solution initiated inside the ball always remains in it and 2) the radius of the ball tends to infinity as the mesh size tends to zero. A priori estimates of solutions are obtained showing that the restriction of dynamic scheme operators to balls mentioned has a compact uniformly attracting set which remains bounded when the grid step tends to zero. For the transition to the original spatial variables the balls are introduced whose position does not depend on the variable coefficients of the system and the additive semigroup of time is determined by with respect to which the new balls have the properties 1) and 2). On the basis of the theory of families of discrete-time semiprocesses it is established that for sufficiently small grid steps the semiexplicit scheme has a local uniform attractor, the basin of attraction of which is infinitely expands as the discretization step tends to zero. The investigation of the convergence of the scheme attractor reduces to the verification of the conditions of the theorem on upper semicontinuous dependence on a parameter of uniform attractors for families of semiprocesses. In this case the zero value of a parameter corresponds to a family of continuous-time semiprocesses generated by the Lorenz system. The grid steps are taken as positive values of a parameter which correspond to a family of discrete-time semiprocesses generated by the finite-difference scheme. It is proved that the uniform attractors of the scheme lie in an arbitrarily small neighborhood of the true attractor of the Lorenz system as the grid step tends to zero.

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