Lorenz System, The Long Time Behavior Of The Finite Difference Scheme

In this paper, Lorenz system is considered. The existence of an attractor is one of themost important characteristics for a dissipative system. The long-time dynamics is com-pletely determined by the attractors of the system. At first, characteristics, domestic andinternational research results of Lorenz system are introduced, and the physical meaningof the parameters of the equations are given. Because the equations is one of nolinearODES, it is difficult to obtain the solutions. We have to study the numerical computingmethods. In chapter two, a backward Euler difference scheme and a Crank-Nicolson dif-ference scheme are given for the transformed equations. The local truncation errors of thedifference schemes are first order accurate and second order accurate respectively. Theexistence of the difference solution is proved by fixed point theorem. In chapter four andfive, the stability and convergence of the two schemes are obtained. Finally, In chaptersix, the long time behavior of the two difference schemes are discussed. The existence ofabsorbing sets and global attractors of the dynamic system which generated by the twodifference shemes are proved. In chapter seven, numerical simulation are given.