Abstract
Generally, it is difficult to obtain convergent chaotic solution in an arbitrarily given finite interval of time. Some researchers even believe that all chaotic responses are simply numerical noise and have nothing to do with solutions of differential equations. However, using 1200 CPUs of the National Supercomputer TH-A1 at Tianjin and a parallel integration algorithm of the so-called "Clean Numerical Simulation" (CNS) based on the 3500th-order Taylor expansion and data in 4180-digit multiple precision, one can obtain reliable convergent chaotic solution of the Lorenz equation in a rather long time interval [0, 10 000]. This supports Lorenz's optimistic viewpoint [Lorenz, 2008] that "numerical approximations can converge to a chaotic true solution throughout any finite range of time". It also supports Tucker's proof [Tucker, 1999, 2002] for the famous Smale's 14th problem that the strange attractor of the Lorenz equation indeed exists.