pThis book describes the basic principles of the Clean Numerical Simulation CNS, proposed by the author in 2009, and several of its applications. Unlike conventional algorithms, CNS can give bconvergentb chaotic trajectory in a sufficiently long interval of time, whose numerical noise is much lower than the true physical solution iso that one can gain accurately. So, CNS provides us, b for the first timeb, with an ability to check statistics stability of chaos, which leads to a completely bnewb concept ultra-chaos that has both btrajectory instabilityb and bstatistics instability band thus is of a higher disorder.ippNotably it is bimpossibleb to repeat experimental results of an ultra-chaos even in the statistical senses. However, the reproducibility of physical experiments forms a bcornerstoneb of modern science. So, ultra-chaos reveals an bincompletenessb of modern science bparadigmb. Besides, it also reveals that statistics stability is a bpreconditionb for the use of conventional a