On the reliability of computed chaotic solutions of nonlinear differential equations

In this paper a new concept, namely the critical predictable time Tc, is introduced to give a more precise description of computed chaotic solutions of nonlinear differential equations: it is suggested that computed chaotic solutions are unreliable and doubtable when t > Tc. This provides us a strategy to detect reliable solution from a given computed result. In this way, the computational phenomena, such as computational chaos (CC), computational periodicity (CP) and computational prediction uncertainty, which are mainly based on long-term properties of computed time series, can be completely avoided. Using this concept, the famous conclusion “accurate long-term prediction of chaos is impossible” should be replaced by a more precise conclusion that “accurate prediction of chaos beyond the critical predictable time Tc is impossible”. So, this concept also provides us a time-scale to determine whether or not a particular time is long enough for a given nonlinear dynamic system. Besides, the influence of data inaccuracy and various numerical schemes on the critical predictable time is investigated in details by using symbolic computation software as a tool. A reliable chaotic solution of Lorenz equation in a rather large interval 0 ≤ t < 1200 non-dimensional Lorenz time units is obtained for the first time. It is found that the precision of initial condition and computed data at each time-step, which is mathematically necessary to get such a reliable chaotic solution in such a long time, is so high that it is physically impossible due to the Heisenberg uncertainty principle in quantum physics. This however provides us a so-called “precision paradox of chaos”, which suggests that the prediction uncertainty of chaos is physically unavoidable, and that even the macroscopical phenomena might be essentially stochastic and thus could be described by probability more economically.