Ultra-chaotic motion in the hexagonal Beltrami flow

In this paper, we investigate the influence of small disturbance on the statistical behaviors of fluid particles of the three-dimensional divergence-free hexagonal Beltrami flow from a Lagrangian point of view. Due to the butterfly-effect, numerical noise increases exponentially for chaotic dynamic systems. Thus, a powerful strategy, namely, the clean numerical simulation, is used to gain reliable/convergent trajectory in a long enough interval of time. It is found that the statistics of chaotic trajectory of fluid particles are stable in some cases, corresponding to the so-called “normal-chaos,” but unstable in some cases, i.e., rather sensitive to small disturbances, corresponding to the so-called “ultra-chaos,” which is a new concept proposed currently. Obviously, an ultra-chaotic trajectory of fluid particles is at a higher disorder than a normal chaotic trajectory. In theory, it is impossible to repeat any experimental/numerical results of an ultra-chaotic system even by means of statistics, but reproducibility is a corner-stone of our modern science paradigm. Hence, the wide existence or non-existence of ultra-chaos has a very important meaning. In this paper, we illustrate that the ultra-chaotic trajectories of fluid particles indeed widely exist in a hexagonal Beltrami flow field. This fact is important for deepening our understanding of chaotic dynamic systems and revealing the limitations of our paradigm of modern science.