Abstract

Chaos theory is widely regarded as the third greatest scientific revolution in physics in 20th century, comparable to Einstein’s theory of relativity and the quantum mechanics. It is widely believed that, although trajectories of a chaotic system are unstable due to “butterfly-effect”, its statistics should be stable and thus its results should be reproducible in the meaning of statistics. Note that reproducibility is the cornerstone of modern science. Unfortunately, more and more experiments have been reported to be not reproducible, which leads to the so-called “crisis of reproducibility”. By means of a new strategy of numerical technique, namely the “clean numerical simulation” (CNS), we found that there exist the so-called “ultra-chaos” whose statistics are unstable to small disturbances, corresponding to statistics instability. Since random tiny disturbances are unavoidable, in practice any experimental and/or numerical results of an ultra-chaotic system cannot be repeated even in the meaning of statistics. Therefore, even if all artificial disturbances are excluded, the crisis of reproducibility still cannot be avoided in practice. Ultra-chaos as a new concept reveals the incompleteness of the modern paradigm of scientific researches, which is based on the reproducibility of experiments. In this talk, we briefly introduce the basic ideas of the “clean numerical simulation” (CNS), describe the definition of ultra-chaos, give a few examples of ultra-chaos, and suggest some conjectures and open questions, including the so-called “modified fourth Clay millennium problem”.