Analytical Prediction and Representation of Chaos

The concept of randomness entered Newtonian dynamics almost a century ago: in 1926, Synge, J. introduced a new type of instability - orbital instability- in classical mechanics, [1], that can be considered as a precursor of chaos formulated a couple of decades later, [2]. The theory of chaos was inspired by the fact that in recent years, in many different domains of science (physics, chemistry, biology, engineering), systems with a similar strange behavior were frequently encountered displaying irregular and unpredictable behavior called chaotic. Currently the theory of chaos that describes such systems is well established. However there are still two unsolved problem remain: prediction of chaos (without numerical runs), and analytical description of chaos in term of the probability density that would formally follow from the original ODE. This paper proposes a contribution to the solution of these problems.