How Can We Observe and Describe Chaos?

There exist several trials to describe chaos appeared in classical or quantum dynamical systems [?]. One of the present authors introduced Information Dynamics(ID for short) [18] as a frame to discuss complexity and chaos appeared in various fields, in which he tried to find a common basis by synthesizing the state change (dynamics) and the complexity associated with dynamical systems. Since then, ID has been applied to several different topics [?], among which a chaos degree, a quantity measuring the degree of chaos associated with a dynamics, was introduced by means of the complexities in ID and its entropic version (called Entropic Chaos Degree (EDC for short)) has been computed numerically for rather famous chaotic dynamics such as logistic map, baker’s transformation, Tinkerbel map. It is surprised that the result of the ECD exactly macthes to that of Lyapunov exponent in the case that the later can be computed. Moreover the algorithm computing the ECD is much easier than that of Lyapunov exponent, so that the ECD can be almost always computable even when the Lyapunov exponent can not be so. However there are some unclear points in both conceptually and mathematically why the ECD could be so successful for computational experiments. In this paper we study these points and propose a new description of chaos. In Section 2, we briefly review Information Dynamics and Chaos Degree, and in Section 3 the entropic chaos degree and its algorithm are recalled with a computational result. In Section 4, a new way judging chaos from a given dynamics is discussed based on the ECD, that is, we propose a new view to define chaos of dynamical systems.