Generalization of the discrete and continuous Shannon entropy by positive definite functions related to the constant of motion of the non-linear Lotka-Volterra system

Abstract

In cybernetics, the Shannon formula is a positive function that measures the entropy of discrete probability distributions. The conversion of this formula to continuous probability distributions gives the infinity and negativity catastrophes, similar to the problem of renormalisation in physics. This paper suggests a solution to these two open problems, in starting from the derivation of positive definite functions from the Shannon entropy. These definite positive functions are related to the constant of motion of the non-linear Lotka-Volterra system. Then these positive definite functions are generalized for exchange of information with both discrete and continuous probability distributions. These functions generalized the Kullback-Leibler and Wiener information gain. A practical example is presented which shows a remarkable result with a scale invariance of the information gain. Then in this paper, we recall the properties of the differential equations of the non-linear Volterra predator-prey system and of the autocatalytic system of Lotka. As this paper deals with the conversion of discrete information to continuous information, we have developed the presentation of the discrete Lotka-Volterra equations. The numerical simulation of the hyperincursive discrete Lotka-Volterra equations shows an orbital stability. Moreover it is demonstrated that the hyperincursive discrete Lotka-Volterra equations are separable into two alternating discrete incursive Lotka-Volterra equations, similar to the hyperincursive discrete harmonic oscillator.