Dynamics of quadratic operators generated by China's Five elements philosophy

TL;DR

This paper models China's Five Elements philosophy using a quadratic stochastic operator, analyzing its dynamics to reveal fixed points, periodic orbits, and complex boundary behaviors within a four-dimensional simplex.

Contribution

It introduces a novel quadratic operator based on CFEP, studies its dynamical properties, and provides a detailed classification of trajectory behaviors and fixed points.

Findings

Unique fixed point that is a repeller

Existence of two 5-periodic orbits for non-zero parameters

Partition of the simplex into regions with different convergence behaviors

Abstract

Motivating by the China's five element philosophy (CFEP) we construct a permuted Volterra quadratic stochastic operator acting on the four dimensional simplex. This operator (depending on 10 parameters) is considered as an evolution operator for CFEP. We study the discrete time dynamical system generated by this operator. Mainly our results related to a symmetric operator (depending on one parameter). We show that this operator has a unique fixed point, which is repeller. Moreover, in the case of non-zero parameter, it has two 5-periodic orbits. We divide the simplex to four subsets: the first set consists a single point (the fixed point); the second (resp. third) set is the set of initial points trajectories of which converge to the first (resp. second) 5-periodic orbit; the fourth subset is the set of initial points trajectories of which do not converge and their sets of limit points are infinite and lie on the boundary of the simplex. We give interpretations of our results to CFEP.