Dynamics of quadratic stochastic operators generated by China’s five element philosophy. (English) Zbl 1482.37045

Summary: Motivated 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 point trajectories of which converge to the first (resp. second) 5-periodic orbit; the fourth subset is the set of initial point 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.


37H10 Generation, random and stochastic difference and differential equations
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
60H25 Random operators and equations (aspects of stochastic analysis)
47B80 Random linear operators
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