Nonlinear mapping between continuous- and discrete-time dynamics

¹ School of Physics, Northwest University - Xi'an 710069, China
² Shaanxi Key Laboratory for Theoretical Physics Frontiers - Xi'an 710069, China
³ School of Physics and Electronic-Electrical Engineering, Ningxia University - Yinchuan 750021, China
⁴ School of Science, East China University of Technology - Nanchang, Jiangxi 330013, China
⁵ Yangtze Delta Region Institute of University of Electronic Science and Technology of China Huzhou, Zhejiang 313000, China
⁶ Institute of Computational Physics and Complex Systems, Lanzhou University - Lanzhou, Gansu 730000, China

^(a) ccr@nwu.edu.cn (corresponding author)

Received: 22 January 2022
Accepted: 22 March 2022

Abstract

Linear mapping is widely used in dynamic modeling and empirical data analysis, but it suffers from the serious shortcoming that it does not work in the common case of time intervals being large. The fundamental cause of the failure is that the linear mapping does not take into account the coupling effects of multiple events within a discrete time interval. Here, we develop a theoretical framework to provide a nonlinear mapping between continuous- and discrete-time dynamics by accounting for the coupling effect. We have verified the effectiveness of our mapping by exploring classical susceptible-infected-susceptible and susceptible-infected-recovered models. In particular, we give a quantitative criterion that the sum of two transition probabilities —from one state to the other and vice versa— must be strictly less than 1 for binary-state dynamics.