Eur. Phys. J. Spec. Top. (2022) 231: 2103-2114
https://doi.org/10.1140/epjs/s11734-021-00412-6

Regular Article

Double Hopf bifurcation and stability of Koren–Feingold cloud–rain system with rain production delay

¹ School of Mathematics and Statistics, Zhengzhou University, 450001, Zhengzhou, Henan, People’s Republic of China
² College of Science, Henan University of Engineering, 451191, Zhengzhou, People’s Republic of China

^b peilijun@zzu.edu.cn

Received: 5 June 2021
Accepted: 14 December 2021
Published online: 24 December 2021

Abstract

In this paper, we study the dynamical behaviors of a basic delayed physical model for aerosol–cloud–precipitation system, which was proposed by Koren and Feingold. The rain production delay is crucial for this system. The main work of the present paper is to study the effect of cloud and rain interaction on climate by the mathematical methods, such as double Hopf bifurcation analysis. First, we prove the uniqueness of the equilibrium of interest, i.e., the positive equilibrium, which has never been reported. Next, we use DDE-BIFTOOL to plot the bifurcation diagrams with respect to two bifurcation parameters, i.e., a and , and then find its double Hopf bifurcation points. Then, we study their unfolding and classification by the methods of multiple scales and normal form. Finally, we verify the results by numerical simulations. We find some complicated dynamic behaviors of the system via analytical method, such as stable equilibrium, stable periodic, and quasi-periodic solutions. For the stable equilibrium, the numerical simulations agree well with the analytical results. The stable equilibrium implies a balance between cloud formation and depletion (drizzle), and stable periodic solution is related to cycles of formation of thicker clouds that are later consumed by stronger rain (the moderately drizzle). These complex dynamical phenomena can be very helpful for the researchers to understand the mechanism of aerosol–cloud–precipitation system. And these results can also be used to predict rainfall and climate change, thereby reducing the impact of extreme weather, and its subsequent natural disasters and accidents on economic and social development.