Complex Dynamics Of Koren-Feingold Cloud-Rain System With Rain Production Delay

Cloud-rain system is an important model to study the formation of rainfall,and the study of its complex dynamics can help reveal the complex phenomena and mechanisms of rainfall,which is of great significance to meteorology and human production and life.Supported by NSFC(Nos.11972327 and 11372282),in this dissertation,the double Hopf bifurcation,stability and complex dynamics of Koren-Feingold cloud-rain system with rain production delay and its improved system are studied.Koren-Feingold cloud-rain system,as a basic model of aerosolcloud-precipitation interaction,was firstly proposed by Koren and Feingold.The rain production delay is critical for this system.The main work of this dissertation is to study the mechanism of cloud and rain interaction through mathematical methods,such as double Hopf bifurcation analysis.Firstly,we prove the uniqueness of the equilibrium of interest,i.e.,the positive equilibrium.Then DDE-BIFTOOL is used to plot the bifurcation diagrams with respect to two bifurcation parameters,i.e.,(6and ,and the double Hopf bifurcation points of the system are found.Then,at these double Hopf bifurcation points,the method of multiple scale and normal form are used to study their unfolding and classification.Finally,the software Win PP is used for numerical verification.Through analysis,we find the dynamical behaviors of stable equilibrium,periodic and quasi-periodic solutions.There is an issue here:the numerical verification of the equilibrium is easy,but those of periodic and quasiperiodic solutions are difficult.And both the trajectory near them are attracted to the meaningless negative equilibrium.We guess it’s because the time delayed term appears in the denominator,which induces the singularity easy.And then further improve the model.The denominator is removed by time transformation,and the original cloud-rain model is transformed into a simpler polynomial system under certain conditions,namely,the improved Koren-Feingold cloud-rain system.The double Hopf bifurcation of the improved system is studied by using the methods of multiple scales,and the complex and rich dynamics such as quasi-periodic solution and chaos are found.The numerical simulation results are consistent with the theoretical analysis results,which also verifies our judgment.These rich dynamic phenomena are of great significance for understanding the mechanism of aerosol-cloud-precipitation system,ecosystem and climate control.The features and innovations of this dissertation are as follows: Firstly,we prove the uniqueness of the equilibrium of interest,i.e.,the positive equilibrium of KorenFeingold cloud-rain system.Secondly,by removing the denominator through time transformation,the original cloud-rain model is transformed into a simpler model under certain conditions.Thirdly,based on the improved cloud-rain system,we find two routes to chaos: the Period-Doubling bifurcation of Period-2 and Period-3solutions to chaos.