Research On The Bifurcation And Integrability Of Segmented Disc Dynamo

Since the 1980 s E.Knobloch studied the chaotic phenomenon of segmented disk dynamo,the dynamic properties of segmented disk dynamo have been largely and extensively studied,and have achieved great and far-reaching influences.In this paper,we use the Routh-Hurwitz criterion,central manifold theorem,the normal form theory and Darboux integral theory to study the complex dynamic behavior of a threedimensional segmented disk dynamo with mechanical friction.It's complex dynamic behavior,bifurcation and integrability will be studied.In addition,the integrability of four-dimensional disk dynamo has also been studied in depth.It is divided into four chapters in detail:The first chapter discusses the research background and research significance of this paper.This paper briefly introduces the background knowledge and theoretical research status of Hopf bifurcation,Pitchfork bifurcation and Bogdanov-Takens bifurcation.The research history and development status of Darboux integrable are descripted too.The development history and research status of segmented disk dynamo are also illustrated.The second chapter studies the complex dynamic behavior of segmented disk dynamo with mechanical friction.Using the central manifold theory and RouthHurwitz criteria,the equilibrium point of the system and its linear stability are determined.Using the normal theory,the Hopf bifurcation and Pitchfork bifurcation of the segmented disk dynamo with mechanical friction are studied.The direction of the Hopf bifurcation is analyzed in detail,and the stability of the Hopf bifurcation limit cycle is determined.It is found that when the system experiences Hopf bifurcation,its bifurcation parameters will be change,and a formula for determining the supercritical or subcritical Hopf bifurcation of the system is obtained.The third chapter mainly studies the integrability of segmented disk dynamo with mechanical friction.In the theory of polynomial differential equations,the integral plays a crucial role.The Darboux integral of the system is studied by applying algebraic invariant surfaces,exponential factors and Darboux polynomial theory.The results show that this system does not have a non-zero factor Darboux polynomial,no polynomial first integral and exponential factor.In addition,its first Darboux integral does not exist.In the fourth chapter,based on the segmented disk dynamo proposed by Moffatt,a four-dimensional segmented disk dynamo is proposed and its Darboux integrability is studied in depth.It is found that the non-zero factor Darboux polynomial of the fourdimensional segmented disk dynamo does not exist,and it has no polynomial first integral and Darboux first integral.