Bifurcation Of Limit Cycles For Two Classes Of Planar Dynamical Systems

He qingApplied MathematicsDirected by Hong XiaochunSince the differential dynamical system has a wide range of practical application values,such as the stability state of the dynamical system can be applied to simulate the balance of ecosystem,the spread of epidemic diseases,the spread of radar and radio sound waves,the trajectory of celestial bodies,etc.,the weakened Hilbert 16-th problem has been a hot issue in the world since it was first proposed in 1977.Scholars all over the world have done a lot of research work and made some progress,but there are still a lot of problems to be solved about the maximum number and position relationship of limit cycles.Based on this background,this paper will study the bifurcation of limit cycles of two kinds of planar dynamical systems from the following four aspects.In the first part,this paper mainly introduces the historical background,research significance and main research results of the weakened Hilbert 16-th problem.In the second part,this paper introduces two methods to solve the weakened Hilbert 16-th problem,that is,when the number of disturbances is low,the number of limit cycles is given by using the detection function method,and the possible location of limit cycles is given by using the numerical detection method;when the disturbance times are high,the upper bound of the number of limit cycles is given by using Picard-Fuchs equation method and Riccati equation method.In the third part,the bifurcation of limit cycles of a class of quintic Hamiltonian systems under quartic polynomial perturbation is studied by using the decision function method and numerical detection method.It is found that the system can branch out six limit cycles at the same time,and the six limit cycles have two distributions,namely((3,0),3)and((0,3),3).At the same time,we also give the bifurcation of the limit cycle of the system under several other parameter combinations,and get another five distributions,which are((2,2),1),((1,2),1),((2,1),1),((2,0),2)and((0,2),2).In the fourth part,by using the Picard-Fuchs equation method and Riccati equation method,this paper studies the upper bound of the number of limit cycles of a class of quadratic integrable non-Hamiltonian systems under arbitrary9)polynomial perturbation.The result shows that the upper bound of the number of Abelian integral zeros is 15 [(9)+ 1)/3] + 35(9)? 5).