Limit Cycles Of A Class Of Quadratic Reversible Systems And A Class Of Quintic Systems

Because Hilbert's 16 problem has important theoretical and practical significance in modern mathematics and real life,mathematicians all over the world have never stopped their research on it and made some corresponding progress.In1977,V.I.Arnold raised the weakened Hilbert's 16 th problem,everyone scrambled to study it.In this academic background,we consider the number of limit cycles for a class of quadratic reversible systems and a class of quintic systems under different polynomial disturbances by applying two different research methods.Under the guidance of qualitative analysis,when the degree of perturbation polynomial is n,the upper bound of the number of the limit cycles for a class of quadratic reversible systems is studied by using the Picard-Fuchs equation method and the Riccati equation method.Firstly,the Hamiltonian function of the quadratic reversible system is numerically transformed to get the standard form.Secondly,the representation Abelian integrals of the quadratic reversible system is obtained by using the Picard-Fuchs equation method and Riccati equation method.Finally,the upper bound of the number of zeros of Abelian integrals is estimated by using the correlation theorem,and the upper bound of the number of limit cycles of the system can be obtained.When the degree of perturbation polynomial is 5,the number and position of limit cycles for a class of quintic systems are studied by using the combination of detection function and numerical exploration method.At first,the detection function of the system is given by using the relevant definition of the detection function.Then the number of limit cycles is obtained by assigning value to the detection function,and finally the exact position of the limit cycles is found by numerical exploration.The results show that Picard-Fuchs equation method and Riccati equation method can be used to estimate the upper bound of the number of zeros of the Abelian integrals when the degree of the disturbance polynomial is high or n.When the degree of the perturbation polynomial is low,the number and position of limit cycles can be studied by using the combination of detection function and numerical exploration method.The conclusion shows that when the degree of polynomial perturbation is n(n ? 5),the upper bound of the number of zeros of Abelian integrals for a class of quadratic reversible systems is 7n-12.In other words,the upper bound of the number of limit cycles for this quadratic reversible systems is 7n-12 when n ? 5.When the degree of polynomial perturbation is 5and it contains 4 arbitrary parameters,the quintic system obtain 5 limit cycles.By using computer simulation software,we find the positions of the 5 limit cycles.