Study On Limit Cycles And Abel Integrals For Two Kinds Of Quadratic Reversible Systems

So far, the Hilbert sixteenth problem is still the most famous and most challenging problem of nonlinear differential equations. In 1977, V. I. Arnold put forward a weakened form of the problem, then the weakened Hilbert’s 16 problem research became one of the frontier and the hot issues in the study of differential equations today. But so far, most of the results of this problem are on the Hamiltonian system and few researches for the integrabel non Hamiltonian system. In recent years, the reversible system which is one of the integrabel non Hamiltonian systems is in people’s attention. But due to the lack of methods, the researches is still relatively difficult.Based on this background and the qualitative theory, using two different methods, two classes of quadratic reversible systems are considered when the perturbation frequency is different in this thesis. When the number of perturbation polynomial is 4, firstly, through the relevant studies on the behavior of the quadratic reversible system trajectory and the definition of the detection function, we got a detection function of the system. Secondly, the conclusion is obtained that the number of limit cycles can be obtained by proper assignment of the corresponding parameters in the detection function. Finally, the exact location of these limit cycles is determined by the numerical simulation method, and the conclusion is further verified; When the number of perturbation polynomial is an arbitrary n, by means of the Hamilton quantity of the system, the relevant Picard-Fuchs equation and Riccati equation are obtained. Then find the relationship between the Picard-Fuchs equation and the Riccati equation, and finally get a linear estimate of the number of zeros of Abelian integrals for the system.Research results show that when the number of perturbation polynomial is low, we can use the detection function and numerical simulation method to get the number of limit cycles and the exact location of each limit cycle. Furthermore, when the number of perturbation polynomial is high, we can choose the way that using Picard-Fuchs equation and Riccati equation to study the upper bound of the number of zeros of Abelian integral. The conclusion of this thesis we get is that a class of quadratic reversible systems can generate 3 limit cycles in the case of 4 disturbances, and the exact location of each limit cycle is determined; What’s more, another class of quadratic reversible systems under the perturbation of any arbitrary n degree of polynomial, the result is that the upper bound of the number zeros of Abelian integrals is 7[n/2]-4 when n≥3.