Isochronicity And Bifurcation Of Limit Cycles For Planar Polynomial Autonomous Differential Systems

This Ph.D. thesis is devoted to the problems of isochronicity and bifurcation of limit cycles for planar polynomial autonomous differential systems. It is composed of seven chapters.In Chapter 1, the historical background and the present progress of problems con-cerned with centers, integrability, isochronous centers, linearizability and bifurcation of limit cycles for planar polynomial autonomous differential systems are introduced and summarized. Meanwhile, the main work of this paper is simply concluded.In Chapter 2, the problems of bifurcation of limit cycles and center conditions at higher-order singular point and infinity for a class of planar septic polynomial system with two small parameters and nine normal parameters are investigated. The origin of the system is a higher-order singular point and the equator contains no real critical point. With the help of computer algebra system-Mathematica, we firstly derive the first 9 singular point quantities at higher-order singular point and the first 7 singular point quantities at infinity. Then we discuss the conditions under which the higher-order singular point and infinity are centers. Finally, we obtain the limit cycle configurations of{(8),3} and{(3),6} under synchronous perturbation at higher-order singular point and infinity.In Chapter 3, the center conditions, pseudo-isochronous center conditions and bi-furcation of limit cycles at higher-order singular point for a class of septic system are investigated. Firstly, the higher-order singular point is transferred into the origin by a homeomorphic transformation and a complex transformation. Then the first 45 singular point quantities at the origin are calculated and conditions for higher-order singular point to be a center and highest order fine focus are deduced as well. On the basis, a septic system which allows the appearance of 8 limit cycles in the neighborhood of higher-order singular point is constructed. Finally, a new algorithm is applied to find necessary con-ditions for pseudo-isochronous centers, then the sufficiency of these conditions is proved by some effective methods.In Chapter 4, we consider the linearizability problem of the Lotka-Volterra system in the neighborhood of a singular point with eigenvalues in the ratio 4:-m and 5:-m by computing carefully and strict proof. The calculation of generalized period constants is an effective way to find necessary conditions for linearizable systems with any rational resonance ratio. A new method is applied to find the necessary conditions for linearizability, needless to solve firstly the problem of integrability. In the end, the sufficiency of such conditions are proved by various methods.In Chapter 5, we are interested in the linearizability problem of p:-q resonant singular point for polynomial differential systems. Firstly, we transform singular point into the origin via a homeomorphism. Moreover, we establish a new recursive algorithm to compute the so-called generalized period constants for the origin of the transformed system. The algorithm is linear and recursive. With forcing only addition, subtraction, multiplication and division, the period constants can be deduced. Compared with the known methods, complex integrating calculations and operations of trigonometric func-tions are avoided. It is also easy to realize with computer algebra system. Finally, to illustrate the effectiveness of our algorithm, we discuss the linearizability problems of 1:-1 resonant degenerate singular point for a septic system and the Lotka-Volterra system with 4:-5 resonance.In Chapter 6, we deal with the problem of characterizing isochronous centers for real planar quasi-cubic analytic system. The technique is based on transforming the quasi-cubic analytic system into an analytic system. With the help of the computer algebra system-Mathematica, we compute the period constants of the origin and obtain the necessary isochronous center conditions for the transformed system. Finally, we give a proof of the sufficiency by various methods. Our work consists of the existing results related to cubic polynomial system as a special case.In Chapter 7, for the three-order nilpotent critical point of a cubic Lyapunov sys-tem, the center problem and bifurcation of limit cycles are investigated. With the help of computer algebra system-Mathematica, the first 7 quasi-Lyapunov constants are de-duced. As a result, sufficient and necessary conditions in order to have a center are obtained. The fact of there exist 7 small amplitude limit cycles created from the three-order nilpotent critical point is also proved.