Critical Point Quantities And Integrability Conditions For Complex Planar Resonant Polynomial Differential Systems

This thesis is devoted to singular point quantities and integrability conditions of complex polynomial differential systems with resonant singular point, and center-focus determination and bifurcation of limit cycles of planar polynomial differential systems. It is composed of five chapters.In chapter 1, the historical background and the present progress of problems that concern with center-focus determination and bifurcation of limit cycles of planar polynomial differential systems are introduced and summarized. At the same time, the main work of this paper is simply concluded.In chapter 2, the concept of singular point quantity in [103] is generalized to the case of complex polynomial differential systems with p: -q resonant singular point:Focus values and saddle quantities, which play an important role in center-focus determination and bifurcation of limit cycles of real planar polynomial differential systems, are particular examples of the generalized singular point quantities. Furthermore, the sufficient and necessary conditions of existence of first integrals of system (1) are discussed and a new linear recursive formula to compute generalized singular point quantity is given. Only addition, subtraction, multiplication and division of the right-hand side coefficients of system (1) are needed to compute the generalized singular point quantity. Therefore, complex nonlinear integral operation and solving equations are avoided in computation, which is easily realized with computer symbol operation systems.In chapter 3, integrability conditions of Lotka-Volterra systems with p: -q resonanceare studied. We give and prove the first generalized singular point quantities of Lotka-Volterra systems with 1:-q resonance and 2:-q resonance. Sufficientconditions for integrable Lotka-Volterra systems with 3: -q resonance are given. In the particular cases from 3:-4 resonance to 3:-17 resonance, sufficient and necessary integrability conditions are given. For Lotka-Volterra systems 4: —q and 5:-q resonances, sufficient aad necessary integrability conditions for some particular systems are given.In chapter 4, we study integrability conditions for a class of quadratic systems with p:-q resonance— = pz+az2 +bzw+fiv2Idw ,2— = -qw - czw - dwdT Hthe first two generalized singular pout quantities of systems (3) with I'.-q resonance are given and proved. For systems (3) with 1: -q and 2: -q resonance, sufficient integrability conditions are given. In the particular cases from 2:-3 resonance to 2: -11 resonance, sufficient and necessary integrability conditions are given.In chapter S, the first 18 singular point quantities for a class of quasi cubic systems (liS) ()A-\i 2w+apv2— = (l-iS)z + (zw)A-\43Qzi + a2lz2w+al2pv Z^ = -(1 + W)w - (zwf^b^w2 + b2yZ + bl2wz2 + V3)' dTare obtained, and for some kinds of real planar autonomous differential systems, the problems of center-focus determination and bifurcation of limit cycles for elementary critical point, higher critical point and infinite point are solved.