题目：Results and open problems on the algebraic limit cycles of polynomial vector fields in R2
演讲者：Jaume Llibre ，西班牙皇家科学院院士
时间：2017年10月9日下午3:00-4:00
地点：理科楼112
摘要： We shall present a survey on some resent results on the algebraic limit cycles of the real planar polynomial vector fields, and also a list of related open problems. The survey is based in the following articles.
1. N.N. Bautin Estimation of the number of algebraic limit cycles of the system X’ = P(x; y), y’ = Q(x; y), with algebraic right-hand sides (Russian), Differentsial'nye Uravneniya 16(2) (1980), 362-383.
2. A. Campillo and M.M. Carnicer, Proximity inequalities and bounds for the degree of invariant curves by foliations of P2C, Trans. Amer. Math. Soc. 349 (1997), 2211-2228.
3. M.M. Carnicer, The Poincare problem in the nondicritical case, Annals of Math.140 (1994), 289-294.
4. D. Cerveau and A. Lins Neto, Holomorphic foliations in CP(2) having an invariant algebraic curve, Ann. Inst. Fourier 41 (1991), 883-903.
5. J. Chavarriga, H. Giacomini and J. Llibre, Uniqueness of algebraic limit cycles for quadratic systems, J. Math. Anal. and Appl. 261 (2001), 85-99.
6. J. Chavarriga, J. Llibre and J. Sorolla, Algebraic limit cycles of degree 4 for quadratic systems, J. Differential Equations 200 (2004), 206-244.
7. C. Christopher, J. Llibre and G. ¶Swirszcs, Invariant algebraic curves of large degree for quadratic system, J. Math. Anal. Appl. 303 (2005), 450-461.
8. C. Christopher, Polynomial vector fields with prescribed algebraic limit cycles, Geometria Dedicata 88 (2001), 255-258.
9. C. Christopher and J. Llibre, Algebraic aspects of integrability for polynomial systems, Qualitative Theory of Planar Differential Equations 1 (1999), 71-95.
10. C. Christopher and J. Llibre, Integrability via invariant algebraic curves for planar polynomial differential systems, Annals of Differential Equations 16 (2000), 5-19.
11. C. Christopher and J. Llibre, A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals, Proc. Amer. Math. Soc. 130 (2002), 2025-2030.
12. G. Darboux, Memoire sur les equations differentielles algebriques du premier ordre et du premier degre (Melanges), Bull. Sci. math. 2eme serie 2 (1878), 60-96; 123-144; 151-200.
13. M.V. Dolov and R.V. Kuzmin, On limit cycles of a class of systems, Differential Equations 29(9) (1993), 1282-1285.
14. M.V. Dolov and R.V. Kuzmin, Limit cycles of systems with a given particular integral, Differential Equations 30(7) (1994), 1044-1050.
15. R.M Evdokimenco, Construction of algebraic paths and the qualitative investigation in the large of the properties of integral curves of a system of differential equations, Differential Equations 6 (1970), 1349-1358.
