關於極限環的唯一性問題要比存在性問題難些,直到20世紀四五十年代才有N.萊文森(Levinson),G.桑索內(Sansone),R.孔蒂(Conti),J.I.馬賽拉(Massera)等人的惟一性定理,而他們得到的充分條件都加在函式g(x),f(x),或F(x)的對稱性或它們零點的對稱性上。1957年張芷芬在副博士論文中第一次指出,阻尼函式的凹凸性是影響極限環唯一性的更本質的性質,實際上f(x)的星形性就能保證唯一性。她在1958年和1986年發表的文章中,對廣義李納系統在常規條件下,證明了若導函式,(0,+∞)),則(4)的極限環唯一。這一結果一直被國內外同行廣泛地引用。如見秦元勛的“微分方程所定義的積分曲線”(下冊)(1959),葉彥謙的“極限環論”(1984),桑索內和孔蒂的書“非線性微分方程”(“Non-linear Differential Equations”)(1964),L.佩柯(Perko)的書“微分方程和動力系統”(“Differential Equations and Dynamical Systems”)(1993)。在二次多項式系統和生物數學等領域中的極限環唯一性問題,很多都是利用這個唯一性定理證明的。 1982年張芷芬的學生和同事曾憲武對系統(1)的唯一性定理作了本質性推進,在阻尼函式沒有對稱性和凸凹性的限制下,他對發散量積分用分段估算、相互補償的辦法作了更精細的估計。接著張芷芬和曾憲武、高素志又將此結果從系統(1)推廣到系統(4)。他們總結了二三十年來的相關結果,經深入研究,發表了論文:“On the uniqueness of the limit cycle of the generalized Lienard equation”,它不是一篇簡單的綜合文章,文中最前面的11條引理揭示了方程(4)的發散量積分的最本質特性,每個定理後面的推論都指出了定理的要點和如何套用,已有的很多唯一性都是本文推論的特例。
1 Zhifen Zhang. On the uniqueness of limit cycles of certain equations of nonlinear oscillations. Dokl. Akad. Nauk SSSR, 1958, 119,659—662
2 Zhang Zhifen, Ding Tongren, Huang Wenzao. Answer to some questions on topological dynamical systems posed by Nemytskii and the others. Kexue Tongbao, 1980, 25 (11), 895—899參與編著的書籍
3 Zhang Zhifen. Theorem of existence of exact n limit cycles in |χ| ≤(n+1) π for the differential equation +μsin +χ= 0. Scientia Sinica,1980, 23 (12): 1502—1510
4 Zhang Zhifen. On the existence of exact two limit cycles of Lienard equation. Acta Math. Sinica, 1981, 24 (5): 710—716
5 Zhang Zhifen. An example of compact nonhomogeneous minimal set.Acta Math. Sinica, 1982, 25 (3): 354—364
6 Zhang Zhifen. A topological dynamical system in R3 with antonie's necklace as a minimal set. Scientia Sinica, 1982, 25 (9) : 932—941
8 Li Weigu, Zhang Zhifen. The “Blue sky Catastrophe” on closed surfaces. Proceeding of the International Conference “Dynamical System and Related Topics”. Nagoya, Japan, World Scientific, 1990. 316—332
9 Zhang Zhifen, Ding Tongren, Huang Wenzao, Dong Zhenxi. Qualitative theory of differential equations. Translations of Mathematical Monographs, AMS, 1992, Vol. 101
10 Zeng Xianwu, Zhang Zhifen, Gao Suzhi. On the uniqueness of the limitcycle of the generalized Lienard equation. Bull London Math. Soc. ,1994, 26: 213—247
11 Li Baoyi, Zhang Zhifen. A note on a result of G. S. Petrov about the weakened 16th Hilbert problem. J. Math. Anal. Appl. , 1995, 190 (2): 489—516
12 Dumortier Freddy, Li Chengzhi, Zhang Zhifen. Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops. J. Differential equations, 1997, 139 (1): 146—193
13 Li Baoyi, Zhang Zhifen. Bifurcation phenomenon of a class of planar codimension 3 polycycle S(2) with two saddles resonating. Science China (Ser. A), 1997, 40 (12): 1259—1271
15 Zhao Liqin, Li Weigu, Zhang Zhifen. Cyclicity of elementary polycycles and ensembles with codimension 3 degeneration. Chinese Sci.Bull., 1998, 43 (22): 1849—1864
16 Zhao Yulin, Zhang Zhifen. Linear estimate of the number of zeros of Abelian integrals for a kind of quartic Hemiltonians. J. Differential Equations, 1999, 155 (1): 73—88
17 Li Chengzhi, Li Weigu, Llibre Jaume, Zhang Zhifen. Linear estimate for the number of zeros of Abelian integrals for quadratic isochronous centers. Nonlinearity, 2000, 13: 1775—1800
18 Li Chengzhi, Li Weigu. Llibre Jaume, Zhang Zhifen. Linear estimation for the number of zeros of Abelian integrals for some cubic isochronous centers. J. Differential Equations, 2002, 180: 307—333
19 Gasull Armengol, Li Weigu, Llibre Jaume, Zhang Zhifen. Chebyshev property of complete elliptic integrals and its application to abelian integrals. Pacific Journal of Mathematics. 2002, 202 (2): 341—361
20 Zhao Yulin, Li Weigu, Li Chengzhi, Zhang Zhifen. Linear estimate for the number of zeros of Abelian integrals for quadratic centers having almost all their orbits formed by cubics. Science China (Ser. A) , 2002,45 (8): 964—974
21 Li Weigu, Zhao Yulin, Li Chengzhi, Zhang Zhifen. Abelian integrals for quadratic centers having almost all their orbits formed by quartics.Nonlinearity, 2002, 15: 863—885