by Meirong Zhang, Tsinghua University, Beijing

Peking University is one of the most influential universities in China. Many excellent people in its School of Mathematical Sciences were working on a variety of subjects of dynamical systems and had a tremendous influence on the study of dynamical systems in China.

The late Professor Shantao Liao, Member of the Chinese Academy of Sciences, began his systematical study on differentiable dynamical systems in the late 1950s. In the early 1960s, he obtained a theorem which is now called Oseledets's Multiplicative Ergodic Theorem [1]; his formulation is in terms of the associated systems on the bundles of frames, not completely the same as the tangent systems in modern terminology. His work on this subject is now widely recognized in the world. His main contribution is on the Structural Stability Conjecture, which remained as the main thesis of differentiable dynamical systems for a long time. He systematically developed two important notions, namely Systems of Standard Equations and Obstruction Sets, and then gave the first proof of the Conjecture for the case of three-dimensional flows without singularities. The approach developed by Liao is somewhat different from those in the Western World, but also fruitful. His complete work is available in English [4]. His research is continued by his students, including Lan Wen, another Member of the Chinese Academy of Sciences, Zhenxi Dong, Shaobo Gan, Wenxiang Sun (all in Peking U.), and Chengbo Yue (Academia Sinica) and Meirong Zhang (Tsinghua U.), with main interests in the C¹ Closing Lemma, C¹ Connecting Lemma, C¹ Stability Conjecture [6], C¹ Density Conjecture [7], rigidity and Anosov systems, etcetera.

Professor Zhifen Zhang obtained her Ph.D. from Moscow State University and from the start of the 1950s she was interested in the Qualitative Theory of Differential Equations [8]. Since then her group has been working on Bifurcation Theory and Normal Forms. Their research on the weak Hilbert 16th Problem and bifurcation problems (see for example [3]) has close connections with many groups in North America and Europe, in particular the United States, Russia, Belgium and Spain. Their results are widely circulated in the world. Zhifen Zhang, Wenzhao Huang, Chengzhi Li, Weigu Li, Duo Wang, Jiazhong Yang, Zhiming Zheng are in Peking U., Lanyu Wang is in Academia Sinica and Yun Tang in Tsinghua U.

Professor Min Qian is an expert in Probability Theory. He began his study on dynamical systems in 1970s and has been involved in many subjects of this interesting area. His group produced many important results on ergodic theory, random dynamical systems, infinite-dimensional systems, mathematical physics and their applications [5]. Students of him and his colleagues are now spread over many parts of the world, while some of them, Peidong Liu, Guanxiang Wang and Zhengdong Wang, are still working in Peking U.

Professor Tongren Ding and his students mainly work on nonlinear oscillations and Hamiltonian systems. A remarkable contribution is on KAM Theory for Hamiltonian and reversible systems and its applications to ODEs and PDEs. Ding himself has systematically developed the time-mapping approach and found their connections with the Moser Twist Theorem [2]. He also has many interesting results on some fundamental problems of dynamical systems, such as the refinement of a construction of Birkhoff which resulted in finding an analytical example of a system with the properties that it has a 2-torus as its minimal set and has no almost periodic motion at all. His students, for example, Bin Liu (Peking U.), Rong Yuan (Beijing Normal U.) and Zaihong Wang (Capital Normal U.), have been working in Beijing, while many others are working elsewhere and have important influences on their institutions. The recent work of Meirong Zhang on the rotation number approach to eigenvalues and spectrum and its applications to nonlinear oscillations is closely related to this group.

Because of its history, and of its graduates from this group, the dynamical systems group at Peking University has a long-standing influence on the developments of dynamical systems in all of China and in all important areas, including topological dynamical systems, differentiable dynamical systems, Hamiltonian systems, and ergodic theory. Peking University is the most important dynamical systems center in China and has many connections with other parts of the dynamical systems community in the world. The University has hosted the 1996, 1998 and 2001 International Conference on Dynamical Systems and Differential Equations. In recent years, there are international dynamical systems conferences and some regular programs every year in China. Some of them are jointly organized with the United States.

All international cooperation on dynamical systems and their applications are welcome by the group.

References

[1]	L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
[2]	T. Ding, R. Iannacci, and F. Zanolin, "Existence and multiplicity results for periodic solutions of semilinear Duffing equations," J. Differential Equations105 (1993), 364-409.
[3]	F. Dumortier, C. Li, and Z. Zhang, "Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops," J. Differential Equations 139 (1997), 146-193.
[4]	S. Liao, Qualitative Theory of Differentiable Dynamical Systems, Science Press, Beijing, 1996. [English; translated from the Chinese]
[5]	P.-D. Liu and M. Qian, Smooth Ergodic Theory of Random Dynamical Systems, Lecture Notes Math. Vol. 1606, Springer-Verlag, Berlin, 1995.
[6]	L. Wen, "On the C1 stability conjecture for flows," J. Differential Equations 129 (1996), 334-357.
[7]	L. Wen, "Homoclinic tangencies and dominated splittings," Nonlinearity 15 (2002), 1445-1469.
[8]	Z. Zhang, T. Ding, W. Huang, and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monographs Vol. 101, Amer. Math. Soc., Providence, RI, 1992. [English; translated from the Chinese by Anthony Wing Kwok Leung]