1. On the Uniqueness of Limit Cycle of Lena Equation

Uniqueness of limit cycles is more difficult than existence. It was not until the forties and fifties of the 20th century that the uniqueness theorems of N. Levinson, G. Sansone, R. Kong Di (Conti) and J. I. Macaira (Massera) were obtained, and their sufficient conditions were added to the functions g(x) and f(x). In 1957, Zhang Zhifen pointed out for the first time that the concavity and convexity of damping function is a more essential property that affects the uniqueness of limit cycle. In fact, the star-shaped nature of f(x) can guarantee uniqueness. In her articles published in 1958 and 1986, she proved the Jordan function of the generalized Lena system, (0,+) under normal conditions.

∞)), the limit cycle of (4) is unique. This result is widely quoted by domestic and foreign counterparts. For example, see Qin Yuanxun's Integral Curve Defined by Differential Equation (Volume II) (1959), Ye's Limit Cycle Theory (1984) and Sang Songhe's Nonlinear Differential Equation (1968). L. Perko's book Differential Equations and Dynamical Systems (1993). This uniqueness theorem proves many problems about the uniqueness of limit cycles in quadratic polynomial systems and biomathematics. In 1982, Zhang Zhifen's students and colleagues have made essential progress in the uniqueness theorem of the system (1). Under the constraint that the damping function does not have symmetry and convexity, he estimated the divergence integral more carefully by piecewise estimation and mutual compensation. Then Zhang Zhifen and Zeng Hegao extended this result from system (1) to system (4). They summarized the related achievements in the past twenty or thirty years, and after in-depth research, published a paper: On the Uniqueness of Limit Cycle of Generalized lien ard Equation, which is not a simple comprehensive article. In this paper, the former 1 1 lemma reveals the most essential characteristics of divergence integral of equation (4), and the inference behind each theorem points out the main points of the theorem and how to apply it. Many existing uniqueness is a special case of this inference.

2. The N-uniqueness of limit cycles for a class of periodically damped Lena equations.

Zhang Zhifen's First Proof Equation in 1980

For all μ≠0, there are exactly n limit cycles in the band of phase space (x,) |≤ (n+ 1) π, which has a long history (n = 0, 1, 2, ...). This result has attracted the attention of colleagues at home and abroad. Not only because it is an unsolved conjecture for many years, but also because it is related to Hilbert's 16 problem. As we all know, the number of limit cycles of analytical systems is limited. Equation (5) is an analytical system, but it has an infinite number of limit cycles at infinity. It is revealed by examples that the analytic property can only guarantee the local finiteness of the number of limit cycles, but not the global finiteness. Only the number of limit cycles of polynomial system is finite in the whole plane.

topological dynamic system

1. Non-homogeneous minimum set

The almost periodic minimal set defined on a complete metric space is a compact topological group, and the operation of this group can be uniformly extended to a closed package, so it is homogeneous, that is, the dimension of each point is the same. The discrete dynamic system defined by E.E. Floyd on the closed subset of R2 square is non-homogeneous, and it has zero-dimensional points and 65438+ zero-dimensional points. A discrete dynamic system defined by Zhang Zhifen on a closed subset of an n-dimensional square has 0, 1, …, n- 1 dimensional points. Imitating this, we can define an n-dimensional compact heterogeneous minimal set with only 0, k 1, k2, …, kj, where 0≤k 1≤k2≤…≤kj≤n- 1. This shows the difference between almost periodic minimal set and minimal set. G.D. Birkhoff conjecture defines that the minimal set on the n-dimensional manifold is homogeneous. A Markov proves that this conjecture is correct for the minimal set of finite-dimensional continuous flows.

2. Antonie necklace

In 1950s, W.H. Gottschalk proposed whether a topological dynamical system with Anthony necklace A as the minimum set could be defined. In the article published in China Science 1982, Zhang Zhifen defined the topological mapping φ from R3 to itself, so that A is a completely disconnected, compact and completely invariant set of (R3, φ) (equivalent to Cantor set), while R3/A is not simply connected (hence the name of the necklace) and A is a discrete dynamic system (R3, φ). Furthermore, A is also an almost periodic minimal set of (R3, φ), so it is homogeneous, and the dimension of each point is 0, so A is not only a compact topological group, but also a simple topological group, that is, it has a dense cyclic subgroup. The dynamics of A is extremely simple, but the geometry of A is not simple, and A is obviously not the union of finite manifolds.

On bifurcation theory of vector field

Zhang Zhifen began to pay attention to the bifurcation theory of vector field in 1980s, mainly the bifurcation problem of Hamiltonian vector field, that is, the number of limit cycles of system (2), also known as weak Hilbert problem 16.

Let H=h0 and H=h 1 respectively correspond to the singularity and odd closed orbit of Hamiltonian vector field dH=0. Let the closed orbit г h be H- 1 (h) (H0

△Pε=△Pε(h)-h=εM 1+o(ε)

Is Abel integral, which is the first-order Merini function.

The necessary and sufficient condition for the perturbed system (2) to have a closed orbit is that the displacement function △Pε=0, and M 1(h) is the first-order approximation of the displacement function to ε, so its number of isolated zeros (counting times) N(m, n) on (h0, h 1) and the number of limit cycles of the system (2).

1. For m=n=2, an accurate estimate of N(m, n) is given.

When m=2 and dH=0***, there are five general cases and eight unusual cases. It has been proved that n (2 2,2) = 2 or 3. Eight of them were solved by Ilief, Li Chengzhi and Zhao Yulin. Zhang Zhifen and Li Chengzhi solved one of the five common situations. Recently, Li Chengzhi and his student Chen Fengde gave a unified proof of five general cases in real number field.

2. On the generalization of Pontryagin's theorem.

Pontryagin proved in 1934 that when the right side of system (2) is smooth enough and M 1(h*)=0. M(h*)≠0, then the system (2) has a unique limit cycle Lh. It continuously depends on ε, LH→г h *, when ε→ 0; Lh is stable (unstable) when ε m1(h *); 0，δ0 & gt； 0。 When |ε|ε0, the system (2) has at most n limit cycles in δ (г h *) = u г h This result was quoted in Forty Years of Mathematics in the Soviet Union.

3. Cyclicity of polygonal rings

Polygonal rings can be divided into two categories: infinite codimension and finite k codimension.

For the first kind of rings, Zhang Zhifen and her student Li Baoyi proved that the cyclicity of S(2) is the second kind under certain nondegenerate conditions. For a ring with codimension k, it is known that its cyclicity E(k)≤k, when k = 1, 2; E(k)>k, when k≥4. Zhang's doctoral student answered this question satisfactorily in her thesis. She proved that E(k)≤k if and only if k = 1, 2,3.

4. "Blue Sky Catastrophe" on a closed surface, a global bifurcation.

In an article in volume 1975 of handout Math.468, scholars such as J. Palis put forward fifty unsolved problems in dynamical systems, among which the thirty-seventh problem is: Is there a "blue sky catastrophe" in a single-parameter universal vector field family, that is, on a C∞ compact manifold M, define a continuous vector field family Xμ(μ∈R). μ 0) → 【 closed orbit l (μ) of xμ 】, when μ→μ0, the period T(μ)→∞ of L(μ) does not tend to any singularity of x μ, which is called "blue sky catastrophe", that is, the closed orbit L(μ) suddenly disappears because the period T(μ) tends to infinity, but this is not due to. Li and Zhang Zhifen solved this problem completely on closed surfaces. They proved that the "blue sky catastrophe" can happen on any closed surface except S2 and P2, but it can only happen on Klein bottle K2 with one parameter, and it happens in a specific way.

5. Integrable non-Hamiltonian system

There are many problems about the sixteenth problem of weak Hilbert, among which the integrable non-Hamiltonian system is worth mentioning. Because the integration factor is generally irregular. Multiplying Abel integers by such a factor will be difficult to move, and there are only a handful of existing works. However, if the integrable system has a rational center, that is, the closed curve of rational algebra is around the center, the integration factor of the system is a rational function according to Darbough's quantization. For the case that the center is surrounded by a low-order algebraic closed curve, Zhang Zhifen and her colleagues proved that for all systems, when the center is surrounded by a quadratic algebraic curve, then N(n)=O(n). For all quadratic polynomial systems, when there is a cubic algebraic curve or a quartic algebraic curve near the center, there is also N(n)=O(n). These works can be regarded as a step towards this difficult problem.

In the above three research directions, Zhang Zhifen, students and colleagues have published more than 50 papers in domestic and foreign magazines. "The number of limit cycles of Lena equation and several examples of topological dynamic systems" won the second prize of scientific and technological progress of the State Education Commission 1988.

Teaching and postgraduate training

Starting from 1957, in the work of teaching and educating people, Zhang Zhifen focused on cultivating senior college students and graduate students. She realized that it is a very arduous task to train high-quality talents for the country, so that they can keep forging ahead in their future posts and gradually stand at the forefront of discipline development.

Since 1960s, Zhang Zhifen has offered several specialized courses on qualitative theory of differential equations for senior college students and graduate students. Later, based on this lecture, she collaborated with Ding Tongren, Huang Wenzao and Dong Zhenxi to compile a textbook, which was published by Science Press 1985 as a series of Basic Modern Mathematics, reprinted by 1997, and translated by American Mathematical Society Press 1992.

At the same time, Zhang Zhifen cooperated with Ding Tongren and Huang Wenzao to open a seminar on topological dynamic system for senior students and young teachers. The basic teaching material is the relevant chapters and two comprehensive articles of the book Qualitative Theory by Zhang Zhifen's tutors Nemekki and V V stepanov, and two six-year college students have been trained, and more than ten graduation theses have been completed, some of which have reached the master's thesis level. Together with the teachers' papers, these papers * * * answered half of the unresolved questions listed in Nemecky's comprehensive article.

From 198 1, Zhang Zhifen, Li, etc. Organized a discussion class on bifurcation theory of vector field and dynamic system, and systematically read some basic documents and important new results.

The academic activities of the seminar greatly broadened the horizons of teachers and students. Regarding postgraduate training, apart from the source of students, Zhang Zhifen realized that for teachers, the first thing is to choose a topic, and to make the direction of the thesis closer to the forefront according to the actual situation of students as much as possible, so that it is worth continuing to explore after graduation. Secondly, we should train from reading literature, asking questions to solving problems. Every paper must have a critical point for students to overcome by themselves, so that after this kind of training, they can improve their ability and confidence, and they still have the courage to carry out research independently after graduation. The seminar she led also plays an important role in postgraduate training. During this period, Zhang Zhifen * * * trained 8 master students and 1 1 doctoral students. Today, most of them have become experts and professors in relevant research institutes, including Zheng Zhiming, Li, Zhang Weinian, Xiao Dongmei, Cao, Qi,,, Li Baoyi and Wang Tianxi.