Zhang, Zhifen

Kyoto University Research Information Repository

Title Infinitesimal Hilbert 16th Problem (Dynamics of FunctionalEquations and Related Topics)Author(s)Zhang, ZhifenCitation数理解析研究所講究録 (2002), 1254: 119-121Issue Date2002-04URL http://hdl.handle.net/2433/41880RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityInfinitesimal Hilbert 16th ProblemZhifen Zhang*(Peking University)Hilbert 16th Problem (Second Part) is the following:Given$\frac{dy}{dx}=\frac{P(x,y)}{Q(x,y)}$ , (1)where $P$ and $Q$ are real polynomials of $x$ and $y$ , what may be said about the numberof limit cycles and their relative positions.Infinitesimal Hilbert 16th problem is one of the restricted version of Hilbert 16thproblem, it was posted by V. I. Arnold in 1977 [1] as follows:Given$H(x, y)=h$ , $\omega$ $=Q_{n}dx-P_{n}dy$ ,where $H$ , $P_{n}$ and $Q_{n}$ are real polynomials of $x$ and $y$ ,Fig. 1$\max(\deg P_{n}, \deg Q_{n})=n$ , $\deg H=m+1$ . Then$\Gamma_{h}\subset(H=h)$ ,$I(h)= \int_{\Gamma_{n}}\omega$ .Problem: What is the number of isolated zeros of $I(h)$ ?Author would like to express her gratitute to Professors T. Hara and J. Sugie for their help to hervisit to Japan. She was invited to Japan by Grant-in-Aid for Scientific Research (A) N0.11304008,Japanese Ministry of Education, Science, Sports and Culture1254 2002 119-121119This problem is closely related to the problem of estimating the number of limitcycles of the following Hamiltonian System with small perturbations:$\frac{dx}{dt}=\frac{\partial H}{\partial y}+\epsilon P_{n}$ ,(2)$\frac{dy}{d\mathrm{t}}=-\frac{\partial H}{\partial x}+\epsilon Q_{n}$ ,where $0<|\epsilon|\ll 1$ .$H=h$ ,Fig. 2Then we have$\Delta H=\int_{0}^{T(P,e)}\frac{dH}{dt}|_{(2)}d\mathrm{t}$$= \epsilon\int_{0}^{T(P,\epsilon)}(\frac{\partial H}{\partial x}P_{n}+\frac{\partial H}{\partial y}Q_{n})dt$$= \epsilon\int_{\Gamma_{h}}(Q_{n}dx-P_{n}dy)+O(\epsilon^{2})$$=\epsilon I(h)+O(\epsilon^{2})$ , $h_{1}<h<h_{2}$ ,where $T(P, \epsilon)$ is the time along $\overline{PP_{1}}$ . Hence, there exists aperiodic orbit of (2) passingthrough $P\in\Gamma_{h}$ if and only if $\Delta H(h)=0$ , i.e. $I(h)$ is the first approximation of thePoincar\‘e return map $\Delta H$ with respect to $”\epsilon"$ .Relationship:(i) If $I(h)=0$ , $I’(h)\neq 0$ $\Rightarrow$ $\exists hyp$ . limt cycle $L_{h(\epsilon)}arrow\Gamma_{h}(\epsilonarrow 0)$ ;conversely, if3 $hyp$.limt cycle $L_{h(\epsilon)}arrow\Gamma_{h}(\epsilonarrow 0)\Rightarrow I(h)=0$ .(ii) If $I(h)=I’(h)=\cdots=I^{(k-1)}(h)=0$ , $I^{(k)}(h)\neq 0\Rightarrow$ there are at most $k$ limitcycles bifurcated from $L_{h}$ .(iii) $\neq$ {limit cycle $s.t$ . $L_{h(\epsilon)}arrow\Gamma_{h}(\epsilonarrow 0)$} $\leq\#\{I(h)=0\}=Z(m, n)$ .Here multiplicity taking into account120In this talk it will be introduced the main results and main methods of how tostimate the number of isolated zeros of $I(h)$ .The main methods are as follows:1. Argument principle [2], [3];2. Generalized Rolle Theorem [4], [5];3. Estimate zeros of analytic functions [6], [7].References[1] V. I. Arnold, Loss of stability of self-0scillations close to resonance and versaldeformations of equivariant vector fields, Fund. Anal. Appl. 11 (1977), 85-92.[2] G. S. Petrov, Number of zeros of complete elliptic integrals, Fund. Anal. Appl.18 (1984), 148-149.[3] Baoyi Li and Zhifen Zhang, Anote on aresult of G. S. Petrov about the weakened16th Hilbert problem, J. Math. Anal. Appl. 190 (1995), 489-516.[4] E. Horozov and I. D. Iliev, Linear estimate for the number of zeros of Abelianintegrals with cubic Hamiltonians, Nonlinearity 11 (1998), 1521-1537.[5] Yulin Zhao and Zhifen Zhang, Linear estimate of the number of zeros of Abelianintegrals for akind of quartic Hamiltonians, J. Differential Equations 155 (1999),73-88.[6] Yu. S. Il’yashenko and S. Yakovenko, Double exponential estimate for the numberof zeros of complete Abelian integrals, Invent. Math. 121 (1995), 613-650.[7] D. Novikov and S. Yakovenko, Simple exponential estimate for the number of realzeros of complete Abelian integrals, Ann. Inst. Fourier (Grenoble) 45 (1995),897-927.121

Infinitesimal Hilbert 16th Problem (Dynamics of Functional Equations and Related Topics)

http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/41880/1/1254_15.pdf

Infinitesimal Hilbert 16th Problem (Dynamics of Functional Equations and Related Topics)

Abstract

Similar works

Full text

Available Versions

Kyoto University Research Information Repository