弱化的XVI Hilbert问题：Hamilton无扰部分有九个平衡点的周期系统的不变环面
Weakened XVI Hilbert's Problem: Invariant Tori of the Periodic Systems with the Nine Equilibrium Points in Hamiltonian Unperturbed Part
摘要：
研究了两类二维时间周期常微分方程组，其扰动部分具有一个小参数e，e是连续的，且对于$\varepsilon=0$，e是零解析的.根据公共因子$\varepsilon$的存在或不存在，这些类包含具有“快”或“慢”时间的系统。这些系统的未扰动部分由哈密顿量$H=2x^2-x^4+γ(2y^2-y^4)$ $(γ\in(0,1])$产生。通用的方法，这意味着它可以适用于任何哈密顿量，称为生成环面分裂（GTS方法）的方法，开发和应用于此类系统的研究。对于任意一类系统，该方法可以求出相应的未扰动系统解的初值集，并对每个初值集给出系统扰动的显式条件，而与参数无关。满足上述条件的任何这样的集合，确定未扰动系统的解，其参数化生成周期。所得到的环是不变柱面的母线。证明了该系统有两个周期不变的表面，同胚的环面，如果时间是因式分解的周期方面，在小的方面，该表面的e邻域。一个例子的一组系统的11个不变环面和一个扰动，其平均值是一个三项多项式的三次，被构造作为演示的GTS方法的实际使用。GTS方法是一个通用的替代所谓的方法的检测功能和Melnikov功能的方法，这是用于研究有关的削弱十六希尔伯特的问题的评价的一些极限环的自治系统的哈密顿未扰动的一部分。
英文摘要：
Two classes of two-dimensional time-periodic systems of ordinary differential equations with a small parameter e in the perturbed part, which is continuous and, for $\varepsilon=0$, analytic in zero, are studied. Depending on the presence or absence of the common factor $\varepsilon$, these classes contain the system with "fast" or "slow" time. The unperturbed part of these systems is generated by the hamiltonian $H=2x^2-x^4+γ(2y^2-y^4)$ $(γ\in(0,1])$. The universal method, meaning it can be applied to any hamiltonian, called the method of the generating tori splitting (GTS method), is developed and applied to the research of such systems. For arbitrary system of any class, this method allows to find the sets of the initial values for the solutions of the corresponding unperturbed system, and for each such set, to provide the explicit conditions on the system perturbations independent of the parameter. Any such set, that satisfies the aforementioned conditions, determines the solution of the unperturbed system, which parametrizes the generating cycle. The obtained cycle is a generatrix of an invariant cylindrical surface. It is proven that the system has two-periodic invariant surface, homeomorphic to torus, if time is factored with the respect to the period, in the small with the respect to e neighbourhood of this surface. An example of the set of systems with eleven invariant tori and a perturbation, which average value is a three-term polynomial of the third degree, is constructed as a demonstration of the practical use of the GTS method. The GTS method is an universal alternative to the so-called method of detection functions and the Melnikov function method, which are used in studies concerning the weakened XVI Hilbert's problem on the evaluation of a number of limit cycles of autonomous systems with the hamiltonian unperturbed part.
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