迭代泛函微分方程中的解析理论与小除数问题

In this project, we will discuss the analytic theory of iterative functional differential equation by small divisor theory in dynamical system, in particular, the qualitative property in complex field. Consider: .(1) Analytic theory of iterative functional differential equation with higher derivative; compare it with the conclusions of iterative functional differential equation with first derivative or second derivative;.(2) Existence interval of analytic solutions of iterative functional differential equation..We will use the small divisor theory and linearization to study the analytic solutions of iterative functional differential equation under Diophantine conditions, Brjuno condition or weaker arithmetic condition. In method we require that the eigenvalues of the solutions at their fixed point be off the unit or lie on the unit circle under certain conditions. For condition (1) it is hard to discuss further because of complicated computation for higher order derivatives of high order iteration, so we try to handle this problem in a simplified method and discuss it with small divisor theory. For condition (2) there are few results about existence interval of analytic solutions. So it is worth to study the maximum existence interval of analytic solutions.

本项目利用动力系统中的小除数理论，对迭代泛函微分方程解析理论进行探讨，特别是在复数域内研究这类方程的定性性质。讨论：.(1) 具有高阶导数的迭代泛函微分方程的解析理论，并考虑与具有低阶导数的迭代泛函微分方程有关结果的联系；(2) 一般迭代泛函微分方程解析解的存在区间。.我们用小除数理论及线性化思想，在Diophantine条件或Brjuno条件，甚至更弱的算术性条件下研究迭代泛函微分方程的解析解。在方法上要求解在不动点处的特征值不在单位圆上或在单位圆上但满足适当的条件。对于情况(1)，由于函数高次迭代的高阶导数涉及复杂的运算，从而使讨论难以进一步进行。为了解决该问题，我们试图寻找一个较简便的方法克服这个困难，再利用小除数理论对其进行讨论。对于情况(2)，关于解析解存在区间的结论不多。因此，如何确定解析解的最大存在区间是一个值得考虑的问题。

本项目的研究结果包括三个方面：(1)迭代泛函微分方程解析理论及平面映射的不变曲线方面，(2)利用KAM方法讨论反转系统，(3)函数方程在群上的解。. 在第一方面，对一类二阶泛函微分方程解析解的存在性进行讨论，此外又考虑了一类标准映射的解析不变曲线。. 在第二方面，通过KAM方法讨论了两类具有周期强迫的反转系统的解的有界性问题。. 在第三方面，得到了三阶柯西差分方程在几类特殊群上的解。