2复齐次ODE系统的定性性质：与多边形台球的联系
Qualitative properties of systems of 2 complex homogeneous ODE's: a connection to polygonal billiards
摘要：
一个系统的2个复杂的，齐次的，多项式常微分方程的实系数和多边形台球的轨道之间的对应关系显示。这种对应是普遍的，因为它适用于特定类型的常微分方程组的开放集合。这使得转移结果众所周知的多边形台球理论，如遍历性，周期轨道的存在，指数发散的情况下，存在额外的守恒定律，并存在不连续的动态，相应的常微分方程系统。它还表明，多边形台球已知存在的相当复杂性，也参加这些显然更简单的常微分方程系统。
英文摘要：
A correspondence between the orbits of a system of 2 complex, homogeneous, polynomial ordinary differential equations with real coefficients and those of a polygonal billiard is displayed. This correspondence is general, in the sense that it applies to an open set of systems of ordinary differential equations of the specified kind. This allows to transfer results well-known from the theory of polygonal billiards, such as ergodicity, the existence of periodic orbits, the absence of exponential divergence, the existence of additional conservation laws, and the presence of discontinuities in the dynamics, to the corresponding systems of ordinary differential equations. It also shows that the considerable intricacy known to exist for polygonal billiards, also attends these apparently simpler systems of ordinary differential equations.
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