In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems. These systems present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms and unique evolution. They correspond to a class of systems for which a transfinite method exist to compute the solution. We also presented several examples including a nontrivial one whose solution yields, at an integer time, a real encoding of the halting set for Turing machines; therefore showcasing that the behavior of solvable systems might describe ordinal Turing computations. In the current article, we study in more depth solvable systems, using tools from descriptive set theory. By establishing a correspondence with the class of well-founded trees, we construct a coanalytic ranking over the set of solvable functions and discuss its relation with other existing rankings for differentiable functions, in particular with the Kechris-Woodin, Denjoy and Zalcwasser ranking. We prove that our ranking is unbounded below the first uncountable ordinal.

翻译：在最近的一篇文章中，我们引入并研究了一类精确的动力学系统，称为可解系统。这类系统呈现由具有可解右端项且演化唯一的间断常微分方程所支配的动力学。它们对应着一类存在超限方法可计算其解的系统。我们还给出了若干示例，包括一个非平凡的例子，其解在整数时刻产生图灵机停机集的一个实数编码；从而展示了可解系统的行为可能描述了序数图灵计算。在本文中，我们利用描述集合论的工具，更深入地研究可解系统。通过建立与良基树类之间的对应关系，我们在可解函数集上构造了一个共解析的秩，并讨论了该秩与其他现有可微函数秩（特别是Kechris-Woodin秩、Denjoy秩和Zalcwasser秩）的关系。我们证明了我们的秩在第一个不可数序数之下是无界的。