标题： 对于长笛类系统的迭代映射 显示英文标题

标题： Iterated maps for clarinet-like systems

摘要： 克拉纳特类系统的动力学方程在合理的近似下已知可以简化为非线性迭代映射。 这导致了时间振荡，这些振荡由方波信号表示，类似于弦乐器的拉曼区域。 在本文中，我们更详细地研究了相应非线性迭代的特性，重点在于可用于对各种解（例如有或没有簧片拍打）进行分类的几何构造，以及在混沌区域内出现的周期性窗口。 特别是，我们发现了一个周期三重化发生的区域，并研究了间歇性的条件。 我们还表明，虽然直接观察迭代函数并不能揭示乐器振荡区域的太多信息，但高阶迭代的图形可以直接提供关于振荡区域的可见信息（如周期倍增的数量、混沌行为等）。 显示英文摘要

摘要： The dynamical equations of clarinet-like systems are known to be reducible to a non-linear iterated map within reasonable approximations. This leads to time oscillations that are represented by square signals, analogous to the Raman regime for string instruments. In this article, we study in more detail the properties of the corresponding non-linear iterations, with emphasis on the geometrical constructions that can be used to classify the various solutions (for instance with or without reed beating) as well as on the periodicity windows that occur within the chaotic region. In particular, we find a regime where period tripling occurs and examine the conditions for intermittency. We also show that, while the direct observation of the iteration function does not reveal much on the oscillation regime of the instrument, the graph of the high order iterates directly gives visible information on the oscillation regime (characterization of the number of period doubligs, chaotic behaviour, etc.).