鸭解是法国数学家(Benoit,1981)在研究van der Pol振子时首次发现和研究，经典的鸭解现象解释参数快速变化时, 解从小振幅的极限环(鸭环)变化到大振幅的松弛环.这个快速转化称为鸭爆破,对应一个很小的指数变化的控制范围. 所以,鸭解现象的判断十分困难,更进一步,鸭环在相平面的形态更像一个鸭子.鸭解概念来源于这个动物形态,起源于 标准(Eckhaus [1983])和非标准的方法.(Benoît et al [1981]). 方法: 鸭解的主要分析方法是非标准分析,渐进展开,爆破技术和几何奇异摄动(从Fenichel理论到非双曲点).更复杂 的分析,鸭解现象可以被理解为几何观点和Gevrey理论-可以方便的研究鸭解的渐进展开. 定义: 鸭解现象出现在奇异扰动系统(快慢系统),鸭解是奇异摄动系统的解,在吸引慢流形之后,经过临界流形分岔点, 最后长时间的排斥慢流形.用几何语言,鸭解对应非双曲点附近吸引慢流形和排斥慢流形的交叉.几何体是最大鸭解. 在各级问题中的Hopf分岔,出现的现象不同,会产生时滞失稳.经典的鸭解现象是van der Pol振子.鸭解出现的最低维数 是三维.在三维系统中会出现三类不同的鸭解.更一般的鸭解现象是混合模式振子(Mixed Mode Oscillations-MMOs),典型 的例子是Belousov-Zhabotinsky化学反应.MMOs correspond to switching between small amplitude oscillations and relaxation oscillations. These patterns were first discovered in the famous Belousov-Zhabotinsky reaction and, since then, have been frequently observed in experiments and models of chemical and biological rhythms. One way to explain these patterns is based on canards of folded node type. The reason is that canards of folded node type can be responsible for small amplitude oscillations (Wechselberger [2005]). A good intuition for MMOs is that a system moves dynamically from a small amplitude oscillatory state to a relaxation oscillatory state and the feature of the large relaxation oscillation is to bring the system back to the basin of attraction of the small amplitude oscillatory state. Other proposed mechanisms for MMOs are break-up/loss of stability of a Shilnikov homoclinic orbit (Koper [1995]), break-up of an invariant torus (Larter and Steinmetz [1991]) or slow passage through a delayed Hopf bifurcation (Larter et al. [1988]).