We construct a dynamical system based on the Källén–Crafoord–Ghil conceptual climate model which includes the ice–albedo and precipitation–temperature feedbacks. Further, we classify the stability of various critical points of the system and identify a parameter which change generates a Hopf bifurcation. This gives rise to a stable limit cycle around a physically interesting critical point. Moreover, it follows from the general theory that the periodic orbit exhibits relaxation-oscillations that are a characteristic feature of the Pleistocene ice ages. We provide an asymptotic analysis of their behaviour and derive a formula for the period along with several estimates. They, in turn, are in a decent agreement with paleoclimatic data and are independent of any parametrization used. Whence, our simple but robust model shows that a climate may exhibit internal relaxation oscillations without any external forcing and for a wide range of parameters.

中文翻译：

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概念气候模型中内部弛豫振荡的渐近分析

我们基于Källén–Crafoord–Ghil概念性气候模型构建了一个动力系统，该模型包括冰–反照率和降水–温度反馈。此外，我们对系统各个关键点的稳定性进行分类，并确定一个变化会产生霍普夫分支的参数。这会在物理上有趣的临界点附近产生一个稳定的极限循环。而且，根据一般理论，周期轨道表现出弛豫振荡，这是更新世冰期的特征。我们提供了它们的行为的渐近分析，并得出了该时期的公式以及一些估计值。反过来，它们与古气候数据具有良好的一致性，并且与所使用的任何参数化无关。何处，