illion years, i.e. the duration of a couple of Milankovitch cycles. Running such a simulation is like having to bridge a thousand kilometers by foot (with steps of about one meter). Fortunately, there is a more efficient way to solve this problem. As illustrated in Fig. 5.6(a), small forces can only cause substantial changes when they act in the same direction for a long time. Fig. 5.6(b) shows that force components that quickly vary in direction over time can never generate large changes as they counteract themselves. This suggests to first filter the equations governing the system's state-changing behav- ior. Hereby, we only retain the force components that act in the same direction over a sufficiently long time. Next the filtered equations can be solved for the long-term be- havior. This procedure is illustrated in Fig. 5.5(b). The filtering process is also called averaging [Bogo61]4. The resulting equations, indicated with a bar on top of them, are called the averaged equations. Since solutions of the averaged equations are, by con- struction, void of fast-varying components, they can be solved using a large integration time step and, therefore, in a much shorted CPU time. Hence, averaging allows us to focus attention to the important long-term system behavior while neglecting small fast-varying fluctuations. 4 Averaging, in turn, is related to the theory of continuous transformation (or Lie) groups [Cohn65, Firk00]. 5.1 THE STORY BEHIND THE MATH 5.1.5 How does electronic oscillator dynamics fit in? Fig. 5.7(a) summarizes the procedure outlined above. Firstly, we partition the system into a core and a set of perturbation terms. Next, the core is characterized by identifying all its states, i.e. all possible steady-state solutions of the corresponding unperturbed system equations. The perturbation terms slowly drive the core system from one state to another. Finally, averaging helps us to identify the relevant part of the equations that govern the system's state-changing behavior. Fig. 5.7(b) redraws Fig. 5.7(