 changing solve Figure 5.5: In obtaining the long-term (dashed) characteristic in Fig. 5.4, there are two ways to proceed: (a) first solve the original set of state-changing equations and remove what's unimportant afterwards by filtering; (b) first filter (average) the state-changing equations and then solve these averaged equations for the desired characteristic. of the ice ages. Hence, small forces can have significant long-term consequences. 5.1.4 Averaging: focusing on what's important Before linking the material above to electronic oscillators, there is one aspect about the earth's state-changing behavior that still needs to be addressed. Zooming in on the (extrapolated) position of the earth's perihelion with respect to time (expressed as an angle with respect to some reference position), we typically observe the behavior in Fig. 5.4. Here, the solid line represents the exact position, showing a slow but steady increase with small fast-varying fluctuation on top. In many cases, the latter fluctuation is of no interest and we would like to focus on the long-term progress. It is, therefore, the dashed line in Fig. 5.4 that is of interest. One way to obtain this dashed curve is illustrated in Fig. 5.5(a). Here, we first compute the perihelion's exact motion after which a lowpass filtering operation removes all fast-varying components. This, 5.1.4 Averaging: focusing on what's important ef(t2) ef(t3) ef(t1) ef(t2) ef(t3) ef(t1) ef(t5) ef(t4) ef(t4) ef(t5) Figure 5.6: (a) A small force acting during a very long time in the same direction may, over time, cause significant changes to occur. (b) A small force that continuously changes direction counteracts itself and can never cause more than small fluctuations. however, is very time-consuming as the computation of the perihelion's exact motion requires us to use a small simulation time step: we need a couple of samples per year in order to capture the perihelion's fast-varying fluctuations. Using this time step, we need to cover a simulation time interval over a m