scillator's phase behavior, discarding amplitude variations and common-mode behavior. Furthermore, in [Dem00a, Dem02], the equation (5.4) is solved by constructing a (complicated) Fokker-Planck equation [Risk89]. In this chapter, we develop a more general math- ematical framework for the analysis of an oscillator's behavior. It embeds the phase noise analysis in [Dem00a, Dem00b, Dem02] as a particular application. Moreover, the proof of these existing results is significantly simplified by means of the averaging methods here presented. 5.2.3 Numerical simulation Finally, there is the possibility to study oscillator behavior by means of numerical simu- lation. Hereby, one can either use general-purpose algorithms, like SPICE [Nag75], or dedicated methods [Bra96, Nar99, Ngo96, Riz96]. SPICE, however, is troubled by the presence of the widely spaced time constants that characterize the behavior of many oscillators. Capturing the slow-varying transients requires a long overall simulation time interval. In contrast, the presence of fast oscillations forces the simulator to take a small simulation time step. Hence, covering the long simulation time interval comes at the expense of a huge number of time steps. It often takes minutes or even hours for a single simulation run to complete. Recent developments brought about simulation methods that try to separate the compu- tation of fast- and slow-varying behavior [Bra96, Nar99, Ngo96, Riz96]. These meth- ods build on the multi-variate formulation of the circuit simulation problem as intro- duced in [Royc01]. However, although very efficient for the verification of transistor- level implementations, these algorithms do not provide a lead to construct compact oscillator models that can be used for system-level exploration and verification. 5.2.4 Contributions of this work This chapter presents methods that allow us to efficiently handle the slow-varying be- havior of oscillation amplitudes, phases, phase differences and common-mode lev- els. It presents a significant genera