- age levels in the input-free system (i.e. for u(T) = 0). If not, u(t) can no longer be considered a perturbation. As usual, a perturbation variable E is introduced to mark the perturbation terms, labeling them as being small. 5.7.1 Capturing oscillator phase behavior eb(x)u(t) Figure 5.22: (a) With regard to the impact of a small external input signal eu(T) on the oscillator's phase behavior, we partition the system into the input-free oscillator (the core system) and the set of perturbing input signals. (b) The solutions of the input-free oscillator are attracted to a 1-dimensional orbit. The perturbing input signal causes the oscillator to (phase) shift along its own orbit. 5.7.1 Capturing oscillator phase behavior If the input-free oscillator (the core system) is mono-stable, then all its steady-state solutions are phase-shifted copies of each other, i.e. (5.103) Here, Xosc (T) is a T-periodic solution of (5.102) for E = 0. All other processes that solve this unperturbed equation are attracted to the 1-dimensional orbit M={Yose(T)TER} (5.104) traced by the steady-state solution xosc(T). In (5.103), p = [0] = is the core system's 1-dimensional - state vector. Hereby, 0 represents the oscillator's phase which is, for instance, a measure for the positions in time of the oscillation signal's zero crossings. As discussed in section 5.5, the perturbation term eb(x)u(t) forces the oscillator to move over the manifold M. In this case, this is translated as: the oscillator starts to shift along its own orbit. This behavior is captured by means of slow-varying changes in the oscillator's phase O(T). As illustrated in Fig. 5.22(b), the equation that governs the oscillator's phase behavior is obtained by projecting the perturbation term eb(x)u(t) onto the tangent line to the orbit M. This yields dt = evT(T+0)b(osc(t+0))u(t) . (5.105) 5.7 OSCILLATOR PHASE (NOISE) ANALYSIS Here, [(++0) E R1xN corresponds to the projector VT (t,p) discussed in the sections 5.5.2 and 5.5.3. In the literature on phase noise analysis [Dem00a,