By Henry M. Paynter

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This example explains the general properties of the phase that are independent of the shape of the limit cycle. We complete this section with a demonstration of how phase can be deﬁned for an arbitrary cycle. 1 Phase and amplitude of a quasilinear oscillator The limit cycle of a quasilinear oscillator is nearly a circle, and the oscillation itself can be taken as the sine wave, x(t) = A sin(ω0 t + φ0 ). Here ω0 denotes the angular frequency which is related to the oscillation period by ω0 = 2π/T and should be distinguished from the cyclic frequency of oscillation f 0 = 1/T .

It is not enough to know the value of x: indeed, for the same x the process may either increase or decrease. 3 Quite often two variables, x and y, are enough, and we proceed here with this simplest case. , the angle of the pendulum with respect to the vertical and its angular velocity. Thus, the behavior of the system can be completely described by the time evolution of a pair (x, y) (Fig. 1). These variables are called coordinates in the phase space (state space) and the y(t) vs. x(t) plot is called the phase portrait of the system; the point with coordinates (x, y) is often denoted the phase point.

Correspondingly, a single observation is not sufﬁcient to conclude on synchronization. Synchronization is a complex dynamical process, not a state. 14. An example of what cannot be considered as weak coupling. 3 Synchronization: an overview of different cases Thus far we have mentioned many examples of cooperative behavior of diverse oscillating objects, from simple mechanical or electrical devices to living systems. Now we want to list different forms of synchronization. , whether the oscillations are periodic or irregular, or whether the coupling is bi- or unidirectional.

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