Harmonic entrainment of van der Pol oscillations: Phaselocking and asynchronous quenching

The mathematical description of entrainment of van der Pol oscillations, though not simple, is at present reasonably well understood. In contrast to this, the physical mechanisms responsible for the phenomenon have remained completely elusive. This paper will show that there are in fact three physically different processes which can cause synchronism, or entrainment. They will be designated as phaselocking (PL), passive asynchronous quenching (PAQ), and active asynchronous quenching (AAQ). The purpose is to demonstrate the above physical interpretation and to derive criteria to distinguish among the three processes. In doing so, we hopefully resolve the present controversy over entrainment in this TRANSACTIONS. The mathematical criteria will be given in the form of a set of regions mapped on the resonance diagram. This is done because this representation is the one most familiar to readers acquainted with the subject. The key to the physics of entrainment, however, is the
b_{1} - b_{2}
parameter plane description of transient behavior originally due to Andronov and Witt (A-W plane). For this reason, the connection between the A-W plane and the resonance diagram is discussed first. Next, the physics of PL is explained by using the limiting case of "pure PL" seen in relaxation oscillations. After this, a hydro-mechanical model of the A-W plane description is developed by means of the generalization of Adler's "pendulum model." This is then utilized to show that the presence of a "saddle point" is the condition necessary for PL to occur. The model will also be used to obtain the criteria for AAQ and PAQ. These results are relevant to the fields of control, communication, and biology.