Energy Conversion and Phase Regulation in Transient States of Frequency Entrainment Described by van der Pol and Phase-Locked Loop Equations

We study the role of energy conversion in phase regulation of frequency entrainment. For an open dynamical system that interacts with its environment, energy conversion in the system is the key to a wide variety of nonlinear phenomena including frequency entrainment. In this paper, using the standard notion of energy, we study the phenomena of frequency entrainment by periodic forces in two different types of oscillations: libration and rotation. Theoretical analysis shows a relationship between phase regulation and energy conversion in the entrainment phenomena. Both of them are explained as a common phase regulation. On the other hand, no common relationship between transient behaviors and energy conversion is identified for the two different types of oscillations. For libration, the development of frequency entrainment does not depend on the energy conversion. The energy input to the oscillator affects the amplitude of libration. For the rotation, the development of frequency entrainment is governed by the amount of energy conversion. The energy input to the system directly regulates the phase of rotation, in other words, controls the entrainment phenomenon. These results suggest a different dynamical and control origin behind the two types of entrainment phenomena as the energy conversion in the systems. key words: synchronization, energy conversion, phase regulation, libration, rotation