This paper analyzes frequency entrainment described by van der Pol and phase-locked loop (PLL) equations. The PLL equation represents the dynamics of a PLL circuit that appear in typical phase-locking phenomena. These two equations describe frequency entrainment by a periodic force. The entrainment originates from two different types of limit cycles: libration for the van der Pol equation and rotation for the PLL one. To explore the relationship between the geometry of limit cycles and the mechanism of entrainment, we investigate the entrainment using an energy balance relation. This relation is equivalent to the energy conservation law of dynamical systems with dissipation and input terms. We show response curves for the dc component, harmonic amplitude, phase difference, and energy supplied by a periodic force. The obtained curves indicate that the entrainments for the two equations have different features of supplied energy, and that the entrainment for the PLL equation possibly has the same mechanism as does the regulation of the phase difference for the van der Pol equation.

Topics
Chaotic maps, Dynamical systems, Resonance phenomena, Oscillators, Comparators, Energy conversion, Energy conservation, Energy production, transmission and distribution, Frequency measurement, Phase lock loop
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This condition of A12 partly shows a region of stable responses in Fig. 12.1 in Ref. 2 with a0=4∕γ⁠, r12=Av12∕a02⁠, and detuning σ1=(1−ν2)∕(μν)⁠.

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© 2007 American Institute of Physics.
2007
American Institute of Physics
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