The critical variational equation governing the stability of phase-locked modes for a pair of diffusively coupled van der Pol oscillators is presented in the form of a linear oscillator with a periodic damping coefficient that involves the van der Pol limit cycle. The variational equation is transformed into a Hill’s equation, and stability boundaries are obtained by analytical and numerical methods. We identify a countable set of resonances and obtain expressions for the associated stability boundaries as power series expansions of the associated Hill determinants. We establish an additional “zero mean damping” condition and express it as a Pade´ approximant describing a surface that combines with the Hill determinant surfaces to complete the stability boundary. The expansions obtained are evaluated to visualize the first three resonant surfaces which are compared with numerically determined slices through the stability boundaries computed over the range 0.4<ε<5. [S0739-3717(00)00502-X]
Phase-Locked Mode Stability for Coupled van der Pol Oscillators
Contributed by the Technical Committee on Vibration and Sound for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received Nov. 1999. Associate Technical Editor: A. Vakakis.
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Storti, D. W., and Reinhall, P. G. (November 1, 1999). "Phase-Locked Mode Stability for Coupled van der Pol Oscillators ." ASME. J. Vib. Acoust. July 2000; 122(3): 318–323. https://doi.org/10.1115/1.1302314
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