On the collective behavior of parametric oscillators

I. Bena
Limburgs Universitair Centrum, B-3590 Diepenbeck, Belgium

R. Kawai
Department of Physics, University of Alabama at Birmingham, Birmingham, AL 35294

C. Van den Broek
Limburgs Universitair Centrum, B-3590 Diepenbeck, Belgium

Mauro Copelli and Katja Lindenburg
Department of Chemistry and Biochemistry 0340 and Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA 92093

We revisit the mean field model of globally and harmonically coupled parametric oscillators subject to periodic block pulses with initially random phases. The phase diagram of regions of collective parametric instability is presented, as is a detailed characterization of the motions underlying these instabilities. This presentation includes regimes not indentified in earlier work [I. Bena and C. Van den Broek, Europhys. Lett. 48,498 (1999)]. In addition to the familiar parametric instability of individual oscillators, two kinds of collective instabilities are identified. In one the mean amplitude diverges monotonically while in the other the divergence is oscillatory. The frequencies of collective oscillatory instabilities in general bear no simple relation to the eigenfrequencies of the individual oscillators nor to the frequency of the external modulation. Numerical simulations show that systems with only nearest neighbor coupling have collective instabilities similar to those of the mean field model. Many of the mean field results are already apparent in a simple dimer [M. Copelli and K. Lindenberg, to appear in Phys. Rev. E].
PACS number: 05.90.+m, 64.60.-i

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