Observation of chaotic dynamics of coupled nonlinear oscillators

The nonlinear charge storage property of driven Si p-n junction passive resonators gives rise to chaotic dynamics: period doubling, chaos, periodic windows, and an extended period-adding sequence corresponding to entrainment of the resonator by successive subharmonics of the driving frequency. The physical system is described; equations of motion and iterative maps are reviewed. Computed behavior is compared to data, with reasonable agreement for Poincaré sections, bifurcation diagrams, and phase diagrams in parameter space (drive voltage, drive frequency). N=2 symmetrically coupled resonators are found to display period doubling, Hopf bifurcations, entrainment horns (‘‘Arnol’d tongues’’), breakup of the torus, and chaos. This behavior is in reasonable agreement with theoretical models based on the characteristics of single-junction resonators. The breakup of the torus is studied in detail, by Poincaré sections and by power spectra. Also studied are oscillations of the torus and cyclic crises. A phase diagram of the coupled resonators can be understood from the model. Poincaré sections show self-similarity and fractal structure, with measured values of fractal dimension d=2.03 and d=2.23 for N=1 and N=2 resonators, respectively. Two line-coupled resonators display first a Hopf bifurcation as the drive parameter is increased, in agreement with the model. For N=4 and N=12 line-coupled resonators complex quasiperiodic behavior is observed with up to 3 and 4 incommensurate frequencies, respectively.