Mathematical Sciences: Nonlinear Dynamics of Oscillator Arrays

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DMS_9500948 Strogatz The proposal is to study the nonlinear dynamics of oscillator arrays, using mathematical methods of dynamical systems theory, bifurcation theory, and perturbation theory, along with numerical simulation. One project concerns arrays of superconducting Josephson junctions, and will be carried out in collaboration with an experimental group at MIT. This work is aimed at understanding the phase-locking properties of series arrays; the propagation and instabilities of kinks in one-dimensional parallel arrays; and the behavior of vortices and row-switched states in two-dimensional arrays. A second project (in collaboration with an experimental group at Georgia Tech) will address the dynamics of laser arrays. Areas of investigation will include phase-locking in slightly detuned laser arrays; controlling chaos in laser arrays; and mutual synchronization of chaotic lasers, with possible applications to private communication. The Principal Investigator and his experimental collaborators will study the dynamics of superconducting Josephson arrays and laser arrays. Josephson arrays can be used as very sensitive detectors of magnetic fields (with biomedical and astronomical applications), voltage standards, power generators at submillimeter wavelengths (a part of the spectrum that is difficult to access by other technologies), and as ultrafast switching devices and transmission lines. All of these technologies would benefit from a deeper mathematical understanding of synchronization and vortex propagation in Josephson arrays. The goals of the laser array project are to find methods for improving the phase-locking characteristics of laser arrays (to increase their power output), and to develop novel encryption schemes for private communications, based on mutual synchronization of chaotic lasers.