Coupling nonlinear oscillators for fun and profit

Adi Bulsara, Space and Naval Warfare Systems Center, San Diego

It is well known that overdamped unforced dynamical systems do not oscillate. However, well-designed coupling schemes, together with the appropriate choice of initial conditions, can induce oscillations when a control parameter exceeds a threshold value. We demonstrate this behavior in a specific prototype system, a soft-potential mean-field description of the dynamics in a hysteretic ``single-domain'' ferromagnetic sample. Our analysis shows that N (odd) unidirectionally coupled elements with cyclic boundary conditions will, in fact, oscillate when a control parameter -- in this case the coupling strength -- exceeds a critical value. These oscillations are now finding utility in the detection of very weak ``target'' signals, via their effect on the oscillation characteristics, e.g. the frequency and asymmetry of the oscillation waveforms. In this talk we explore the underlying dynamics of this system. Scaling laws that govern the oscillation frequency in the vicinity of the critical point, as well as the zero-crossing intervals in the presence of a symmetry-breaking target dc signal, are derived; these quantities are germane to signal detection and analysis.