Design of multiple delay connections in a pair of oscillators for amplitude death

This report describes amplitude death in a pair of two-dimensional limit cycle oscillators coupled by multiple delay connections. Amplitude death is considered as a diffusive-connection-induced stabilization of an unstable fixed point of coupled oscillators. The stability of a linearized system around the fixed point in the coupled oscillators is analyzed. Our stability analysis focuses on the root locus movement of its characteristic equation. The analysis provides us with a numerical procedure to design the coupling strength and the multiple delay times required for amplitude death. The main advantage of the multiple delay connections is that if the time-difference of the multiple delay times is kept at the constant value designed by the procedure, any long delay times can be employed. The numerical example verifies our stability analysis and the design procedure.