MURPHYS-HSFS-2014 - 7th International Workshop on Multi-Rate Processes & Hysteresis, 2nd International Workshop on Hysteresis and Slow-Fast Systems, April 7-11, 2014 - Abstract

Erneux, Thomas

Square-wave delay dynamics

Nonlinear delay dynamics have found during the last ten years a renewed interest in both the fundamental and applied sciences. In particular, several photonic devices used in applications are now studied in the laboratory to identify delay-induced bifurcation phenomena. Here, we concentrate on square-wave oscillations that appear if the delay is sufficiently large. They typically exhibit slowly varying plateaus connected by fast transition layers. Motivated by recent experiments, we show that square-wave periodic solutions of rate-equation models may have different plateau lengths or different periods. These periodic regimes may emerge from either a Hopf bifurcation or from a saddle-node of limit-cycles. In the latter case, the timing of the fast transition layers between the present and the delayed state variable plays a crucial role. All our numerical simulations are substantiated analytically by using asymptotic techniques.