Distribution of residence times in bistable noisy systems with time-delayed feedback

We analyze theoretically and experimentally the residence time distribution of bistable systems in the presence of noise and time-delayed feedback. We explain various nonexponential features of the residence time distribution using a two-state model and obtain a quantitative agreement with an experiment based on a Schmitt trigger. The limitations of the two-state model are also analyzed theoretically and experimentally using a semiconductor laser with optoelectronic feedback.

Figure 1

Left: Circuit diagram of a Schmitt trigger. For a standard trigger, point A is grounded. For the modified circuit, the potential at A is proportional to the trigger output at a time $\tau$ previously. Right: Transfer function of a Schmitt trigger. The time taken by the trigger to transition from one saturation voltage to the opposite voltage takes several hundreds of nanoseconds. The typical residence time is of order $10\mathrm{ms}$.

Figure 2

Upper: Residence time distribution of the Schmitt trigger with both negative (solid line) and positive (dashed line) feedback. The feedback time is $20\mathrm{ms}$. Lower: Transition rates as a function of time after transition for same data. Rates $p_1$ and $p_2$ are shown by dotted lines. The rate $\sqrt{p_1{p}_2}$ is shown by a dotted line and is calculated from average values.

Figure 3

Graphic illustrating the potentials and escape rates for the continuous system with a switch at $t=0$ and no switch for several $\tau$ subsequently. The major point to note is that the potential barrier height and the positions of the extrema change between $\tau <t<2\tau$ (dashed) and $2\tau <t<3\tau$ (dots), leading to different switching rates. Solid line is the potential barrier in the range $0<t<\tau$.

Figure 4

(a) Experimental RTD of VCSEL system described in Sec. 3B. A large discontinuity at $T=\tau$ can be seen as well as a smaller discontinuity at $T=2\tau$. (b) Simulations of continuous model of Eq. (2) showing discontinuities at $T=\tau$ and $T=2\tau$. Inset contains closeup of the discontinuity at $T=2\tau$. Here $D=0.07$.