Delayed feedback control of stochastic spatiotemporal dynamics in a resonant tunneling diode

The influence of time-delayed feedback upon the spatiotemporal current density patterns is investigated in a model of a semiconductor nanostructure, namely a double-barrier resonant tunneling diode. The parameters are chosen below the Hopf bifurcation, where the only stable state of the system is a spatially inhomogeneous “filamentary” steady state. The addition of weak Gaussian white noise to the system gives rise to spatially inhomogeneous self-sustained temporal oscillations that can be quite coherent. We show that applying a time-delayed feedback can either increase or decrease the regularity of the noise-induced dynamics in this spatially extended system. Using linear stability analysis, we can explain these effects, depending on the length of the delay interval. Furthermore, we study the influence of this additional control term upon the deterministic behavior of the system, which can change significantly depending on the choice of parameters.

Figure 1

Current-voltage characteristic of the DBRT model. The null isoclines for the dynamical variables $u$ (which is denoted by the load line, dash-dotted) and $a$ in the case of a homogeneous $a\left(x\right)$ (solid) and in the case of inhomogeneous $a\left(x\right)$ (dotted) are shown. The inset shows an enlargement, where $I$ and $H$ mark the inhomogeneous and the homogeneous fixed points of the system, respectively. $U_0=-84.2895$, $r=-35$. Other parameters as in [19, 23]. This gives typical units of $500\mathrm{A}∕{\mathrm{cm}}^2$ for the current density and $0.35\mathrm{mV}$ for the voltage.

Figure 2

Spatiotemporal patterns $a(x,t)$ induced by different noise intensities $D_u=0.1,1.0,2.0$. At $t=0$, the system is prepared in the spatially inhomogeneous steady state “$I$” and with the parameters of Fig. 1. The system is then simulated with $D_a={10}^{-4}$ and $D_u$ as indicated. $U_0=-84.2895$, $r=-35$, $\epsilon =6.2$. Time $t$ and space $x$ are measured in units of the tunneling time ${\tau }_a$ and the diffusion length $l_a$, respectively. Typical values at $4\mathrm{K}$ are ${\tau }_a=3.3\mathrm{ps}$ and $l_a=100\mathrm{nm}$ [23].

Figure 3

Correlation time vs noise intensity $D_u$ without control $(K=0)$ and with control and two different values of $\tau$ as indicated. Averages from 100 time series of length $T=10000$, parameters as in Fig. 2. The inset shows an enlargement.

Figure 4

Correlation time vs feedback strength $K$ for $\tau =5$ and $\tau =7$. $D_u=0.1$, $D_a={10}^{-4}$. Calculated as in Fig. 3.

Figure 5

Spatiotemporal patterns $a(x,t)$ and voltage time series $u\left(t\right)$ for different values of the control strength $K$ and delay time $\tau$: (a) $\tau =4.0$, $K=0.4$; (b) $\tau =13.4$, $K=0.1$. $D_u=0.1$, $D_a={10}^{-4}$, and other parameters as in Fig. 2.

Figure 6

(a) Correlation time [Eq. (3)] for two different noise intensities in dependence of the feedback delay $\tau$. (b) Real parts of the eigenvalues ${\Lambda }_i$ of the linearized deterministic system $(D_a=D_u=0)$ calculated at the spatially inhomogeneous fixed point for $K=0.1$. The black dots are calculated from the spatially discretized system (set of ODEs) whereas the squares are calculated from Eq. (17) (see text). The vertical dotted lines mark values of $\tau$ at which the leading eigenvalue (i.e., the one with the largest real part) changes. (c) Eigenperiods $2\pi ∕\mathrm{Im}\left({\Lambda }_i\right)$ of the deterministic system and basic periods $T_0≔1∕f_{\max }$ of the noise-induced oscillations, where $f_{\max }$ denotes the frequency of the highest peak in the Fourier power spectral density of the noisy system with $D_u=0.1$, $K=0.1$.

Figure 7

(Color online) Fourier power spectral density of the dynamical variable $u\left(t\right)$. The different curves are shifted vertically for clarity. Averages from 100 time series of length $T=10000$ with $K=0.1$, $D_u=0.1$, $D_a={10}^{-4}$, other parameters as in Fig. 2. Dashed (green) curves correspond to analytical approximation from Eq. (33).

Figure 8

Eigenperiods of the deterministic full system at the inhomogeneous fixed point [$2\pi ∕\mathrm{Im}\left({\Lambda }_i\right)$, same as in Fig. 6c], eigenperiods of the reduced homogeneous system at the homogeneous fixed point [$2\pi ∕\mathrm{Im}\left({\Lambda }_i^{\prime }\right)$], and basic periods $T_0≔1∕f_{\max }$ of the noise-induced oscillations, where $f_{\max }$ denotes the frequency of the highest peak in the Fourier power spectral density of the noisy system with $D_u=1.0$, $K=0.1$.

Figure 9

(Color online) Stability regime in the $\tau \text{−}K$ plane of the deterministic system (4), $D_u=D_a=0$. The area below the curve marks the regime within which the inhomogeneous fixed point is still stable under the influence of the control loop. Within the upper regime, the fixed point is unstable. The shaded (green) area is calculated from the space-discretized system of ODEs, whereas the black curve is calculated from Eq. (17). The dotted line corresponds to the upper bound of $K$ given in Eq. (30).

Figure 10

Real parts of the eigenvalues ${\Lambda }_i$ of the linearized deterministic system with (a) $\tau =5$ and (b) $\tau =7$. The arrows mark Hopf bifurcations.

Figure 11

Delay-induced periodic patterns: Spatiotemporal patterns $a(x,t)$ of the controlled deterministic system (4) with $\tau =7$ for $K=1$ and $K=3$. $D_u=D_a=0$, other parameters as in Fig. 2.

Figure 12

Spatiotemporal patterns $a(x,t)$ of the controlled system (4) with noise, $K=1.0$, $\tau =7.0$, $D_a={10}^{-4}$, $D_u$ as indicated, other parameters as in Fig. 2.