Dynamical singularities in adaptive delayed-feedback control

We demonstrate the dynamical characteristics of adaptive delayed-feedback control systems, exploiting a discrete-time adaptive control method derived for carrying out detailed analysis. In particular, the systems exhibit singularities such as power-law decay of the distribution of transient times and almost zero finite-time Lyapunov exponents. We can explain these results by characterizing such systems as having (1) a Jacobian matrix with unity eigenvalue in the whole phase space, and (2) parameters approaching a stability boundary proven to be identical with that of (nonadaptive) delayed-feedback control.

Figure 1

Initial conditions with success, plotted as black dots, for (a) adaptive DFC and (b) nonadaptive DFC.

Figure 2

Distributions of control times for (a) adaptive DFC (log-log plot) and (b) nonadaptive DFC (semilog plot).

Figure 3

Time evolution for the adaptive DFC. (a) $x_1\left(t\right)$ vs $t$. The dashed line for the position of the $x_1$ component of ${\mathbf{x}}_{*}$ (cf. footnote 6). (b) The time-6 Lyapunov exponent vs $t$. Inset: A magnification for $2000\le t\le 8000$.

Figure 4

Trajectories in parameter space for two-parameter adaptive DFC. All the trajectories originate from $[k_{11}\left(0\right),k_{12}\left(0\right)]=(-0.8,0)$ labeled as the plus sign. Different colors (gray levels) indicate different initial conditions. The closed curve represents the stability boundary given by Eq. (7).