Topology-free design for amplitude death in time-delayed oscillators coupled by a delayed connection

This paper deals with amplitude death in time-delayed oscillators coupled by a delayed connection with topology uncertainty. A systematic procedure without trial-and-error testing for designing connection parameters is proposed from a robust control theory viewpoint. This procedure has the following two advantages: the designed connection parameters are valid for any network topology and the procedure is valid even for long-delay oscillators. These analytical results are verified by some numerical simulations.

Figure 1

Illustration of a network system consisting of delayed oscillators (1) coupled by delay connection (2) with topology uncertainty and a time-delay linear system with parameter uncertainty. The stability of steady state (4) in the network is equivalent to that of the linear system.

Figure 2

Sketches of segment $\Omega$ and vectors $\overline{g}(j\omega ,0)$ and $\overline{g}(j\omega ,2)$: (a) $\Omega$ with vertices $\overline{g}(s,0)$ and $\overline{g}(s,2)$ in the coefficient space and (b) vectors $\overline{g}(j\omega ,0)$ and $\overline{g}(j\omega ,2)$ on the complex plane.

Figure 3

Flow chart of our systematic procedure for designing $k$ and $T$. This procedure is based on a sufficient condition for the steady state to be stable.

Figure 4

Roots of $\overline{g}(s,\rho )=0$ $(\rho =0,2)$ ($\tau =5,T=2.5,k=-2$).

Figure 5

Vector loci of $\overline{g}(j\omega ,0)$ and $\overline{g}(j\omega ,2)$ ($\tau =5,T=2.5,k=-2$): (a) $\overline{g}(j\omega ,0)$ and (b) $\overline{g}(j\omega ,2)$.

Figure 6

Marginal stability curves and time-series data of the complete network ($N=200,k=-2$). (a) Marginal stability curves: bold (thin) lines indicate the curves with negative (positive) direction denoted in Eq. (A3). Dashed line indicates $T=\tau /2$ where the sufficient condition for the steady state to be stable always holds. (b) Time-series data of state variable $x_1$ and coupling signal $u_1$ at point (A) ($\tau =10,T=5$) in (a).

Figure 7

Marginal stability curves of the network system on (a) ring topology ($N=100,k=-4$) and (b) small-world topology ($N=50,N_C=20,k=-10$). Dashed line $T=\tau /2$ indicates the sufficient condition for the steady state to be stable.

Figure 8

Root loci of $\overline{g}(s,\rho )=0$ ($\rho =0,2$) for $k=-2$: (a) first rightmost root loci, (b) second rightmost root loci.