Amplitude death of identical oscillators in networks with direct coupling

It is known that amplitude death can occur in networks of coupled identical oscillators if they interact via diffusive time-delayed coupling links. Here we consider networks of oscillators that interact via direct time-delayed coupling links. It is shown analytically that amplitude death is impossible for directly coupled Stuart-Landau oscillators, in contradistinction to the case of diffusive coupling. We demonstrate that amplitude death in the strict sense does become possible in directly coupled networks if the node dynamics is governed by second-order delay differential equations. Finally, we analyze in detail directly coupled nodes whose dynamics are described by first-order delay differential equations and find that, while amplitude death in the strict sense is impossible, other interesting oscillation quenching scenarios exist.

Figure 1

Steady-state stability region: (a) ${\mathrm{\Lambda }}_{\text{max}}^{\prime }<0$ below gray transparent surface and (b) slice defined by $a\tau =0.5$, corresponding to a rotated and scaled version of the MSF with ${\mathrm{\Lambda }}_{\text{max}}^{\prime }\left(\kappa \right)<0$ inside (red) boundary (c) $\mathrm{Im}\left(\zeta \right)=0$ slice (d) $\mathrm{Re}\left(\zeta \right)=0$ slice

Figure 2

[(a)–(e)] MSF for $\tau =0.2$ and $(b,a)$ parameters values: (a) $(-1.7,-0.1)$; (b) $(-1.5,1)$; (c) $(-0.5,1)$; (d) $(-0.5,-0.5)$; (e) $(-1,1.2)$. Unstable dimension is zero in blue (dark gray) shaded region. Unstable dimension increases (decreases) by one as the blue dashed (red solid) boundary line is crossed in the positive ${\kappa }^{\prime }$ direction. (f) Stability diagram of uncoupled system with unstable dimension and region of stable steady state indicated; Andronov-Hopf bifurcation, solid line (blue); $\mathrm{\Lambda }=0$-bifurcation, dashed line (red and gray); Takens-Bogdanov bifurcation, TB (diamond); and transition from a complex conjugate pair to a pair of (positive) real roots, dotted line (gray).

Figure 3

MSF for $b=-1.7,a=-0.1$, corresponding to $\mathrm{A}$ in Fig. 2, with $\tau$ equal [(a) and (e)] 0.01, [(b) and (f)] 0.33, [(c) and (g)] 1.8, and (d) 20.

Figure 4

Stability region for optimal networks. (a) Upper boundary (transparent green) of the $a-b-\tau$ parameter region where the zero steady state is stabilized by coupling. Uncoupled networks are stable in the (blue) shaded region (shown here for $\tau =0$ only, the corresponding volume would extend “upward” to all values of $\tau$). (b) Projection onto $a-b$ plane with the boundary (thick red line) corresponding to $\tau =0$. AH (Andronov-Hopf) and $\mathrm{\Lambda }=0$ label bifurcations of the uncoupled systems [compare to Fig. 2].

Figure 5

Oscillation quenching for Eq. (11) with $a-b$ parameters corresponding to point B in Fig. 2. (a) Fixed-point bifurcations of uncoupled system as function of $b$: PF (triangle), pitchfork at $b=1$; SN (circle), saddle node at $b\approx 0.2984$; AH (square), Andronov-Hopf at $b\approx 0.6523$; yellow shaded region, oscillation amplitude; red dash-dotted line, $b=1.2$. (b) MSF of coupled system with $\tau =0.2$ and $b=1.2$. Red triangle and circle, network eigenvalues ${\gamma }_1={\gamma }_2^{*}=ic$ with $c=3.5$. (c) Time evolution of Eq. (11) started from random initial conditions close to zero for uncoupled and (d) coupled system.

Figure 6

Excitatory coupling with $c=2, \tau =0.33$. (a) $a-b$ plane as in Fig. 2. Coupled: red triangle and circle at $a=-0.1,b=-1.7$ correspond to MSF in panel (b); blue triangle and circle at $a=-0.7,b=-1.7$ correspond to panel (c). (b) MSF for zero steady state and (c) nonzero steady state. Triangle (circle) is the in-phase (antiphase) network eigenmode. (d) Time evolution for the uncoupled and (e) coupled systems.

Figure 7

Inhibitory coupling with $a=-0.1,b=-1.7,c=-2, \tau =0.33$: $t<0$ uncoupled, $t>0$ coupled.

Figure 8

Second-order DDE, Eq. (15): (a) Region of stable steady state for uncoupled [hatched, below the thick dashed (black) line] and directly coupled [shaded, below the thick solid (blue) line] system. AD: amplitude death region [dark (red) shade]. (b) Time evolution of $u\left(t\right)$ with coupling turned on at $t=0$. (c) Zoom for $t$ close to $t=0$. (Initial conditions: $\left|u\right|<0.001$, transient not shown). Parameters used: $T_0=11.47, d=1.71, g_{\text{sf}}=1.01{\tau }_c=0.124, {\tau }_{\text{sf}}=12.435$. This implies ratios ${\tau }_c/{\tau }_{\text{sf}}\approx 0.01$ and ${\tau }_{\text{sf}}/T_0\approx 1.08$.