Amplitude death in a pair of one-dimensional complex Ginzburg-Landau systems coupled by diffusive connections

This paper shows that, in a pair of one-dimensional complex Ginzburg-Landau (CGL) systems, diffusive connections can induce amplitude death. Stability analysis of a spatially uniform steady state in coupled CGL systems reveals that amplitude death never occurs in a pair of identical CGL systems coupled by no-delay connection, but can occur in the case of delay connection. Moreover, amplitude death never occurs in coupled identical CGL systems with zero nominal frequency. Based on these analytical results, we propose a procedure for designing the connection delay time and the coupling strength to induce spatial-robust stabilization, that is, a stabilization of the steady state for any system size and any boundary condition. Numerical simulations are performed to confirm the analytical results.

Figure 1

Sketch of CGL systems (1) coupled by the time delay connection (4).

Figure 2

Spatiotemporal behavior of amplitude $|W_1(t,x)|$ in CGL systems (1) coupled by the no-delay connection (4) without delay ($\tau =0, \alpha =3.0, \beta =-1.2, {\omega }_0=6.0, \varepsilon =5.0, L=64$): (a) identical systems ($\mathrm{\Delta }\omega =0.0$), (b) nonidentical systems ($\mathrm{\Delta }\omega =4.0$). These systems are coupled at $t=50$.

Figure 3

Spatiotemporal behavior of amplitude $|W_1(t,x)|$ in identical CGL systems (1) coupled by the time delay connection (4) ($\tau =0.25, \mathrm{\Delta }\omega =0.0, \beta =-1.2, {\omega }_0=6.0, \varepsilon =5.0, L=64$): (a) $\alpha =3.0$, (b) $\alpha =7.0$. These systems are coupled at $t=50$.

Figure 4

Stability region of uniform steady state (3) in parameter space $(\varepsilon ,\mathrm{\Delta }\omega )$ for coupled CGL systems (1) and (4) without time delay ($\tau =0$). Steady state (3) is spatial-robust stable in the gray region ($\gamma =0$).

Figure 5

Stability region and boundary curves in the $\varepsilon -\tau$ plane ($w_0=6.0$).

Figure 6

Curves of $\mathrm{Re}\left[f_{+}(i{\lambda }_I,\gamma )\right]=\mathrm{Im}\left[f_{+}(i{\lambda }_I,\gamma )\right]=0$ and $\mathrm{Re}\left[f_{-}(i{\lambda }_I,\gamma )\right]=\mathrm{Im}\left[f_{-}(i{\lambda }_I,\gamma )\right]=0$ on the $\gamma -{\lambda }_I$ plane ($\tau =0.25, {\omega }_0=6.0, \varepsilon =5.0$): (a) $\alpha =3.0$, (b) $\alpha =7.0$.

Figure 7

Boundary curve of $f_{-}(i{\lambda }_{\mathrm{I}},0)=0$ with $\varepsilon =5.0$ and $\tau =0.25$, below which both of $f_{\pm }(s,\gamma )$ are stable.