Spatial patterns of desynchronization bursts in networks

We adapt a previous model and analysis method (the master stability function), extensively used for studying the stability of the synchronous state of networks of identical chaotic oscillators, to the case of oscillators that are similar but not exactly identical. We find that bubbling induced desynchronization bursts occur for some parameter values. These bursts have spatial patterns, which can be predicted from the network connectivity matrix and the unstable periodic orbits embedded in the attractor. We test the analysis of bursts by comparison with numerical experiments. In the case that no bursting occurs, we discuss the deviations from the exactly synchronous state caused by the mismatch between oscillators.

Figure 1

Rössler attractor (projection onto $x-y$ plane) and embedded period 1 orbit, displayed as a thick white curve inside the attractor. The parameters are $a=b=0.2$, $c=7$.

Figure 2

Master stability function $\Psi \left(\alpha \right)$ for a typical trajectory in the attractor (thick continuous curve), for the period 1 orbit (thick dashed curve), and for periodic orbits up to period 4 (thin curves). The curves for the four period 5 orbits are similar to the latter and were left out for clarity.

Figure 3

$x_1-x_6$ as a function of time for $N=12$ Rössler systems connected in a ring with $g=0.71$. Note the desynchronization burst which starts at $t\approx 1380$.

Figure 4

$x_1$ vs $y_1$ for $1372⩽t⩽1392$. During this period, which corresponds approximately to the starting point of the burst in Fig. 3, the trajectory follows closely the transversally unstable period 1 orbit embedded in the attractor (see Fig. 1).

Figure 5

$x_j-x_{j-1}$ vs the node index $j$ for $t=1360$ (open triangles), $t=1385$ (open circles), and $t=1410$ (open squares). Note that the burst is absent first and grows with a long wavelength pattern.

Figure 6

$y_1-y_2$ as a function of time for 8 Rössler systems in a ring. The coupling strength $g$ was $1.09$. The desynchronization burst develops at $t\approx \mathrm{15\hspace{0.17em}000}$, although it is not as sharp due in part to the smaller magnitude of the transversal Lyapunov exponents [$\Psi \left(4.36\right)$ in Fig. 2].

Figure 7

$y_j-y_{j-1}$ vs the node index $j$ for $t=\mathrm{15\hspace{0.17em}000}$ (open triangles), $t=\mathrm{15\hspace{0.17em}200}$ (open circles), and $t=\mathrm{15\hspace{0.17em}400}$ (open squares). The desynchronization burst has a short wavelength spatial pattern.

Figure 8

${\xi }_k^2$ as a function of time for $k=1,2,3,4$. The shortest wavelength component corresponds to $k=4$ (top curve). The curves corresponding to $k=1,2,3$ are close to the horizontal axis.

Figure 9

Arrangement of the dynamical units in a ring with the strength of the connection between nodes 4 and 5 doubled. The matrix $G$ corresponding to this network is in Eq. (10).

Figure 10

$x_5-x_4$ as a function of time for $N=8$ Rössler oscillators in a ring with the strength of the connection between nodes 4 and 5 doubled. The coupling strength is $g=0.79$. A desynchronization burst starts approximately at $t\approx 9000$.

Figure 11

(a) $x_j-\overline{x}$ for $t=8750$ (open triangles), $t=9000$ (open circles), and $t=9250$ (open squares), for the configuration in Fig. 9. The burst develops with the spatial pattern of the localized eigenvector in (b). (b) Localized eigenvector of matrix $G$ in Eq. (10).

Figure 12

${\left[{\eta }_k\right]}_x^2$ as a function of time for $k=4$ (top curve) corresponding to the localized mode, and for $k\ne 4$ (bottom curves, close to zero), corresponding to other modes. In the burst, the localized mode is excited first and only after some time are the other modes also somewhat excited. The localized mode is dominant during the burst.

Figure 13

$y_j-y_{j-1}$ for different times during a burst (filled symbols), and a scaled version of $\sin \left[\left(2\pi j∕12\right)+\varphi \right]-\sin \mathbf{\left(}\left[2\pi (j-1)∕12\right]+\varphi \mathbf{\right)}$ with $\varphi$ as given in the text (open circles). The phase of the burst spatial pattern coincides with the phase of the long wavelength component of the mismatch.

Figure 14

$x_1-x_2$ as a function of time for a configuration of oscillators with a large (top curve) and with a small (curve closer to zero) short wavelength component of the mismatch. The quality of the synchronization is much better in the second case.

Figure 15

$⟨\mid {\eta }_k\mid ⟩$ (open squares), $⟨\mid {\left(QL\right)}_k\mid ⟩$ (open triangles), and $⟨\mid {\left(QL\right)}_k\mid ⟩∕h_k$ (open circles) for $N=8$, $g=0.6$, $k=1,\dots ,7$. The forcing term (open triangles) roughly determines the response (open squares). The corrected forcing term (open circles) matches well the response (open squares).