Observability coefficients for predicting the class of synchronizability from the algebraic structure of the local oscillators

Understanding the conditions under which a collective dynamics emerges in a complex network is still an open problem. A useful approach is the master stability function—and its related classes of synchronization—which offers a necessary condition to assess when a network successfully synchronizes. Observability coefficients, on the other hand, quantify how well the original state space of a system can be observed given only the access to a measured variable. The question is therefore pertinent: Given a generic dynamical system (represented by a state variable $\mathit{\text{x}}$) and given a generic measure on it $h\left(\mathit{\text{x}}\right)$ (which may be either an observation of an external agent, or an output function through which the units of a network interact), are classes of synchronization and observability actually related to each other? We explicitly address this issue, and show a series of nontrivial relationships for networks of different popular chaotic systems (Rössler, Lorenz, and Hindmarsh-Rose oscillators). Our results suggest that specific dynamical properties can be evoked for explaining the classes of synchronizability.

Figure 1

ROC curve for distinguishing between good and poor observabilities.

Figure 2

Evolution of the manifold observability coefficients ${\eta }_s^{\mathrm{man}}$ for the Rössler system versus $a$ parameter values.

Figure 3

Master stability functions ${\mathrm{\Lambda }}_s\left(\mu \right)$ versus the normalized coupling parameter $\mu$ computed for Rössler systems coupled via each one of the three variables (a)–(c) and via a combination of two of its variables (d)–(f) and for different $a$ values as shown in the legend of (a). Other parameter values are $b=2$ and $c=4$.

Figure 4

Master stability function for Rössler systems coupled through the three variables with an output function $\mathit{\text{h}}\left(\mathit{\text{x}}\right)=(x,y,z)$.

Figure 5

Three chaotic attractors produced by the Lorenz system for some parameter values. Two coexisting asymmetric attractors (one in black, one in red) are shown in (b).

Figure 6

Evolution of the manifold observability coefficients for the Lorenz system when $\sigma =10$ and $b=\frac{8}{3}$ (top panel) and $\sigma =30$ and $b=1$ (bottom panel).

Figure 7

Master stability functions ${\mathrm{\Lambda }}_s\left(\mu \right)$ versus the normalized coupling parameter $\mu$ computed for Lorenz systems coupled via each of the three variables (a)–(c) and via a combination of two of its variables (d)–(f) and for different sets of parameter values [see legend in (a)].

Figure 8

Distance between two bidirectionally coupled Lorenz systems (slightly detuned) versus the coupling parameter ${\epsilon }_s$.

Figure 9

The chaotic attractors produced by the Lorenz system for two different sets of parameter values. The singular observability manifold ${\mathcal{M}}_z^{\mathrm{obs}}$ is plotted in red.

Figure 10

Master stability functions ${\mathrm{\Lambda }}_s\left(\mu \right)$ versus the normalized coupling parameter $\mu$ for the Hindmarsh-Rose systems coupled via each one of the three variables (top panel) and via a combination of two and three of its variables (bottom panel). Parameter values are as follows: $a=1, b=3, c=1, d=5, r=0.001, s=4, X_c=-\frac{1+\sqrt{5}}{2}$, and $I=3.318$.

Figure 11

Distance between two bidirectionally coupled Hindmarsh-Rose systems (slightly detuned) versus the coupling parameter ${\rho }_s$. Same parameter values as in Fig. 10.

Figure 12

Time series for the three variables of the Hindmarsh-Rose system. Same parameter values as in Fig. 10.