Controlling cluster synchronization by adapting the topology

We suggest an adaptive control scheme for the control of in-phase and cluster synchronization in delay-coupled networks. Based on the speed-gradient method, our scheme adapts the topology of a network such that the target state is realized. It is robust towards different initial conditions as well as changes in the coupling parameters. The emerging topology is characterized by a delicate interplay of excitatory and inhibitory links leading to the stabilization of the desired cluster state. As a crucial parameter determining this interplay we identify the delay time. Furthermore, we show how to construct networks such that they exhibit not only a given cluster state but also with a given oscillation frequency. We apply our method to coupled Stuart-Landau oscillators, a paradigmatic normal form that naturally arises in an expansion of systems close to a Hopf bifurcation. The successful and robust control of this generic model opens up possible applications in a wide range of systems in physics, chemistry, technology, and life science.

Figure 1

Schematic view of (a) in-phase synchronization $\left(M=1\right)$, (b) a three-cluster $\left(M=3\right)$, and (c) a splay state $\left(M=N\right)$. Each cluster consists of the same number of nodes.

Figure 2

Evolution of the network topology with the goal to achieve an eight-cluster state. Black, excitatory weighted links; gray (blue), inhibitory weighted links. Node colors denote phase differences with respect to the first node. Parameters: $\lambda =0.1,\omega =1,c=0.01,K=0.1,\tau =\pi ,{\gamma }_G=10,N=40,M=8$.

Figure 3

Control of an eight-cluster state: (a) radii $r_j$, (b) phase difference ${\phi }_j-{\phi }_0$ with respect to the first node, (c) coupling weights $G_{ij}$, and (d) goal function $Q_8$ [dashed gray (turquoise) line] and its reduced part $q_8$ (black (red) line) which excludes the unity-row-sum term. Control starts at $t=0$. The vertical dotted line at $t=80$ is explained in Sec. 5b. Parameters as in Fig. 2.

Figure 4

Average link strength $\overline{G}\left(\frac{2\pi }{M}n\right)$ versus phase differences ${\Delta }_{ij}={lim}_{t\to \infty }[{\phi }_i\left(t\right)-{\phi }_j\left(t\right)]$ in the final state as defined in Eq. (8). $N=30,M=10$. Average over 100 realizations. Other parameters as in Fig. 2.

Figure 5

Control of an eight-cluster state: Fourier coefficients as defined in Eq. (12): (a) $b_1^j$, (b) $b_2^j$, (c) $b_3^j$, and (d) $b_4^j$. At $t=80,b_2^j,b_3^j$, and $b_4^j$ are set to random values. Parameters as in Fig. 2.

Figure 6

Oscillation quenching of a splay state: (a) radii $r_j$, (b) phases ${\phi }_j$, (c) coupling weights $G_{ij}$, and (d) goal function $Q_{12}$ [black (red) line], its reduced part $q_{12}$ [gray (turquoise) line], and the collective frequency $\Omega$ [black (blue) dotted line]. $N=12,M=12$. At $t=40$ the adaptive control is switched off and $a$ is set to $a=\frac{2\omega }{NK}$. Other parameters as in Fig. 2.

Figure 7

Topology (a) before $\left(t=0\right)$ and (b) after control $\left(t=t_c\right)$. Black links, fixed links; gray (green), adapted links. Color code of nodes, phase difference $\Delta \phi ={\phi }_j-{\phi }_0$ with respect to the first node. Parameters: $\tau =3,A/L=0.3,P/L=0.4,N=15,M=3$. Other parameters as in Fig. 2. Initial conditions: directed random network.

Figure 8

Restricted control of a three-cluster state: (a) radii $r_j$, (b) phase difference ${\phi }_j-{\phi }_0$ with respect to the first node, (c) coupling weights $G_{ij}$, and (d) goal function $Q_3$ [gray (turquoise) dashed line] and it its reduced part $q_3$ [black (red) line]. Parameters as in Fig. 7.

Figure 9

Fraction $f_c$ of successfully controlled networks as a function of number of random links $P$ and number of controlled links $A$, normalized by $L=N(N-1)$. Inset: times $t_c$ needed to reach the control goal. $K=0.2$. We simulated ten realizations for each parameter combination. Other parameters as in Fig. 7.