Pinning control of chimera states in systems with higher-order interactions

TL;DR

This paper addresses how to control chimera states in systems with higher-order interactions by applying pinning control to a small subset of nodes. Using a model of identical Stuart-Landau oscillators coupled via nonlocal $d$-hyperrings, it compares higher-order topologies with their clique-projected (pairwise) counterparts and shows that chimera states can be induced and sustained with significantly fewer pinned nodes in the higher-order setting. Two pinning protocols are explored—additive and parametric—alongside a phase-reduction interpretation that explains the observed ease and robustness of control in higher-order networks. The results indicate that higher-order interactions not only promote chimera states but also enable efficient control, offering potential implications for energy-aware controllability and the design of directed or more complex higher-order networks.

Abstract

Understanding and controlling the mechanisms behind synchronization phenomena is of paramount importance in nonlinear science. In particular, the emergence of chimera states, patterns in which order and disorder coexist simultaneously, continues to puzzle scholars, due to its elusive nature. Recently, it has been shown that higher-order (many-body) interactions greatly enhance the presence of chimera states, which are easier to be found and more persistent. In this work, we show that the higher-order framework is fertile not only for the emergence of chimera states, but also for its control. Via pinning control, a technique consisting in applying a forcing to a subset of the nodes, we are able to trigger the emergence of chimera states with only a small fraction of controlled nodes, at striking contrast with the case without higher-order interactions. We show that our setting is robust for different higher-order topologies and types of pinning control and, finally, we give a heuristic interpretation of the results via phase reduction theory. Our numerical and theoretical results provide further understanding on how higher-order interactions shape collective behaviors in nonlinear dynamics.

Figures (11)

• Figure 1: In panel a), a $3$-hyperring of $24$ nodes, together with its corresponding clique-projected network, depicted in panel b). We will call the nodes which are part of two hyperedges (resp. cliques) junction nodes, while all the others will be called non-junction nodes.
• Figure 2: Scheme of the pinning control for a $3$-hyperring of $24$ nodes. The system starts in the synchronized state and the control input is applied to a subset $\mathcal{I}_p = \{1, \ldots, N_p\}$ of the nodes to induce the emergence of a chimera state.
• Figure 3: Additive pinning induces phase chimera states on a $3$-hyperring (i.e., $4$-body interactions) of $204$ nodes. Panel a) depicts the whole time series of variables $y_i(t)$ with $i=1,...,N$, panel b) the hyperedge-based local order parameter, panel c) a snapshot of variables $y_i(t)$ with $i=1,...,N$ for $t_{final}=1000$ time units and panel d) shows a zoom of the time series of variables $y_i(t)$ with $i=1,...,N$. The results for the variables $x_i(t)$ are analogous and, hence, not shown. The model parameters are $\alpha=1$ and $\omega=1$ and the coupling strength is $\varepsilon=0.01$. Pinning control is applied to $N_p=40$ consecutive nodes for $t_p=100$ time units. The parameters $\lambda_{i_p}$ are drawn from a uniform distribution in the interval $[-2,2]$.
• Figure 4: Additive pinning on a clique-projected network of $204$ nodes. Chimera states do not emerge in this setting, at contrast with the previous Fig. \ref{['fig:additive']}. Panel a) depicts the whole time series of variables $y_i(t)$ with $i=1,...,N$, panel b) the clique-based local order parameter, panel c) a snapshot of variables $y_i(t)$ with $i=1,...,N$ for $t_{final}=1000$ time units and panel d) shows a zoom of the time series of variables $y_i(t)$ with $i=1,...,N$. The results for the variables $x_i(t)$ are analogous and, hence, not shown. The model parameters are $\alpha=1$ and $\omega=1$ and the coupling strength is $\varepsilon=0.01$. Pinning control is applied to $N_p=40$ consecutive nodes for $t_p=100$ time units. The parameters $\lambda_{i_p}$ are the same of the previous figure.
• Figure 5: Parametric pinning induces phase chimera states on a $3$-hyperring (i.e., $4$-body interactions) of $204$ nodes. Panel a) depicts the whole time series of variables $y_i(t)$ with $i=1,...,N$, panel b) the hyperedge-based local order parameter, panel c) a snapshot of variables $y_i(t)$ with $i=1,...,N$ for $t_{final}=1000$ time units and panel d) shows a zoom of the time series of variables $y_i(t)$ with $i=1,...,N$. The results for the variables $x_i(t)$ are analogous and, hence, not shown. The model parameters are $\alpha=1$ and $\omega=1$ and the coupling strength is $\varepsilon=0.01$. Pinning control is applied to $N_p=40$ consecutive nodes for $t_p=100$ time units. The parameters $\omega_{i_p}$ are drawn from a uniform distribution in the interval $[0.5,2.5]$.