Chimera states on m-directed hypergraphs

TL;DR

The paper addresses how chimera states can arise in systems of identical oscillators when interactions are both non-reciprocal and higher-order, using $m$-directed hypergraphs and their clique projections. It couples $N$ Stuart-Landau oscillators through $1$-directed $d$-hyperedges and analyzes frequency, amplitude, and phase via Fourier methods, complemented by a normalized total-variation metric. The study demonstrates that directionality, together with higher-order interactions, promotes amplitude-mediated and phase chimera states, with phase reduction confirming the phase-chimera regime and providing a reduced description. This work extends the understanding of chimera patterns to more realistic, nonlocal, directed higher-order couplings, with potential implications for brain networks and other complex systems.

Abstract

Chimera states are synchronization patterns in which coherent and incoherent regions coexist in systems of identical oscillators. This elusive phenomenon has attracted significant interest and has been widely analyzed, revealing several types of dynamical states. Most studies involve reciprocal pairwise couplings, where each oscillator exerts and receives the same interaction from neighboring ones, thus being modeled via symmetric networks. However, real-world systems often exhibit non-reciprocal, non-pairwise (many-body) interactions. Previous studies have shown that chimera states are more elusive in the presence of non-reciprocal pairwise interactions, while they are easier to observe when the interactions are reciprocal and higher-order (many-body). In this work, we investigate the emergence of chimera states on non-reciprocal higher-order structures, called mdirected hypergraphs, which we compare with their corresponding networks, and we observe that chimera state and specifically amplitude-mediated chimeras can emerge due to directionality, which had not been previously observed in the absence of directionality. We also compare the effect of non-reciprocal interactions between higher-order and pairwise couplings, and we find numerically that chimera states appear over a broader parameter range when considering higher-order interactions than in the corresponding network case, demonstrating the impact of directionality and the effect of higher-order interactions. Finally, the nature of phase chimeras has been further validated through phase reduction theory.

Figures (19)

• Figure 1: An example of symmetric nonlocal $2$-hyperring with $10$ nodes, as introduced in muolo2024persistence.
• Figure 2: We schematically represent a family of $1$-directed $2$-hyperrings obtained as a weighted "combination" of three base hyperrings. Each hyperring is made of $1$-directed $2$-hyperedges, weighted by the coefficients $(q_1, q_2, q_3)$. When $q_1 = q_2 = q_3 = 1$, the resulting structure is a symmetric $2$-hyperring, as shown in Fig. \ref{['symetrie_hypergraph']}. On the other hand, different weights allow to generate a family of $1$-directed $2$-hyperrings, where some directions are favored. The heads of the hyperedges are highlighted in light blue, while arrows help to identify the directionality of the hyperedges, illustrating thus the direction of the interactions in the hyperring. In Appendix \ref{['appB']}, we show an analogous construction for the case of $2$-directed $2$-hyperrings.
• Figure 3: We schematically represent a family of clique-projected networks, obtained from the $1$-directed $2$-hyperring presented in Fig. \ref{['hypergraphe_1_diridé_2_hyperring']}. Each directed clique is obtained from the directed hyperedge and it is weighted by the coefficients $(q_1, q_2, q_3)$. When $q_1 = q_2 = q_3 = 1$, the resulting structure is a symmetric network.
• Figure 4: Analysis of the dynamics on a $1$-directed $2$-hyperring of $204$ nodes. The first row shows the spatiotemporal diagrams for the $y$ variable (the behavior of the $x$ variable is analogous), the second row the average amplitudes, the third row the average frequencies, and the last row the average phases. The directionality parameter $p$ is varied with the columns: (a1, a2, a3, a4) coherent behavior for $p = 1$ (with $V(\langle a \rangle) \approx 2.45\, 10^{-4}$, $V(\langle \omega \rangle) = 1.28 \, 10^{-3}$, and $V(\langle \theta \rangle) \approx 1.92 \, 10^{-2}$) ; (b1, b2, b3, b4) traveling amplitude-mediated chimera state for $p = 0.2$ (with $V(\langle a \rangle) \approx 0.0125$, $V(\langle \omega \rangle) = 0.025$, and $V(\langle \theta \rangle) \approx 0.076$); (c1, c2, c3, c4) traveling amplitude-mediated chimera state for $p = 0.1$ (with $V(\langle a \rangle) \approx 0.0148$, $V(\langle \omega \rangle) = 0.044$, and $V(\langle \theta \rangle) \approx 0.201$); (d1, d2, d3, d4) incoherent behavior for $p = 0$ (with $V(\langle a \rangle) \approx 0.0348$, $V(\langle \omega \rangle) = 0.138$, and $V(\langle \theta \rangle) \approx 0.527$). The model parameters are $\alpha=1$ and $\omega=1$, and the coupling strength is $\epsilon = 0.2$. The shaded light blue area represents the standard deviation of the quantity under scrutiny, computed over $10$ consecutive sub-intervals, and quantifies the temporal variability of the node dynamics around its mean value.
• Figure 5: Analysis of the dynamics on a clique-projected network of $204$ nodes. The first row shows the spatiotemporal diagrams for the $y$ variable (the behavior of the $x$ variable is analogous), the second row the average amplitudes, the third row the average frequencies, and the last row the average phases. The directionality parameter $p$ is varied with the columns: (a1, a2, a3, a4) coherent behavior for $p = 1$ (with $V(\langle a \rangle) \approx 5.0\, 10^{-4}$, $V(\langle \omega \rangle) = 1.13\, 10^{-4}$, and $V(\langle \theta \rangle) \approx 0.0185$) ; (b1, b2, b3, b4) coherent behavior for $p = 0.2$ (with $V(\langle a \rangle) \approx 5.09\,10^{-4}$, $V(\langle \omega \rangle) = 2.32\,10^{-4}$, and $V(\langle \theta \rangle) \approx 0.0174$); (c1, c2, c3, c4) traveling amplitude-mediated chimera state for $p = 0.1$ (with $V(\langle a \rangle) \approx 6.16\,10^{-3}$, $V(\langle \omega \rangle) = 0.008$, and $V(\langle \theta \rangle) \approx 0.0571$); (d1, d2, d3, d4) incoherent behavior for $p = 0$ (with $V(\langle a \rangle) \approx 0.0477$, $V(\langle \omega \rangle) = 0.1306$, and $V(\langle \theta \rangle) \approx 0.5765$). The model parameters are $\alpha=1$ and $\omega=1$, and the coupling strength is $\epsilon = 0.2$. The shaded light blue area represents the standard deviation of the quantity under study, computed over $10$ consecutive sub-intervals, and quantifies the temporal variability of the node dynamics around its mean value.