Phase chimera states on non-local hyperrings

TL;DR

The paper investigates whether phase chimera states can arise in purely higher-order, many-body interacting systems organized as non-local hyperrings. It models Stuart-Landau oscillators coupled through $d$-hyperedges encoded by adjacency tensors and compares the dynamics to a clique-projected pairwise network, using the normalized total phase variation $V_\theta$ as a coherence metric. Results show robust phase chimeras across 2-, 3-, 4-, and 5-body interactions on the hyperring, with longer-lived incoherence than in pairwise projections; alternative projections to pairwise networks do not sustain phase chimera, although amplitude chimeras and chimera death also appear. The work demonstrates that higher-order interactions can markedly enhance chimera patterns and provides a framework for exploring higher-order synchronization phenomena in complex systems.

Abstract

Chimera states are dynamical states where regions of synchronous trajectories coexist with incoherent ones. A significant amount of research has been devoted to study chimera states in systems of identical oscillators, non-locally coupled through pairwise interactions. Nevertheless, there is an increasing evidence, also supported by available data, that complex systems are composed by multiple units experiencing many-body interactions, that can be modeled by using higher-order structures beyond the paradigm of classic pairwise networks. In this work we investigate whether phase chimera states appear in this framework, by focusing on a novel topology solely involving many-body, non-local and non-regular interactions, hereby named non-local d-hyperring, being (d+1) the order of the interactions. We present the theory by using the paradigmatic Stuart-Landau oscillators as node dynamics, and show that phase chimera states emerge in a variety of structures and with different coupling functions. For comparison, we show that, when higher-order interactions are "flattened" to pairwise ones, the chimera behavior is weaker and more elusive.

Figures (13)

• Figure 1: Non-local hyperring vs. clique-projected network. a) A non-local $3$-hyperring with $m=5$ hyperedges and $15$ nodes. b) Corresponding clique-projected network obtained by transforming each hyperedge into a clique.
• Figure 2: Non-local hyperring vs. clique-projected network. Left panels show spatiotemporal patterns a) and time series b) for variables $y_j(t)$ of the SL oscillators coupled with a non-local $3$-hyperrings with $n=204$ nodes and $m=68$ hyperedges; the emerging dynamical behavior is a phase chimera state with two heads, i.e., there are regions of regular behavior, separated by two regions of decoherence. Right panels c) and d) show the analogous quantities on the clique-projected network. Panels b) and d) show the time series for nodes 50 (blue) and 101 (green). For panels a) and c), on the horizontal axis we set the node index $j$ while in the vertical one, the time. The coupling strength is fixed at $\epsilon=0.01$ and model parameters are $\lambda=1$ and $\omega=1$.
• Figure 3: Dynamical quantities associated to non-local hyperring and clique-projected network. Left panels show frequency a), amplitude b) and phase c) of the Stuart-Landau oscillators computed by using the Fast Fourier Transform (FFT) on the complex variables $z_j(t)=a_j(t)\mathrm{exp}\left[i(2\pi \Omega_j t + \theta_j)\right]$, as function of the node index, $j$, at $2500$ time units. The left panels refer to quantities whose evolution is determined by a non-local $3$-hyperring with $n=204$ nodes and $m=68$ hyperedges. The right panels show the analogous quantities for Stuart-Landau oscillators coupled via the clique-projected network. The coupling strength is $\epsilon=0.01$ and the model parameters are $\lambda=1$ and $\omega=1$.
• Figure SM1: Equations for junction vs. non-junction nodes A non-local $3$-hyperring with $m=5$ hyperedges and $15$ nodes. b) Corresponding clique-projected network obtained by transforming each hyperedge into a clique. The nodes in two hyperedges and cliques are labeled, so that the explicit form of the coupling can be displayed for a junction and a non-junction node, nodes $1$ and $2$ respectively, at the bottom of the Figure. Note that the $6!$ comes from the permutations of the indices in the coupling function, the hypergraphs being symmetric.
• Figure SM2: Non-local $3$-hyperring: persistence of chimera patterns vs. number of nodes Spatiotemporal patterns for $y(t)$ and frequencies, amplitudes and phases obtained through the FFT at $2500$ time units: we observe that a chimera state with two heads is obtained with $408$ oscillators (panels a-d), and persists for $204$ (Figs. 2 and 3 of the main text), $102$ nodes (panels e-h) and even for $48$ nodes (panel i-l), but the coherent regions are becoming smaller. On the other hand, the case with $24$ oscillators (panel m-p) exhibits incoherence, because the chosen time frame is longer than the life of the chimera state, which has decayed. $j$ is the node index. For all cases, the coupling strength is $\epsilon=0.01$ and the model parameters are $\lambda=1$ and $\omega=1$.