Higher-order interactions shape collective dynamics, but how they affect transitions between different states in swarmalator systems is yet to be determined. To that effect, we here study an analytically tractable swarmalator model that incorporates both pairwise and higher-order interactions, resulting in four distinct collective states: async, phase wave, mixed, and sync states. We show that even a minute fraction of higher-order interactions induces abrupt transitions from the async state to the phase wave and the sync state. We also show that higher-order interactions facilitate an abrupt transition from the phase wave to the sync state bypassing the intermediate mixed state. Moreover, elevated levels of higher-order interactions can sustain the presence of phase wave and sync state, even when pairwise interactions lean towards repulsion. The insights gained from these findings unveil self-organizing processes that hold the potential to explain sudden transitions between various collective states in numerous real-world systems.

Recent developments in complex systems have witnessed that many real-world scenarios, successfully represented as networks, are not always restricted to binary interactions but often include higher-order interactions among the nodes. These beyond pairwise interactions are preferably modeled by hypergraphs, where hyperedges represent higher-order interactions between a set of nodes. In this work, we consider a multiplex network where the intralayer connections are represented by hypergraphs, called the multiplex hypergraph. The hypergraph is constructed by mapping the maximal cliques of a scale-free network to hyperedges of suitable sizes. We investigate the intralayer and interlayer synchronizations of such multiplex structures. Our study unveils that the intralayer synchronization appreciably enhances when a higher-order structure is taken into consideration in spite of only pairwise connections. We derive the necessary condition for stable synchronization states by the master stability function approach, which perfectly agrees with the numerical results. We also explore the robustness of interlayer synchronization and find that for the multiplex structures with many-body interaction, the interlayer synchronization is more persistent than the multiplex networks with solely pairwise interaction.

Networks incorporating higher-order interactions are increasingly recognized for their ability to introduce novel dynamics into various processes, including synchronization. Previous studies on synchronization within multilayer networks have often been limited to specific models, such as the Kuramoto model, or have focused solely on higher-order interactions within individual layers. Here, we present a comprehensive framework for investigating synchronization, particularly global synchronization, in multilayer networks with higher-order interactions. Our framework considers interactions beyond pairwise connections, both within and across layers. We demonstrate the existence of a stable global synchronous state, with a condition resembling the master stability function, contingent on the choice of coupling functions. Our theoretical findings are supported by simulations using Hindmarsh-Rose neuronal and Rössler oscillators. These simulations illustrate how synchronization is facilitated by higher-order interactions, both within and across layers, highlighting the advantages over scenarios involving interactions within single layers. Published by the American Physical Society 2024

Understanding how the interplay between higher-order and multilayer structures of interconnections influences the synchronization behaviors of dynamical systems is a feasible problem of interest, with possible application in essential topics such as neuronal dynamics. Here, we provide a comprehensive approach for analyzing the stability of the complete synchronization state in simplicial complexes with numerous interaction layers. We show that the synchronization state exists as an invariant solution and derive the necessary condition for a stable synchronization state in the presence of general coupling functions. It generalizes the well-known master stability function scheme to the higher-order structures with multiple interaction layers. We verify our theoretical results by employing them on networks of paradigmatic Rössler oscillators and Sherman neuronal models, and we demonstrate that the presence of group interactions considerably improves the synchronization phenomenon in the multilayer framework.

A potential issue of interest is figuring out how the combination of temporal and higher-order interactions influences the collective dynamics of the brain, specifically, neuronal synchronization. Motivated by this, here we consider an ensemble of neurons interacting with each other through gap junctions, modeled by temporal higher-order networks (simplicial complexes), and study the emergence of complete neuronal synchronization. We find that the critical synaptic strength for achieving neuronal synchronization with time-varying higher-order interaction is relatively lower than that with temporal pairwise interactions or static many-body interactions. Our study shows that neuronal synchronization can occur even in the sole presence of higher-order, time-varying interactions. We also find that the enhancement in neuronal synchronization in temporal higher-order structure is highly related to the density of group interactions among the neurons. Furthermore, to characterize the local stability of the synchronous solution, we use the master stability function approach, which shows that the numerical findings are in good agreement with the analytically derived conditions.

Self-organized bistability (SOB) stands as a critical behavior for the systems delicately adjusting themselves to the brink of bistability, characterized by a first-order transition. Its essence lies in the inherent ability of the system to undergo enduring shifts between the coexisting states, achieved through the self-regulation of a controlling parameter. Recently, SOB has been established in a scale-free network as a recurrent transition to a short-living state of global synchronization. Here, we embark on a theoretical exploration that extends the boundaries of the SOB concept on a higher-order network (implicitly embedded microscopically within a simplicial complex) while considering the limitations imposed by coupling constraints. By applying Ott-Antonsen dimensionality reduction in the thermodynamic limit to the higher-order network, we derive SOB requirements under coupling limits that are in good agreement with numerical simulations on systems of finite size. We use continuous synchronization diagrams and statistical data from spontaneous synchronized events to demonstrate the crucial role SOB plays in initiating and terminating temporary synchronized events. We show that under weak-coupling consumption, these spontaneous occurrences closely resemble the statistical traits of the epileptic brain functioning.