Riccardo Muolo

Synchronization is a fundamental phenomenon in complex systems, observed across a wide range of natural and engineered contexts. The Kuramoto model provides a foundational framework for understanding synchronization among coupled oscillators, traditionally assuming pairwise interactions. However, many real-world systems exhibit group and many-body interactions, which can be effectively modeled through hypergraphs. Here, we show that the effect of such higher-order interactions on synchronization is non-monotonic. Through a numerical study of higher-order Kuramoto models on random hypergraphs and on globally coupled systems, we find that the degree of synchronization reached from incoherent initial conditions is maximized at small but nonzero higher-order coupling strength: Weak higher-order interactions enhance synchronization when added to pairwise ones, whereas strong ones work against it, in line with earlier reports of reduced basins and of cluster states. We further show, through a cost-constrained allocation analysis, that under a constrained budget for interactions, a mixed allocation of pairwise and higher-order couplings consistently achieves higher synchronization than relying on either type alone. These findings clarify the role of higher-order interactions in shaping collective dynamics and point to design principles for optimizing synchronization in complex systems.

Chimera states are one of the most intriguing phenomena in nonlinear dynamics, characterized by the coexistence of coherent and incoherent behavior in systems of coupled identical oscillators. Despite extensive studies and numerous observations in different settings, the development of reliable and systematic methods to classify chimera states and distinguish them from other dynamical patterns remains a challenging task. Existing approaches are often limited in scope and lack robustness. In this work we propose a method based on Fourier analysis combined with statistical classification to identify chimera behavior. The method is applied to a system of topological signals coupled via the Dirac operator, where it successfully captures the rich dynamical regimes exhibited by the model. We demonstrate that the proposed approach is robust with respect to variations in network topology and system parameters. Beyond the specific model considered, the framework provides a general and automated tool for distinguishing different dynamical regimes in complex systems.

Synchronization of coupled oscillators is observed in many natural and engineered systems and emerges due to the interactions within the system. It can be both beneficial, e.g., in power grids, and harmful, e.g., in epileptic seizures. In the latter case, efficient control methods to desynchronize the systems are crucial. Recent studies have shown that interactions are not always pairwise, but higher-order, i.e., many-body, and this greatly affects the dynamics. For instance, higher-order interactions increase the linear stability of synchronized states but simultaneously shrink their attraction basin, with potentially opposite effects on control methods. Here, we use a minimally invasive pairwise control based on Hamiltonian control theory and investigate its efficiency on phase oscillators with higher-order interactions. We show that, if the initial phases are close to the synchronized state, higher-order interactions make desynchronization more difficult to achieve. Otherwise, a non-monotonic effect appears: intermediate strengths of higher-order interactions impede desynchronization while larger ones facilitate it. In all cases, the control can desynchronize the system with a sufficient number of controlled nodes and intensity.

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 nonreciprocal, nonpairwise (many-body) interactions. Previous studies have shown that chimera states are more elusive in the presence of nonreciprocal 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 nonreciprocal higher-order structures, called m-directed hypergraphs, which we compare with their corresponding networks, and we observe that some types of chimera states can emerge due to directionality, which had not been previously observed in its absence. We also compare the effect of nonreciprocal 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.

The dynamics of coupled Stuart–Landau (SL) oscillators plays a central role in the study of synchronization phenomena. Previous works have focused on linearly coupled oscillators in different configurations, such as all-to-all or generic complex networks, allowing for both reciprocal or non-reciprocal links. The emergence of synchronization can be deduced by proving the linear stability of the limit-cycle solution for the SL model; the linear coupling assumption allows for a complete analytical treatment of the problem, mostly because the linearized system turns out to be autonomous. In this work, we analyze SL oscillators coupled through nonlinear functions on both undirected and directed networks. Synchronization now depends on the study of a non-autonomous linear system and thus novel tools are required to tackle the problem. We provide a complete analytical description of the system behaviour for some choices of the nonlinear coupling, e.g. in the resonant case. Otherwise, we develop a semi-analytical framework based on Jacobi–Anger expansion and Floquet theory, which allows us to derive precise conditions for the emergence of complete synchronization. The obtained results extend the classical theory of coupled oscillators and pave the way for future studies of nonlinear interactions in networks of oscillators and beyond.

In dynamical systems on networks, one assigns the dynamics to nodes, which are then coupled via links. This approach does not account for group interactions and dynamics on links and other higher dimensional structures. Higher-order network theory addresses this by considering variables defined on nodes, links, triangles, and higher-order simplices, called topological signals (or cochains). Moreover, topological signals of different dimensions can interact through the Dirac–Bianconi operator, which allows coupling between topological signals defined, for example, on nodes and links. Such interactions can induce various dynamical behaviors, for example, periodic oscillations. The oscillating system consists of topological signals on nodes and links whose dynamics are driven by the Dirac–Bianconi coupling, hence, which we call it Dirac–Bianconi driven oscillator. Using the phase reduction method, we obtain a phase description of this system and apply it to the study of synchronization between two such oscillators. This approach offers a way to analyze oscillatory behaviors in higher-order networks beyond the node-based paradigm, while providing a ductile modeling tool for node- and edge-signals.

Flutter suppression facilitates the improvement of structural reliability to ensure the flight safety of an aircraft. In this study, we propose a novel strategy for enlarging the amplitude death (AD) regime to enhance flutter suppression in two coupled identical airfoils with structural nonlinearity. Specifically, we introduce an intermittent mixed coupling strategy, i.e., a linear combination of intermittent instantaneous coupling and intermittent time-delayed coupling between two airfoils. Numerical simulations are performed to reveal the influence mechanisms of different coupling scenarios on the dynamic behaviors of the coupled airfoil systems. The obtained results indicate that the coupled airfoil systems experience the expected AD behaviors within a certain range of the coupling strength and time-delayed parameters. The continuous mixed coupling favors the onset of AD over a larger parameter set of coupling strength than the continuous purely time-delayed coupling. Moreover, the presence of intermittent interactions can lead to a further enlargement of the AD regions, i.e., flutter suppression enhancement. Our findings support the structural design and optimization of an aircraft wing for mitigating the unwanted aeroelastic instability behaviors.

The Kuramoto model is the simplest case of globally coupled phase oscillators with a purely sinusoidal fundamental-harmonic phase coupling function, whose dynamical properties have been extensively studied. While coupled phase oscillators are derived from weakly interacting limit-cycle oscillators via phase reduction, this procedure does not necessarily yield the Kuramoto model or its higher-order extensions exactly for general limit-cycle oscillators and interaction functions, except in the special case of interacting Stuart–Landau oscillators. In this study, we artificially design optimal pairwise and higher-order interaction functions between limit-cycle oscillators, from which higher-order Kuramoto models can be exactly derived via phase reduction for arbitrary smooth limit-cycle oscillators. We validate the results through numerical simulations of FitzHugh–Nagumo oscillators, demonstrating that the collective synchronization dynamics predicted by the reduced higher-order Kuramoto models are realized. Control of the collective phase of the FitzHugh–Nagumo oscillators based on Ott–Antonsen reduction of the higher-order Kuramoto model is also demonstrated.