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We propose a method to reconstruct the phase dynamics in rhythmical interacting systems from macroscopic responses to weak inputs by developing linear and nonlinear response theories, which predict the responses in a given system. By solving an inverse problem, the method infers an unknown system: the natural frequency distribution, the coupling function, and the time delay which is inevitable in real systems. In contrast to previous methods, our method requires neither strong invasiveness nor microscopic observations. We demonstrate that the method reconstructs two phase systems from observed responses accurately. The qualitative methodological advantages demonstrated by our quantitative numerical examinations suggest its broad applicability in various fields, including brain systems, which are often observed through macroscopic signals such as electroencephalograms and functional magnetic response imaging.

Nonlocally coupled oscillators with a phase lag self-organize into various patterns, such as global synchronization, the twisted state, and the chimera state. In this paper, we consider nonlocally coupled oscillators that move on a ring by randomly exchanging their positions with the neighbors and investigate the combined effects of phase lag and mobility on the collective phase dynamics. Spanning the whole range of phase lag and mobility, we show that mobility promotes synchronization for an attractive coupling, whereas it destroys coherence for a repulsive coupling. The transition behaviors are discussed in terms of the timescales of synchronization and diffusion of the oscillators. We also find a novel spatiotemporal pattern at the border between coherent and incoherent states.

The goal of this article is to identify and understand the fundamental role of spatial symmetries in the emergence of undulatory swimming using an anguilliform robot. Here, the local torque at the joints of the robot is controlled by a chain of oscillators forming a central pattern generator (CPG). By implementing a symmetric CPG with respect to the transverse plane, motor activation waves are inhibited, preventing the emergence of undulatory swimming and resulting in an oscillatory gait. We show experimentally that the swimmer can recover from the traveling wave inhibition by using distributed fluid force feedback to modulate the phase dynamics of each oscillator. This transition from oscillatory to undulating swimming is characterized by a symmetry breaking in the CPG and the body dynamics. By studying the stability of the oscillator chain, we show that the sensory feedback produces a frequency detuning gradient along the CPG chain while preserving its stability. To explain the origin of the instability, we introduce a toy model where the couplings between the dynamics of the oscillators and the body deformation reinforce the symmetry breaking.

Neuromodulatory processes in the brain can critically change signal processing on a cellular level, leading to dramatic changes in network level reorganization. Here, we use coupled nonidentical Kuramoto oscillators to investigate how changes in the shape of phase response curves from Type 1 to Type 2, mediated by varying ACh levels, coupled with activity-dependent plasticity may alter network reorganization. We first show that, when plasticity is absent, the Type 1 networks with symmetric adjacency matrix, as expected, exhibit asynchronous dynamics with oscillators of the highest natural frequency robustly evolving faster in terms of their phase dynamics. However, interestingly, Type 1 networks with an asymmetric connectivity matrix can produce stable synchrony (so-called splay states) with complex phase relationships. At the same time, Type 2 networks synchronize independent of the symmetry of their connectivity matrix, with oscillators locked so that those with higher natural frequency have a constant phase lead as compared to those with lower natural frequency. This relationship establishes a robust mapping between the frequency and oscillators' phases in the network, leading to structure and frequency mapping when plasticity is present. Finally, we show that biologically realistic, phase-locking dependent, connection plasticity naturally produces splay states in Type 1 networks that do not display the structure-frequency reorganization observed in synchronized Type II networks. These results indicate that the formation of splay states in the brain could be a common phenomenon.

We consider the one-dimensional deterministic complex Ginzburg-Landau equation in the regime of phase turbulence, where the order parameter displays a defect-free chaotic phase dynamics, which maps to the Kuramoto-Sivashinsky equation, characterized by negative viscosity and a modulational instability at linear level. In this regime, the dynamical behavior of the large wavelength modes is captured by the Kardar-Parisi-Zhang (KPZ) universality class, determining their universal scaling and their statistical properties. These modes exhibit the characteristic KPZ superdiffusive scaling with the dynamical critical exponent z=3/2. We present numerical evidence of the existence of an additional scale-invariant regime, with the dynamical exponent z=1, emerging at scales which are intermediate between the microscopic ones, intrinsic to the modulational instability, and the macroscopic ones. We argue that this new scaling regime belongs to the universality class corresponding to the inviscid limit of the KPZ equation.

We investigate the finite-size origin of the coherence time (or equivalently of its inverse, the emission linewidth) of a spatially extended, one-dimensional nonequilibrium condensate. We show that the well-known Schawlow-Townes scaling of laser theory, possibly including the Henry broadening factor, only holds for small system sizes, while in larger systems the linewidth displays a novel scaling determined by Kardar-Parisi-Zhang physics. This is shown to lead to an opposite dependence of the coherence time on the optical nonlinearity in the two cases. We then study how subuniversal properties of the phase dynamics such as the higher moments of the phase-phase correlator are affected by the finite size and discuss the relation between the field coherence and the exponential of the phase-phase correlator. We finally identify a configuration with enhanced open boundary conditions, which supports a spatially uniform steady state and facilitates experimental studies of the coherence time scaling.

Cognition involves the global integration of distributed brain regions that are known to work cohesively as cognitive subsystems during brain functioning. Empirical evidence has suggested that spatiotemporal phase relationships between brain regions, measured as synchronization and metastability, may encode important task-relevant information. However, it remains largely unknown how phase relationships aggregate at the level of cognitive subsystems under different cognitive processing. Here, we probe this question by simulating task-relevant brain dynamics through regional stimulation of a whole-brain dynamical network model operating in the resting-state dynamical regime. The model is constructed with structurally embedded Stuart-Laudon oscillators and then fitted with human resting-state functional magnetic resonance imaging data. Based on this framework, we first demonstrate the plausibility of introducing the cognitive system partition into the modeling analysis framework by showing that the clustering of regions across functional networks is better circumscribed by the predefined partition. At the cognitive subsystem level, we focus on how task-relevant phase dynamics are organized in terms of synchronization and metastability. We found that patterns of cognitive synchronization are more task specific, whereas patterns of cognitive metastability are more consistent across different states, suggesting it may encode a more task-general property during cognitive processing, an inherent property conferred by brain organization. This consistent network architecture in cognitive metastability may be related to the distinct functional responses of realistic cognitive systems. We also provide empirical evidence to partially support our computational results. Our paper may provide insights for the mechanisms underlying task-relevant brain dynamics, and establish a model-based link between brain structure, dynamics, and cognition, a fundamental step for computationally aided brain interventions.

Swarmalators are oscillatory systems endowed with a spatial component, whose spatial and phase dynamics affect each other. Such systems can demonstrate fascinating collective dynamics resembling many real-world processes. Through this work, we study a population of swarmalators where they are divided into different communities. The strengths of spatial attraction, repulsion, as well as phase interaction differ from one group to another. Also, they vary from intercommunity to intracommunity. We encounter, as a result of variation in the phase coupling strength, different routes to achieve the static synchronization state by choosing several parameter combinations. We observe that when the intercommunity phase coupling strength is sufficiently large, swarmalators settle in the static synchronization state. However, with a significant small phase coupling strength the state of antiphase synchronization as well as chimeralike coexistence of sync and async are realized. Apart from rigorous numerical results, we have been successful to provide semianalytical treatment for the existence and stability of global static sync and the antiphase sync states.

Partial synchronization is an important dynamical process of coupled oscillators on various natural and artificial networks, which can remain undetected due to the system complexity. With an analogy between pairwise asynchrony of oscillators and topological defects, i.e., vortices, in the two-dimensional XY model, we propose a robust and data-driven method to identify the partial synchronization on complex networks. The proposed method is based on an integer matrix whose element is pseudovorticity that discretely quantifies asynchronous phase dynamics in every two oscillators, which results in graphical and entropic representations of partial synchrony. As a first trial, we apply our method to 200 FitzHugh-Nagumo neurons on a complex small-world network. Partially synchronized chimera states are revealed by discriminating synchronized states even with phase lags. Such phase lags also appear in partial synchronization in chimera states. Our topological, graphical, and entropic method is implemented solely with measurable phase dynamics data, which will lead to a straightforward application to general oscillatory networks including neural networks in the brain.

Swarmalators are phase oscillators that cluster in space, like fireflies flashing in a swarm to attract mates. Interactions between particles, which tend to synchronize their phases and align their motion, decrease with the distance and phase difference between them, coupling the spatial and phase dynamics. In this work, we explore the effects of inducing phase frustration on a system of swarmalators that move on a one-dimensional ring. Our model is inspired by the well-known Kuramoto-Sakaguchi equations. We find, numerically and analytically, the ordered and disordered states that emerge in the system. The active states, not present in the model without frustration, resemble states found previously in numerical studies for the two-dimensional swarmalators system. One of these states, in particular, shows similarities to turbulence generated in a flattened media. We show that all ordered states can be generated for any values of the coupling constants by tuning the phase frustration parameters only. Moreover, many of these combinations display multistability.

A density oscillator is a fluid system in which oscillatory flow occurs between different density fluids through the pore connecting them. We investigate the synchronization in coupled density oscillators using two-dimensional hydrodynamic simulation and analyze the stability of the synchronous state based on the phase reduction theory. Our results show that the antiphase, three-phase, and 2-2 partial-in-phase synchronization modes spontaneously appear as stable states in two, three, and four coupled oscillators, respectively. The phase dynamics of coupled density oscillators is interpreted with their sufficiently large first Fourier components of the phase coupling function.

The present paper proposes an adaptive control law for inducing in-phase and antiphase synchronization in a pair of relaxation oscillators. We analytically show that the phase dynamics of the oscillators coupled by the control law is equivalent to that of Kuramoto phase oscillators and then extend the results for a pair of oscillators to three or more oscillators. We also provide a systematic procedure for designing the controller parameters for oscillator networks with all-to-all and ring topologies. Our numerical simulations demonstrate that these analytical results can be used to solve a dispatching problem encountered by automated guided vehicles (AGVs) in factories. AGV congestion can be avoided and the peak value of the amount of materials or parts in buffers can be suppressed.