Papers

Fluctuation–response relations (FRRs) provide a fundamental relation between spontaneous fluctuations and the linear response to external perturbations, yet their validity in non-equilibrium systems remains an open problem. In a general class of noisy, finite-size, mean-field models of coupled phase oscillators, we reveal two types of FRRs in the incoherent phase below the synchronization transition. The first, corresponding to the infinite-size limit (I-FRR), involves the convolution between the external force and the correlation function in the time domain. The second, the finite-size counterpart (F-FRR), is governed by finite-size fluctuations and carries a correction factor: a spectrum function whose roots correspond to eigenvalues, or Landau poles, of the linear operator. The finite-size fluctuations are suppressed in the weak-coupling or strong-noise limits, and F-FRR reduces to I-FRR. Our analytical findings are verified by extensive numerical simulations.

This work introduces a spectral transfer formalism for nonlinear synchronisation dynamics to resolve how internal dynamical activity redistributes across network scales, moving beyond traditional global observables like the Kuramoto order parameter. By projecting Kuramoto phase dynamics onto the eigenbasis of the graph Laplacian, we construct a time-dependent transfer matrix and derive novel network metrics, including the spectral flux, which quantifies directional redistribution between structural dynamical scales rather than physical transport across nodes. Numerical simulations on modular and hierarchical networks reveal highly structured, intermittent spectral transfer episodes characterised by forward and inverse spectral cascades, directional reversals, and complex interaction geometries. Crucially, we demonstrate a clear separation between macroscopic coherence and microscopic spectral interactions, showing that highly organised, fluctuating internal redistribution dynamics persist even beneath nearly stationary and featureless global synchronisation states. This framework extends conventional synchronisation analysis and establishes a multiscale approach for interpreting network dynamics as an evolving process of spectral reorganisation.

The danger of collective risk in a public goods game could provide an effective escape route from the tragedy of the commons. Existing studies usually embed risk mitigation in the contribution itself, whereas many real systems rely on a separate costly and non-excludable protective activity that requires a participation quorum. Whether such protection can evolve and persist remains unclear. Here, we formulate an $N$-player public-goods game with ordinary cooperators, defectors, and protective cooperators. Defectors endogenously increase a saturating probability of collective failure, whereas protective cooperators incur an additional cost and reduce this risk for all group members once their number reaches the quorum. Replicator analysis and numerical bifurcation calculations yield three main findings. First, protective cooperation gains an advantage when the focal participant completes the quorum and the resulting expected loss reduction exceeds its additional cost. This advantage varies non-monotonically with group composition. Second, when a single protective cooperator cannot activate protection, protection effectiveness does not enter its first-order invasion fitness at full defection; greater effectiveness cannot overcome the rare-invasion barrier when full defection is locally asymptotically stable. Third, a boundary saddle–node bifurcation creates a saddle and a locally asymptotically stable coexistence equilibrium of defectors and protective cooperators. When full cooperation and full defection are also locally asymptotically stable, three attractors coexist, and the numerically observed evolutionary outcome depends on the initial population composition. These results distinguish the effectiveness of activated protection from its ability to become established from rarity.

Adaptive regulation can itself perturb the state it is intended to stabilize. In replicated simulations of an adaptive agent, we compare external disturbance, persistent internally generated disturbance, and control-generated disturbance under regulation-first and disturbance-first ordering. Persistent internal disturbance produces the largest exposure and regulatory burden within the tested parameter grid. When positive controller updates generate an immediate disturbance cost, increasing that cost produces a nonmonotonic response: effective disturbance initially rises, variability across stochastic runs increases over an intermediate range, and corrective activity becomes strongly suppressed at higher costs. The results show how disturbance source and timing shape exposure and controller burden in this model. They motivate testing adaptive agents with distinct disturbance sources and assessing regulatory activity alongside exposure.

The idea that emotions in biological systems are analogous to intrinsic colored noise is advanced and justified. A model describing the dynamics of operation of biological networks under the influence of colored noise is suggested. The agents of a biological network can be represented either by biological species, such as humans and animals, or by neurons of the brain, or by the nodes of a neural network. Operational actions, or decisions, in a biological noisy network are based not only on the evaluation of utility of alternatives, but also on the agents emotions. The model is probabilistic, with the choice of alternatives characterized by the related probabilities. At the initial step, the agents make decisions individually and then start exchanging information with each other and imitating the actions of other agents, thus forming a biological network of interacting agents. Numerical simulations are accomplished for a heterogeneous society consisting of three groups of agents, one group possessing long-range memory, the other group, short-range memory, and the third group of super-rational agents acting strictly on the basis of utility, being deprived of emotions. Dynamics of opinions in different groups can be smooth, oscillatory, or chaotic. Altogether, eight types of operation are found, depending on the dynamics of group decisions. Under strong imitation effect, there appears chaotic motion in the evolution of decision choice. It is shown how the Ellsberg paradox can be resolved and how the exchange of information influences the dynamics of this paradox.

Previous studies of network dynamics have suggested that heterogeneity among nodes inhibits stability, at odds with the ubiquity of inherently heterogeneous natural and engineered systems. Here, we show that this conclusion arises from model reductions introduced for mathematical tractability and breaks down when nodal dynamics are higher-dimensional, yielding non-Hermitian Jacobians. In such systems, including neural, power-grid, and material networks, nodal heterogeneity can instead enhance stability, even when parameters are randomly disordered. Non-Hermiticity also underlies the stabilizing effects of network heterogeneity, which can arise even in one-dimensional nodal dynamics through nonreciprocal interactions, as shown for ecological networks. Our framework reveals disorder not as a liability but as a general resource for stabilizing complex systems.

We report on frequency bursting oscillations in a system of phase oscillators with adaptive and delayed coupling. Adaptation of the coupling strengths is considered slow and depends on the phase shift between the oscillators. We find due to the combined chain of adaptation, collective dynamics, and time delays, the system robustly achieves a state in which the oscillator's frequencies are nearly synchronized but detuned by an integer number of small adaptation frequencies. We demonstrate that this quantization of the detuning is caused by alternating slow and fast transitions. Moreover, the observed motions take the form of bursts of instantaneous frequency, and the number of spikes in each burst corresponds to the quantization level of the detuning. We provide a fast-slow analysis of this phenomenon and explain the mechanisms behind the emergence of bursts. Our findings indicate that these frequency bursting oscillations are robust and exist stably within finite parameter regions.

We investigate the discrete-time version of the Kuramoto model with phase lag, which comprises globally-coupled phase oscillators of distributed frequencies that are evolving under a nonlinear map. In the continuum limit of an infinite number of oscillators ($N\to \infty$), we derive the exact Frobenius-Perron equation for the time evolution of the single-oscillator probability density, and study linear stability of the incoherent state. Instability signals onset of synchronization. The corresponding synchronization threshold is obtained analytically for the case of a Lorentzian distribution of the oscillator frequencies. The threshold differs from that of the continuous-time Kuramoto model, reflecting the fundamentally different stability conditions for discrete-time maps and continuous-time flows. Beyond synchronization threshold, we observe several interesting nonlinear phenomena: Unlike the classical Kuramoto model, the discrete-time version exhibits periodic and chaotic states. Numerical simulations of the finite-$N$ system confirm the analytical prediction for the synchronization threshold, while highlighting breakdown of the celebrated Ott-Antonsen ansatz invoked to conveniently study the continuous-time Kuramoto model in terms of a low-dimensional description.

Empirical regularities described by parametrized families of probability distributions can shape how complex systems are interpreted and explained. A recurring problem is to determine the extent to which these descriptions are supported by observations, and how they can be used to characterize the underlying system. Here, we introduce a general framework for constructing constrained surrogates capable of representing arbitrary families of continuous distributions. We demonstrate its utility by providing explicit algorithms for several canonical distributions in statistical physics, including continuous powerlaw, lognormal, exponential, and truncated Gaussian distributions. These constrained surrogates enable unbiased estimation and accurate tests of model validity based on arbitrary (physically meaningful) sample statistics, without requiring specification or estimation of model parameters. We show how these theoretical advantages translate into more informative and nuanced conclusions in applications ranging from bushfire magnitudes and word usage in texts to brain connectomes and city populations.

A wide range of physical and biological systems are adaptive networks, in which the dynamics of the nodes and of the edges connecting them co-evolve. Mean-field reductions of such systems typically track only the average coupling strength, and therefore cannot determine when the coupling stays effectively homogeneous and when structured connectivity emerges. Here, we present a second-order moment closure that allows us to derive mean-field equations for the coupling-weight variance in networks of heterogeneous phase oscillators with adaptive coupling, starting from uniform coupling weights. In agreement with network simulations, we find a nonlinear, non-monotonic dependence of the relative weight variance on the oscillator heterogeneity that is mediated by the phase coherence. Moreover, we find that deviations from global coupling strongly depend on an interaction between the oscillator heterogeneity and the adaptation rule. Whereas symmetric adaptation causes a strongly coupled core of coherent oscillators to emerge and creates a bistable regime that is absent without adaptation, antisymmetric adaptation leads to antisymmetric coupling within the same core, thereby destabilizing it. Our equations therefore delineate the regimes in which adaptive networks behave like globally coupled systems from those in which more complex coupling patterns form.

Dendritic silhouettes are cheap. Pond ice neurons, diffusion-limited aggregates, wiring-minimizing arbors, and biological neurons can share a branching statistic because they share a growth instability, not because they are functionally similar. We take that observation as a modeling constraint. The ice neuron only structurally resembles a biological neuron. A biological neuron carries an internal state and a field that it helps generate, letting geometry, dynamics, and field co-sculpt one another. In our construction, the ice neuron is the zero-gain limit of an interfacial system in which a FitzHugh–Nagumo state lives on the moving front and writes back into the field that drives it, with dimensionless gain $\gamma$. At $\gamma=0$, morphology and internal dynamics decouple into a Mullins–Sekerka growth band and a damped oscillator. Finite $\gamma$ deforms the spectrum, shifts the selected instability, and opens an oscillatory region below the isolated Hopf threshold. On a dendrite grown at $\gamma=0$ and then frozen, field closure increases distal persistence without changing the geometry; its first-order effect remains positive across repeated realizations of the same growth rule and localizes to a small set of non-cable field shortcuts. Nonlinear dynamics remain finite beyond the linear instability: below threshold, closure is almost the transmission; at spike amplitude, it is a correction to a regenerative cable event. A dendritic tree therefore is not intrinsically a structural spandrel. Shape alone is morphology without an internal state or a source term. That is the dendritic silhouette of the ice neuron. Excitability adds an active cable. Field closure is the biophysical mélange in which the neuron helps write the field that acts back on it, sculpting its dynamics and, during growth, its structure. That is a biological neuron.

Phase reduction is a powerful technique to obtain phase models from highly dimensional oscillatory systems. Starting from pairwise interactions, a first order approximation yields Kuramoto and Winfree phase models, while nonpairwise interactions emerge at the second order. Recently, phase models have been extended to account for nonpairwise interactions, notably, the higher-order Kuramoto model. Is there an intrinsic difference between physical and emergent nonpairwise interactions? And can we make use of the former to even out the effects of the latter? In this work, we exploit a recently developed parametrization method to compute the phase reduction and answer these questions. After a comparison of the new method with the classic Kuramoto-style phase reduction, we solve the network motifs of the emergent nonpairwise interactions and compare them with the physical once. Lastly, we exploit the motifs to adopt a coupling design approach. Our framework paves the way for further exploitations of physical nonpairwise interactions for applications in synchronization engineering.

The great transoceanic voyages of the Age of Discovery imposed severe human costs on their crews. Long travel durations, deteriorating sanitary conditions, and finite food and water supplies made disease, exhaustion, and mortality central features of maritime expansion. We propose a minimal compartmental model describing the voyage itself as a dynamical system coupling crew health and resource depletion. The population is divided into healthy, ill, and dead individuals, while an additional variable represents the effective level of provisions on board. Illness incidence depends nonlinearly on resource availability, capturing the increasing health risk associated with deteriorating food, water, and sanitary conditions. The coupled ordinary differential equations are analyzed in the transient regime relevant to finite-duration voyages. Rather than reconstructing specific historical expeditions, the model identifies generic mechanisms linking provisioning, health deterioration, mortality, and voyage duration. Numerical results show that resource limitations can substantially reduce the healthy and operational fraction of the crew even when overall survival remains relatively high, revealing a distinction between demographic survival and operational viability. Increasing initial provisioning delays the crossover at which ill individuals become as numerous as healthy individuals and extends the period over which the crew remains predominantly operational. We also introduce a critical provisioning time, defined as the time required for resources to reach a prescribed stress threshold, and find an approximately linear dependence on the initial resource buffer. The framework provides a quantitative perspective on the internal dynamics of early transoceanic navigation and illustrates how dynamical-systems approaches can contribute to the study of historical processes.

The most energetic coherent structure of a hovering full-scale quadcopter is associated with a self-organizing process in which the individual rotor vortices synchronize their frequencies while undergoing merging events, yielding a globally correlated structure. We identify and characterize this phenomenon by applying spectral modal and conditional analyses to assimilated three-dimensional velocity data acquired via Shake-The-Box Lagrangian particle tracking. The dataset captures a high-Re, turbulent flow further complicated by time-varying rotor speeds stemming from active flight control, low-frequency vehicle drift, finite spatio-temporal resolution, and measurement uncertainty. Most coherent structures recover established single-rotor features such as tip vortices and their subharmonic pairing. The globally synchronized vortex merging manifests as a spectral peak at an incommensurate frequency below the rotor band, which cannot be explained by single-rotor aerodynamics, subharmonic instabilities, or band-to-band triadic interactions. Instead, conditional averaging provides evidence of the aforementioned intermittent, distinctly non-subharmonic vortex-merging process involving all four rotor wakes. Establishing whether or not this phenomenon is observed across different flight conditions and configurations remains speculative; however, the consistent characterization of the globally synchronized vortex merging using complementary frequency- and time-domain analyses despite experimental complexities, in particular rotor speed variations, demonstrates its robustness.

Crises and population responses can form coupled dynamical systems, with crisis conditions shaping protective behaviours and collective responses altering the crisis trajectory. Existing models rarely capture both evolving crisis conditions and the reinforcement-dependent spread of competing behaviours. We propose a complex contagion with feedback (CCF) model that couples competing complex contagion with crisis dynamics through bidirectional feedback. Agents stochastically switch between competing states according to social reinforcement and adoption complexity, which is influenced by crisis conditions and information campaigns, while population-level behavioural change feeds back into the crisis. We formulate and analyse the model for a well-mixed population, characterising its equilibria and associated stability conditions. As a case study for model validation, we integrate CCF model with a census-calibrated agent-based model of COVID-19 transmission in Australia to represent social distancing adoption and discontinuation. We compare the CCF model with an existing opinion dynamics model and find that, despite using fewer parameters, it achieves comparable performance in reproducing recurrent infection waves. The resulting framework is parsimonious, analytically tractable, and can be adapted to different types of crises.

Whether human or large language model (LLM), an agent in a discussion reads only a few of the others' contributions, bounded by cognition, context, or cost. LLM collectives can settle on a wrong consensus even when a majority starts out correct; we ask how far that reading bound alone decides the outcome. We model the bound with one number, the message capacity, which sets how many of the others' messages an agent reads, and generate the communication network from it. Over 31,824 randomized queries, we found that an 8-billion-parameter model's judgment of a claim effectively reduces to a logistic function of a weighted sum of its inbox, the update rule of a stochastic binary neuron with divisively normalized weights. From these weights and the network's degree statistics alone, the wrong consensus should become unreachable from any start once agents read, on average, fewer than 6.4 of their 31 sources. In 1,414 episodes with assigned starts the prediction failed: the correct side won in fewer than 50% of episodes from every start, and in only 28-45% when 75% of agents started correct. The failure traces to the field, the threshold that a claim's wording sets for the agent's answer before any message is read: the experimental claims' fields lay below the calibration mean, and with each claim's own field the same weights reproduce the outcomes. Reversing the wording showed that the threshold follows what a claim asserts, not whether it is true. On a second 8B model the pipeline predicts claim-dependent bistability; transition points appeared where computed, and an eight-claim calibration matched in 15 of 16 conditions. At 70B the assertion bias is not detected. Thus a collective's fate is largely set by two single-agent measurements: the threshold a claim's wording sets, and the message capacity that sets the transition point.

Labor strikes do not necessarily lead to complete institutional disruption, particularly when non-striking sectors maintain part of normal activities. We propose a minimal dynamical model to investigate how such residual activity affects the persistence of labor conflicts. The model couples strike participation, institutional response, and overall functionality through two competing mechanisms: a direct suppressive effect of residual activity on strike participation and an indirect effect arising from the reduced institutional pressure to resolve the conflict. Analytical calculations reveal a counterintuitive regime in which increasing residual activity increases, rather than decreases, stationary strike participation. We characterize this behavior through a strike-response susceptibility and derive an exact compensation threshold separating the conventional regime from a resilience-paradox regime. At the threshold, the two mechanisms exactly balance, making stationary strike participation independent of residual activity. Numerical results confirm the analytical predictions and illustrate the underlying mechanism. Our results show that institutional resilience can have an unintended dynamical consequence: by absorbing part of the disruption generated by a strike, continued operation may reduce the urgency of institutional response and thereby favor conflict persistence. This provides a simple example of how individually stabilizing processes can collectively generate counterintuitive outcomes in complex social systems.

The hulusi is a wind instrument that was invented in Yunnan Province, China, and has become tremendously popular in recent years. It consists of a mouthpiece, a gourd, and three bamboo tubes, all with free reeds made of copper. The main bamboo tube in the middle has seven finger holes. In this instrument, the pipe length, not the free reed's eigenfrequency, determines the instrument's pitch, unlike, for example, with the Western accordion or the blues harp. In this study, a machine learning model implemented in the COMSAR framework (https://github.com/ifsm) was used to investigate the timbre characteristics of the hulusi to cluster different instruments and pitches. The measured hulusi pitches C, B, A, G, and F were analyzed according to seven psychoacoustic features, among which only the spectral centroid, sharpness, and fractal correlation dimension are shown to form pitch clusters. These timbre features were used to train Kohonen self-organizing maps (SOMs) for clustering. Brightness and sharpness analysis revealed that the highest pitches were less bright and less sharp than mid- and low-range pitches were. Furthermore, the fractal correlation dimension, which mainly determines the chaoticity of the initial transients, was the best-clustering timbre feature for the hulusi, with the highest pitches showing the least chaoticity. This result is supported by defining a cluster quality index for the SOMs.

Ultrashort laser pulses can induce order out of chaos. A femtosecond pulse melts a metal surface for about a hundred picoseconds; under photoexcitation with no net polarization axis, the resolidifying film nonetheless organizes into regular nanoscale patterns. Here we derive from first principles the nonlinear hydrodynamics responsible for it and show it accounts for the full phase diagram of morphologies from measurable laser and material parameters. Starting from the Navier-Stokes equations in the lubrication limit, the molten film obeys a damped Kuramoto-Sivashinsky equation that reduces to a Swift-Hohenberg formalism near threshold, with every coefficient fixed by the laser fluence, melt thickness, optical skin depth, and thermocapillary response. The ratio of melt depth to optical skin depth sets the pattern wavelength, while the inter-pulse delay, via post-pulse damping, gates the route from flat surface to ordered pattern to chaos. We validate the phase diagram (hexagonal cavity arrays, bistable mixed states, labyrinths) numerically and confront it with pulse-resolved experiments on two metals, Ni and Fe3Cr. The agreement between theory and experiment quantitatively links the microscopic laser-material response to the macroscopic pattern selected after many pulses. This order is self-limiting: the same corrugation the instability builds scatters the modes that build it, and simulations, with every scale fixed by material parameters, place the coherence breakdown near a roughness of 30 nm after a few tens of pulses, a finite window confirmed by pulse-by-pulse measurements of orientational order on the experimental surfaces; at stronger coupling, disorder does not destroy the pattern but saturates and pins it instead. This yields a closed, predictive description of laser-driven self-organization, from single-pulse melt dynamics to the arrest of nanoscale order.

Synchronization is a collective phenomenon in which interacting nonlinear oscillators develop a persistent relation between their phases and frequencies. In quantum systems, its description requires connecting the classical dynamics of limit-cycle oscillators with quantum fluctuations, dissipation, and the notion of phase. In this tutorial review, we offer a systematic framework for analyzing synchronization in classical and quantum dynamical systems. Starting from amplitude-phase dynamics, we introduce successively topics of relevance, such as phase locking, the Adler equation, Arnold tongues, and the Kuramoto model, and use these concepts to analyze coupled oscillators and their stability. We then formulate the corresponding description for quantum limit-cycle oscillators using Lindblad master equations and phase-space methods, with explicit examples of two and many coupled oscillators. We further discuss noise, synchronization measures, and the extreme quantum regime, including synchronization in spin systems. The emphasis throughout is on analytical derivations and a practical methodology for identifying synchronized states, determining their stability, and characterizing synchronization in quantum systems. The review is designed as a self-contained guide from which we hope that students can learn the essential concepts and techniques, while researchers can readily find the analytical tools and methodological insights needed to approach new synchronization problems.

We observe the emergence of extreme events in an active fluid involving two distinct chemical species that regulate active stress. One species is slow diffusing and the other is fast diffusing, and the growth of the fast-diffusing species is modelled using a nonlinear logistic term. We demonstrate the presence of extreme events in the temporal evolution of the concentrations, as well as in the spatial profile of the system,in regimes of merging-emerging soliton-like dynamics and spatio-temporal chaos, through analysis of the time-series, bifurcation diagrams, probability distribution functions of the concentration of the two species, return maps and distributions of inter-event intervals. Interestingly, we also find evidence of pronounced bunching of extreme events and super-extreme events in the slow chemical species in the soliton-like regime. We go on to systematically explore the dependence of the extreme event occurrences on the Péclet number and the strength of the nonlinear growth term, and find that the probability of extreme events increases after a critical Péclet number, while increasing the nonlinearity suppresses extreme events. Lastly, in order to gain further insight, we investigate a modified mode-truncated reduced order model comprising of coupled differential equations mimicking this active fluid system. We find that this reduced order model also exhibits extreme events whose emergence is correlated with a sudden expansion in attractor size due to a crisis arising from attractor collision.So these results demonstrate the existence of extreme events in an active fluid system, and are of potential relevance to biological phenomena where active transport plays an important role.

How can a neural circuit rhythmically burst when none of its constituent neurons can endogenously do so? We address this question through a bottom-up reconstruction of a 4-cell neural circuit modeled after the swim central pattern generator (CPG) of the sea slug Dendronotus iris. We first map the intrinsic regimes of a swim interneuron (SiN) model neuron and show that slow mutual inhibition can generate anti-phase bursting in a half-center oscillator (HCO) assembled from tonic-spiking or quiescent cells. A slow–fast phase-space deconstruction explains this pairwise rhythm as a release mechanism. We then move on to CPG parametrization, in which cells 1 and 2 are quiescent, while cells 3 and 4 are tonic spikers but their isolated HCO can only exhibit tonic spiking or suppression. Bursting therefore does not arise at either the cellular or HCO level. It appears only after the two modules are assembled into the complete 4-cell network, where cross excitation, cross inhibition, and rectified electrical coupling act together to support a robust network rhythm-generation. Event-based symbolic encoding and GPU-parallel parameter sweeps show that this higher-order collective state occupies extended parameter domains rather than a single tuned point, and they quantify how the network interactions reshape those domains and the spike counts per burst. Symbolic sweeps identify the healthy range of activity regimes (periodic sequences) and transition boundaries; Lempel–Ziv complexity is used as a descriptor of aperiodic symbolic output rather than as proof of chaos or dynamical instability. Our results establish a hierarchy of rhythm generation—from intrinsic cell dynamics, through conditional HCO bursting, to bursting that emerges only in the fully assembled 4-cell CPG—and provide experimentally accessible predictions for perturbing its chemical and electrical couplings.

Reducing the complexity of ride-pooling paths is a central challenge in systems with distributed demand. Here we show that such dynamics admit an exact coarse-grained representation: for a broad class of routing algorithms whose decisions depend only on path lengths, the full network can be reduced to an effective network of active nodes weighted by shortest-path distances without altering the resulting trajectories, up to stochastic degeneracy breaking. The reduction therefore defines an equivalence class of network representations generating identical path dynamics. We further demonstrate that for globally optimizing dispatchers this equivalence is systematically violated through degeneracy amplification, yet remains quantitatively accurate beyond the exactly solvable regime. Our results identify when spatial structure can be integrated out without loss of dynamical fidelity, providing a general framework for the analysis of interacting path processes.

We propose a learning procedure for system identification in interacting particle systems from single-snapshot observations of collective behaviors, unlike existing approaches that rely on observations of trajectories. This setting leads to a fundamentally ill-posed inverse problem, which we solve by using a regularization strategy based on the empirical distribution of observed configurations, drawn from different, unobserved initial conditions. We test our learning procedure on a variety of representative models with steady-state and quasi-stationary patterns, where collective behaviors encode implicit information about the interaction mechanisms, demonstrating that our approach enables stable and accurate recovery of the underlying interaction laws, leading to faithful reproduction of the collective behavior, and in many cases even of the dynamics leading up to it.

Driven dynamical systems can compute when their transient states encode complex transformations of past inputs. The information processing capacity (IPC) framework allows for a detailed accounting of these computational properties, however its interpretation in noisy systems has remained incomplete. In this work, we clarify how noise affects the IPC and how one can reconstruct the noiseless IPC. First, we show how to distinguish the dynamics of an unperturbed system from the noise-free component of the stochastic dynamics: The IPC measured for responses averaged over noise realizations is in general not the same as the IPC of the unperturbed system. We explicitly demonstrate that noise can redistribute computational capacity and sometimes even enhance performance on particular tasks, rather than merely degrading a fixed computation. We then introduce covariance reconstruction by orthogonal projection (CROP), which reconstructs the covariance and IPC of the noise-free component directly from noisy observations, without requiring a detailed model of either the system or the noise. At fixed total measurement budget, numerical tests on a classical nonlinear reservoir show that CROP estimates the noise-free IPC more accurately than the standard practice of ensemble averaging over repeated trials. We find the same advantage in a quantum reservoir subject to unavoidable measurement noise. Our results provide a general route to recovering the computational structure of noisy physical systems from finite observations.

Whether human motor and brain lateralization arises from fundamentally distinct neural architectures or emerges from conserved network dynamics remains a central question at the intersection of network science and neurobiology. Conventional measures of cortical activation often fail to resolve how directed information exchange adapts to manual preference during complex motor tasks. This ambiguity leaves it unclear whether left-handed individuals possess atypical neural organization or follow shared dynamical principles. Here, we combine Permutation Transfer Entropy (PTE) with network-based indices to map directed cortical signal flow using electroencephalography (EEG) during high-precision motor execution (handwriting) in both right- and left-handed individuals. We demonstrate that cortical communication is strictly scale-dependent regardless of handedness: ipsilateral dominance occurs primarily in general, network-wide interactions, whereas inter-hemispheric interactions maintain a contralateral profile. Crucially, left-handed individuals exhibit inter-hemispheric information dynamics that are functionally equivalent to right-handers when executing tasks with the same anatomical hand, refuting assumptions of mirrored or atypical lateralization. By demonstrating that human cortical lateralization relies on dynamical principles conserved across manual preferences, these findings suggest that functional asymmetries reflect adaptive optimizations governed by epigenetic or social flexibility, rather than fixed structural divergences.

Spectral gaps are widely interpreted as signatures of collective organization, yet it is rarely clear whether the resulting modes correspond to physical variables or merely to convenient mathematical coordinates. Here we show that, in resistor-coupled LC networks, the graph Laplacian defines an experimentally accessible hierarchy of electrical descriptions. Its eigenvectors become measurable voltage patterns, while its eigenvalues quantify the coupling-induced resistive damping of those patterns. When coupling is strong within regions and weak between them, microscopic voltages collapse into nested collective variables. A network of 1002 resonators reduces first to 42 regional voltages and then to four geometry-controlled global modes. Summing the microscopic Kirchhoff equations produces a directly constructible 42-node RLC circuit, without parameter fitting, that reproduces regional dynamics with 1.4 to 4.6% global RMS error. Finer modal coordinates improve accuracy but no longer correspond generally to simple resistor graphs. Robustness tests show that the regional level survives substantial conductance disorder and internal rewiring, but weakens when perturbations impair mixing within a region; changes in interregional connectivity selectively disrupt the global level. Thus the Laplacian spectrum becomes a physical design principle: it identifies which collective electrical variables emerge, how they can be constructed, and when they provide an adequate reduced description.

We consider a geometric mechanochemical model of vesicles which couples the Helfrich flow for the shape of a lipid bilayer vesicle membrane $X$ with a reaction-diffusion equations for a single “morphogen” $\phi$ on $X$. The Helfrich flow is the $L^2$ gradient flow of the elastic bending energy $E(X)=\int_X (H-c_0)^2 dS$ of $X$, typically supplemented by area or volume constraints, or both. The morphogen $\phi$ adsorbs/desorbs at places of high/low mean curvature $H$, i.e., the kinetics of $\phi$ depend on $H$, and conversely $\phi$ modifies the spontaneous curvature $c_0$ on $X$. The flow is no longer gradient, and hence allows for more complicated dynamics, including time periodic orbits, e.g., “breathing and moving” vesicle shapes. We show how to compute bifurcation diagrams for such solution branches via numerical continuation and bifurcation methods. We mostly focus on “planar” vesicles (1D closed curves) but also give an outlook on 3D vesicles (2D closed membranes).

Can asymmetry between two interacting networks fundamentally change how they synchronize? We identify double explosive transitions in the forward direction, backward direction, or a combination thereof, with single or double hysteresis loops in an adaptive bilayer multiplex network of Kuramoto oscillators with pairwise and three-body interactions and asymmetric phase lags. Using the Ott-Antonsen reduction, we derive a low-dimensional system and perform a stability analysis. The reduced model accurately captures the full microscopic dynamics and enables analytical expressions for the bifurcation points. Systematic mapping across multiple parameter planes reveals eight distinct synchronization regimes. The relative ordering of saddle-node and pitchfork bifurcation points—controlled by phase-lag asymmetry, cross-layer adaptation, and the higher-order interaction strength—creates two distinct coherent branches (weak and strong), giving rise to double explosive transitions. Crucially, phase-lag asymmetry acts as a robust control knob: while symmetric phase lags suppress explosive transitions, layer-specific differences promote multistability and double explosive transitions. The higher-order interaction strength $K_2$ and adaptation strengths $q$, $p$, and $h$ further modulate these transitions in a complex, parameter-dependent manner. Excellent agreement between analytical predictions and numerical simulations confirms the robustness of our reduced description.

Finite-range alignment can end either in a single flock or in several noninteracting clusters, yet convergence results rarely determine which outcome follows from a given finite-particle state. We study a one-dimensional normalized alignment model with a hard interaction cut-off and obtain exact predictions in an expansive regime. Velocity order is invariant, so pair separations are nondecreasing and the communication graph evolves through finitely many irreversible edge deletions. On each fixed graph, the group inverse of the random-walk Laplacian gives the total relative displacement remaining before relaxation. Comparing this displacement with the available interaction slack selects the next deletion and yields a finite recursion for the complete switching sequence, terminal partition, limiting cluster velocities, and internal geometry. For path configurations, an explicit Green kernel gives a necessary-and-sufficient fragmentation criterion and a sharp critical alignment rate, including asymptotic boundary contact at criticality. A spectral-geometric condition extends the theory to an open set of initially nonordered velocities, while a common self-weight extension covers both self-excluding and self-including local averages. Numerical computations reproduce the thresholds and multi-event cascades. The results provide an exact finite-size theory of fragmentation and terminal state selection for a class of finite-range interacting particle systems.