Robustness to perturbation is a key topic in the study of complex systems occurring across a wide variety of applications from epidemiology to biochemistry. Here we analyze the eigenspectrum of the Jacobian matrices associated to a general class of networked dynamical systems, which contains information on how perturbations to a stationary state develop over time. We find that stability is always determined by a spectral outlier, but with pronounced differences to the corresponding eigenvector in different regimes. We show that, depending on model details, instability may originate in nodes of anomalously low or high degree, or may occur everywhere in the network at once. Importantly, the dependence on extremal degrees results in considerable finite-size effects with different scaling depending on the ensemble degree distribution. Our results have potentially useful applications in network monitoring to predict or prevent catastrophic failures, and we validate our analytical findings through applications to epidemic dynamics and gene regulatory systems.

Most real-world networks exhibit a significant degree of modularity. Understanding the effects of such topology on dynamical processes is pivotal for advances in social and natural sciences. In this work we consider the dynamics of Kuramoto oscillators on modular networks and propose a simple coarse-graining procedure where modules are replaced by effective single oscillators. The method is inspired by EEG measurements, where very large groups of neurons under each electrode are interpreted as single nodes in a correlation network. We expose the interplay between intra-module and inter-module coupling strengths in keeping the coarse-graining process meaningful and show that its accuracy depends on the degree of intra-module synchronization. We show that, when modules are well synchronized, the phase transition from asynchronous to synchronous motion in networks with 2 and 3 modules is very well described by their respective reduced systems, regardless of the network structure connecting the modules. Application of the method to real networks with small modularity coefficients, on the other hand, reveals that the approximation is not accurate, although it still allows for the computation of the critical coupling and the qualitative behavior of the order parameter if the inter-module coupling is large enough.

Self-induced stochastic resonance (SISR) is the emergence of coherent oscillations in slow-fast excitable systems driven solely by noise, without external periodic forcing or proximity to a bifurcation. This work presents a physics-informed machine learning framework for modeling and predicting SISR in the stochastic FitzHugh-Nagumo neuron. We embed the governing stochastic differential equations and SISR-asymptotic timescale-matching constraints directly into a Physics-Informed Neural Network (PINN) based on a Noise-Augmented State Predictor architecture. The composite loss integrates data fidelity, dynamical residuals, and barrier-based physical constraints derived from Kramers' escape theory. The trained PINN accurately predicts the dependence of spike-train coherence on noise intensity, excitability, and timescale separation, matching results from direct stochastic simulations with substantial improvements in accuracy and generalization compared with purely data-driven methods, while requiring significantly less computation. The framework provides a data-efficient and interpretable surrogate model for simulating and analyzing noise-induced coherence in multiscale stochastic systems.

We study the opinion dynamics in a population by considering a variant of Kuramoto model where the phase of an oscillator represents the opinion of an individual on a single topic. Two extreme phases separated by $\pi$ represent opposing views. Any other phase is considered as an intermediate opinion between the two extremes. The interaction (or attitude) between two individuals depends on the difference between their opinions and can be positive (attractive) or negative (repulsive) based on the defined thresholds. We investigate the opinion dynamics when these thresholds are varied. We observe explosive transition from a bipolarized state to a consensus state with the existence of scattered and tri-polarized states at low values of threshold parameter. The system exhibits multistability between various states in a sizeable parameter region. These transitions and multistability are studied in populations with different degrees of diversity represented by the width of conviction distribution. We found that a more homogeneous population has greater tendency to exhibit bipolarized, tri-polarized and clustered states while a diverse population helps mitigate polarization among individuals by reaching to a consensus sooner. Ott-Antonsen analysis is used to analyse the system's long term macroscopic behaviour and verify the numerical results. We also study the effects of neutral individuals that do not interact with others or do not change their attitude on opinion formation, nature of transitions and multistability. Furthermore, we apply our model to language data to study the assimilation of diverse languages in India and compare the results with those obtained from model equations.

We demonstrate the deterministic coherence and anti-coherence resonance phenomena in two coupled identical chaotic Lorenz oscillators. Both effects are found to occur simultaneously when varying the coupling strength. In particular, the occurrence of deterministic coherence resonance is revealed by analysing time realizations $x(t)$ and $y(t)$ of both oscillators, whereas the anti-coherence resonance is identified when considering oscillations $z(t)$ at the same parameter values. Both resonances are observed when the coupling strength does not exceed a threshold value corresponding to complete synchronization of the interacting chaotic oscillators. In such a case, the coupled oscillators exhibit the hyperchaotic dynamics associated with the on-off intermittency. The highlighted effects are studied in numerical simulations and confirmed in physical experiments, showing an excellent correspondence and disclosing thereby the robustness of the observed phenomena.

This research investigates the complex spatiotemporal behaviors of Chialvo neuron maps under the influence of Levy noise on three different network topologies that is a ring network, a two dimensional lattice affected by electromagnetic flux, and a delayed coupled lattice. On the ring structure, we show that adding non uniform Levy noise induces the formation of new collective dynamics like standing and traveling waves. The frequency and type of these emergent patterns depend sensitively on the intrinsic excitability parameter and the noise intensity, revealing new pathways to control synchronization behavior through noise modulation. In the 2D lattice network, we show that electromagnetic flux and noise together induce a diverse range of behaviors, from synchronized waves to desynchronized states. Most strikingly, spiral wave chimeras emerge under moderate noise, with coherent and incoherent regions coexisting, highlighting the fine balance between external forcing and stochastic perturbations. Finally, upon introducing delay in the lattice structure, the system displays a rich variety of dynamical regimes such as labyrinth patterns, rotating spirals, and target waves whose stability and transitions are greatly affected by both delay and coupling strength.

In multi-cellular organisms, cells differentiate into multiple types as they divide. States of these cell types, as well as their numbers, are known to be robust to external perturbations; as conceptualized by Waddington's epigenetic landscape where cells embed themselves in valleys corresponding to final cell types. How is such robustness achieved by developmental dynamics and evolution? To address this question, we consider a model of cells with gene expression dynamics and epigenetic feedback, governed by a gene regulation network. By evolving the network to achieve more cell types, we identified three major differentiation processes exhibiting different properties regarding their variance, attractors, stability, and robustness. The first of these, type A, exhibits chaos and long-lived oscillatory dynamics that slowly transition until reaching a steady state. The second, type B, follows a channeled annealing process where the epigenetic changes in combination with noise shift the cells towards varying final cell states that increase the stability. Lastly, type C exhibits a quenching process where cell fate is quickly decided by falling into pre-existing fixed points while cell trajectories are separated through periodic attractors or saddle points. We find types A and B to correspond well with Waddington's landscape while being robust. Finally, the dynamics of type B demonstrate a differentiation process that uses a directed shifting of fixed points, visualized through the dimensional reduction of gene-expression states. Correspondence with the experimental data of gene expression variance through differentiation is also discussed.

The renormalization group (RG) constitutes a fundamental framework in modern theoretical physics. It allows the study of many systems showing states with large-scale correlations and their classification in a relatively small set of universality classes. RG is the most powerful tool for investigating organizational scales within dynamic systems. However, the application of RG techniques to complex networks has presented significant challenges, primarily due to the intricate interplay of correlations on multiple scales. Existing approaches have relied on hypotheses involving hidden geometries and based on embedding complex networks into hidden metric spaces. Here, we present a practical overview of the recently introduced Laplacian Renormalization Group for heterogeneous networks. First, we present a brief overview that justifies the use of the Laplacian as a natural extension for well-known field theories to analyze spatial disorder. We then draw an analogy to traditional real-space renormalization group procedures, explaining how the LRG generalizes the concept of "Kadanoff supernodes" as block nodes that span multiple scales. These supernodes help mitigate the effects of cross-scale correlations due to small-world properties. Additionally, we rigorously define the LRG procedure in momentum space in the spirit of Wilson RG. Finally, we show different analyses for the evolution of network properties along the LRG flow following structural changes when the network is properly reduced.

Effective frequency control in power grids has become increasingly important with the increasing demand for renewable energy sources. Here, we propose a novel strategy for resolving this challenge using graph convolutional proximal policy optimization (GC-PPO). The GC-PPO method can optimally determine how much power individual buses dispatch to reduce frequency fluctuations across a power grid. We demonstrate its efficacy in controlling disturbances by applying the GC-PPO to the power grid of the UK. The performance of GC-PPO is outstanding compared to the classical methods. This result highlights the promising role of GC-PPO in enhancing the stability and reliability of power systems by switching lines or decentralizing grid topology.

Light propagation in semiconductors is the cornerstone of emerging disruptive technologies holding considerable potential to revolutionize telecommunications, sensors, quantum engineering, healthcare, and artificial intelligence. Sky-high optical nonlinearities make these materials ideal platforms for photonic integrated circuits. The fabrication of such complex devices could greatly benefit from in-volume ultrafast laser writing for monolithic and contactless integration. Ironically, as exemplified for Si, nonlinearities act as an efficient immune system self-protecting the material from internal permanent modifications that ultrashort laser pulses could potentially produce. While nonlinear propagation of high-intensity ultrashort laser pulses has been extensively investigated in Si, other semiconductors remain uncharted. In this work, we demonstrate that filamentation universally dictates ultrashort laser pulse propagation in various semiconductors. The effective key nonlinear parameters obtained strongly differ from standard measurements with low-intensity pulses. Furthermore, the temporal scaling laws for these key parameters are extracted. Temporal-spectral shaping is finally proposed to optimize energy deposition inside semiconductors. The whole set of results lays the foundations for future improvements, up to the point where semiconductors can be selectively tailored internally by ultrafast laser writing, thus leading to countless applications for in-chip processing and functionalization, and opening new markets in various sectors including technology, photonics, and semiconductors.

The number $\mathcal{N}$ of stable fixed points of locally coupled Kuramoto models depends on the topology of the network on which the model is defined. It has been shown that cycles in meshed networks play a crucial role in determining $\mathcal{N}$, because any two different stable fixed points differ by a collection of loop flows on those cycles. Since the number of different loop flows increases with the length of the cycle that carries them, one expects $\mathcal{N}$ to be larger in meshed networks with longer cycles. Simultaneously, the existence of more cycles in a network means more freedom to choose the location of loop flows differentiating between two stable fixed points. Therefore, $\mathcal{N}$ should also be larger in networks with more cycles. We derive an algebraic upper bound for the number of stable fixed points of the Kuramoto model with identical frequencies, under the assumption that angle differences between connected nodes do not exceed $\pi/2$. We obtain $\mathcal{N}\leq\prod_{k=1}^c\left[2\cdot{\rm Int}(n_k/4)+1\right]$, which depends both on the number $c$ of cycles and on the spectrum of their lengths $\{n_k\}$. We further identify network topologies carrying stable fixed points with angle differences larger than $\pi/2$, which leads us to conjecture an upper bound for the number of stable fixed points for Kuramoto models on any planar network. Compared to earlier approaches that give exponential upper bounds in the total number of vertices, our bounds are much lower and therefore much closer to the true number of stable fixed points.

In dynamical systems, the full stability of fixed point solutions is determined by their basin of attraction. Characterizing the structure of these basins is, in general, a complicated task, especially in high dimensionality. Recent works have advocated to quantify the non-linear stability of fixed points of dynamical systems through the relative volumes of the associated basins of attraction [D. A. Wiley {\it et al.} Chaos {\bf 16}, 015103 (2006), P. J. Menck {\it et al.} Nat. Phys. {\bf 9}, 89 (2013)]. Here we revisit this issue and propose an efficient numerical method to estimate these volumes. The algorithm first identifies stable fixed points. Second, a set of initial conditions is considered that are randomly distributed at the surface of hypercubes centered on each fixed point. These initial conditions are dynamically evolved. The linear size of each basin of attraction is finally determined by the proportion of initial conditions which converge back to the fixed point. Armed with this algorithm, we revisit the problem considered by Wiley et al. in a seminal paper [D. A. Wiley {\it et al.} Chaos {\bf 16}, 015103 (2006)] that inspired the title of the present manuscript, and consider the equal-frequency Kuramoto model on a cycle. Fixed points of this model are characterized by an integer winding number $q$ and the number $n$ of oscillators. We find that the basin volumes scale as $(1-4q/n)^n$, contrasting with the Gaussian behavior postulated in Wiley et al.'s paper. Finally, we show the applicability of our method to complex models of coupled oscillators with different natural frequencies and on meshed networks.

Determining the number of stable phase-locked solutions for locally coupled Kuramoto models is a long-standing mathematical problem with important implications in biology, condensed matter physics and electrical engineering among others. We investigate Kuramoto models on networks with various topologies and show that different phase-locked solutions are related to one another by loop currents. The latter take only discrete values, as they are characterized by topological winding numbers. This result is generically valid for any network, and also applies beyond the Kuramoto model, as long as the coupling between oscillators is antisymmetric in the oscillators' coordinates. Motivated by these results we further investigate loop currents in Kuramoto-like models. We consider loop currents in nonoriented $n$-node cycle networks with nearest-neighbor coupling. Amplifying on earlier works, we give an algebraic upper bound $\mathcal{N} \le 2 \, {\rm Int}[n/4]+1$ for the number $\cal N$ of different, linearly stable phase-locked solutions. We show that the number of different stable solutions monotonically decreases as the coupling strength is decreased. Furthermore stable solutions with a single angle difference exceeding $\pi/2$ emerge as the coupling constant $K$ is reduced, as smooth continuations of solutions with all angle differences smaller than $\pi/2$ at higher $K$. In a cycle network with nearest-neighbor coupling we further show that phase-locked solutions with two or more angle differences larger than $\pi/2$ are all linearly unstable. We point out similarities between loop currents and vortices in superfluids and superconductors as well as persistent currents in superconducting rings and two-dimensional Josephson junction arrays.

In this work, we propose a comprehensive theoretical framework combining percolation theory with nonlinear dynamics in order to study hypergraphs with a time-varying giant component. We consider in particular hypergraphs with higher-order triadic interactions that can upregulate or downregulate the hyperedges. Triadic interactions are a general type of signed regulatory interaction that occurs when a third node regulates the interaction between two other nodes. For example, in brain networks, the glia can facilitate or inhibit synaptic interactions between neurons. However, the regulatory interactions may not only occur between regulator nodes and pairwise interactions but also between regulator nodes and higher-order interactions (hyperedges), leading to higher-order triadic interactions. For instance, in biochemical reaction networks, the enzymes regulate the reactions involving multiple reactants. Here we propose and investigate higher-order triadic percolation on hypergraphs showing that the giant component can have a non-trivial dynamics. Specifically, we demonstrate that, under suitable conditions, the order parameter of this percolation problem, i.e., the fraction of nodes in the giant component, undergoes a route to chaos in the universality class of the logistic map. In hierarchical higher-order triadic percolation, we extend this paradigm in order to treat hierarchically nested triadic interactions demonstrating the non-trivial effect of their increased combinatorial complexity on the critical phenomena and the dynamical properties of the process. Finally, we consider other generalizations of the model studying the effect of considering interdependencies and node regulation instead of hyperedge regulation.

We investigate the scaling properties of the order parameter and the largest nonvanishing Lyapunov exponent for the fully locked state in the Kuramoto model with a finite number $N$ of oscillators. We show that, for any finite value of $N$, both quantities scale as $(K-K_L)^{1/2}$ with the coupling strength $K$ sufficiently close to the locking threshold $K_L$. We confirm numerically these predictions for oscillator frequencies evenly spaced in the interval $[-1, 1]$ and additionally find that the coupling range $\delta K$ over which this scaling is valid shrinks like $\delta K \sim N^{-\alpha}$ with $\alpha\approx1.5$ as $N \rightarrow \infty$. Away from this interval, the order parameter exhibits the infinite-$N$ behavior $r-r_L \sim (K-K_L)^{2/3}$ proposed by Paz\'o [Phys. Rev. E 72, 046211 (2005)]. We argue that the crossover between the two behaviors occurs because at the locking threshold, the upper bound of the continuous part of the spectrum of the fully locked state approaches zero as $N$ increases. Our results clarify the convergence to the $N \rightarrow \infty$ limit in the Kuramoto model.

Geographical features such as mountain ranges or big lakes and inland seas often result in large closed loops in high voltage AC power grids. Sizable circulating power flows have been recorded around such loops, which take up transmission line capacity and dissipate but do not deliver electric power. Power flows in high voltage AC transmission grids are dominantly governed by voltage angle differences between connected buses, much in the same way as Josephson currents depend on phase differences between tunnel-coupled superconductors. From this previously overlooked similarity we argue here that circulating power flows in AC power grids are analogous to supercurrents flowing in superconducting rings and in rings of Josephson junctions. We investigate how circulating power flows can be created and how they behave in the presence of ohmic dissipation. We show how changing operating conditions may generate them, how significantly more power is ohmically dissipated in their presence and how they are topologically protected, even in the presence of dissipation, so that they persist when operating conditions are returned to their original values. We identify three mechanisms for creating circulating power flows, (i) by loss of stability of the equilibrium state carrying no circulating loop flow, (ii) by tripping of a line traversing a large loop in the network and (iii) by reclosing a loop that tripped or was open earlier. Because voltage angles are uniquely defined, circulating power flows can take on only discrete values, much in the same way as circulation around vortices is quantized in superfluids.

We present a model of the central pattern generator (CPG) network that can control gait transitions in hexapod robots in a simple manner based on phase reduction. The CPG network consists of six weakly coupled limit-cycle oscillators, whose synchronization dynamics can be described by six phase equations through phase reduction. Focusing on the transitions between the hexapod gaits with specific symmetries, the six phase equations of the CPG network can further be reduced to two independent equations for the phase differences. By choosing appropriate coupling functions for the network, we can achieve desired synchronization dynamics regardless of the detailed properties of the limit-cycle oscillators used for the CPG. The effectiveness of our CPG network is demonstrated by numerical simulations of gait transitions between the wave, tetrapod, and tripod gaits, using the FitzHugh-Nagumo oscillator as the CPG unit.

Establishing a human settlement on Mars is an incredibly complex engineering problem. The inhospitable nature of the Martian environment requires any habitat to be largely self-sustaining. Beyond mining a few basic minerals and water, the colonizers will be dependent on Earth resupply and replenishment of necessities via technological means, i.e., splitting Martian water into oxygen for breathing and hydrogen for fuel. Beyond the technical and engineering challenges, future colonists will also face psychological and human behavior challenges. Our goal is to better understand the behavioral and psychological interactions of future Martian colonists through an Agent-Based Modeling (ABM simulation) approach. We seek to identify areas of consideration for planning a colony as well as propose a minimum initial population size required to create a stable colony. Accounting for engineering and technological limitations, we draw on research regarding high performing teams in isolated and high stress environments (ex: submarines, Arctic exploration, ISS, war) to include the 4 basic personality types within the ABM. Interactions between agents with different psychological profiles are modeled at the individual level, while global events such as accidents or delays in Earth resupply affect the colony as a whole. From our multiple simulations and scenarios (up to 28 Earth years), we found that an initial population of 22 was the minimum required to maintain a viable colony size over the long run. We also found that the agreeable personality type was the one more likely to survive. We find, contrary to other literature, that the minimum number of people with all personality types that can lead to a sustainable settlement is in the tens and not hundreds.