The templex is a topological object bridging homologies and templates for chaotic dynamics. This article places the templex within category theory, introducing a directed path algebra, an edge operator on directed paths, and an equivalence relation for directed cycles that is distinct from directed homologies. The resulting functorial invariants are of two kinds: Abelian-group invariants, namely the homology groups, and semigroup invariants, namely the generatex semigroups. These invariants are separable through forgetful functors and constitute a robust framework for identifying tipping points, disambiguating physical mechanisms, and benchmarking data-driven models against observations or simulations. The formulation sets forth a nonmetric criterion for chaos from finite-time data and reveals that the concatenable nature of topological modes of variability is a direct consequence of the semigroup structure of the directed path algebra. Two applications are presented: an experimental speech signal and a climatic numerical simulation.

Due to the deterministic nature of chaotic systems, fluctuations in their trajectories arise solely from the choice of initial conditions. Some of these dynamical fluctuations may lead to extremely unlikely scenarios. Understanding the impact of such rare events and the trajectories that give rise to them is of significant interest across disciplines. Yet, identifying the initial conditions responsible for those events is a challenging task due to the inherent sensitivity to small perturbations of chaotic dynamics. In a recent paper [Gutiérrez et al., Phys. Rev. Lett. 131, 227201 (2023)], this challenge was addressed by finding the effective dynamics that make rare events typical for one-dimensional chaotic maps. Here we extend such large-deviation framework to $d$-dimensional chaotic maps. Specifically, for any such map, we propose a method to find an effective topologically conjugate map which reproduces the rare-event statistics in the long-time limit. We demonstrate the applicability of this result using several observables of paradigmatic examples of two-dimensional chaos, namely, the two-dimensional tent map and Arnold's cat map.

Behavioral feedback plays a crucial role in shaping epidemic dynamics. In this work, a contagion framework that combines higher-order interactions with a dynamic activity-feedback mechanism is developed and analyzed. An evolution equation describing the temporal dynamics of the activity rate is formulated under the assumption that group behavior is regulated by the current prevalence. Two different bistable states are identified, and the conditions for their appearance and disappearance are clarified. To assess the generality of these findings, we numerically simulated the framework on networks with different levels of hyperedge overlap. Numerical simulations further show that hyperedge overlap modulates the bistable window in a feedback-dependent manner and, under strong behavioral feedback, low-overlap networks can trigger a rapid decline in activity rate, thereby suppressing sustained outbreaks. These results provide a qualitative perspective for understanding behavioral-response patterns reported during the COVID-19 pandemic.

Entrainment is an important collective behavior in complex networks, where the output period of the coupled neuronal oscillator matches the external input period and maintains a stable phase relationship. The typical feature of entrainment is the regions in the input period and input amplitude parameter space, known as the Arnold tongue. In this article, we introduced a higher-order Kuramoto model and found that the transition process of the coupled neuronal oscillators from the unentrainment state to the entrainment state is a discontinuous first-order phase transition process accompanied by the hysteresis phenomenon. Interestingly, this discontinuous first-order phase transition process results in the appearance of two Arnold tongue structures in the parameter space. Our results provide more insights into the emergence of the explosive entrainment and have been confirmed by the theoretical analysis.

Disordered materials occur naturally and also provide a broader design space than ordered or crystalline structures. We investigate a two-dimensional disordered network metamaterial constructed from a Delaunay triangulation of an underlying point cloud. Small perturbations in the point cloud induce discrete topological changes. One such change we identify is a Delaunay flip, in which two neighboring Delaunay triangles that form a convex quadrilateral structure with their common edge being one of the two quadrilateral diagonals exchange this diagonal for the other diagonal. These topological changes can cause substantial jumps in the effective resistance measured diagonally across the network, when the change is located near the source or the sink node. The jumps are explained analytically by showing that the change in effective resistance from edge removal or addition depends on the voltage drop across that edge. However, Delaunay flips have less impact on global resistance measurements and in larger networks. These local topological changes are relevant for finite-sized samples and experimentally measurable properties such as electrical transport. Global characterizations of the network disorder or topology lack the location-specificity of our observed effects on network transport, and thus may be inadequate for predicting certain experimentally measurable transport properties in disordered network metamaterials, highlighting the importance of localized regions in material design.

A scale-by-scale analysis of energy flux in the turbulent cascade can be performed using the spatially filtered magnetohydrodynamic (MHD) equations, while the gradient tensor invariants are widely used to characterize the structure of velocity and magnetic fields. Physical mechanisms responsible for energy flux require specific field configurations whose strength is quantified by these tensor invariants. We explore this requirement, showing that the tensor invariants act as proxies for mechanistic energy fluxes under quantifiable conditions. As a special case, the purely hydrodynamic contributions to energy flux can be expressed exactly in terms of the invariants of the velocity gradient tensor. We also show that the invariants bound the available energy flux for distinct physical mechanisms, formalizing the idea that each transfer mechanism requires field configurations with gradients of sufficient strength to support a given energy flux. Results are illustrated using three-dimensional simulations of freely decaying MHD turbulence. These findings advance our understanding of MHD turbulence by revealing key relationships between field topology and energy transfer across scales due to distinct physical mechanisms. They further provide a tool for probing the turbulent cascade using multispacecraft data where local tensor invariants are directly accessible.

This study presents an analytical solution of the two-dimensional concentration distribution of a contaminant in a channel with a prismatic cross-section and asymmetric velocity distribution, influenced by reversible and irreversible reactions, along with the bulk chemical reaction. Recent works by Zhan et al. [J. Hydrol. 632, 130855 (2024)] have shown that weak desorption leads to complex transient dispersion behavior in open-channel flows, with a strong dependence on the initial distribution of contaminants and a delayed response compared to tube flows. Similarly, the present study examines contaminant transport in a channel with a prismatic cross-section and asymmetric velocity profile, incorporating reversible adsorption-desorption and irreversible absorption at the boundaries in both fluid and solid phases. Using Mei homogenization, analytical expressions for two-dimensional concentration up to second order are derived, and the influence of key transport parameters on both mean and two-dimensional concentrations is examined. The findings reveal that increasing the velocity parameters ($\alpha , \beta$) sharpens or skews the velocity profile, thus enhancing shear and increasing dispersion. Consequently, the dispersion coefficient $D_T^{*}$ varies nonmonotonically, while the mean concentration consistently decreases. Increased boundary absorption and bulk reaction parameters significantly reduce the two-dimensional concentration, while increasing the adsorption or desorption parameters raises the two-dimensional concentration. Increasing adsorption-desorption at the boundaries increases the two-dimensional concentration variation in both symmetric and asymmetric cases. It causes persistent nonuniformity when $\alpha \ne \beta$, while symmetry ($\alpha =\beta$) leads to uniform concentration profiles over time. The findings are crucial to improving the quality of the natural stream, reducing pollution, and mitigating the effects of reactions.

Odd fluids are a class of fluids characterized by nonzero antisymmetric transport coefficient tensors induced by broken time-reversal symmetry. In our previous work, a mesoscale simulation model for two-dimensional isotropic odd fluids was developed. Here, we extend the model to the three-dimensional case that corresponds to an anisotropic odd fluid with cylindrical symmetry. Using kinetic theory, we analytically derive the viscosity tensor and Navier-Stokes equation for the three-dimensional mesoscale odd fluid, which we quantitatively verify with simulations. Furthermore, through both simulation and hydrodynamic theory, we demonstrate that the planar Poiseuille flow of the three-dimensional odd fluid exhibits exotic transport behavior. This work thus paves the way for performing large-scale simulations to explore and exploit intriguing phenomena of odd fluids.

Collective motion is a fundamental mode of organization in systems ranging from biological assemblies to driven physical media. Reproducing and controlling such behavior in soft-matter systems remains challenging, as soliton-soliton interactions are often symmetric, which tends to confine collective dynamics to predominantly repulsive motion. Here, we demonstrate that soliton ensembles can be reversibly switched among disordered, linear, and circular motion by electrical control, enabling regulation of soliton population, velocity, and trajectories. Circular motion stabilizes soliton populations while retaining information about earlier trajectories, allowing partial recovery of the original motional state. Under suitably designed electric-field configurations, localized many-body motion can also be achieved. These results establish a controllable route to collective dynamics based on localized solitonic textures and suggest general design principles for programmable self-organization in soft matter, with potential relevance to reconfigurable photonic structures and other driven many-body systems.

A particle on a substrate supporting a surface acoustic wave can experience horizontal drift excited by the dry friction force. The effect is referred to as vibrational transportation, or as a surface acoustic wave motor, and is used in a number of industrial applications. A traditional theory of vibrational transportation considers a particle as a material point moving on a rigid substrate. A more realistic representation is a contact model based on Cattaneo-Mindlin (also called Hertz-Mindlin) mechanics applicable to an axisymmetric deformable particle. The contact zone in this case is not a point but rather a circle that generally contains a smaller circle of stick and a surrounding annulus of slip. A recent semianalytical extension of the Cattaneo-Mindlin solution called the method of memory diagrams allows one to compute the hysteretic friction force for an arbitrary loading history in terms of contact displacements and, subsequently, to numerically solve the equations of motion. Depending on the materials' and excitation parameters, the particle can stay in permanent contact with the substrate or experience multiple jumps. In the former case, the particle can drift in a horizontal direction due to asymmetric sliding condition created by oscillating normal and tangential contact forces. In other words, during each wave period, the particle advances and recedes with different efficiencies, which finally results in a drift. The drift can occur in the wave propagation direction and against it and require a specific choice of system's parameters. In the regime of multiple jumps, directed horizontal motion is also possible. However, it is governed by a completely different mechanism based on synchronization between the wave period and rebounding events. There exist cases where the rebound occurs once per period and consistently at the same phase. During each rebound, the particle gets horizontal momentum of the same sign (against the wave propagation direction). The value of this momentum depends on the horizontal velocity mismatch between the particle and the substrate; therefore, at the beginning of the process, the particle moves with an acceleration that decreases and finally disappears. Exactly the same type of motion against the wave has been observed in our preliminary experiments. In other cases, the time of flight of the particle and the wave period are uncorrelated, thus resulting in a chaotic motion. We also demonstrate that a point mass in the same situation behaves differently. In particular, in a regime of permanent contact, negative and positive sliding are equilibrated, which produces no drift. In addition, multiple rebounds of a point mass are always chaotic, at least for fully conservative collisions. In conclusion, the deformable particle model can be a better guide for various applications, such as particle micropositioning or acoustic dust cleaning.

Understanding fluctuations of observables across stochastic trajectories is essential for various fields of research, from quantum thermal machines to biological motors. We introduce a framework to analyze the statistics of counting observables in subtrajectories, dubbed stochastic excursions, of processes out of equilibrium. Given a partition of the state space into two sets $A$ and $B$, an excursion is defined as the segment of the trajectory that starts with a transition from $A$ to $B$ and ends upon the first return from $B$ to $A$. Our approach offers analytical expressions for the full distribution of counting observables (such as currents, heat, work, entropy production, and dynamical activity) and the excursion duration, capturing their correlations and finite-time fluctuations. As our main result, we uncover a nontrivial fundamental relation between fluctuations of counting observables at the single-excursion level and the steady-state noise obtained from full counting statistics, offering a tool to inspect noise sources. Interestingly, individual excursions preserve a fluctuation theorem structure and satisfy a thermodynamic uncertainty relation, whose insights into precision-cost trade-offs are explored. We discuss examples from distinct fields in which the excursion framework naturally addresses relevant questions and explore in more detail how analyzing excursions yields additional insights into the operation of the three-qubit absorption refrigerator.

We consider the mean first-passage time (MFPT) through a target of interest for a diffusive particle of Langevin type, with the added condition that the particle is reset to its original position with some rate $r$. We study both smooth and nonsmooth, nonconvex potentials, focusing on the case where the reset rate depends on the space coordinate. For quadratic and piecewise-quadratic potentials, we show that the benefits of resetting depend on the ratio between drift and noise, and become more important as the drift potential becomes smaller compared to the noise. When the target is a local optimum of the potential, we further show that it is beneficial to use a space-dependent resetting where the reset rate is lower when the particle is closer to the target.

Understanding information transfer among individuals is fundamental to revealing the collective dynamics of complex systems. Information transfer has been quantified using various information-theoretic tools and assigned the concept of influence. However, information-theoretic measures are inherently statistical, not causal, and influence in the context of causal inference implies a causal relation, so equating influence with information transfer creates conceptual confusion and interpretational challenges. Here, we introduce an influence-based Vicsek model with nonreciprocal interactions to distinguish influence from information transfer and examine their relationship. At the pairwise level, for fixed noise strengths, influence and transfer entropy exhibit quasilinear relations; for fixed interaction weights, influence and transfer entropy exhibit nonlinear relations because noise on influencers enhances information transfer, whereas noise on followers suppresses the overall information transfer. At the collective level, we find that both influence and normalized transfer entropy form two-branched relations that clearly identify the transition points across three distinct phase transitions. These transition points reveal a different aspect of the collective dynamics not captured by classical order parameters: phase transitions are associated with changes in the relative importance of influencers' presents or followers' presents on followers' futures. Finally, we use our model to assess partial information decomposition methods and identify two methods most suitable for analyzing our system, one based on pointwise surprisal changes and the other on secret key agreement. Our work is a first step in distinguishing the concept of influence from information transfer in a physical model system, provides a concrete testbed for methods emerging from the growing field of information theory-based causal inference, and offers new insights into the dynamics of complex systems.

The Fredrickson-Andersen model with hyperparameter $K=1$ is a severely constrained kinetic lattice spin system, such that any site is temporarily blocked from changing its packing state (empty or occupied) if there is one or more occupied nearest neighbors. Starting from a completely random initial configuration with a fraction $\rho$ of sites being occupied, some of the sites may be permanently frozen to their initial state under this severe kinetic constraint. The remaining sites can switch states at least occasionally, and they form the unfrozen subsystem associated with the given initial configuration. In the present work we investigate thermodynamic phase transitions in such unfrozen subsystems of the two-dimensional square lattice and the three-dimensional cubic lattice by extensive numerical simulations. We demonstrate that the giant connected component of the unfrozen subsystem collapses at a certain critical value ${\rho }_c$ of initial packing density, with ${\rho }_c=0.2475$ for the square lattice and ${\rho }_c=0.2809$ for the cubic lattice. This phase transition belongs to the same universality class as conventional site percolation. We also observe that the ground states (densest packing configurations) experience a continuous crystal-to-glass phase transition at the critical value ${\rho }^{*}=0.1423$ of the initial packing density for the cubic lattice. For the two-dimensional square lattice, we argue that long-range crystalline order is destroyed in the ground states as long as the initial packing density $\rho$ is positive.

We develop a motif-based framework for spatiotemporal chaos in spatial evolutionary games and use it to map the dynamical phase diagram in the payoff plane. Using Boolean linearization of the imitate-the-best rule, we derive analytical instability thresholds for local motifs including invaders, cooperative pairs, stripe interfaces, and cooperative cores. These thresholds are obtained from payoff balance at contested motif interfaces and recover classical invasion thresholds of spatial evolutionary games, which emerge here as boundaries of the chaotic phase. Combining the Derrida slope with the asymptotic Hamming distance, we obtain a four-region cartography: ordered, transient-chaotic, sustained-chaotic, and subcritical-chaotic dynamics. The phase diagram is organized by density-dependent motif selection: Different initial cooperator densities activate different instability mechanisms, yet a small set of motif-instability lines consistently bounds the sustained-chaos region across densities. This cartography reveals a subcritical chaotic phase (Derrida slope $s<1$ but asymptotic Hamming distance $d_{\infty }>0$), where infinitesimal perturbations decay while finite-amplitude perturbations sustain chaos. The motif-based framework is anchored by an exact benchmark: For homogeneous backgrounds, the Boolean Jacobian yields an exact correspondence between the Derrida slope and spectral radius, linking damage spreading to deterministic instability.

Finding feasible coils for stellarator fusion devices is a critical challenge of realizing this concept for future power plants. Current design efforts struggle to navigate the highly nonconvex optimization landscape, spend considerable resources scanning the parameter space, and may produce suboptimal coils. In this work, we present an augmented Lagrangian approach to tackle the ill-posed problem of coil optimization. We illustrate its effectiveness and versatility by generating coils for five stellarators with very different symmetries and magnetic-field shaping. In all cases, we find Pareto-optimal coil solutions that in various ways outperform published coil sets.

Multiscale periodic metamaterials have been designed for numerous applications, such as impact absorption, acoustic cloaking, photonic band gaps, and mechanical logic gates. This prior work has focused on optimizing mesoscale structure for desired bulk isotropic properties. In contrast, we seek to develop materials with highly anisotropic elastic properties. To quantify elastic anisotropy, we introduce two rotationally invariant, normalized quantities that characterize the anisotropic response to shear and compression, respectively, $A_G$ and $A_C$. $A_G \left(A_C\right)$ is the normalized variance of the shear (uniaxial compressive) modulus obtained by changing the coordinate axes. We find that typical crystalline solids possess average elastic anisotropy ${\overline{A}}_G\approx 0.15$ and ${\overline{A}}_C\approx 0.09$. Compared to atomic crystals, jammed granular materials can attain elastic anisotropies that are several orders of magnitude larger. Since grain rearrangements reduce anisotropy in granular materials, to preserve strong elastic anisotropy, we design tessellated granular materials that consist of multiple connected grain-filled voxels, which limit rearrangements and enable highly anisotropic elastic properties. Bulk granular packings with $N$ grains prepared at pressure $p$ have maximal anisotropy for $p{N}^2\sim 1$ and become isotropic in the large-$p{N}^2$ limit. We show that homogeneously tessellated granular systems can inherit the elastic response of the constituent voxel configurations with elastic anisotropy up to 100 times that of crystalline compounds over a range of $p{N}^2$. We show further methods to enhance the elastic anisotropy of tessellations by designing heterogeneously patterned voxel configurations and tessellations that allow large boundary deformations.

Period doubling is a robust feature of systems subjected to time-dependent parameter modulation, which manifests among a variety of fields including physics, chemistry, and biology. When the parameter is modulated in space, an analogous behavior can occur, characterized by the emergence of patterns with a wavelength twice that of the underlying modulation. Here, we consider the case of a system depending both on time and space. Using a simple mechanical model, here a vibrating strip, we show that such a modulation yields a dynamics richer than in the classical (i.e., purely temporal or spatial) case. In particular, instead of standard period doubling, we report a dual subharmonic response exhibiting occurrences of tristability. These features are captured within a unified framework, which further reveals that the frequency content of the response can be tuned via the ratio of the system size to the modulation length scale, a control parameter that can be adjusted through the strip's material properties, length, or tension.

When the time-reversal and parity symmetries in a fluid are broken, transverse transport coefficients can arise in response to perturbations, an example being odd viscosity. We refer to these systems as odd fluids. While much progress has been made in the continuum theory of odd-viscous fluids, and noncollisional models for odd viscous fluids have been proposed, a classical microscopic description in which the transverse responses originate from collisions is lacking. In this paper, we show that a dilute granular gas of rough and inelastic particles driven by a constant torque is a minimal microscopic model of an odd fluid. By applying the methods of Boltzmann kinetic theory, we obtain a hydrodynamic description of the microscopic model. Then, using the method of adiabatic elimination, we numerically compute all the response coefficients of the model, explicitly showing that the model has many odd response terms. Our theory predicts that certain odd response coefficients can change sign even when the direction of the external torque is fixed. While we choose a particular case, the procedure we present can be applied to any collisional model. We also present a semiquantitative method to determine the hydrodynamic variables of the theory by observing the eigenvalue spectrum of the linear collision operator. In the end, we repeat the analysis to study a system with rotational drag. We observe that including rotational drag can affect the choice of hydrodynamic variables and numerical values of response coefficients.

We study relations between three inequality indices, namely the Gini ($g$), Pietra ($p$) and Kolkata ($k$) introduced in 1912, 1915, and 2014 respectively, and all are derived from the Lorenz function $L\left(x\right)$ introduced in 1905. The Kolkata index [which corresponds to a fixed point of the complementary Lorenz function $L_c\left(x\right)\equiv 1-L\left(x\right)$] gives the fraction $k$ of wealth possessed by the richest $1-k$ fraction of people ($k$ = 0.8 corresponds to Pareto's 80-20 law from 1896). We show rigorously that while the Pietra index value $p$ should be greater than or equal to $2k-1$, the Robin Hood index should strictly be equal to the excess wealth fraction $2k-1$ possessed by the richest $1-k$ fraction of people. Our numerical data analysis for U.S. IRS income data (1983–2022), Bollywood (India) movie income data (1999–2024), and the citation inequalities across the publications by forty Nobel laureates (2020–2025) in economics, physics, chemistry, and medicine clearly show that $p/(2k-1)$ is always greater than unity but the deviation is never more than 5%. Assuming some simple analytic form for the Lorenz function, we also derived the relations $k=(1/2)+(3/8)g$ for small $g$ values and $p/g=3/4$. However, by considering the Lorenz function appropriate for the generalized Pareto power law distribution, we obtain an extended range ($1_{+}<p/g\le e/2$) that captures most of the empirical data.