Papers

We study the effect of boundary conditions on the selection of wavenumbers in pattern-forming systems. Prior work established that, for a large class of boundary conditions, equilibria only within a narrow band of wavenumbers are compatible with the boundary. We demonstrate here that for a large class of initial conditions, solutions evolve towards an extremal member of this narrow band of equilibria. In fact, this extremal member is neutrally stable, nonlinearly unstable, and the convergence is realized through a series of snapping events at the boundary. We observe this behavior in simulations of the Swift-Hohenberg equation when the pattern forming region is confined to a large spatial interval through a parameter quench. We prove the convergence in a phase-diffusion approximation using comparison principles and astutely chosen subsolutions that mimic the snapping behavior.

Mechanical localized modes generally retain exponentially decaying tails, whereas confinement to a single structural unit is difficult to achieve without resonators, flat-band lattices or topological design. To address this need, an asymmetric linkage system is introduced in which geometry tunes a boundary mode from finite localization to its extreme limit. Each cell comprises two unequal rigid bars connected by a torsional spring and is described by one generalized coordinate. Linearization and coordinate transformation yield a tridiagonal model and closed-form expressions for the edge mode frequency and spatial decay ratio. The analysis identifies the configurations supporting a singular edge mode at which its decay ratio vanishes, leaving only the boundary cell active in the linear regime. Simulations show near-complete localization under finite-amplitude excitation, and experiments reproduce the configuration-dependent transition from bulk transmission to boundary confinement. These results establish a minimal and analytically tunable platform for mechanical localization

Higher-order semi-rational solutions are constructed for the (2+1)-dimensional nonlocal multi-component Yajima-Oikawa system. This system is obtained from its local \(2M\)-component counterpart through a spatial-reflection reduction. Using Hirota's bilinear method and the Kadomtsev-Petviashvili hierarchy reduction, we derive a unified \(6\times6\) block Gram determinant. Its entries involve Schur polynomials and exponential functions. The complex-conjugation and spatial-reflection reductions are imposed through algebraic constraints on the spectral parameters and determinant blocks, with the former ensuring a real-valued denominator tau function. Separate spectral parameter sets are assigned to the rational, breather, and dark-soliton blocks. These solutions describe paired lumps or line rogue waves on breather, dark-soliton, or combined backgrounds. Pure rational, breather, and dark-soliton solutions follow by setting the appropriate block sizes to zero. Asymptotic analysis establishes the elastic interactions of the fundamental breather and dark-soliton pairs and determines their position and phase shifts.

We investigate signal transmission in a network of unidirectionally coupled plasma oscillators, where only the first oscillator is driven by a weak periodic input. As the signal propagates through the network, nonlinear interactions generate higher-order harmonic components. Spectral analysis identifies five distinct transmission patterns for the fundamental frequency and odd harmonics, and four patterns for even harmonics, encompassing damped propagation, undamped transmission, and localized amplification. The transmission characteristics are governed by both the coupling strength and system parameters. Furthermore, we analyze how variations in these key parameters shape the overall signal dynamics.

We investigate signal transmission in a one-dimensional chain of $n$ locally coupled plasma oscillators with asymmetric bidirectional coupling $(\epsilon_1,\epsilon_2)$. Our results reveal a clear contrast between symmetric and asymmetric interactions: while symmetric coupling suppresses signal propagation, coupling asymmetry enables efficient nonlinear transmission across the network. In particular, sustained long range transmission is achieved when the forward coupling strength exceeds the backward coupling strength ($\epsilon_1 > \epsilon_2$), as demonstrated through a comprehensive phase diagram in the $(\epsilon_1,\epsilon_2)$ parameter space. Furthermore, under asymmetric diffusive coupling, signal transmission can persist even when the first few oscillators in the chain undergo oscillation death. A central finding of this study is the intrinsic generation of higher harmonics induced by nonlinear interactions, leading to a pronounced reversal of vibrational resonance. This nonlinear redistribution of energy facilitates selective amplification and robust propagation of specific frequency components.

This article presents an electronic experimental validation of discontinuity-induced bifurcations, including period-adding cascades, in a wake-induced dynamical system subjected to soft impacts. A cylindrical structure immersed in a cross-flow undergoes vortex-induced vibrations and interacts with a movable barrier during the upper and lower branches of the response. The structural motion is modelled as a harmonic oscillator, while the fluctuating lift is represented by a Van der Pol oscillator. The soft contact interaction between the structure and the movable barrier is modelled using a bilinear restoring force, resulting in a piecewise-smooth wake-oscillator system. A systematic numerical investigation reveals period-adding cascades of periodic orbits interspersed with chaotic responses as the barrier distance and fluid-flow velocity are varied. The results are experimentally validated using an inductorless electronic analogue of the piecewise-smooth wake model constructed from operational amplifiers and implemented on a printed-circuit board to improve experimental robustness. The route to instability of the periodic and aperiodic responses is further characterized using Floquet multipliers and Lyapunov exponents. A zero-time transverse discontinuity mapping is implemented to account for the evolution of perturbations in the local neighborhood of the discontinuity boundary during stability analysis. The combined numerical, experimental, and stability analyses demonstrate the role of soft impacts in generating complex nonlinear responses in the wake-induced system.

In these lecture notes we review some constructions and results on the relation between topology of the moduli space of stable curves and integrable systems of evolutionary PDEs.

Belief propagation in social networks depends on how trust and distrust jointly shape interpersonal influence. We develop a network model that treats trust and distrust as distinct but statistically coupled quantities. Their joint values determine the net trust and uncertainty of each relationship, and a neighbor can influence an agent only when both threshold conditions are satisfied. We show that the correlation between trust and distrust reshapes the transition between consensus and fragmentation by changing the availability of influential neighbors. Network connectivity also affects this transition, whereas degree heterogeneity has a more limited effect when networks have comparable mean degree. Moreover, the availability of influential neighbors alone does not determine the final consensus: correlations between relational properties and network degree can select agents with different degree compositions, altering the direction and magnitude of collective belief. Finally, we introduce a comfort index that quantifies the expected availability of influential neighbors and analytically links the joint trust-distrust structure to collective dynamics. Our results establish a mechanistic connection between relational structure, effective social influence, and emergent collective belief states.

We establish an exact correspondence between the quasi-Wronskian and quasi-Grammian constructions for the anti-self-dual Yang–Mills (ASDYM) equations with complex \(2\times2\) matrix-valued fields. Starting from vacuum wave functions associated with \(2M\) pairwise distinct nonzero constant spectral parameters, we restrict attention to a nonempty local domain on which the corresponding interpolation matrix is invertible. We show that both constructions are governed by the same uniquely solvable matrix-polynomial interpolation problem. In particular, the quasi-Wronskian Darboux polynomial \(T(\lambda)\) and the quasi-Grammian rational dressing matrix \(W(\lambda)\) satisfy \[ T(\lambda)=q_\xi(\lambda)W(\lambda), \] where \(q_\xi(\lambda)\) is the monic scalar polynomial whose zeros are the prescribed poles of \(W(\lambda)\). This identity yields explicit relations between the corresponding Yang and potential matrices and shows that the two constructions generate identical ASDYM gauge potentials in the same gauge.

We find a robust scenario for the emergence of the Golden Ratio (GR) dynamical universality class, characterized by the dynamical exponent $z \equiv \varphi=\frac{1}{2}(1+\sqrt{5})$. Previously, its emergence required fine-tuning of system parameters. Here, the GR universality class appears generically in a wide class of Markov processes as a consequence of a global, circulant symmetry of the flux Jacobian. This symmetry leads to GR universality in systems where the total number of conservation laws is odd and larger than three. We introduce a multilane exclusion process with Bernoulli stationary measure as a minimal, parameter-free model, which features GR universality for all conserved modes, most of which are doubly degenerate. Our predictions, based on mode-coupling theory generalized to treat degenerate conserved modes, are corroborated by large-scale Monte Carlo simulations.

Phononic frequency combs are equidistant spectral ladders generated when a driven vibrational mode transfers energy to an autoparametrically excited subharmonic mode through anharmonic intermode coupling. In previous studies, the coupling strength governing this energy transfer has typically been treated as a free parameter, extracted from a fabricated device, or adopted from an earlier calculation for a specific mode pair. As a result, whether a given crystal can support a phononic frequency comb, and the drive strength required to generate it, has not been predictable a priori. Here we obtain the complete parameter set from density functional theory. Twisted bilayers of \MoSSe{} and \MoSMoSe{} at the commensurate angle $21.787^{\circ}$ adopt space group $P3$ in a 42-atom \rootseven{} moiré supercell. The second-order force constants provide the mode frequencies; the third-order constants provide the linewidths through the Boltzmann transport equation and hence the quality factors; and the same third-order tensor projected onto the phonon eigenvectors of a chosen mode pair provides the anharmonic couplings $\kt$ and $\kss$. Both bilayers constitute a mode pair within $0.8\%$ of the $2\!:\!1$ resonance with $\omega_1/\omega_2 = 1.9923$ in \MoSSe{} and $1.9943$ in \MoSMoSe{}. By integrating the resulting two-mode equations, both modes generate combs through a discontinuous transition: at $701\,\fth$ in \MoSSe{} the envelope variance rises from $2.1\times10^{3}$ to $1.37\times10^{4}$ and the tooth count of mode 1 from 7 to 16 within a single $5\,\fth$ increment, identifying the onset as subcritical. The spectral entropy peaks before this jump, at $341\,\fth$. The two compounds carry comparable coupling, differing by a factor of $1.2$. However, the comb excitation threshold is set by the quality-factors of signal modes: $Q_2 = 672$ for \MoSSe{} and $Q_2 = 3844$ for \MoSMoSe{}.

We establish that bond Hamiltonian densities, up to local difference terms whose sum vanishes under periodic boundary conditions, lying in the commutant of bond operators, $w_j\pm w_{j+1}$, is a necessary and sufficient condition for a one-dimensional nearest-neighbour Hamiltonian to conserve a uniform or staggered sum of the one-site operators $w_j$. Starting from these local commutants, we show that every such Hamiltonian and its conserved charge can be obtained from monodromy matrices constructed using regular and non-regular members of scalene Yang-Baxter triples. The results hold on periodic chains of length $L\geq3$, with even length required in the staggered case. The results do not require translation invariance, and hence the interaction on each bond can be chosen independently from the full corresponding commutant. Our representation-independent construction applies in arbitrary finite local dimension and exhausts the Hamiltonians with either prescribed additive symmetry, without requiring ordinary Yang-Baxter integrability. We illustrate the construction with spin-$\tfrac12$ and spin-1 systems and discuss its relation to the double-commutant theorem. Classifying the one-site generators $w$ and their associated bond commutants provides a route to classifying such Hamiltonians with additive conserved charges.

Transport networks are key to living and engineered systems. How can transport networks adapt efficiently using only local rules, without a global objective or specific boundary conditions? Inspired by observations in biological transport networks, we adapt networks by subsequently pruning edges by relative resistance, thus removing edges whose resistance is higher than the resistance for routing flow through the rest of the network. We find that adaptation by such a local, boundary-condition-free measure of redundancy, set by topology alone, yields sparse, loop-rich, low-dissipation networks.

We describe a family of two-dimensional metric deformations that preserve integrability. The seeds are (classically) conformal integrable field theories and the deformations are directly related to different classes of auxiliary field sigma models. We introduce a master action connecting deformations by Lie-algebra-valued auxiliary fields, the so-called $\nu$-frame, to deformations by two Lorentz-charged scalars. These auxiliary scalar fields can be identified as Beltrami differentials of an auxiliary two-dimensional metric, providing a direct connection with the metric approach to $T\bar{T}$-like deformations. For Weyl-invariant seed matter theories, the auxiliary metric sector is described by a reduced gravitational potential. We show that the resulting framework preserves classical Lax integrability whenever the seed matter theory admits a Lax representation of its equations of motion on a generic curved two-dimensional spacetime. Finally, within the Lorentz-invariant family and the zweibein deformations of conformal matter theories considered here, we identify $T\bar{T}$ as the only deformation that can be interpreted as a field-dependent coordinate transformation.

Quantum reservoir computing and its special case, quantum extreme learning, are powerful tools for predicting classical discrete time-series with quantum devices. Forecasting continuous signals, however, requires careful discretization as oversampling artificially boosts short-term accuracy but fails over realistic intervals, while undersampling captures too few features and yields poor predictions. Here, we experimentally generate four continuous time-series signals from a chaotic electronic circuit for forecasting with quantum extreme learning. Since simplified circuit models do not fully capture real behaviour, it is unclear whether algorithms that predict simulated time-series also work for experimental signals. We address discretization via two approaches: (i) sampling local peaks/troughs; and (ii) uniform sampling at the autocorrelation timescale. The former captures resonant behaviour, while the latter fully captures chaotic features as well as allowing the reconstruction of chaotic attractors. Our algorithm achieves high forecasting accuracy for both simulated and experimental signals, demonstrating its power for continuous signal processing on near-term quantum simulators, and remains robust against moderate decoherence.

We introduce a nonconservative nonlinear Schrödinger equation (NLSE) which includes additional nonlinear terms accounting for the pseudo-stimulated-Raman-scattering (pseudo-SRS) effect and cubic loss. An essentially novel feature of the model is the linear anti-diffusion gain, which leads to the obvious blow-up in the infinite or spatially periodic system. We consider the absorptive boundary conditions, which suppress the blow-up driven by effective convective instability in the setup. The NSLE model is derived, in the extended adiabatic approximation, from the underlying Zakharov system, which includes the viscosity term acting on its low-frequency component and the anti-diffusion gain in the high-frequency (HF) one. The latter term appears if the expansion, with respect to the deviation of the wavenumber, is performed not around the minimum-loss point (as in the usual situation), but around a maximum-loss point. Dynamics of moving solitons in the model is addressed by means of analytical and numerical methods. The sech approximation for the soliton predicts it as a stable fixed point (FP) of the system of balance equations for the HF norm and wave momentum, derived in the moving reference frame. The FP attraction basin is identified too. The predictions are corroborated by direct simulations of the full NSLE with the absorptive boundary conditions.

Nonlinear optics expands the polarization in powers of the drive and retains only the lowest-order terms. This truncation is safe at optical frequencies, where the convergence field lies far above the damage threshold. In the terahertz regime, phonon polaritons enhance the nonlinear susceptibilities by orders of magnitude, lowering the convergence field into the sub-damage range and opening a window in which the expansion fails while the crystal remains recoverable. We term this window silent breakdown. At resonance, the nearest singularities of the driven lattice mode are a complex-conjugate pair of branch points below the multistability threshold, so the response remains smooth and single-valued even though the $\chi^{\left( n \right)}$ description fails. The convergence field scales as $Q^{-3/2}$, so sharpening a resonance enhances the nonlinearity but narrows the range over which the expansion applies. It follows from the mode shift $S$ alone, $E_{\mathrm{conv}}=E_{\mathrm{ref}}(3\sqrt{3}QS)^{-1/2}$, giving 2$\sim$9 MV/cm for $\mathrm{LiNbO}_{3}$. Since resonant enhancement is central to terahertz nonlinear optics, this boundary defines the field range over which susceptibility-based descriptions are quantitatively reliable.

The staggered order parameter of the gapped XYZ spin-$1/2$ chain follows from the eight-vertex solution, but in elliptic variables that fix the Hamiltonian couplings only implicitly. We derive a closed form for the coefficient governing the leading quadratic response of the spontaneous staggered magnetization to the XY anisotropy at fixed longitudinal anisotropy in the gapped antiferromagnetic regime. Starting from the Baxter–Kelland–Jimbo–Miwa–Nakayashiki polarization formula, we show that at fixed longitudinal anisotropy both the modular nome and the crossing-parameter variable of the formula move with the XY anisotropy, and that the motion of the crossing-parameter variable reverses the sign of the expansion at fixed elliptic variables. The coefficient is a rational term plus a Lambert series. It is strictly negative and bounded above by a rational function of the longitudinal anisotropy, to which it is asymptotic in the Ising limit. At the isotropic point it diverges as the inverse five-halves power of the distance to that point, and since the Lambert sum has an asymptotic expansion with exponentially small remainder, the subleading terms of the divergence also follow in closed form. When the XY anisotropy vanishes as the five-fourths power of the distance to the isotropic point, the ratio of the perturbed to the unperturbed order parameter tends to a Gaussian in the scaling variable, and the quadratic truncation is controlled when that variable is small. A fourth-order perturbative calculation in the spin chain, with no input from the eight-vertex solution, reproduces the leading strong-coupling behaviour, and infinite-DMRG calculations at four anisotropies in the gapped regime are consistent with the closed form.

We construct a family of self-dual $(1+1)$-dimensional $\mathbb{CP}^1$ sigma models whose potentials $V$ are determined from pre-potentials $U$ by Platonic geometry. Using the round metric on $\mathbb{S}^2\simeq\mathbb{CP}^1$ and the generalized self-duality formalism, we build symmetry-invariant pre-potentials $U$ from the unit vertices of each Platonic solid intersecting the unit circumsphere. The regular-lattice sites associated with vertex, face-center, and edge-midpoint directions are critical points of $U$ and degenerate vacua of the scalar potential $V$. Dual solids share the same regular lattice but generate different pre-potentials, charges, potentials, and metric-gradient flows. Numerical integration reveals distinct self-dual trajectories and energy-density profiles, including multi-peak structures for paths passing near saddle points, while solutions with the same endpoint difference $\Delta U$ have the same total energy. The positive maxima of $V$ are also organized into symmetry orbits. This construction provides a geometric prescription for controlling the vacuum structure and self-dual sectors of $\mathbb{CP}^1$ models with a potential.

In deep learning, approximation theory motivates increasing representation size. We ask whether this benefit extends to dynamics learning through autoregressive prediction. We analyze the learned time evolution through the eigenstructure of Koopman operators, using relative residuals to detect spurious eigenpairs arising even as one-step error falls. For bounded Koopman operators, we show that minimal residuals over learned dictionary spaces converge pointwise to their full-space counterparts as these spaces approximate the observable space in $L^2$. Our hypothesis is that Koopman spectral reliability helps explain how consistently rollout error decreases with increasing dimension. We compare two models of a shared Koopman autoencoder trained alternately for reconstruction and latent evolution, using latent-prediction loss (one-step prediction errors in latent coordinates) or spectral-residual loss (relative residuals of candidate eigenpairs). Across six chaotic systems, both models reduced median windowed rollout error from smallest to largest dimension. The spectral-residual model achieved lower medians than the latent-prediction model for all systems and dimensions, and its median fell by a larger factor in every system. Its median decreased monotonically with dimension in four systems, against one for latent prediction. Against four baseline families, its mean valid prediction times were nearly always longer. At the largest dimension under two-stage training, we compared eigenvalue positions with each learned dictionary's residual contours. Spectral-residual eigenvalues concentrated in low-residual regions, whereas latent-prediction eigenvalues also appeared in high-residual regions, consistent with the hypothesis.

A lazy cellular automaton $\tau:A^G\to A^G$ is determined by a finite neighborhood $S\subseteq G$ containing the group identity $e$, a pattern $p\in A^S$, and a writing symbol $a\in A\setminus\{p(e)\}$: its local rule is equal to the projection to $e$ except when it reads the pattern $p$, in which case it writes $a$. For a nonempty subset $T$ of the fiber $S_a:= p^{-1}(a)$, we introduce compatibility conditions on $p$ along the right translates $St$, $t\in T$. We show that, over every group, the existence of a compatible set is necessary for $\tau$ to be non-idempotent, and that it is also sufficient whenever $st = ts$ for all $t \in S_a$, $s \in S_{p(e)}$. In particular, compatibility characterizes non-idempotence over abelian groups, which reduces deciding idempotence to a search over subsets of $S_a$. Finally, for every dihedral group $D_n$ with $n\geq 4$, we construct an idempotent lazy cellular automaton that admits a compatible set, showing that the commutation hypothesis cannot simply be dropped.

The evolution of matter-wave vector solitons in two dimensions under nonlinearity management is studied. The averaged over strong and rapid modulations in time of the inter-species interactions vector Gross-Pitaevskii equation is derived. The averaging gives the appearance of the effective nonlinear quantum pressure depending on the population of the other component. Using this system of equations, the dynamical stabilization of the two-dimensional vector matter-wave solitons under the management of the mean-field nonlinearity is investigated. It is shown that the nonlinear quantum pressure arrests the collapse in the averaged system, giving rise to stable vector solitons. Full numerical simulations of the original, time-dependent system of Gross-Pitaevskii equations confirm that the management strongly prolongs the lifetime of these solitons compared to the unmanaged case.

We report the experimental and numerical demonstration of stimulated Brillouin scattering-induced stabilization of Kerr-driven modulation instability in fiber Fabry–Perot resonators. We show that Brillouin scattering shapes the modulation instability spectrum into a stable multi–free-spectral-range cascade that is otherwise unstable in the absence of Brillouin effects. We further identify that this stabilization originates from the real part of the Brillouin response, leading to an asymmetric Kerr-Brillouin interaction governed by the relative position of the Brillouin frequency shift with respect to the cavity resonances. These results establish stimulated Brillouin scattering as an additional degree of freedom for controlling nonlinear dynamics in Kerr resonators.

The role of adaptive coupling in controlling collective noise-induced dynamics is demonstrated on an example of stochastic resonance in an ensemble of coupled overdamped bistable oscillators. Specifically, it is found that tuning the adaptive coupling parameters allows one to either enhance or suppress stochastic resonance and to shift the optimal noise intensity that yields the most regular stochastic oscillations. All the revealed effects are studied in numerical simulations for two kinds of adaptive coupling models. The first one is a simplified, phenomenological model designed such that the coupling becomes stronger for oscillators with larger differences between their dynamical variables (two configurations for local interaction and one option for global coupling are under consideration). The second one is a local memristive coupling model that provides a physically motivated realization of adaptive interactions with similar properties. In all the considered cases, adaptive coupling is shown to be a flexible stochastic resonance control tool, both in the absence and in the presence of the static coupling component.

The second-order Kuramoto model captures rotor dynamics relevant to synchronization in power grids. In actual power grids, violation of angle-stability limits can trigger protective actions that are not included in the model, restricting its validity to trajectories that remain within these limits. We therefore define the secured basin as the region of disturbance space comprising disturbances whose trajectories remain within the angle-stability limits throughout the transient. We then characterize the secured basin through energy barriers, quantifying their characteristic energy scale and heterogeneity across perturbation directions. Across seven power-grid models, this characterization revealed angle-stability vulnerabilities overlooked when stability is assessed solely by the final synchronization state. It also showed that the two energy-barrier measures characterizing the secured basin exhibit distinct associations with structural connectivity and dynamical parameters. This characterization further offers practical advantages, as energy thresholds derived from state-space perturbations remain effective in distinguishing secured and unsecured outcomes under short-duration power disturbances and can be estimated efficiently by concentrating simulations near the secured-basin boundary. Overall, the secured-basin framework provides a useful basis for distinguishing node-level structural and dynamical influences on transient stability in oscillator networks with synchronization constraints.

Theories of how the brain processes information and returns intelligent outputs are numerous and often difficult to conclusively test. Here, we consider an approach grounded in informatics and algorithmic thermodynamics for how neural systems respond to different patterns of information in different ways, with a focus on the relationship between the information entropy of a signal and a new proposed quantity we term information enthalpy, which represents the internal resource of structured information within a system available to be used for predictions and other forms of work. To evaluate the capacity of the external signals to increase information enthalpy in the system, we introduce the information enthalpy potential (IEP) as a defined metric. We propose and implement a method for quantifying the amount of potential information enthalpy - the IEP - in a given signal across a range of example signals. We offer a conjecture of how information enthalpy may be treated by neural systems within a biological framework before showing how it may integrate and offer falsifiability to existing approaches such as the Free Energy Principle. This framework is also positioned in light of the role of neural criticality. Finally, we postulate a testable and falsifiable framework where the distinct mechanisms within each driver interact through a described n-body-inspired model to govern the "motion" of a neural system through a high dimensional state-space. Through these processes, it is proposed that these features form the most fundamental basis of the neural drivers that give rise to the complexity of adaptive behaviors that are commonly called: intelligence.

A challenge in nonlinear filtering is to incorporate nonlinear effects into approximations of the conditional mean while retaining a finite-dimensional recursive algorithm. We derive a second-order asymptotic expansion of the filtering mean with respect to the system-noise intensity. We apply this expansion to the Lorenz–63 model, a prototypical three-dimensional chaotic dynamical system. The first-order approximation is computed using a Kalman–Bucy filter for the system linearized around the deterministic reference trajectory. The second-order correction incorporates the Hessian of the Lorenz–63 drift and is computed through a closed recursive system with 3+9+18=30 independent scalar components. The full system, including the reference trajectory and the first-order mean and covariance, has 42 independent scalar components. To support stable tracking of chaotic trajectories over long time intervals, we introduce recentering of the expansion and redistribution of Gaussian components between expansion orders. In the numerical experiments, we compare the proposed algorithm with an ensemble transform Kalman filter (ETKF) as a practical benchmark and a particle filter (PF) as a numerical reference. Under the full-observation settings, the proposed algorithm achieves a lower root-mean-square error relative to the truth and a smaller distance to the 10,000-particle PF reference than the 20-member ETKF. These results suggest that the proposed second-order extension of Kalman filtering has the potential to balance filtering accuracy and computational cost.

The purpose of this work is to introduce and discuss a general framework for performing a linear instability analysis on multiplex networks, in the absence of inter-layer diffusion, with the specific motivation of detecting the Turing instability. The formalism itself is presented in full generality for $m$-unknowns, each of which arise as solutions of non-autonomous reaction-diffusion equations, on $n$-node temporal multiplex networks. In addition, we identify a class of multiplex networks for which the Laplacian matrices commute. In this setting analytical progress can be made and the explicit inequalities, governing instability, are derived. In order to explore our formalism we apply it to the specific example of a two-species predator-prey model on archipelagos. Here nodes are interpreted as islands and edges correspond to the transportation network for each species. Moreover, the non-autonomous setting is used to model a variety of natural phenomena.

Complex Network (CN) classification requires high-level structural characterizations that are both scale-invariant and computationally efficient. Methods based on Life-Like Network Automata (LLNA) offer an interesting way to extract network descriptors by leveraging emergent temporal patterns without requiring provided features, but their efficacy is bottlenecked by a high-cost combinatorial optimization problem: the selection of the automaton transition rule. While current literature relies on exhaustive searches that are unfeasible for large-scale applications, this work reveals that the rule space is fundamentally structured by a property we term “jaggedness”, that quantifies the resemblance of a LLNA transition function with a sawtooth shape. We demonstrate that this metric acts as a theoretical proxy for chaoticity and sensitivity – properties essential for generating discriminative dynamic behaviors among network categories. Moreover, we introduce a heuristic search strategy that uses jaggedness to guide the rule selection. Experimental results show that our approach achieves classification accuracies within 5% of the global optimum while reducing computational overhead by 90% compared to exhaustive approach. Our findings provide a novel, efficient, framework for optimizing automata-based methods for pattern recognition.

We study the properties of matrix coupled oscillators in ring networks and show that the system supports several families of synchronized clustered solutions. One of these families is characterized by consecutive oscillators belonging to distinct clusters, called alternate ordering, with two classes of solutions. The first consists of one or two static clusters, that lock on phases that depend on the coupling parameters, despite the non-zero natural oscillation frequency. The second one is comprised of clusters that rotate rigidly. The number of clusters depends on the neighbor connectivity. We also investigated sequential ordered clusters and showed that the constraints imposed by the interface oscillators imposes additional restrictions to equilibrium configurations. Although some of these solutions are stable, their basins of attraction depend on model parameters: when the system is initialized with random phases, the oscillators tend to organize themselves into dynamical clusters whose interfaces move along the ring. The oscillators within each cluster remain close to one of the stable phase configurations, while particles are exchanged at the interfaces. When interfaces moving in opposite directions collide, the corresponding clusters annihilate, until a state with interfaces moving in a common direction is reached.