Papers

In this work, we study finite-time effects in ultracold atomic systems by considering time-dependent modulations with variable waveforms and durations. These two characteristics can be controlled by adjusting only a single parameter. For arbitrarily short pulses, our model recovers the paradigmatic kicked rotor while maintaining the impulse transmitted per period and unit amplitude constant. Furthermore, we demonstrate that finite-time effects have a profound impact on dynamical localization, a result that cannot be captured by the {\delta}-kicked-rotor model. Through a detailed analysis of the effects of different modulation amplitudes, periods, and waveforms, we identify the conditions for which dynamical localization is significantly enhanced. We show that the strength of dynamical localization increases sharply as the system approaches the {\delta}-kicked-rotor limiting case. Moreover, we establish the existence of an optimal value of the period that maximizes dynamical localization for given values of the amplitude and shape parameter.

We prove that a scalar tau-symmetric Hamiltonian deformation of the Riemann hierarchy in generalized standard form is uniquely determined by the nonzero coefficient of $u_x^2$ and the coefficients of $u_{xx}^g$, $g\ge2$, in its first Hamiltonian density. This determines the original hierarchy up to normal Miura transformations. Combined with the construction of Hodge-class double ramification hierarchies by Buryak–Rossi, this establishes Hodge universality at nonzero dispersion.

We construct the first fully spatially local fault-tolerant quantum computer based on topological codes in fewer than four spatial dimensions. Our construction is a two-dimensional architecture that uses only geometrically local quantum and classical operations, bounded-speed classical communication and computation, and a constant density of quantum and classical resources. The core component is a new time-translation-invariant cellular-automaton decoder for the surface code. This decoder preserves logical information for a time stretched-exponential in the code distance and operates continuously during state injection, stabilizer-state preparation, lattice surgery, and transversal readout. We also prove that every translation-invariant topological Pauli stabilizer code is locally decodable under phenomenological noise.

We study current fluctuations in a temporally random integrable quantum circuit. Commutativity reduces every drive history exactly to its layer composition, turning annealed fluctuations into a competition between current gain and the large-deviation cost of rare compositions. For each fixed composition, homogeneous thermodynamic Bethe ansatz dressing supplemented by ballistic fluctuation theory yields the conditional current statistics. Their annealed large-deviation contraction predicts a first-order switch between two dominant history classes, terminating on a line of regular cusp endpoints. Finite-time analysis shows how the switch is rounded. Thus temporal randomness can act as an emergent order-parameter-like coordinate in trajectory space.

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.

We prove the existence of two kinds of self-similar vector solitons in an inhomogeneous optical fiber medium, where light propagation is governed by a pair of coupled higher-order nonlinear Schrodinger equations with varying second- and third-order dispersions, self- and cross-phase modulation non linearities, self-steepening, and linear gain/loss effects. The newly found self-similar waves comprise bright-W-shaped and kink-antikink waveforms with nonvanishing amplitudes. As a practical exam ple, we discuss the propagation dynamics of these soliton structures in a periodically distributed fiber system as well as an exponential dispersion-decreasing fiber. The results demonstrate that the parameter functions of gain/loss and third-order dispersion serve as a key factor in determining the nonlinear dynamics of self-similar vector solitons. In particular, we find that precise control over the shape and dynamic evolution of self-similar pulses can be achieved through a proper choice of the distributed third-order dispersion parameter, while the gain/loss coefficient controls their intensity.

The characteristic intersection angle of the desiccation crack relaxes from near \ang{90} toward \ang{120} under repeated drying–wetting cycles. However, the theoretical understanding of this relaxation is insufficient, especially the modeling of the drying–wetting cycles. Here we introduce a phase-field model of desiccation fracture, extending the model proposed in previous studies by adding crack healing and a scar effect left by past cracks. By repeating drying–wetting cycles in a finite element simulation, we find that the angle distribution develops a growing peak near \ang{120} as the cycle number increases, consistent with experiments. The standard deviation of the intersection angle from \ang{120} relaxes exponentially with a characteristic time of about 2.85 cycles. These results are consistent with experiments, except that the characteristic time is slightly smaller than the experimental value. Crack energy dominates the total energy and also relaxes exponentially with nearly the same characteristic cycle as the angle relaxation. This decay is driven mainly by a shortening of the effective crack length rather than a change in effective fracture toughness.

We develop a framework for offline full-window branch discovery, optionally followed by online continuation with an ensemble Kalman filter (EnKF). Three mechanisms drive the branch search: adjoint path-kernel (APK) differentiation balances kernel differentiation and correction-stabilized path perturbation, shifting the optimization from exploration to exploitation; an optimized Gaussian initial law broadens the search over initial-state basins; and loss-weighted mixing across independent runs recombines successful path components. We may then select an interior state using a local loss and continue online with an EnKF. In 40-dimensional Lorenz-96 experiments, the mean offline path RMSE of APK is 4.3 times smaller than that of population weak-$\mathrm{4D\text{-}Var}_x$. The resulting APK-EnKF method has a mean online RMSE 64 times smaller than that of ordinary EnKF.

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.

Internal waves often propagate in the presence of currents. Internal waves and currents are widespread in the world's oceans. The equatorial internal waves and the equatorial undercurrent (EUC) are perhaps the most representative illustration of this phenomenon. The current profile in this study is a quadratic function of depth, which is in agreement with the known results of the EUC. With this assumption the integrable nonlinear Gardner's equation is derived as a model for the wave motion of the thermocline, separating the two layers of slightly different densities. In addition, we derive the generalised Gardner's equation in the case of variable bottom. We demonstrate that the coefficients of the equation are very sensitive with respect to the parameters of the EUC.

Interactions of two solitary waves with an up and a down orientation in the modified Korteweg-de Vries equation are shown to produce rogue-like waves. For waves that asymptotically vanish, the maximum ratio between the height of the interaction profile and the height of the tallest incoming wave is 2.41 when the waves have approximately equal speeds, and this ratio decreases to 2 when the speed ratio is 1.5. For waves that approach a non-zero constant at infinity, the same ratio reaches a maximum of 2.65 when the speed ratio of the waves is 6.32.

Complex patterns in physical and biological systems often emerge through slow collective dynamics governed by a small number of key variables. In nonlinear optical resonators, dissipative Kerr solitons provide an important example, where interactions between well-separated solitons can evolve over timescales far longer than the characteristic loss and gain timescales. Direct numerical simulation of these dynamics is challenging because stiffness forces conventional methods to resolve many rapidly damped degrees-of-freedom with very small time steps. We present a numerical scheme, the synergetic method, that eliminates these rapidly damped degrees-of-freedom and retains the slowly evolving modes, enabling time steps many orders of magnitude larger than those used in conventional approaches. Applied to soliton molecules in driven Kerr cavities, the method achieves speedups of $10^3$ to $10^5$ while capturing dynamics on laboratory timescales. We use it to model the full interaction dynamics of a three-soliton molecule and the evolution of an eight-soliton molecule. The approach provides an efficient framework for studying slow pattern formation in nonlinear systems with widely separated timescales.

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.

The potential of crossing climate tipping points (TP) has reached the attention of many researchers and the general public. On the one hand, the basis for this concern is strengthening, with simulations showing that abrupt transitions might occur even for moderate emission scenarios. On the other hand, our understanding of what constitutes such transitions mathematically is becoming more nuanced. This leads to challenges for the fidelity of early-warning signals (EWS), which accompany bifurcations in systems that closely track a slowly changing steady state. Different kinds of out-of-equilibrium dynamics in a rapidly changing climate, as well as chaos, blur the critical tipping threshold and imply limits in predictability. Using EWS to detect loss of local stability from data requires careful evaluation in the high-dimensional system, and may have limited predictive power in a highly multistable climate, where there is more than a single, well-known alternative state. This calls for new methods using non-equilibrium statistical mechanics and dynamical systems to efficiently probe the global stability properties of climate models and observations.

Vortex light fields carry orbital angular momentum and have attracted significant attention because of their wide applications in light field manipulations, optical communications, and quantum information processing. In nonlinear media, the balance between diffraction and self-action enables the formation of vortex solitons. However, achieving the stability of vortex solitons remains challenging due to radial and azimuthal modulation instabilities. Here, we report stable vortex solitons and vortex-soliton arrays in disclination-fractal configurations with different rotational symmetries, constructed by applying disclination operations to fractal lattice structures. The stable vortex-soliton arrays—composed of several vortex solitons—can exist along domain walls, resulting from the disclination operation applied to the topologically trivial phase. Their stability is verified through both linear stability analysis and direct simulations of perturbed propagation. For comparison, vortex-soliton arrays in conventional disclination configurations are found to be completely unstable, demonstrating the significance of the fractal configuration in stabilizing these nonlinear states. These findings reveal a new nonlinear excitation mechanism arising from the interplay between fractal geometry and disclination defects, thereby providing theoretical underpinnings for both the improved understanding of multi-field excitation phenomena in complex geometric systems and opening new avenues for the design of photonic devices based on fractal disclination structures.

Data-driven machine learning has established a robust foundation for reconstructing nonlinear dynamical systems from observations, primarily for the purposes of forecasting and control. However, most existing efforts focus on recovering specific observed dynamics rather than the generative synthesis of new ones. Inspired by image fusion and style transfer, we introduce a neural network framework termed Neural Dynamical Transfer Learning (NDTL) to create new systems with prescribed dynamics from pairs of parent nonlinear dynamical systems. By computing fundamental dynamical signatures, including the intrinsic dimension, the Kaplan-Yorke dimension, the invariant measure statistics, and the Lyapunov spectrum, we demonstrate that NDTL preserves key features inherited from the parent models while simultaneously generating novel dynamics. Beyond these validation examples, NDTL induces a criterion for dynamics classification, creates stable oscillatory coexistence in the Hastings-Powell food chain model, produces interpretable epidemiological models, and provides a chaotic source for image encryption.

Symbols can provide compact representations of complex sensory information. Mathematically, chaotic attractors can provide a basis for such representations, with object identity encoded in the topological organization of the attractor. In three-dimensional phase space, this organization can be characterized by the linking matrix of unstable periodic orbits, which serves as a topological fingerprint. Because the linking matrix has no closed-form dependence on the learnable parameters, we learn the chaotic series generating the target attractors rather than optimizing the invariant directly; the reconstructed attractor then determines its unstable periodic orbits and linking matrix through its topology. In this paper, we propose a third-order tensor that maps inputs to connection matrices that generate the corresponding chaotic series. Exploiting the duality between a vector and its sequential representation as a series, we lift inputs into a higher-dimensional space and reorder their components, allowing the input to be presented sequentially rather than simultaneously. We first demonstrate that this representation can discriminate between classes in the \href{https://archive.ics.uci.edu/dataset/151/connectionist+bench+sonar+mines+vs+rocks}{Sonar} dataset. We then examine perceptual constancy using the \href{https://www.csc.kth.se/cvap/databases/kth-tips/index.html}{KTH-TIPS2-a} dataset, where variations in color and texture within a class are mapped to distinct chaotic series associated with the same topological attractor class. The results demonstrate that the proposed tensor-based mapping can preserve attractor class while accommodating variations in the sensory realization of an object.

For every right half we determine the unique left half whose Rule 30 central trace is constant: an alternating tail selected by the leading one of the right half when the initial center is 0, and a single universal checkerboard when it is 1. Every nonzero member of either fiber carries infinitely many ones, so the zero row is the only finite configuration with a constant trace. For support radius w the sharp maximum constant-prefix length is 2*ceil(w/2)+1 when the initial center is 0 and 2*floor(w/2)+2 when it is 1, so the maximum over both is w+2, attained by exactly 2^w configurations for even w and 2^w - 1 for odd w. Hence no column of a nonzero finite Rule 30 orbit is eventually constant.

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).

Quantum reservoir computing (QRC) is a machine learning approach which employs the internal dynamics of a physical system (the reservoir) to encode and process information. In this work, we explore the use of dual-unitary circuits in a brickwork architecture as a platform for QRC, well suited to current noisy intermediate-scale quantum devices. Dual-unitary circuits present both practical and conceptual advantages. Our results indicate that, under appropriate conditions, dual-unitarity can lead to an enhanced regime of operation: we numerically verify that it improves memory effects and nonlinear processing, and shields against finite-shot noise, mitigating exponential concentration. Moreover, dual unitarity offers an intuitive picture of how operator dynamics gives rise to memory and nonlinear processing in circuit-based reservoirs.

We present an analytical, statistical solution to the binary-binary (2+2) outcome of the chaotic non-hierarchical four-body problem. The solution is based on the density-of-states formulation pioneered by J. J. Monaghan. The method skips the computationally expensive integration of the equations of motion, and instead samples the outcome from the chaotic phase-space, subject to conservation of energy, momentum, and angular momentum. From the joint distribution, we extract marginal distributions of several key parameters using Monte-Carlo integration, and numerically verify them by comparing to an identical ensemble of scattering experiments produced by the FEWBODY code. From the comparison, we identify a regime not represented by the density-of-states formulation: the hard binary regime, where one binary is much harder then the four-body energy scale, and the system acts as an effective three-body system. We hypothesize that at this regime the system's probability distribution spreads over an effective reduced three-body chaotic phase space. The process of transfer from four-body to three-body phase-space is still not understood, and represents the next natural extension of density-of-states methods.

Nonlinear wave equations exhibit rich interplay among dispersion, nonlinearity, coupling, dissipation, and external forcing, producing diverse coherent and complex wave structures. Although exact travelling-wave solutions provide analytical benchmarks, their construction alone does not reveal underlying phase-space, bifurcations, stability, or responses to perturbations. This review presents a reduction-based methodological framework connecting nonlinear partial differential equations(PDEs) to reduced dynamical systems, invariant phase-space structures, exact waveform reconstruction, stability analysis, and verification in PDEs. Its applicability and limitations are examined across Schrödinger-type, coupled, nonparaxial, dissipative, magnetic, shallow-water, and fractional wave equations. Particular emphasis is placed on correspondence between analytical waveforms and invariant orbits. Equilibria, periodic trajectories, homoclinic orbits, and heteroclinic connections geometrically represent constant, periodic, localized, and front-like structures, respectively. This review distinguishes existence and stability of invariant structures from onset of chaos, emphasizing complementary diagnostics rather than reliance on phase-portraits or finite-time indicators alone. Major methodological gaps include incomplete parameter-space characterization, weak correspondence between reduced models and full-PDE dynamics, ambiguities in generalized and fractional formulations, limited robustness analysis, inadequate numerical reproducibility, and insufficient links to experimental observables. The framework prioritizes physical admissibility, stability, robustness, and predictive relevance over generation of additional formal solutions. It supports reliable nonlinear wave prediction, stability assessment, control, and system design in fluid, optical, plasma, and other nonlinear physical systems.

We consider a minimal pressure-based model of heterogeneous cell populations consisting of proliferative and non-proliferative cells with different mobilities. The model is formulated as a system of reaction–cross–diffusion equations describing the spatio-temporal dynamics of the cell densities. The model is known to admit one-dimensional travelling wave solutions with strictly segregated components: non-proliferative cells occupy a finite region at the leading edge, while proliferative cells remain at the rear. However, the speed, parameter dependence, and stability of these waves remain poorly understood. In this work, we derive an almost explicit variational bound on the wave speed by reformulating the problem as a free-boundary problem for a generalised porous–Fisher equation. The estimates we obtain apply to general pressure laws and growth kinetics, agree closely with the results of numerical simulations, and become sharp in the incompressible limit, where we formally recover a fully explicit characterisation of the wave speed. We then analyse the stability of the waves to show that segregated waves are stable only when non-proliferative cells are more mobile than proliferative cells. Finally, motivated by numerical observations of finger-like protrusions, we investigate the stability of incompressible segregated circular waves through asymptotic shape-perturbation analysis. This yields explicit expressions for the pressure, interface velocity, and growth rates of angular modes, thereby making evident the destabilisation mechanisms that may lead to the emergence of fingering instability. Interestingly, we find that, in contrast with the one-dimensional case, the stability of such circular waves is not determined solely by the relative value of the mobility coefficients, and thus instabilities may arise irrespective of which cell type has the larger mobility.

One of the central challenges in evolutionary dynamics is understanding why some species combinations persist while others disappear. Although cyclic-interaction models have provided fundamental insights into biodiversity maintenance, much less is known about how hierarchical competitive interactions shape long-term community organization. Here, we investigate a hierarchical extension of the May-Leonard model, in which species interact through a directed predation chain while undergoing reproduction and mortality. Combining mean-field analysis with Monte Carlo simulations, we show that the fully coexisting state is generically unstable, causing the dynamics to evolve toward lower-dimensional coexistence states. The simulations further reveal stochastic extinctions dominating small populations with the dynamics progressively approaching the mean-field predictions as the system size increases. Rather than permitting arbitrary species combinations, the hierarchical-interaction structure dynamically constrains coexistence by selecting only specific subsets of species for long-term persistence. We show that these admissible coexistence states have a natural graph-theoretic interpretation as independent sets in the hierarchical interaction network, thereby providing general constraints on coexistence in hierarchical communities. Together, these results establish a theoretical framework linking hierarchical interactions, dynamical selection, graph topology, and biodiversity organization, extending the classical May-Leonard model beyond cyclic competition.

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.

We consider Thouless pumping of two-dimensional quadratic solitons composed from coherently interacting fundamental frequency (FF) and second harmonic (SH) components propagating in a periodic $\chi^{(2)}$ material. The pumping is induced by two mutually sliding two-dimensional lattices defined by shallow transverse and longitudinal periodic refractive index modulation. Focusing on solitons in the semi-infinite gap, we find three distinct pumping scenarios: the absence of transport for small-amplitude solitons, non-quantized transport in a transient regime at intermediate amplitudes, and stable quantized transport for solitons with relatively large amplitudes. Different directions of the sliding velocity were investigated resulting in different trajectories of soliton center. The transition to the regime of quantized transport is found to depend strongly on phase mismatch between FF and SH waves that also influences stability properties of two-dimensional $\chi^{(2)}$ lattice solitons. We also show that pumping with longitudinal periods in the x and y directions, whose ratio approximates an irrational number, induces quantized transport whose direction rapidly converges, with increase of the accuracy of the approximation, to a limiting pumping direction determined by this irrational number and corresponding to truly incommensurate longitudinal periods.

Recently, a soliton mechanism for room-temperature superfluorescence in thin perovskite films has been proposed, with a fundamental soliton predicted to remain stable under LO phonon–exciton interactions. At the same time, superlattice architectures offer a route to enhancing superfluorescence in perovskites. Motivated by recent observations of room-temperature superfluorescence in periodic superlattices of quasi-2D metal-halide perovskites, we extend the 2D nonlocal nonlinear Schrödinger equation describing Wannier exciton–LO phonon interactions to superlattice structures, obtaining a 3D nonlocal nonlinear Schrödinger equation. We show that interlayer tunnelling gives rise to breather dynamics corresponding to a stable fundamental soliton in mixed coordinate–momentum space, with the coordinate parallel to the layers and the momentum perpendicular to them. The breather dynamics originate from miniband formation, which induces a momentum-dependent phase modulation of the soliton. In the absence of interlayer tunnelling, the breather dynamics disappear and the soliton becomes stationary. These results establish a direct connection between interlayer tunnelling, miniband formation and soliton dynamics, suggesting that breather behavior can provide a signature of interlayer tunnelling in quasi-2D perovskite superlattices.

Artificial feedback spin masers exhibit rich nonlinear dynamics, yet a systematic framework for understanding and controlling their behaviors remains elusive. In this paper, we consider the polarity and geometric configuration of the feedback field as important control parameters for studying the nonlinear dynamics of spin masers. By cooperatively tuning the strength and phase delay of feedback field, we investigate the spin dynamics at various configurations of the feedback field. Periodic, quasiperiodic and chaotic spin oscillations are uncovered. Specifically, a unique spin oscillation mode akin to the time crystal effects emerges within a certain parameter regime. This work presents a different viewpoint understanding the nonlinear dynamics of spin masers including the time crystal.

Quantized vortices are ubiquitous in physics, spanning superconductivity, astrophysics, superfluid condensed matter systems, and nonlinear optics. Yet embedding vorticity into topologically protected nonlinear states has remained a major challenge, with all previously observed corner solitons in higher-order topological insulators (HOTIs) exhibiting only trivial phase distributions. Here, we report on the first realization of stable topological corner vortex solitons in a photonic fractal HOTI. Using an array of laser-written waveguides in the shape of Sierpiński gasket with a controllable distortion, we design linear topological vortex modes, from which nonlinear corner vortex solitons bifurcate. Moreover, we demonstrate that these solitons exhibit exceptional robustness across a broad power range and, unlike vortex solitons in topologically trivial lattices, form without a power threshold. Our results introduce the angular momentum degree of freedom into the physics of topological corner modes, opening prospects for topologically protected vortex-based photonics.

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.