Discovering governing equations from data is crucial for understanding complex systems in many diverse fields from science to engineering. Yet, there still is a lack of versatile computational toolbox to deal with this long standing challenge due to the inherent non-autonomicity and unknowability of the underlying dynamics. Here, we introduce a data-driven approach for inferring non-autonomous dynamical equations by identifying an optimal set of basis functions within the model space, enabling the reconstruction of complex systems behavior under simplified prior specifications. Our method demonstrates effectiveness in equation discovery on canonical synthetic systems such as cusp bifurcation and coupled Kuramoto oscillators. Furthermore, we extend the application of this approach to leaf cellular energy, unmanned aerial vehicle navigation, chick-heart aggregates, and marine fish community under simple basis function libraries. Leveraging the inferred equations, we accurately predict the evolution of these empirical systems and further uncover their governing laws. Our approach offers a novel paradigm to reveal the underlying dynamics of a wide range of real-world systems.

The extended KdV equation is a nonlinear dispersive wave model that is asymptotically or variationally derived from the full dispersive Euler shallow water waves equations when gravity-capillary and higher order nonlinear effects are taken into account, under weakly nonlinear and long-wave approximations. This reduction introduces four additional terms beyond the classical KdV equation: a nonlinear term (quadratic nonlinearity), two nonlinear-dispersive terms, and a fully dispersive term (fifth order dispersion). In this paper, we employ a variational approach based on averaged Lagrangians to analyze the accuracy of single solitary wave solutions governed by a particular extended KdV equation where energy conservation is a key feature. Compared with solitary wave solutions previously obtained through higher order asymptotics and algebraic methods, the present variational solutions are notably simpler and more readily applicable to practical problems. The solitary wave solutions obtained through this method are then systematically compared with direct numerical simulations, and the corresponding results are critically discussed. We further demonstrate the applicability of these single solitary waves to problems in the field of non-convex dispersive hydrodynamics. These problems include shallow water classical undular bores, commonly known as dispersive shock waves, and non-classical (resonant) dispersive shocks which are additionally analyzed using the concept of Whitham shocks. Theoretical predictions show excellent agreement with numerical simulations.

We investigate how localized inhomogeneity affects the geometry and stability of migratory bird formations. We use a lifting-line model with a horseshoe-vortex representation to describe the longitudinal dynamics of aerodynamic interactions. As a reference case, we first analyze homogeneous formations and show that their steady states exhibit a U-shaped geometry with hierarchical streamwise spacing, in which adjacent birds become progressively closer toward the leader. We then introduce localized inhomogeneity by modifying the wingspan of a single bird, with its physical properties determined by scaling relations. We determine the range of wingspan variation that preserves a stable formation. The stability range depends strongly on the position of the modified bird, being narrower near the outer wing and broader near the leader. These findings provide a minimal dynamical framework for understanding how local aerodynamic interactions and localized individual differences affect collective flight structures.

This study provides an analytical foundation for adversarial synchronization control in Kuramoto oscillator networks, where small gradient-based perturbations applied repeatedly to oscillator phases can dramatically enhance or suppress collective synchronization. Using the Ott--Antonsen reduction, we derive an exact closed-form expression for the effect of a single adversarial perturbation (kick) on the order parameter. A key finding is that each kick produces a finite, coupling-independent increment in the order parameter even when synchronization is arbitrarily weak, which combined with slow relaxation near the critical coupling and mean-field feedback explains the disproportionate amplification previously observed in numerical simulations. Fixed-point analysis further reveals a fundamental asymmetry between enhancement and suppression, with the latter governed by noise-induced escape in finite systems. Extending the framework to networks via the annealed network approximation, we show that the theory captures the synchronization behavior of representative model networks and identify a decoupling between kick sensitivity and mean-field dominance in scale-free networks. These results offer a tractable theoretical basis for understanding and designing kick-based synchronization control in oscillator networks.

We study Lévy-like and truncated Lévy-like flights with step probability distribution of the form $r^{-1+\nu}$ for negative, positive, and zero $\nu$, focusing on the appearance of fractal geometry characteristics in the generated point sets. Forming ensembles of such point sets with fixed multiplicity, we develop simulation techniques leading to the desired value of correlation dimension in a vast continuous interval of scales. In particular, we demonstrate the possibility to produce ensembles of data sets with a low number of points with the needed properties. Furthermore, we show that the positive $\nu$ distributions, apart from a region near the upper scale limit, show fractal behaviour that extends to infinitesimally low scales. As an example, we apply our findings to producing simulations relevant to the search for critical fluctuations, related to QCD critical endpoint, in heavy-ion collision experiments.

Nondestructive detection of single-electron motion is crucial for quantum information processing with electrons trapped in Paul traps. The standard approach in Penning traps is to detect the image current induced on the trap electrodes by the electron's oscillatory motion. However, applying this approach in Paul traps for single electrons is currently hindered by motional frequency fluctuations arising from trap anharmonicities and instabilities in the rf trapping field. In this work, we propose a robust detection scheme exploiting the transient dynamics of parametric driving to overcome these limitations. Distinct from traditional steady-state approaches, our method focuses on the transient regime to break the temporal constraints imposed by steady-state assumptions, thereby enabling fast readout. We show that a controlled ramp of the parametric drive effectively locks the frequency of the electron motion in the transient regime, rendering the signal highly resilient to realistic experimental noise and inherent micromotion. This work paves the way for the experimental realization of nondestructive detection of single-electron motion in Paul traps.

We develop an ab initio approach to describe the statistical behavior of finite-size fluctuations in the deterministic Kuramoto-Sakaguchi model. We obtain explicit expressions for the covariance function of fluctuations of the complex order parameter and determine the variance of its magnitude entirely in terms of the equation parameters. Our results rely on an explicit complex-valued formula for the solution of the Adler equation. We present analytical results for both the sub- and the super-critical case. Moreover, our framework does not require any prior knowledge about the structure of the partially synchronized state. We corroborate our results with numerical simulations of the full Kuramoto-Sakaguchi model. The proposed methodology is sufficiently general such that it can be applied to other interacting particle systems.

Positive and negative flows of the Chen-Lee-Liu model and its various reductions, including Burgers hierarchy, are formulated within the framework of Riemann-Hilbert-Birkhoff decomposition with the constant grade two generator. Two classes of vacua, namely zero vacuum and constant non-zero vacuum can be realized within a centerless Heisenberg algebra. The tau functions for soliton solutions are obtained by a dressing method and vertex operators are constructed for both types of vacua. We are able to select and classify the soliton solutions in terms of the type of vertices involved. A judicious choice of vertices yields in a closed form a particular set of multi soliton solutions for the Burgers hierarchy. We develop and analyze a class of gauge-Bäcklund transformations that generate further multi soliton solutions from those obtained by dressing method by letting them interact with various integrable defects.

Dissipative structures are open dynamical systems that sustain coherent macroscopic organization by continuously exchanging energy and matter with their environment and generating entropy. A recent thermodynamic analysis of the paradigmatic Malkus--Lorenz waterwheel interpreted the Lorenz system as an engine, deriving an exact formula for its thermodynamic efficiency, and showing that efficiency tends to increase as the system is driven far from equilibrium while displaying sharp drops near the Hopf subcritical bifurcation to chaos. Here, we extend that single-engine framework to coupled dissipative structures. We introduce two canonical couplings -- master-slave coupling (series) and symmetric diffusive coupling (parallel) -- and prove two fundamental association laws allowing us to reduce the composite systems to an equivalent engine with a specified efficiency. We then apply these abstract results to coupled Lorenz waterwheels, deriving efficiency formulas consistent with the underlying power balance. We perform numerical simulations confirming that (a) series coupling induces an increase in thermodynamic efficiency, (b) parallel coupling averages the efficiency of engines and increases total energy flow, (c) synchronization is typically neutral or beneficial for efficiency except in narrow parameter regions, and (d) coupling modifies the curvature of entropy-generation trends. Our theorems suggest a mathematically rigorous and transparent route to define and compute thermodynamic efficiency for generalized flow networks, with potential application to complex systems energetics.

We study collisions between a ferrodark soliton (FDS) and an antiFDS ($\mathbb{Z}_2$ kinks in the spin order) in the easy-plane phase of spin-1 Bose-Einstein condensates (BECs). For a type-I pair (type-I FDS-antiFDS pair) at low incoming velocities, the pair annihilates followed by the formation of an extremely long-lived dissipative breather on a stable background, a spatially localized wave packet with out-of-phase oscillating magnetization and mass superfluid densities. Periodic emissions of spin and density waves cause breather energy dissipation and we find that the breather energy decays logarithmically in time. When the incoming velocity is larger than a critical velocity at which a stationary FDS-antiFDS pair forms, a pair with finite separating velocity is reproduced. When approaching the critical velocity from below, we find that the lifetime of the stationary type-I pair shows a power-law divergence, resembling a critical behavior. In contrast, a type-II pair (type-II FDS-antiFDS pair) never annihilates and only exhibits reflection. For collisions of a mixed type FDS-antiFDS pair, as $\mathbb{Z}_2$ kinks in the spin order, reflection occurs in the topological structure of the magnetization while the mass superfluid density profiles pass through each other, manifesting spin-mass separation.

We study synchronization of two cardiomyocytes mediated by elastic interactions through the substrate. Modeling each cell as an oscillating force dipole governed by a Rayleigh-type equation, we derive an effective mechanical coupling from the elastic response of the surrounding medium. Using phase reduction theory, supported by direct numerical simulations, we obtain a dynamical phase description for two cardiomyocytes that predicts geometry-dependent selection of synchronized states. Depending on the mutual orientation, the cells robustly converge to either in-phase or anti-phase beating, yielding an orientation-dependent state map with a nontrivial state boundary. The synchronization time also depends strongly on the distance and mutual orientation of the cells. These results bridge earlier energetic two-body theory and dynamical single-cell theory, and provide a dynamical framework for elastic synchronization of cardiomyocytes.

Griffiths phases are typically associated with quenched disorder, while frustration gives rise to multistability and spin-glass behavior. Whether extended criticality can arise in other contexts remains an open question. Here, we show that synergistic interactions provide a distinct route to non-conventional critical phenomena. By combining spreading mechanisms that reinforce activity through complementary pathways, we uncover a broad distribution of relaxation rates, leading to Griffiths-like slow dynamics and extended criticality. We demonstrate that this mechanism is robust across networks and emerges both in systems with explicit higher-order interactions and in purely pairwise systems with nonlinear dynamical rules.