Predicting the occurrence of transitions in the qualitative dynamics of many natural systems is crucial, yet it remains a challenging task. Generic early warning signals like variance and lag-1 autocorrelation identify critical slowing down near tipping points but lack practical thresholds for predicting imminent transitions. More recent studies found that the dynamical eigenvalue is rooted in the framework of empirical dynamical modeling and then estimates the dominant eigenvalue of a system from time series, providing a threshold ($|$DEV$|$ = 1) to predict bifurcations and classify their types. However, its application requires careful calibration of the hyperparameters and focuses on reconstructing system dynamics directly from data. Here, we employ Ornstein-Uhlenbeck process to derive analytic approximations for the lag-$\tau$ autocorrelation function prior to period-doubling bifurcation thereby estimating the dominant eigenvalue of dynamical systems, named dominant eigenvalue extracted from autocorrelation (DE-AC), and revealing its dynamic behaviour when approaching a period-doubling bifurcation. Theoretically, dominant eigenvalue tends to $-1$ when the system approaches a period-doubling bifurcation. In particular, we evaluated DE-AC on simulation data from cardiac alternans model and on experimental data from chick heart aggregates undergoing a period-doubling bifurcation. DE-AC reliably detected the beginning of the cardiac arrhythmia (period-doubling bifurcation) in most cases. Moreover, it demonstrated superior sensitivity and specificity as an early warning signal compared to the three widely used indicators -- variance, lag-1 autocorrelation, and dynamical eigenvalue. Our theoretical and empirical results suggest that DE-AC represents a quantitative measure for predicting the onset of potentially dangerous alternating rhythms in the heart.

We introduce a computationally efficient and accurate reduced order modelling approach for the optimization of spatiotemporally chaotic systems. The proposed method combines quantized local reduced order modelling with adjoint-based optimization. We employ the methodology in a variational data assimilation problem for the chaotic Kuramoto-Sivashinsky equation and show that it successfully reconstructs the full trajectory for up to 0.25 Lyapunov times given full state measurements at the final time. The proposed algorithm provides 3.5 times speed-up when compared to the full order model. The proposed method opens up new possibilities for the reduced order modelling of spatiotemporally chaotic systems.

Interacting rotating spiral waves have been observed in complex systems, such as cardiac fibrillation, cognitive processing in the brain cortex and oscillating chemical reactions, during dynamical regimes that are still poorly understood. We present the equivalent of Newton's gravitational attraction law for spiral waves on planar reaction-diffusion systems. The spiral waves' phases and positions determine their regions of influence, separated by collision interfaces. At the collision interfaces, wave front deflections cause spiral drift that pushes the interfaces forward. As a result, the spiral wave drift velocity is proportional to the total force exerted on on it, which can be determined by a boundary integral over its region of influence. The proportionality factor between force and response is akin to the `mass' of the spiral. However, this spiral mass depends on the region of influence of the spiral and thus also varies over time. The forces between spiral wave pairs are not directed along the line connecting their centers, violating Newton's law of action and reaction. Our solution to the N-body interaction problem for spirals in extended excitable media encompasses both pairwise interactions and spiral wave drift in bounded domains, with application to cardiac fibrillation.

We investigate the spectral and dynamical properties of the fractional nonlinear Schrödinger (fNLS) equation with harmonic confinement. In this setting, the classical Laplacian is replaced by its fractional power $(-\partial_x^2)^{\alpha/2}$ with $\alpha\in(1,2]$, introducing nonlocal, Lévy-type dispersion. This modification fundamentally alters the balance between nonlinearity, dispersion, and trapping, reshaping both the structure and stability of stationary states. Using a Fourier pseudo-spectral discretization, we compute stationary branches as functions of the temporal frequency $\Omega$ in focusing ($\sigma=+1$) and defocusing ($\sigma=-1$) regimes, and assess spectral stability via the linearized eigenvalue problem. Direct simulations, performed with split-step and exponential time-differencing integrators, confirm these predictions and reveal $\alpha$-dependent transitions between coherent oscillations, bounded breathing dynamics, and decoherence or fragmentation. Our results show that decreasing $\alpha$ systematically shifts bifurcation curves, fragments stability windows for excited states, and amplifies instability in the focusing regime, while supporting robust coherence in the defocusing case. Beyond clarifying how harmonic confinement mediates the interplay between nonlinearity and fractional dispersion, the study also provides benchmarks for numerical treatments of fractional operators and points toward potential applications in nonlinear optics, Bose--Einstein condensates, and anomalous transport phenomena.

In discrete nonlinear systems, the study of nonlinear waves has revealed intriguing phenomena in various fields such as nonlinear optics, biophysics, and condensed matter physics. Discrete nonlinear Schrödinger (DNLS) equations are often employed to model these dynamics, particularly in the context of Bose-Einstein condensates and optical waveguide arrays. While the classical DNLS with cubic nonlinearity admits well-known solitonic solutions, the introduction of competing nonlinearities, such as quadratic-cubic and cubic-quintic terms, gives rise to new behaviors, including multistability and front formation. One such emergent structure, the Maxwell front, is characterized by stationary interfaces between two energetically equivalent steady states, occurring at a critical parameter known as the Maxwell point. This paper investigates the existence and stability of Maxwell fronts in DNLS models with competing nonlinearities. Specifically, we examine the quadratic-cubic nonlinearity, as found in the discrete quantum droplets equation, and the cubic-quintic nonlinearity, both of which exhibit multistability. We explore the persistence of Maxwell fronts in both the anticontinuum limit (where the coupling between lattice sites is weak) and the continuum limit (where the coupling is strong). The stability of these fronts is analyzed through linear stability analysis, utilizing eigenvalue counting arguments and exponential asymptotic techniques. Our results provide new insights into multistability, front dynamics, and the role of competing nonlinearities in discrete wave systems. The main contributions of this work include the characterization of Maxwell fronts in DNLS equations with competing nonlinearities, the analysis of their stability across different coupling regimes, and the application of novel asymptotic methods to investigate their behavior in the continuum limit.

Collective synchronization is often summarized by a complex order parameter $R e^{i\Psi}$, implicitly treating the global phase $\Psi$ as a meaningful macroscopic coordinate. Here we ask when $\Psi$ becomes \emph{operationally well-defined} in oscillator networks whose coupling varies in time. We study damped (and optionally inertial) phase-oscillator models on graphs with time-dependent coupling $K(t)$, covering standard Kuramoto dynamics as a limit and including network and spatial topologies relevant to engineered settings. We propose an operational emergence criterion: a macroscopic phase is emergent only when it is robustly estimable, which we quantify via gauge-fixed phase-lag fluctuations under weak noise and finite sampling. This yields a quantitative threshold controlled by $NR^2$ and makes explicit why $\Psi$ is ill-posed in incoherent states even when formally definable. Nonautonomous coupling introduces a ramp timescale that competes with relaxation. Using a Laplacian-mode reduction near coherence, we derive a graph-spectral rate criterion: ordering tracks the protocol when $K(t)\lambda_2$ dominates the ramp rate, while faster ramps induce freeze-out. Numerically, we extract an operational freeze-out time from an energy-based tracking diagnostic and show that, for non-spatial networks, the residual incoherence at freeze-out collapses when plotted against the spectral protocol parameter $\lambda_2\tau$ across Erdős--Rényi and small-world graph families. Finally, on periodic lattices we show that topological sectors and defect-mediated ordering obstruct complete alignment, leading to protocol-dependent, long-lived partially synchronized states and systematic deviations from spectral collapse.

The motion of compressible, inviscid fluid under the constant pressure on a rotating sphere is studied. The hodograph equations for the corresponding Euler equation are presented. They provide us with the class of solutions of the Euler equation parameterized by two arbitrary functions of two variables. Several particular explicit solutions are given. The blow-up curves, on which the derivatives of velocitiy blows up, are described. The limiting cases of slowly and rapidly rotating sphere are considered. The equation describing the deformations of elliptic functions modulus is presented.

We investigate nonlinear magnetoelastic wave dynamics and electrically tunable soliton excitations in hexagonal multiferroic media. By varying the magnetoelastic coupling strength and using a coupled magnetoelastic-ferroelectric continuum model, we found that the system evolves from weakly nonlinear quasiperiodic oscillations to strongly anharmonic yet phase-coherent multimode dynamics. Our results suggest that the dynamics remain bounded and approach distorted limit-cycle behavior rather than chaotic motion despite the enhanced nonlinearity. The excitation spectra and the band dispersion relations reveal that this nonlinear evolution originates from strong magnon-phonon hybridization and coupling-induced renormalization of collective excitation branches, leading to coherent energy exchange among magnetic, elastic, and polarization subsystems. In addition, the coupled dynamics can be reduced to an effective magnetoelastic nonlinear Schrödinger equation and support localized excitations such as bright and dark solitons and Kuznetsov-Ma type breathers. Furthermore, it is found that an external electric field modifies both the effective nonlinear coefficient and the dispersion curvature, enabling continuous control of soliton amplitude, width, and stability. The field also induces a saddle-node bifurcation in the magnetization phase space, defining a critical threshold separating multistable and monostable regimes. Our results establish a theoretical framework for electrically tunable nonlinear spin-lattice excitations and soliton engineering in multiferroic systems.

Nonequilibrium conditions fundamentally change how systems undergo phase separation. In systems with temperature gradients, attractive particles have been shown to form periodic patterns and steady convective currents, but a clear theoretical explanation for this behavior is still missing. Here, we present a dynamical mean-field model that describes the mechanism behind this convective phase separation. Using linear stability analysis, we show that the transition from a uniform state to a periodic pattern is driven by the emergence of a dominant unstable mode. Numerical simulations confirm the predicted phase diagram and demonstrate that these convective currents are a robust feature of the steady state, appearing regardless of the initial conditions. These results provide a direct approach for understanding how temperature gradients drive the formation of steady-state convective patterns.

We study the isotropic six-vertex model on $\mathbb{Z}^2$ with spectral parameter $\Delta\in[-1,-1/2]$, that is, with weights $\mathbf{a}=\mathbf{b}=1$ and $\mathbf{c}\in[\sqrt{3},2]$. We show that the associated height function converges, in the scaling limit, to a properly scaled full-plane Gaussian free field. The result extends to anisotropic weights $\mathbf{a}\neq\mathbf{b}$ upon using a suitable embedding of the lattice.

Associated to a convex integral polygon $N$ in the plane are two integrable systems: the cluster integrable system of Goncharov and Kenyon, constructed from the dimer model on bipartite torus graphs, and the Beauville integrable system associated with the toric surface of $N$. These two systems are related by a birational map called the spectral transform. In this paper we study the case when $N$ is the standard triangle of side length $d$, equivalently when the toric surface is $¶^2$, and prove that the spectral transform is a birational isomorphism of integrable systems. Since the Hamiltonians are identified by construction, the essential content is that the spectral transform intertwines the two Poisson structures. In particular, this shows that Beauville integrable systems admit cluster algebra structures.

The Atlantic Meridional Overturning Circulation (AMOC), a crucial ocean current system, could transition to a weak state. Despite severe associated climate impacts, assessing the AMOC's response under global warming and its proximity to possible critical thresholds remains difficult. To understand future Earth system stability, a global dynamical view is needed beyond the local stability analysis underlying classical early-warning methods. Using an intermediate-complexity climate model, we explore the stability landscape of the AMOC for different atmospheric CO$_2$ concentrations. We explicitly compute the edge state (or Melancholia state), a chaotic saddle on the basin boundary separating the strong and weak AMOC attractors found in the model. While being unstable, the edge state can govern the transient climate for centuries, supporting centennial AMOC oscillations driven by atmosphere-ice-ocean interactions in the North Atlantic. At increased CO$_2$ levels projected for the near future, we reveal a boundary crisis where the current AMOC attractor disappears by colliding with the edge state. Under crisis overshoot, long chaotic transients due to "ghost states" lead to diverging ensemble trajectories under time-varying forcing. Rooted in dynamical systems theory, our results offer an explanation of large ensemble variance and apparent "stochastic bifurcations" observed in earth system models under intermediate forcing scenarios.

The main result of this paper is an explicit description of the stratification of the phase space of Calogero--Moser--Sutherland (CMS) integrable systems corresponding to Lie groups $SU(n)$. The phase space decomposes into symplectic strata of dimensions $2s$, where $s = 0, 1, \ldots, n - 1$. On each stratum of the positive dimension, we construct natural action-angle coordinates and compute the symplectic form explicitly, showing that every stratum is symplectomorphic to $\mathbb{R}_{> 0}^s \times \mathbb{T}^s$. The zero-dimensional stratum corresponds to the equilibrium point of the multi-time CMS dynamics.