Papers

We study a novel class of evolution variational inequalities (EVI) driven by bifunctions $\mathsf{b}:\mathrm{D} \times \mathrm{D} \to \mathbb{R}$ defined on a subset $\mathrm{D}$ of a metric space $(\mathrm{X},\mathrm{d})$ and satisfying a natural dissipativity condition $$\mathsf{b}(x,y)+\mathsf{b}(y,x)\le \eta \mathrm{d}^2(x,y).$$ We provide general conditions for existence, stability, regularity, approximation and asymptotic behavior of solutions, by proposing a metric framework that generalizes the classical structure of evolutions driven by monotone operators in Hilbert spaces. A motivating application is the setting of minimax and multispecies coupled gradient flows, in which each of $N$ species evolves in the direction of steepest descent of its own energy functional and the joint system is not a gradient flow of any single energy. To construct solutions, we introduce a variational movement scheme (VMS), a time discretization scheme in which each update is a saddle point of a metrically penalized bifunction, generalizing the classical minimizing movement/ JKO scheme for single species gradient descent. In the coupled multispecies case, the VMS reduces to a Nash equilibrium problem. We prove existence of discrete solutions to the VMS via a general approach which combines dissipativity of $\mathsf{b}$ with a new notion of convexity along suitable $\textit{barycentric interpolations}$ (which is inspired by convexity along generalized geodesics, a crucial notion for gradient flows). Together, these conditions provide a zeroth-order notion of game-theoretic monotonicity applicable to general metric spaces without linear structure.

The interaction of moose (Alces alces) and wolf (Canis lupus) populations in ecosystems such as Isle Royale National Park is a canonical benchmark for ecological modeling of prey-predator dynamics. Mathematical modeling is a useful tool for modeling these interactions. A central challenge in this domain is the inverse problem, recovering governing system parameters from observational data. Although classical parameter estimation methods have seen considerable progress, they mostly rely on data rather than physics and therefore largely fall short in capturing the time-varying nature of ecological interactions driven by environmental fluctuations, seasonal forcing, and habitat change. This study addresses that gap by solving the inverse problem for a non-autonomous prey-predator system that incorporates theta-logistic prey growth with temporally varying intrinsic growth and natural death rates, as well as Holling type-II and ratio-dependent functional responses. A deep learning framework (self-adaptive bc-PINN with transfer learning) is employed to estimate time-dependent and constant parameters directly from population time series data (1959-2019). Before estimating the parameters, we performed a structural identifiability analysis to ensure that the model parameters are identifiable. The framework demonstrates good reconstruction and prediction results across both functional responses. The ratio-dependent model has shown the better prediction trend beyond the training data. Our framework successfully predicts the sudden decline in the moose population in 2020.

Motivated by questions and results of Erdős, Sárközy and Sós (1989), and Khalfalah and Szemerédi (2006), we introduce an ergodic approach to additive problems concerning solutions of equations of the form \[ x+y=\lfloor\alpha(n)\rfloor,\quad x,y\in A,n\in\mathbb{N} \] where $\alpha(t)$ is a sufficiently regular function, such as a polynomial or a logarithmico-exponential function of polynomial growth (e.g. $\alpha(t)=t^{5/2}+t^2\log t$), and $A$ is either a given set of positive upper density, or a cell of a prescribed finite partition of $\mathbb{N}$. We give a complete description of the Hardy field functions of polynomial growth for which these partition and density problems are always solvable. The ergodic approach also applies to the more general problem of finding $x\in A_1,y\in A_2$, where $A_1,A_2$ are two given sets of positive density. This in turn allows us to treat equations of the form $kx+ly=\lfloor\alpha(n)\rfloor$, where $k,l\in\mathbb{Z}$. Some of our results also hold true when $A$ is a subset of the primes with positive relative upper density.

Phase reduction is a powerful technique to obtain phase models from highly dimensional oscillatory systems. Starting from pairwise interactions, a first order approximation yields Kuramoto and Winfree phase models, while nonpairwise interactions emerge at the second order. Recently, phase models have been extended to account for nonpairwise interactions, notably, the higher-order Kuramoto model. Is there an intrinsic difference between physical and emergent nonpairwise interactions? And can we make use of the former to even out the effects of the latter? In this work, we exploit a recently developed parametrization method to compute the phase reduction and answer these questions. After a comparison of the new method with the classic Kuramoto-style phase reduction, we solve the network motifs of the emergent nonpairwise interactions and compare them with the physical once. Lastly, we exploit the motifs to adopt a coupling design approach. Our framework paves the way for further exploitations of physical nonpairwise interactions for applications in synchronization engineering.

We study the growth of the bidegree of an ample irreducible curve in $\mathbb{P}^1 \times \mathbb{P}^1$ under a product polynomial endomorphism $\varphi=(f,g)$, where at least one of $f$ and $g$ is non-exceptional. We prove that, if the curve $C$ is not preperiodic under $(f^a,g^b)$ for any $a,b\geq 1$, then the bidegree of $\varphi^n(C)$ is asymptotic to $(\text{deg}(g)^n,\text{deg}(f)^n)$. As an application of this exponential growth, we prove a geometric analogue of Silverman's theorem on the finiteness of $S$-integral points in orbits. Namely if $C$ is not $(f^a,g^b)$-preperiodic and $C'$ is not totally invariant for $\varphi$, then for any infinite sequence ${n_i}$ of positive integers, the union of the intersections $$ \bigcup_{i \geq 1} \left( \varphi^{n_i}(C)\cap C' \right) $$ is Zariski dense in $C'$.

By introducing new ergodic-theoretic techniques in nilsystems, we determine which infinite sumset configurations occur in $U^k(\Phi)$-uniform and Nil-Bohr sets. To be more precise, our first result associates the degree $k$ of a $U^k(\Phi)$-uniform set with the variety of sumsets it contains, solving a conjecture of Kra, Moreira, Richter and Robertson. Restricting to Nil-Bohr sets we show the existence of infinite sumsets with summands in the shifted primes $\mathbb{P}-1$. As a consequence, we show that for any real polynomial $Q(n)$ with leading irrational coefficient of degree $k$, and any natural numbers $\ell_1, \cdots, \ell_k$ there is an infinite set $P\subset \mathbb{P}$ such that \begin{equation*} Q\Big(\sum_{p \in I} p\Big) \in U \pmod 1 \quad \text{ for all } I \subset P , |I| = \ell_1, \ldots, \ell_k. \end{equation*}

Binary modular Laplacian dynamics is organized by repeated replication, growth, and collapse on dyadic scales. We study how isolated and periodically repeated non-binary updates modify this organization and whether they can sustain densely occupied geometric structures. Computational experiments across multiple finite seeds, neighborhood masks, inserted moduli, and periodic schedules show that constant prime moduli retain the expected \(p\)-adic replication hierarchy. A single non-binary insertion mainly acts as an effective-seed replacement: subsequent evolution remains binary-like, but may resume at an integer phase shift of the collapse–recovery cycle, with only modest finite-time densification. A qualitatively different response occurs for periodic schedules \([2,k,2^s]^\infty\), where \(2^s\) denotes \(s\) consecutive binary updates. Odd-modulus insertions can sustain long-lived, spatially coherent carpet-like regimes with comparatively high and stable occupation, whereas even-modulus insertions remain binary-like. Ternary insertion gives the broadest and most robust high-density response. For \(k=3\), density depends non-monotonically on the binary-tail length, with troughs near \(s=7,15,23\) that reveal phase-sensitive interaction with the binary epoch structure. Cross-modulus synchronization also occurs: for degree-four diagonal Neumann and von Neumann masks, \(k=5,7,9\) repeatedly collapse onto the same complete configuration at subsequent binary phases. A parity argument explains this loss of memory. Increasing seed extent generally raises occupation while reducing tail-length sensitivity. Overall, periodic modular insertions act as phase-selective, geometry-dependent mechanisms transforming \(p\)-adically organized replication into long-lived carpet-like dynamics.

We propose sparse identification of port-Hamiltonian systems (SIPHy), enabling structure-preserving symbolic regression from noisy trajectory observations. The method applies to port-Hamiltonian systems, which provide a general framework for describing dynamical systems in terms of energy exchange, dissipation and control. Our algorithm can jointly identify the Hamiltonian as well as the dissipation and input matrices. Furthermore, we introduce Hamiltonian flow splines to better approximate derivatives of trajectory data corrupted by noise or with missing time points, a major challenge of system identification for differential equations. This method assembles flows of piecewise polynomial Hamiltonians to produce a smooth, differentiable trajectory necessary for sparse regression. Combining flow splines with SIPHy yields interpretable, physically grounded models from noisy and incomplete trajectory data. Because the Hamiltonian, dissipation, and input matrices are identified as separate components, the resulting models can be simulated under control inputs and dissipation regimes never observed during training, thus improving the robustness and generalization capabilities of model discovery methods.

Self-similar solutions to the porous medium equation with dominating spatially inhomogeneous absorption $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, \quad (x,t)\in\real^N\times(0,\infty), \quad N\geq1, $$ with exponents $1<p<m$ and $\sigma>0$, are classified. Looking for solutions in the form $$ u(x,t)=t^{-\alpha}f(|x|t^{\beta}), \quad \alpha=\frac{\sigma+2}{\sigma(m-1)+2(p-1)}, \quad \beta=\frac{m-p}{\sigma(m-1)+2(p-1)}, $$ it is shown that all their profiles satisfy the behavior at infinity given by $$ \lim\limits_{\xi\to\infty}\xi^{\sigma/(p-1)}f(\xi)=\left(\frac{1}{p-1}\right)^{1/(p-1)}, $$ but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with $f(0)>0$, $f'(0)=0$, another unique solution such that $f$ presents a dead-core; that is, $f\equiv0$ for $\xi\in[0,\xi_0]$ for some $\xi_0>0$, and, finally, there exists $K^*\in(0,\infty)$ such that, for any $K\in(0,K^*)$, there is at least a solution such that $$ \lim\limits_{\xi\to0}\xi^{-(\sigma+2)/(m-p)}f(\xi)=K. $$ The large time behavior of general solutions, making strong use of these three types of self-similar solutions, will be addressed in a companion work.

We study reconstruction in autoencoders that apply the same forward map before and after setting the observed coordinates to zero. For equal odd input and hidden dimensions $d\geq 3$, among orientation-preserving diffeomorphisms whose Jacobian singular values lie in $[m,M]$, we show that the least uniform reconstruction-derivative error is $\max\{1-M(M-m)/2,0\}$, with affine maps attaining this sharp bound at every prescribed depth. A translated radial rotation can nevertheless reconstruct any prescribed ball exactly with singular values arbitrarily close to one, motivating additional conditions for a finite-data bound. We test this prediction on a 798,452-point terrestrial LiDAR forest scan. At input scale $0.05$, the mean theoretical bound is $0.155$, about $84\%$ of the mean normalized training error $0.185$ across four spatial regions, two depths, and three seeds. At this scale, adding one hidden coordinate reduces the mean reconstruction error below $6\times10^{-6}$.

The Koopman operator has gained considerable attention in dynamical systems due to its capability to provide a linear viewpoint for nonlinear systems using data-driven methods such as extended dynamic mode decomposition (EDMD). In this work, we suggest an EDMD-variant with a Fourier dictionary on the $d$-dimensional torus, where the data are sampled on an equispaced tensor grid. In this setting, the EDMD regression problem admits a unique closed-form solution, which we show to coincide with trigonometric interpolation of the Koopman image. This identification has two consequences. First, using Koopman invariance of Sobolev spaces, we transfer approximation-theoretic results for trigonometric interpolation to derive error bounds on approximations of the Koopman operator. In particular, the established bounds are of optimal order with explicit constants. Second, the EDMD matrix never has to be assembled, since its action reduces to (nonequispaced) fast Fourier transforms such that the EDMD-approximation may be evaluated matrix-free with quasi-linear cost in the dictionary size. We illustrate the results for the Kuramoto model of coupled oscillators with dictionaries of up to $10^7$ modes.

In this paper, we study the multifractal analysis of Birkhoff averages for parabolic rational maps. We establish a conditional variational principle and prove the real analyticity and strict monotonicity of the Birkhoff spectrum, as well as the existence and uniqueness of the measure attaining the supremum in the conditional variational principle, on a certain region. To this end, we prove the existence and uniqueness of an expanding equilibrium measure and the real analyticity of the pressure function on a suitable domain. For parabolic systems, our approach using thermodynamic formalism provides a unified framework for establishing the conditional variational principle and investigating finer properties of the Birkhoff spectrum, including its real analyticity, strict monotonicity, and the existence and uniqueness of a measure attaining the supremum on a certain region.

We study Lyapunov-type stability for invariant algebraic divisors of codimension-one holomorphic foliations on complex projective spaces. Motivated by the notion of $L$-stability introduced in \cite{LeonScardua2018} for plane singularities, we define a projective stability condition which is compatible with the logarithmic setting and controls leafwise holonomy along the regular part of the invariant divisor, while discarding any stability requirement inside prescribed neighborhoods of its singular locus. Our main result then is a global projective counterpart of the local classification in \cite{LeonScardua2018}: for a maximal invariant algebraic divisor, projective $L$-stability together with non-dicritical non-nodal generalized-curve singularities on a generic plane section forces the ambient foliation to be globally logarithmic. As an application we obtain a topological rigidity consequence on $\mathbb P^2$ for logarithmic foliations under mild generic conditions. The proof of the main theorem is based on propagating the consequences of the stability hypothesis through the reduction tree by means of an explicit Dulac transport argument, together with the classification of $L$-stable groups of germs of one-dimensional complex diffeomorphisms.

For an invertible linear time-invariant system, the Morse normal form reveals that there exist an output injection and an output transformation such that the resulting system possesses a regular relative degree. Based on this fact, this paper develops a method to determine all admissible output injections and output transformations that achieve this property. In contrast to the recursive and implicit computation methods in the existing literature, the proposed determination method for the output injections is explicit and easy to use, since the solutions are obtained by solving linear matrix equations. A rigorous proof of the proposed method is provided, and an illustrative example is presented to demonstrate the applicability of the approach.

We introduce a trigonometric version of the Nosé-Hoover oscillator in which the quadratic mechanical terms and unbounded thermostat coupling are replaced by bounded trigonometric functions. This formulation replaces the harmonic potential by a pendulum-type potential and confines the thermostat interaction to a bounded periodic form. The resulting two-parameter system is naturally defined on the three-dimensional torus and reduces near the origin, to leading order, to the classical polynomial Nosé-Hoover model. We investigate its global dynamics using Poincaré sections, bifurcation diagrams, Lyapunov spectra, Kaplan-Yorke dimensions, and the Lyapunov Integrability Test (LIT). The numerical results reveal the coexistence of regular and chaotic dynamics and characterize changes in dissipative behavior across the parameter plane. We then analyze the two limiting cases associated with the parameter axes. For $a=0$, we construct two functionally independent first integrals on regular domains, whereas for $b=0$ the dynamics reduces to a family of two-dimensional systems on invariant tori, which are analyzed using Darboux polynomials and exponential factors. First-order averaging near the intersection of these integrable limits yields periodic solutions bifurcating from unperturbed periodic orbits and an obstruction to regular $C^1$ first integrals in their neighborhoods. Independently, differential Galois theory applied to the normal variational equation, together with the Ayoul-Zung and Li-Shi criteria, excludes meromorphic $B$-integrability and non-constant meromorphic first integrals near a particular non-equilibrium phase curve for $ab\neq0$. Thus, despite retaining the local structure of the classical Nosé-Hoover oscillator, its trigonometric counterpart exhibits markedly different global dynamics and integrability.

We construct variational integrators to study the dynamics of stochastic Hamiltonian systems using their Wong-Zakai approximation. This approach can be used to interpret and justify previous work of Wang et al (L. Wang, J. Hong, R. Scherer, F. Bai. Dynamics and variational Integrators of stochastic Hamiltonian systems. International Journal of Numerical Analysis and Modeling, 6(4), 2009) on stochastic variational integrators. Numerical examples are provided.

Buckling of thin cylindrical shells under axial compression, a classical example of a subcritical instability, is highly sensitive to small imperfections, with minute geometric variations causing large changes in buckling threshold. To uncover the origin of this sensitivity, we use numerical continuation and bifurcation analysis while systematically varying the depth and size of a single localized Gaussian defect. We show that the instabilities of the imperfect shell originate from localized equilibria already present in the perfect shell. By breaking translation symmetry, the defect pins these equilibria and changes how they connect to the imperfect base state. Small changes in defect geometry can thereby switch the bifurcation that triggers buckling, producing non-monotonic and discontinuous changes in buckling threshold and abrupt changes in buckling mode. Imperfection sensitivity is therefore not simply sensitivity to imperfection magnitude, but sensitivity of the underlying bifurcation structure to imperfection geometry.

We study the Hausdorff dimension of projections of level sets of Lyapunov exponents for affine iterated function systems. Assuming that the tuple of linear parts is typical, we obtain a variational formula in terms of pressure, topological entropy, and the Lyapunov dimensions of invariant measures corresponding to Lyapunov exponents. In $\mathbb{R}^3$, we show that this value equals the Hausdorff dimension of the canonical projection of the level set under the strong open set condition. We further extend the result to arbitrary dimensions, showing that the same formula holds for Lebesgue-almost every choice of translation vectors under a suitable contraction assumption.

We study uniformity of Sobolev mixing estimates for rational maps and uniformly quasiregular mappings. For rational maps of fixed degree, the optimal centered Sobolev trace constant $A_f$ is a continuous proper function on Möbius moduli space. Uniform bounds on $A_f$ therefore characterize relative compactness in moduli, and a minimizing class exists in every degree. In the setting of uniformly quasiregular endomorphisms of degree $d\ge2$ on closed $n$-manifolds with an invariant conformal structure, the $k$th centered transfer operator from critical Sobolev energy into $L^1$ of the equilibrium measure has norm $A_f d^{-k/n}$. On the mean-zero Sobolev space, the spectrum and Fredholm essential spectrum are the closed disk of radius $d^{-1/n}$, with infinite-dimensional eigenspaces throughout its interior. The proofs use the energy scaling of pullback, a bounded equilibrium trace, and an obstruction from atoms of intermediate mass. Combined with conformal barycenter normalization and DeMarco–Faber's degeneration theorem, this obstruction gives the moduli compactness criterion. Explicit families illustrate the distinction between degeneration and concentration caused by changes of coordinates.

Given a family $\mathcal T = (T_\lambda)_{\lambda\in\Lambda}$ of bounded linear operators acting on the same $F$-space $X$ and indexed by a set of parameters $\Lambda \subset \mathbb{R}^d$, $d\geq1$, we explore the existence and non-existence of vectors that simultaneously satisfy, for all elements of $\mathcal T$, one of the following three dynamical properties: hypercyclicity, upper frequent hypercyclicity, and frequent hypercyclicity. Our approach relies on associating to $\mathcal T$ a "Lipschitz constant function” $F$, whose growth encodes the interaction between the operators and the parameter set $\Lambda$. We show that the asymptotic behavior of $F$ determines sharp thresholds between existence and non-existence of common vectors. In the hypercyclicity setting, we identify the divergence of $\sum 1/F(n)^s$, where $s$ is the Hausdorff dimension of $\Lambda$, as a natural condition for existence, and prove complementary non-existence results, which are optimal for weighted shifts. For upper frequent hypercyclicity, we obtain criteria of non-existence in terms of series of the form $\sum 1/F(\gamma^n)$ for parameter sets in $d$-dimensions. Considered in the one-dimensional case, these criteria are optimal and allow us to contrast known existence results of Mestiri for logarithmic-type growth with non-existence beyond this scale. For frequent hypercyclicity, we establish a dichotomy showing that common vectors exist essentially only when $F$ is bounded, while any unbounded growth prevents their existence. Our results apply to a broad class of parameter sets, including self-similar fractals, providing a unified perspective on how growth conditions and geometric features of $\Lambda$ determine common dynamical behavior. Finally, in the specific context of weighted shifts, we prove that two operators do not necessarily share the same frequently hypercyclic vectors if the sequence of their weight products ratios admits two distinct non-zero cluster points, establishing the optimality of a result of Grivaux, Matheron and Menet.

In this paper we affirmatively resolve the singularity conjecture for finitely supported non-degenerate random walks on cocompact Fuchsian groups. The method we use is based on constructing a pair of geodesic currents, one for the Lebesgue measure and one for the random walk measure, and checking that they are different by using Fourier analysis on the boundary.

We define high-order discrete differential and integral calculus following the discrete embedding formalism strategy. Using this framework, we define high-order discrete Lagrangian functionals and the corresponding calculus of variations. This provides a new derivation of Galerkin variational integrators.

We consider the integrable dynamics of a planar Kepler billiard in the plane bounded by a branch of a hyperbola focused at the Kepler center. As a sequel to our previous work \cite{JZ2}, we use the existence of the foci-caustic circle to identify an elliptic curve on which the dynamics is linearized and we analyze Cayley's condition on $n$-periodic orbits in this setting. By taking various limits also we discuss the dynamics of planar Kepler billiards with boundary being a focused parabolic arc, as well as a straight line. In this last case, our approach offers a different way to deduce the elliptic curve as in \cite{Felder}.

We study limit cycles produced by Sotomayor-Teixeira regularizations of piecewise linear vector fields with generic two-fold singularities. We focus on the visible-visible and visible-invisible cases where sliding cycles occur, treating cycles from both sides of the switching manifold in a unified way. In further details, by relating the cyclicity of sliding cycles and zeros of the slow divergence integral, we prove that the cyclicity of compact families of sliding cycles is bounded by two when such integral does not vanish identically. In contrast to previous works, we do not perform a case-by-base study, but instead relate zeros of slow divergence integrals to crossing limit cycles of a suitably defined auxiliary piecewise linear system. For visible folds, we provide necessary and sufficient conditions that assure the existence of a unique simple zero of the slow divergence integral, which implies the existence of two limit cycles. We also show that, when a hyperbolic singularity of the sliding vector field lies at the boundary of the sliding segment, then the cyclicity is bounded by one.

This paper analyzes the stochastic dynamics of a two-link pendulum that models lower limb segments (the thigh and shank) during the swing phase of the gait cycle. We describe the model using a system of non-linear Ito stochastic differential equations in a four-dimensional phase space under both external and internal random perturbations.We perform a spectral analysis of the discrete Kolmogorov operator, compute the stationary invariant measure, and estimate diffusion parameters using Neural Stochastic Differential Equations (Neural SDEs).

We study the minimum adjacency spectral radius in the class $\mathcal{SC}_{m+2}(m)$ of strongly connected digraphs with $m$ vertices and $m+2$ edges. Using generating functions for directed paths, we associate with the relevant digraphs topological polynomials whose smallest positive roots determine the corresponding spectral radii. Based on an ear decomposition, we obtain a complete structural classification of $\mathcal{SC}_{m+2}(m)$ by showing that every digraph in this class can be obtained from a butterfly digraph by attaching a single ear. This reduces the extremal problem to the optimization and comparison of finitely many polynomial families subject to their realizability conditions. We prove that the minimum spectral radius is determined by the polynomial $P_{\min}(z)=1-2z^{m-1}-z^m$. If $R_m\in(0,1)$ denotes its unique root, then $\min_{G\in\mathcal{SC}_{m+2}(m)}\rho(G)=R_m^{-1}$. For $m\geq4$, the minimum is attained, up to isomorphism, uniquely by the cross-chorded cycle $\mathcal{C}_m^\times$. For $m=3$, there are exactly two non-isomorphic minimizers, both with spectral radius $(1+\sqrt5)/2$. Finally, we establish the bounds $2^{1/(m-1)}<\rho\left(\mathcal{C}_m^\times\right)<3^{1/(m-1)}$.

We prove that a complete surface of finite topological type satisfying Condition A is homeomorphic to the two-torus. More generally, we show that an isolated planar end of a complete open surface satisfying Condition A cannot be periodic under the induced action on the space of ends. Condition A includes dense periodic points, a uniformly hyperbolic splitting into one-dimensional subbundles for a complete metric, and invariant line fields that uniquely integrate to transverse stable and unstable foliations. These results give a partial answer to the question of whether a complete surface satisfying Condition A must be compact.

We prove the existence of $C^{r,1-}$-regular stable invariant manifolds and linear conjugacies for analytic maps near a hyperbolic fixed point in the presence of resonances. The regularity exponent $r$ depends on the minimal resonant index, while the Hölder exponent can be chosen arbitrarily close to $1$, i.e., $1-\varepsilon$ for any $\varepsilon>0$. As a consequence, we obtain the stable version of the Hartman conjecture for analytic systems with semisimple linearization: in the fully stable case the local conjugacy can be chosen $C^{1,1-\varepsilon}$ for every $\varepsilon>0$.\\ Our approach introduces a new class of functional expansions based on logarithmic polynomials, which enables the invariance equation to be solved explicitly and algorithmically to arbitrary order. The existence results are obtained via a fixed-point argument in a suitable Banach space. We further present several analytic examples that both illustrate the applicability of the theorem while demonstrating the necessity of its assumptions.

We employ uncertainty quantification methods to investigate how parameter identifiability changes in the vicinity of a bifurcation point. We perform numerical experiments on the non-spatial Klausmeier vegetation model with random coefficients, which describes biomass-water interactions and exhibits a fold bifurcation. We partition the domain around the bifurcation into regions with distinct convergence behaviors. In each region, we follow a UQ workflow that includes sensitivity analysis, Bayesian inference, and Fisher information evaluation to assess parameter identifiability. The results show that proximity to the bifurcation point is decisive for parameter estimation. Reliable joint inference of both model parameters is not possible when the system exhibits bistability. Furthermore, we can reconstruct the model's bifurcation pattern by Fisher information heatmaps, presenting a practical tool for bifurcation detection. Lastly, we highlight the importance of transient data for successful parameter estimation.

We establish three weak averaging principles for distribution-dependent stochastic Navier–Stokes equations with rapidly oscillating coefficients on the torus $\T^2$. First, solution laws are relatively compact on finite time intervals, and every limit point as $\var\to0$ is the path law of a variational martingale solution of the averaged equation. Under a dissipative condition, we can choose bounded complete variational solution laws of the original and averaged equations so that every sequence with $\var\to0$ has a subsequence converging weakly to a bounded complete variational solution law of the averaged equation. At the level of probability laws, the original nonautonomous equation admits a family of weak pullback attractors, while the averaged equation has a weak global attractor. The weak pullback attractors converge upper semicontinuously to the weak global attractor of the averaged equation, uniformly with respect to the coefficient hull.