Papers

Dynamic mode decomposition (DMD) is a data-driven method that computes the best linear approximation of the underlying dynamical system and decomposes the dynamics into a superposition of characteristic spatiotemporal patterns. Originally introduced by the fluid dynamics community, DMD and its extensions have found widespread use in many other research areas such as molecular dynamics, climate science, engineering, finance, and neuroscience. Applications include dimensionality reduction, forecasting, system identification, control, and spectral clustering. In order to apply DMD to partial differential equations, the spatial domain is typically first discretized using finite difference or finite element techniques, thus implicitly rendering the problem finite-dimensional. We extend projected and exact DMD to infinite-dimensional systems. Rather than estimating matrices from vector-valued observations, our DMD variants learn finite-rank operators from functional data such as observables, densities, or wavefunctions. We show that conventional DMD algorithms can be regarded as special cases of their functional DMD counterparts. All results will be illustrated with the aid of guiding examples. We focus in particular on Koopman, Perron-Frobenius, and Koopman-von Neumann operators associated with graphons, ordinary differential equations, and stochastic differential equations.

Let $f$ be a regular endomorphism of $\mathbb{A}_{\mathbb{C}}^N$ and let $C\subseteq\mathbb{A}_{\mathbb{C}}^N$ be an irreducible curve. Suppose $C$ has an infinite intersection with the $f$-orbit of a point $x\in\mathbb{A}^N(\mathbb{C})$. Then the normalization of $C$ is isomorphic to either $\mathbb{A}^1$ or $\mathbb{G}_m$. We prove that $C$ is $f$-periodic in the latter case, as expected by the dynamical Mordell-Lang conjecture.

We prove that a lattice $\Gamma$ in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(\mathbb C)$ has positive trace gap, meaning that its traces are uniformly separated, if and only if it is derived from an admissible quaternion algebra. For cocompact Fuchsian groups, this proves the positive trace gap conjecture attributed to Sarnak by Geninska and Leuzinger in 2008. The same characterization by quaternion algebras holds if the difference set of traces is not dense. Our method also gives a similar result for lattices in $\mathrm{SL}_d(\mathbb R)$, for every $d\ge3$: a lattice $\Gamma<\mathrm{SL}_d(\mathbb R)$ has positive trace gap if and only if all its traces are integers. Equivalently, after conjugation, it has finite index in the norm-one group of an order in a central simple algebra of degree $d$ over $\mathbb Q$ that splits over $\mathbb R$. We also discuss spectral consequences and other applications of the results and techniques.

We examine the nonlinear initial value problem $\dot{y} = C(1 - y^\alpha)$ modeling polytropic gas permeation. For rational $\alpha$, we derive the exact primitive via cyclotomic partial fraction decomposition, establishing its real trigonometric representation. By synthesizing Ritt's theorem with Rosenlicht's differential algebraic residue criterion, we prove the exact step response $y(t)$ is an elementary function if and only if $\alpha \in \{1, 2\}$. Admitting the Lambert $W$ function extends closed-form solvability strictly to $\alpha \in \{-1, 1/2\}$. Finally, the asymptotic limit $\alpha \to 0$ converges to a singular dynamics solved by the logarithmic integral. These results constitute a complete classification of all explicit closed-form solutions admitted by this system.

Communities exhibiting cyclic (i.e., “winnerless”) competition occur throughout nature. These systems can exhibit qualitatively different long-term behavior depending on the strength of inter-species competition and a specific property of the number of interacting species: parity. We consider how ecological communities can adapt to and transition between an odd and an even number of cyclically competing species. Previous studies have shown that, for strong inter-species competition, odd parity communities are dynamically unstable, while species in even parity communities form stable alliances of maximal noncompeting sets. We trace a homotopy between two May-Leonard type models: the odd parity N=3 model and the even parity N=4 system. We identify all physical steady states and analyze their stabilities. We show the existence of stable transitional states that are unique to our symmetric model. We then use a piecewise linear approximation technique to characterize the stability of heteroclinic cycles. We use numerical simulations to map distinct regimes in parameter space. Our computations confirm our analytical results and also reveal a window in parameter space where limit cycles occur. We illustrate that parity transitions can be complex and characterized by diverse parameter-dependent pathways.

We study polygonal projective billiards whose reflection law is constant on each side. We give a single explicit family with an open set of periodic trajectories for every period at least three, including every odd period, and identify the previously known centrally projective examples as a special parameter value. For period five we describe the full scalar-monodromy variety, prove its irreducibility, and show that every configuration preserves a nonzero quadratic form. Under the stated distinct-centre and non-collinearity hypotheses that form is nondegenerate; this yields an explicit unirational classification of the convex five-reflective locus. We also relate the definite members to spherical billiards and prove that, among analytic simply connected constant-curvature surfaces with geodesic walls, only the sphere can carry an open family of periodic trajectories. For general complete analytic ambients with global wall reflections we prove compactness and a common period for the geodesic flow. The arguments reduce the billiard return map to products of harmonic homologies and separate the algebraic closure condition from the strict first-impact inequalities.

We prove that the dynamics on a non-linearizable hedgehog is uniquely ergodic. This solves a conjecture of the author formulated in 1995. Its unique invariant probability measure is the Dirac mass at the indifferent fixed point. The proof relies on the techniques of quasi-invariant curves and the hyperbolic form of the Denjoy-Yoccoz lemma developed by the author.

The sterile insect technique (SIT) is a biological control method that reduces insect pest and disease vector populations through the release of sterile insects (in general only males). In this work, spatial aspects are accounted for through an explicit metapopulation model of $n$ interconnected patches. Each patch includes a non-adult stage and adult stages divided into females and males, with distinct dispersal networks for females, wild males, and sterile males. This framework allows to adopt an area-wide integrated pest management perspective. Focusing on the oriental fruit fly, Bactrocera dorsalis, we consider two types of additional control methods, with life stage and sex specificities. These correspond to entomopathogenic fungi soil treatment (EPFST), with a differentiated effect at every life stage; and to the male annihilation technique (MAT), which attracts and kills wild and sterile males. We derive a sufficient condition ensuring elimination of the wild population in all patches, considering either continuous or impulsive periodic release strategies. Under constraints such as the impossibility of releasing sterile males in some patches, we then study an optimisation problem to minimise a weighted sum of the release rates over all release patches to eliminate the wild population. For illustrative purposes, numerical simulations are used to investigate how dispersal network configurations, local conditions, and the simultaneous use of SIT with EPFST or MAT affect optimal release strategies. The main contribution of this work is to identify efficient spatial release strategies for population elimination under practical constraints, in combination with other control methods.

This paper proves generic structural stability for Riemann solutions to $n \times n$ systems of hyperbolic conservation laws in one spatial dimension. Under assumptions of strict hyperbolicity, genuine nonlinearity, and a regular manifold hypothesis on the Rankine-Hugoniot map, we show that for almost every pair of left and right states, any $n$-wave Riemann solution consisting of Lax-admissible shocks and rarefactions is structurally stable under perturbations of the left state, the right state, and the flux function in the $C^2$ topology. The central new idea is sequential transversality, which chains the $n$ waves through intermediate states and transports their tangent contributions to a common reference point via pushforward maps, reducing the structural stability condition to the invertibility of an $n \times n$ transversality matrix. We apply the results to the $p$-system, polydisperse particle-laden thin films, and machine-learned flux approximations.

We study magnetic flows on closed Riemannian surfaces of genus at least two. Let $(M,g)$ be a surface without focal points and let $\Omega$ be a magnetic field. We prove that, above the Mañé critical value, the absence of conjugate points imposes geometric restrictions on the magnetic flow. First, we obtain a bound for the integral of the Lorentz force in terms of the geodesic curvature of the horocycles of the underlying Riemannian metric. As a consequence, a lower bound on the Gaussian curvature yields an explicit bound in terms of the area of the surface. The proofs combine the global geometry of magnetic geodesics with Liouville's formula for the geodesic curvature in the orthogonal coordinates given by the level sets and geodesics of a Busemann function on the universal cover.

We study a one-dimensional self-consistent transfer operator generated by a repulsive mean-field coupling. The interaction admits a natural interpretation as a myopic relocation dynamics balancing congestion and uniformly distributed resources. Using the quantile representation of probability measures, combined with self-consistent transfer operators methods, we prove the existence and uniqueness of a physically relevant invariant measure for the system and characterize it explicitly through a nonlinear fixed-point equation. We then show that the corresponding finite-particle equilibrium converges to the continuum equilibrium with optimal order in Wasserstein distance and establish a quantitative thermodynamic limit.

Let $A$ be an $\alpha$-Hölder cocycle over a transitive two-sided subshift of finite type, and assume that $A$ is strongly bunched. We prove that if there exist $k\in\{1,\ldots,d-1\}$ and $c>0$ such that \[ \lambda_k(p)-\lambda_{k+1}(p)\ge c \] for every periodic point $p$, then $A$ admits a dominated splitting of index $k$. Thus, under strong bunching, a uniform asymptotic gap in the Lyapunov spectrum of periodic measures implies a uniform geometric splitting along all orbit segments. This extends the periodic-gap criterion of Kassel–Potrie from locally constant cocycles to Hölder cocycles in arbitrary dimension, and extends the two-dimensional fiber-bunched result of Velozo to higher dimensions under the stronger bunching assumption.

We prove a multidimensional Birkhoff theorem for a new class of "$C^0$ Lagrangian subsets" in cotangent bundles, obtained as Hausdorff limits of compact exact Lagrangian submanifolds with control on their Liouville primitives. If the successive images of such a subset $L$ under the flow of a Tonelli Hamiltonian admit convergent subsequences in both positive and negative time, then $L$ and all its images are Lipschitz graphs over the base. This extends Birkhoff's celebrated theorem for twist maps of the annulus and its known higher-dimensional generalizations, and provides a first version of such results for merely continuous objects. The proof combines Floer-theoretic graph selectors, variational solutions of the Hamilton-Jacobi equation, and weak KAM theory. We also investigate the rigidity and uniqueness of limiting primitives for this new class of singular Lagrangian subsets, which may be of independent interest in $C^0$ symplectic topology.

We prove that transcendental entire functions do not have bounded-orbit wandering domains. This answers a long-standing open question. In fact, we prove a more general theorem, establishing the absence of certain wandering domains even for locally defined analytic functions. The proof is based on a recent remarkable argument of Ye, who obtained an elementary proof of Sullivan's classical theorem that rational maps do not have wandering domains.

In [Be17], we introduced the notion of emergence to quantify the statistical complexity of a dynamical system. Roughly speaking, emergence measures the number of probability measures required to describe, up to a given precision $\epsilon$, the statistical behavior of most orbits. A system is said to have high emergence when this number grows super-polynomially as $\epsilon\to0$. We survey recent developments in a program aimed at understanding the prevalence of high emergence in differentiable dynamics. We review several notions of emergence and their connections with quantization, ergodic decompositions, and entropy, and discuss examples exhibiting high or maximal emergence in conservative, symplectic, analytic, and dissipative dynamics, as well as in constrained families such as unimodal, Hénon, and rational maps. We also present new results concerning variants of metric and topological emergence, convexity properties, local emergence, and variational principles, together with a collection of open problems on the typicality of high emergence.

In symbolic dynamics, the fixed-point construction from arXiv:0910.2415 defines shift spaces of finite type whose configurations embed infinite hierarchies of tilings. This article provides a "black box" abstraction of this method phrased in terms of substitutions and $S$-adic limit spaces operating over sequences of increasingly large alphabets. By quantifying the amount of information computed by the substitutions at each level, and using a suitable parallel model of computation, we provide a simple positive criterion of multidimensional soficity that generalizes classical examples from the literature.

In this paper, we study the Abelian maximal pattern complexity $p_{\alpha}^{\ast \mathrm{ab}}(k)$, introduced by Kamae, Widmer and Zamboni, of infinite words $\alpha \in \mathbb{A}^{\mathbb{N}_{0}}$ over finite alphabets $\mathbb{A}$. For recurrent aperiodic words, we determine a lower bound and prove its sharpness. We further give a structural characterization of the words with minimal Abelian maximal pattern complexity. In the general case, we prove that an infinite word $\alpha $ is aperiodic if and only if $\binom{p_{\alpha }^{\ast \mathrm{ab}}(k)}{2}\geq k$ for every $k.$ For aperiodic words over $\ell \geq 2$ letters, each occurring infinitely often, we further prove that $p_{\alpha }^{\ast \mathrm{ab}}(k)\geq m$ whenever $\binom{m}{2}\leq (\ell -1)(k-\ell +2)$, for all $m,k$. Together with a matching construction, this shows that the minimum Abelian maximal pattern complexity in this class is $\sqrt{2(\ell -1)k}+O_{\ell }(1)$. We call a word an Abelian pattern Sturmian word if, at every $k$, its Abelian maximal pattern complexity is the least positive integer $m$ satisfying $\binom{m}{2}\geq k$. We show that a word is Abelian pattern Sturmian if and only if, after relabeling its alphabet, it is the characteristic word of an infinite set $E\subset \mathbb{N}_{0}$ for which the bipartite graph on two disjoint copies of $\mathbb{N}_{0}$, with a left vertex $r$ adjacent to a right vertex $s$ exactly when $r+s\in E$, is a forest.

Data assimilation estimates a dynamical state from partial and noisy observations. Classical ensemble filters are efficient but restrict analysis updates through finite sample covariance and affine Gaussian distribution. We introduce the Flow Ensemble Filter (FlowEF), which uses conditional flow matching to transport the forecast ensemble from a classical baseline filter to an analysis ensemble. FlowEF uses a localized Gaussian source during training, transports forecast ensemble members from a baseline filter at deployment, and conditions its velocity field on ensembles from that baseline filter and the observation. The proposed model therefore learns a nonlinear update while mapping each baseline ensemble independently. For sparsely observed dynamical systems, FlowEF improves both deterministic and probabilistic metrics over all four classical ensemble filters. It also achieves the best performance among the state-of-the-art generative data assimilation models.

We investigate the computation of optimal bias potentials for accelerating transitions between metastable states and for computation of equilibrium properties in molecular dynamics simulations. We formulate optimal biasing as an ergodic optimal control problem (OCP), which can be recast as a linear eigenvalue problem for the infinitesimal generator of the unbiased dynamics. We demonstrate that data-driven learning methods for the generator enable reliable solution of the OCP, computation of biasing potentials, extraction of equilibrium properties, and acceleration of state transitions. We also explore the relation of the control problem to coarse grained representations and learning of coarse grained dynamics.

Rapid growth of data centers and artificial-intelligence services is producing computational loads at scales once associated mainly with largest power plants. Recent grid events show that a routine transmission disturbance can cause several gigawatts of data-center demand to disconnect or transfer to backup nearly at once. This article revisits the classical synchronization and transient-stability theory needed to reason about such events. We organize four lines of work—graph-based synchronization conditions, winding-number descriptions of nonlinear power flow, separable convex network optimization, and direct energy methods—into a single cycle-space certificate framework for the lossless fixed-voltage model. The static layer gives an exact strict-cohesion test within a prescribed winding cell and reveals the widely used Dörfler–Chertkov–Bullo test as a quadratic surrogate of the same convex problem. The dynamic layer converts the critical-energy calculation into a finite family of convex boundary problems. Standard MATPOWER benchmarks illustrate both what the stronger static test gains and where it gains nothing: the 118-bus case admits $16.2\%$ more loading than the sufficient screen, while the 39-bus case is bridge-limited and the thresholds coincide. A stylized $2.7$-GW 39-bus event further shows that transient margin can change by about a factor of two depending on where balancing power is supplied, even when every final balanced operating point remains statically feasible. The result is a tutorial synthesis and an extensible deterministic certificate for emerging gigawatt-scale computational-load contingencies.

Large coupled systems may be chaotic at the microscopic level while exhibiting stable collective behaviour at the macroscopic level. Distinguishing between these two forms of instability has been a problem in the physics literature, and several approaches have been proposed. In this paper, we address this problem by introducing and studying observable Lyapunov exponents, which measure the rate at which perturbations of initial conditions grow or decay when viewed through a chosen observable, rather than in the full phase space. For globally coupled maps, we consider finite-time observable Lyapunov exponents and define their mean-field counterpart by first taking the infinite-system limit and then the long-time limit. Our main result shows that, for a broad class of symmetric macroscopic observables, these exponents turn out to be determined by the linearisation of the self-consistent transfer operator, which is the nonlinear operator governing the evolution of the population distribution. We then apply this result to weakly coupled uniformly expanding maps and prove that their mean-field Lyapunov exponents are negative near a stationary mean-field state. Thus, although the microscopic dynamics is chaotic, perturbations decay at the macroscopic level. Our results provide a rigorous framework for distinguishing microscopic from macroscopic chaos in large interacting systems.

We study bounded operators whose prescribed powers remain uniformly close to the identity, and resolvent conditions sampled along sequences of powers. The displacement bound $\sup_{n\ge1}\|I-T^n\|\le q<2$ forces a finite peripheral spectrum consisting of odd-order roots of unity and makes an explicitly determined odd power of $T$ a Ritt operator. The sharp unconditional threshold for $T$ itself is $\sqrt3$. If $\Phi_*(q)$ denotes the optimal universal Ritt resolvent bound below this threshold, then $\Phi_*(q)=2/(\sqrt3-q)+O(1)$ as $q\uparrow\sqrt3$. At every higher peripheral threshold the optimal finite-peripheral resolvent bound has reciprocal order of growth. We also obtain constructive bounds for general weighted Wiener symbols. For each $1<q<2$, an angular escape invariant characterises exactly the sampling sequences for which a displacement bound by $q$, together with the peripheral spectral condition, forces the Ritt property. We compute this invariant for asymptotically geometric sequences. At the endpoint $q=1$, suitable phase conditions imply the full displacement bound without assuming power boundedness; normal contractions admit an exact scalar criterion. Finally, we establish quantitative Hilbert-space and $L^p$ resolvent estimates, a characterisation of operators with odd Ritt powers on uniformly convex spaces, and strict norm gaps for operator-valued disc-algebra functions, with explicit matrix-polynomial bounds from Fejér–Riesz factorisation.

This paper proves the Gromov–Yomdin property for smooth projective varieties with finite Albanese morphisms. Our approach is to study the orbits of stability conditions under the action of autoequivalences on Bridgeland stability manifolds. For a finite generating set of the derived category, we consider the locus on which all its objects are semistable. Using this locus, we formulate an orbit escape principle, which provides a sufficient condition for an autoequivalence to satisfy the Gromov–Yomdin equality. Together with the stability of simple semihomogeneous bundles with respect to arbitrary numerical stability conditions, this principle readily yields the result for abelian varieties. We then reduce the case of varieties with finite Albanese morphisms to the abelian case by passing to suitable finite étale covers. Finally, we construct autoequivalences with positive categorical entropy on every abelian variety and show that a smooth projective variety with a finite Albanese morphism admits such an autoequivalence if and only if it is not of general type.

We develop a method for constructing minimal subshifts with a prescribed bound on the factor complexity within any strong orbit equivalence (SOE) class. This implies realizations of topological entropies within any SOE class, strengthening a series of classical results by Sugisaki. While prior work established complexity controls for zero-entropy systems, the approach of the present work extends to positive-entropy regimes. Our construction introduces the class of pseudo-Toeplitz subshifts, and we show that they exist in any SOE class. More precisely, for any $\alpha \geq 1$ and sequence $g_n$ growing exponentially at rate $\log(\alpha)$, we construct a system in this class whose topological entropy is $\log(\alpha)$ and whose complexity grows strictly faster, or, under certain conditions, slower than $g_n$. As a consequence, any Choquet simplex can be realized as the set of invariant measures of a Toeplitz subshift with precisely controlled complexity, in either a zero or positive-entropy regime.

We prove that the amplitude of the unique limit cycle of the van der Pol equation $\ddot{x}+\mu(x^2-1)\dot{x}+x=0$ is strictly greater than $2$ for every $\mu>0$. The proof combines orbit comparison in the relaxation oscillation regime, trigonometric comparison curves in the weakly nonlinear regime, and polynomial energy estimates in the intermediate regime. The polynomial sign conditions arising in the latter two regimes are certified using exact rational arithmetic and Bernstein coefficients. The orbit comparison method also applies to a class of symmetric Liénard systems. Under suitable monotonicity assumptions, we establish a sufficient criterion for the limit cycle amplitude to exceed a critical value determined by the damping function.

Self-regulation together with mutual promoter competition provides a simple mechanism for bistability in gene-regulatory models. We study a two-gene system with regulatory terms of Hill exponent one, allowing distinct basal production rates and distinct degradation rates. Each gene product, when bound to its own promoter, may enhance or reduce production relative to the basal rate, while the two products compete through promoter occupancy. We show that the system has at least one and at most three equilibria in the positive quadrant. Exactly two positive equilibria can occur only if one nullcline intersection is degenerate; consequently, a configuration in which all positive nullcline intersections are transverse has either one or three positive equilibria. If there are exactly three distinct positive equilibria, then all three are automatically hyperbolic: the two outer equilibria are asymptotically stable nodes and the middle equilibrium is a saddle. Moreover, every positive solution converges to an equilibrium. Hence the positive quadrant is the disjoint union of the basins of attraction of the two stable nodes and the one-dimensional stable manifold of the saddle, yielding global bistability.

A high dimensional bi-directionally coupled $N$-loop thermosyphon system in a tandem is modeled by $3N$ Lorenz type of differential equations. The fluid flow in each loop is driven by the heat source (Rayleigh numbers) as well as the moment and heat exchanges between the adjacent loops. The flow becomes chaotic when the Rayleigh numbers are large. A decentralized controller design via proportional local state feedback is employed to stabilize the chaotic flows in each loop. As a result of the stability analysis, we show that there exist lower bounds on the controller gains that guarantee global stability of the system. Under the scenario of unknown parameters, we augment the original system with additional dynamic equations of the feedback gains to adaptively determine the feasible gains that stabilize the system. The analysis also sheds insight on the role of thermal coupling in the stability of the control system, which allows us to extend the results to negative $z$-state coupling coefficient as well as a class of nonlinear $z$-state coupling functions. Numerical simulations further demonstrate the effectiveness of the proposed controller design

Naryshkin recently proved that a free action of a countably infinite discrete amenable group $G$ on a compact metrizable space $X$ with the uniform Rokhlin property and comparison (URPC) admits an equivariant embedding into $([0,1]^m)^G$ whenever $\mathrm{mdim}(X,G)<m/2$. We strengthen this result from existence to genericity by a substantially different method: the set of observables $f\in C(X,[0,1]^m)$ whose orbit maps are equivariant embeddings into $([0,1]^m)^G$ is a dense $G_\delta$ subset of $C(X,[0,1]^m)$. The proof combines an adaptation of the Kakutani skyscraper construction used by Gutman–Tsukamoto with a new polynomial perturbation method, departing from the predominantly linear perturbation approaches used in earlier equivariant embedding literature.

We study oscillation of the second-order delay equation \[ x''(t)=x(\tau(t)), \] with a nondecreasing delayed argument $\tau(t)\le t$ satisfying $\tau(t)\overset{t\to\infty}{\to}\infty$. Set \begin{equation*} a=\liminf_{t\to\infty}(t-\tau(t)),\qquad b=\limsup_{t\to\infty}(t-\tau(t)). \end{equation*} We describe a continuous strictly decreasing function \begin{equation} \beta:[0,2/e]\longrightarrow[2/e,\sqrt2],\qquad \beta(0)=\sqrt2,\qquad\beta(2/e)=2/e, \notag \end{equation} such that, if the equation has an eventually positive decreasing solution, then \begin{equation} a\leq2/e,\qquad b\le\beta(a). \notag \end{equation} The bound $b\le\beta(a)$ is sharp and interpolates between the classical $\limsup$ threshold and the $\liminf$ autonomous threshold. A detailed spectral analysis gives a simple interpretation of the result and links it directly to the oscillation theory of first-order delay equations.

This paper studies weak mixing for two classes of dynamical systems with the \(\mathscr{M}_{\alpha}\)-shadowing property using the generic factor characterization of weak scattering. First, for every \(\alpha\in[0,1)\), every \(M\)-system with the \(\mathscr{M}_{\alpha}\)-shadowing property is weakly mixing, thereby answering [P. Oprocha et. al., On partial shadowing of complete pseudo-orbits, J. Math. Anal. Appl., Question~1] negatively. Second, for every \(\alpha\in(0,1)\), the \(\mathscr{M}_{\alpha}\)-shadowing property implies weak mixing for \(E\)-systems and, in particular, for dynamical systems whose measure center is the whole space.Finally, an \(E\)-system with the \(\mathscr{M}_{0}\)-shadowing property which is not weakly mixing is constructed, showing that the parameter range \(\alpha\in(0,1)\) in the \(E\)-system result is optimal.