Papers

The Sprott–Zeraoulia conjecture, formulated on the basis of numerical experiments, states that three-dimensional quadratic continuous-time systems with symmetric Jacobian matrices cannot exhibit chaotic behavior. We give an analytical resolution of this conjecture in terms of the principal precise notions of chaos used for continuous dynamical systems. The starting point is the classical, but decisive, fact that symmetry of the Jacobian on $\mathbb{R}^3$ is equivalent to a global gradient representation. Hence every system in the conjectured class can be written as $\dot{x}~=~\nabla V(x)$, where $V$ is a polynomial of degree at most three, and the conjecture becomes a problem about cubic polynomial gradient flows. For bounded trajectories, the monotonicity property of gradient systems first confines every omega-limit set to a critical level of $V$. When the equilibria on that level are isolated, convergence follows from the connectedness of the omega-limit set. In the general case, which may include curves or surfaces of equilibria, the Łojasiewicz gradient inequality implies finite length and convergence of every bounded forward trajectory to a single equilibrium. Consequently, compact invariant sets support no nonstationary recurrence, positive-entropy measure, topological transitivity, Smale horseshoe, or Devaney, Auslander–Yorke, Li–Yorke, mean Li–Yorke, and distributional chaos. Solutions escaping to infinity are treated through the Poincaré compactification. Finally, we connect the gradient formulation with earlier Darboux-theoretic criteria: invariant algebraic surfaces satisfy $\langle\nabla V,\nabla f\rangle=Kf$, and an invariant affine plane with constant cofactor produces an exact tangential–normal decomposition of the potential and the flow.

We show that for any real, connected, semi-simple matrix group, its Lie algebra of algebraic vector fields can be generated by three complete vector fields.

We describe in detail and provide computer code for constructing optimal geometric braids of three strands from algebraic data encoding the braiding pattern. Our optimality criterion uses the known interpretation of braids as homotopy classes (rel endpoints) of paths in $SL_2(\mathbb{R})$ joining the identity $I$ to some $A \in SL_2(\mathbb{Z})$, i.e., elements of the fundamental group of $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$, a quotient equivalent to the unit tangent bundle of the classical modular surface $\mathbb{H}/PSL_2(\mathbb{Z})$. The main technical result finds the length minimizing geodesic in a prescribed homotopy class. From another perspective, of independent interest, this amounts to shooting the shortest geodesic that connects, with a prescribed number of spins en route, two given unit tangent vectors to the Poincaré (half-)plane $\mathbb{H}$. The length is measured by a Riemannian metric from a family of deformed Sasaki metrics, sometimes called Kaluza-Klein metrics, whereby unit tangent vectors can be interpreted as infinitesimal rotors, called spinners, and the ratio of the mass to the moment of inertia is the deformation parameter. At the universal covering level $\widetilde{SL}_2(\mathbb{R})$, the geometry is one of Thurston's eight model 3D geometries. In the vanishing mass limit, it converges to the better understood Carnot-Carathéodory contact geometry, where our geodesic shooting extends known formulas. The finite mass case is more delicate, requiring numerical solution of a targeting equation. The characterization of the length minimizing geodesics (No-multiplicity Theorem) and the resulting targeting equation are the main original contribution. The exposition is complete and multi-pronged, aimed at a broad spectrum of readers. (Numerous figures are the backbone of the narrative and should be viewed in color.)

We develop a reduction and transfer framework for cohomological, variational, and thermodynamic problems in fiber-contracting extensions of local homeomorphisms. Under uniform contraction along the fibers and the existence of a continuous global section, every Hölder potential admits the explicit decomposition \[ \varphi=\psi\circ\pi+u-u\circ F, \] where $\psi$ is defined on the quotient and $u$ is obtained by a uniformly convergent fiberwise series. We give quantitative criteria for continuity and Hölder regularity and use the reduction to transfer subcohomological and Livšic-type statements from the quotient to the extension. We prove that invariant measures on the extension and quotient are in bijection, invariant averages are preserved, and the fibers have zero relative topological entropy. Consequently, metric entropy, ergodic optimization, pressure, minimizing measures, and equilibrium states correspond. We further establish weak Gibbs transfer under subexponential comparison of Bowen-ball masses and prove blockwise Bowen sequential weak Gibbs transfer along prescribed point-dependent Gibbs times; under uniformly bounded sequential mass distortion, the strong Bowen sequential Gibbs property is preserved. Concrete criteria for Bowen-ball comparability are provided. Finally, affine and symbolic contracting skew-products illustrate the framework, including the explicit reduction \[ \varphi(x,y)=g(x)+by \quad\;\;\;\;\;\;\;\;\quad \psi(x)=g(x)+\frac{b}{1-a}\rho(x) \] for $F(x,y)=(f(x),ay+\rho(x))$, $|a|<1$.

We solve the problem of explicitly constructing a continuous function whose unique maximizing measure for the doubling map is Lebesgue measure. More generally, given a nontrivial compact metrizable abelian group and a continuous surjective endomorphism for which normalised Haar measure is ergodic, we explicitly construct a continuous function on the group for which Haar measure is the unique invariant maximizing measure. The function is the uniform limit of a recursively defined sequence of trigonometric polynomials with rational coefficients; every parameter of the recursion is given by a closed formula, every step is exact, and the rate of convergence is explicit. In specific cases, we further obtain a uniformly convergent Fourier expansion in the classical frequency order, each of whose coefficients is rational and computable exactly, by a finite procedure.

Astrophysical binaries formed by a stellar-mass compact object and a massive black hole are among the most promising sources for future space-based gravitational-wave detectors. These extreme mass-ratio inspiral (EMRI) systems are tracked coherently over more than $10^{5}$ orbits, encoding detailed information about strong-field gravity. Translating a detection into physical parameters requires models of comparable precision. One outstanding difficulty is the treatment of transient orbital resonances. In this work we address the problem at the level of the Koopman operator, where the waveform is promoted to an observable and the evolution over one radial cycle is a linear operator. Removing the fast timescale without a singularity at a crossing is then a matter of choosing a function of that operator. We demonstrate the approach on a Kerr geodesic driven through the 3:2 crossing by a modelled forcing term. There the coefficient of the standard near-identity transformation diverges, while the finite-window generator attains its analytic bound to better than one part in $10^{6}$. Averaging over twice the crossing duration recovers the resonant jump with its predicted magnitude and phase dependence.

Taking classical information geometry as its point of departure, this paper investigates, through gradient flows, how dually flat geometry extends beyond regular convexity, non-degeneracy, and smoothness. The regular theory is developed from the log-determinant potential on positive definite Gram matrices, establishing its Legendre dual, Fisher–Rao metric, Bregman divergence, and generalized Pythagorean theorem. We connect this framework to Craig–Sakamoto deformation, Wolfe duality, and, via Yoshizawa's embedding, Brockett–Bloch–Ratiu double-bracket flows, linking isospectral dynamics, Stiefel optimization, and component learning. The Bures–Wasserstein geometry provides a complementary gradient-flow structure. The singular theory emerges from boundary behavior: difference-of-convex deformations produce indefinite or degenerate Hessians while retaining pseudo-Hessian, dually flat, Legendre-self-dual structures. Newton flows exhibit finite-time collapse or Łojasiewicz-controlled convergence near non-Morse critical sets. Fisher-metric degeneracies on the Birkhoff polytope and elliptic-curve moduli are resolved by explicit blow-ups, yielding a birationally invariant exponential decay law. We further derive a closed-form Kirillov Jacobian and introduce cross curvature as a spectral diagnostic of local escape rates, including a new Box–Cox interpolation. Reproducible numerical experiments support the closed-form results. Rather than claiming a completed theory, the paper provides foundations for singular information geometry centered on degenerate pencils, indefinite dual flatness, blow-up geometry, and Łojasiewicz-type convergence.

In this paper, we study the maximal pattern complexity of two-dimensional words up to Abelian equivalence. We establish a lower bound for the Abelian maximal pattern complexity of two-dimensional words that are non-doubly periodic by projection under the existence of a transverse recurrence direction or strong recurrence. We further show that the bound is attained for every alphabet size. As a consequence, we characterize double periodicity of strongly recurrent binary words by the boundedness of their Abelian maximal pattern complexity.

In this article, we first investigate the box dimension results for the graphs of continuous functions defined on the product of two self-similar fractals ({\mathfrak{F}}s), where the fractal {\mathfrak{F}} satisfies the open set condition. Following this, we manifest the existence of a prevalent surface, defined by a continuous function over the product of two {\mathfrak{F}}s, that satisfies the horizon property (associated with the box dimension), i.e., the box dimension of the prevalent surface is greater than that of its horizon up to a constant {\eta}, where {\eta} is the box dimension of the self-similar fractal {\mathfrak{F}}.

The Trachenko-Zaccone equation provides a compact nonlinear dynamical framework for describing non-Debye relaxation in disordered condensed matter. Originally developed to rationalize stretched- and compressed-exponential relaxation in liquids and glasses from the dynamics of interacting local relaxation events, the same equation has subsequently appeared in broader contexts, including polymer relaxation and nonlinear models of global population dynamics. This review retraces the conceptual development of the equation, with particular emphasis on its physical origin in Kostya Trachenko's feed-forward interaction mechanism, its mathematical structure, and its possible generalizations. We also include personal recollections of the work with Kostya Trachenko at Queen Mary University of London in August 2019, during which the equation emerged in essentially its present form. After reviewing applications to stress relaxation, glassy materials, polymer relaxation and population dynamics, we discuss future directions including time-dependent feedback parameters, coupled order parameters, heterogeneous and spatially resolved formulations, flux terms, network versions and stochastic extensions. The central theme of the review is that the Trachenko-Zaccone equation should be viewed not only as a model of glassy relaxation, but as a general nonlinear feedback equation with potential applications across complex systems.

In this short note we show that certain definition from the work of Kanigowski and Ravotti can be relaxed. More specifically, we show that the definition of good special flows, that is, flows exhibiting a form of mechanism of mixing, shearing of Birkhoff sums can be relaxed. Specifically, we show that the disintegration corresponding to almost partitions, which are uniformly sheared along the direction of the flow, of the base can depend on $k$-tuple of times. This note serves as a complementary material to an article on mixing of higher order in flows of the author (details in the note), where we use the new definition of good special flows to show that flows of Fayad on $n$-torus, $n\geq 3$, are mixing of any order.

Let $G$ be a group, let $\mathbb{K}$ be a field, and let $V$ be a $\mathbb{K}$-vector space. We prove that there exists a bijective linear cellular automaton $V^G\to V^G$ whose inverse is not a cellular automaton if and only if $G$ is not locally finite and $\dim_{\mathbb{K}}V \geq |\mathbb{K}|^{\aleph_0}$. This answers an open problem proposed by T. Ceccherini-Silberstein and M. Coornaert.

We discuss the almost stability theorem of Dicks and Dunwoody in the context of probability measure preserving equivalence relations.

We prove that for every non-elementary torsion-free hyperbolic group $H$ and sufficiently large odd $n>1$, the quotient $H/H^n$ has cost $1$. In particular, when $H$ is the free group of rank $m\ge 2$, we obtain that the free Burnside group $B(m,n)$ has cost $1$ for every odd exponent $n\geq 665$. It follows that $B(m,n)$ is anti-treeable, and that its first $\ell^2$-Betti number $\beta_1^{(2)}(B(m,n))$ vanishes. The latter recovers and extends a result of Feldkamp and Kionke [Proc. Amer. Math. Soc., 2023], who proved that $\beta_1^{(2)}(B(m,p))=0$ for all sufficiently large prime exponents $p$.

The refined Diophantine exponent, recently introduced by the author, is a quantity that measures the periodicity of an infinite word. In this article, we study this exponent from combinatorial and topological viewpoints. First, we show that, over a ternary alphabet, the spectrum of the refined Diophantine exponent is $[1,\infty]$. Second, we show that this exponent has topological properties similar to those of the set of Liouville numbers. Finally, we provide concrete examples with the Champernowne, Rudin–Shapiro, and Thue–Morse words, words coming from coding a rotation by intervals, and bracket words.

We prove that the orbit equivalence class of every free ergodic probability-measure-preserving (pmp) action of a free group contains a totally weak mixing action. Equivalently, every ergodic treeable pmp equivalence relation of cost $n\in\mathbf{N}\cup\{\infty\}$ is generated by a free totally weak mixing action of $\mathbf{F}_n$. This answers a question of Miller and Tserunyan. The proof goes by considering a Polish space of edge slidings along a fixed mixing transformation and proving that for every $w\not=e\in\mathbf{F}_n$ the set of edge slidings that produce an action with $w$ weakly mixing forms a comeager set.

This article introduces and studies the entropy spectrum of a hyperbolic surface, that is the set of entropies of its subsurfaces. The main results are that the entropy spectrum is a reverse well-ordered multiset, with finite multiplicities, and that there is a quantifiable gap around the value $1$. This gap comes from a counting result on the number of non-filling geodesics which in turn comes from explicit estimates on the number of curves on surfaces with boundary in terms of geometric data. The geometric data includes lengths of boundary geodesics and so-called boundary width, which measures maximal distance to the boundary but can also be interpreted in terms of the topology of the surface and the systole.

We study bilateral teleoperation between nonlinear, multi-DOF robotic manipulators in the presence of constant communication delays. Unlike classical wave-transformation architectures that transmit a coordinating force, we consider the case where the environmental force is reflected to the master side to enhance teleoperation transparency. Since direct contact force feedback might destabilize the closed-loop system, we first develop a passivity-shortage characterization for the Euler–Lagrange remote system using a linear matrix inequality (LMI) approach. An upper strictly passive communication law is then employed to compensate for the computed passivity shortage so that the closed-loop stability under delays as well as position and force synchronization are preserved under appropriate conditions. Simulations with nonlinear 2-DOF robotic manipulators in different settings illustrate our approach.

Weather forecasting and climate projection frequently use multi-model ensembles (MMEs) to improve short-term forecasts by averaging across models. However, this practice is often not well justified or validated. Using reservoir computing (RC) as a computationally efficient alternative to large-scale physical models, we assess the validity of the MME approach for chaotic time series. By training multiple randomly constructed RCs on the same dataset, we create a multi-model ensemble in which each model has its own unique error. These model errors lead to very different forecasting performances, with forecast error distributions that exhibit heavy tails. The arithmetic mean across forecasts from multiple models for the same target is usually closer to the ground truth than most individual forecasts, and further improvement is achieved by weighted arithmetic means where the weights are constructed based on each model's test-set performance. We show that iterated forecasts over many time steps deviate from the ground truth along the unstable manifold of the target point, in both directions, so that, if forecast errors were independent and had zero mean, the arithmetic mean forecast should approach the true target like $1/\sqrt{\nens}$ where $\nens$ is the size of the multi-model ensemble. We observe deviations from this behavior, which we attribute to the tails of the error distribution of random RCs.

We study synchronization in the Kuramoto model on finite graphs. We prove that there is an absolute constant $\eta>0$ such that every finite simple graph $G$ on $n$ vertices with minimum degree at least $(3/4-\eta)n$ has no local minima of the Kuramoto energy other than the fully synchronized states. This strictly improves the previous $3/4$ upper bound and refutes a conjecture of Bandeira, Kireeva, Maillard, and Rödder.

When $\omega$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z) = \omega z (1 -z)$ and the entire map $F(z) = \omega z \mathrm{e}^{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $F^{\circ n}(z) = z \bigl( 1 +c z^n +\mathcal{O}(z^{n+1}) \bigr)$ with $c \neq 0$. The classical proof of this fact is transcendental. We present an arithmetic proof which may be extracted from [Towards global models near homoclinic tangencies of dissipative diffeomorphisms; H. Broer, C. Simó, J.C. Tatjer] in the transcendental case and requires working in $\mathbb{Z}/(n -1) \mathbb{Z}$, and which is new in the polynomial case and requires working in the $p$-adic field $\mathbb{Q}_p$ for a suitable prime $p$ such that the order of $2$ in $(\mathbb{Z}/p \mathbb{Z})^\times$ is exactly $n$.

A set of positive integers $R$ is called a set of nice recurrence if for any measure preserving system $(X,\mu,T)$, for all measurable $A\subseteq X$, and each $\varepsilon>0$, there exists $n\in R$ such that $\mu(A\cap T^{-n}A)\geqslant \mu(A)^2 - \varepsilon$. Answering a long-standing question of Bergelson, we show that sets of nice recurrence have the following Ramsey property: any finite colouring of a set of nice recurrence admits a monochromatic set of nice recurrence.

For the planar Newtonian 1+N-body problem when the N masses tend to zero, the corresponding relative equilibria become coorbital around the dominant mass. In this work, we focus on convex central configurations in the planar 1+N coorbital problem. For the 1+5 coorbital problem with the homogeneous potential, we prove that any convex coorbital central configuration with symmetric masses must have an axis of symmetry. Furthermore, under explicit restrictions on the angular variables in a homogeneous potential, we prove the linear stability of both convex symmetric 1+5 and convex 1+N coorbital central configurations.

Freidlin–Wentzell theory converts weak-noise large deviations into a Hamiltonian variational problem. We study how a $k$-dimensional normally hyperbolic invariant manifold (NHIM) $N$ of the deterministic dynamics appears in this Hamiltonian system. Its zero-momentum copy $N_0=N\times\{0\}$ is invariant, but the Hamiltonian dynamics has $2k$ center directions near $N_0$: $k$ tangent to $N$ and $k$ conjugate covector directions. We construct the resulting local symplectic geometry and show that fluctuation extremals approaching $N_0$ backward in time at the strong normal rate form an $n$-dimensional exact Lagrangian invariant manifold carrying a single-valued action. A counterexample shows that a corresponding zero-energy section need not be normally hyperbolic within $H_{\mathrm{FW}}^{-1}(0)$. We then consider reaction dynamics. If a parameter moves a deterministic trajectory toward a codimension-one reactivity boundary, the minimum Freidlin–Wentzell action required to reach the boundary is quadratic in the distance from the threshold parameter. Its coefficient is determined by the relative motion of trajectory and boundary and by how effectively the available noise acts transversely. This deterministic boundary is distinct from a noise-dependent stochastic transition state or a committor surface. In a solvent–solute model, varying solvent mass moves the phase-space reactivity boundary while leaving the potential-energy surface fixed, changing the rare-event cost without changing the potential-energy barrier.

Discovering dynamical systems from trajectory data is a central problem in applied mathematics and engineering. Whilst recent advances in machine learning have led to strong progress in data-driven system identification, much less attention has been given to systems with discontinuous dynamics. These systems are nevertheless highly relevant in applications, including climate dynamics and mechanical systems with friction. In this work, we consider the problem of identifying piecewise-smooth dynamical systems directly from trajectory data. Compared with the smooth setting, this requires recovering the governing equations and detecting the switching hyperplanes that separate different dynamical regimes and characterising their behaviour, such as sliding motion. We present a modular framework for discovering such systems by first estimating switching hyperplanes from data and then learning smooth dynamics within each region using geometry-constrained neural networks. The geometry-learning phase is studied from a statistical perspective, analysing the identifiability of the discontinuities and the robustness of the procedure. We also introduce a novel neural network architecture with a prescribed discontinuity set, and provide a theoretical analysis of its approximation properties. The approach is tested on low-dimensional benchmark problems, including dry-friction oscillators and the PP04 climate model for the ice ages.

This paper investigates the open cover specification property and its role in the study of topological dynamical systems. We show that this property is a topological invariant and a natural extension of the classical topological specification property from uniform spaces to general topological spaces, and that it is preserved under infinite products with compositions. We further prove that on regular spaces, the strong open cover specification property implies Devaney chaos, and that on locally compact, first countable Hausdorff spaces, it guarantees both distributional chaos and Li Yorke chaos. These results demonstrate the significance of the open cover specification property in linking topological structure with dynamical complexity.

Large Language Models (LLMs) are increasingly deployed in high-stakes applications, yet their tendency to generate toxic, harmful, or policy-violating content poses significant risks. Detecting these unsafe outputs efficiently in a black-box manner remains an open challenge. In this paper, we extend a recently proposed dynamical systems framework designed for hallucination detection to LLM safety classification. By projecting both prompts and responses into high-dimensional embedding spaces and fitting separate Koopman-based predictive models for safe and unsafe regimes, we classify new outputs using a new differential residual score that compares prediction errors of the safe and unsafe regimes. A key contribution is the incorporation of the prompt and response embedding dynamics, yielding fitted Koopman operators that capture crucial interaction patterns. We evaluate our black-box method across three safety benchmarks using three embedding models. Our results show that incorporating prompt embeddings yields consistent improvements, particularly for interaction-dependent violations when paired with causal decoders (e.g., in Llama-3), while response-only violations benefit more from dense semantic embedding representations. These findings opens the door for using dynamical systems to analyze AI systems rather than the dominant paradigm of using AI to model dynamical systems.

Zhao and Zhao (2026) established a codimension-four Bogdanov–Takens singularity in a modified Leslie–Gower predator–prey model through a recursive normal form computation. We show that the nonlinear coefficients governing the restriction of the reduced system to the singularity's distinguished eigendirection are not independent. Instead, a single exact rational identity, already present in the transformed equations, generates the entire family of pure Taylor coefficients and proves that their simultaneous vanishing is an exact all orders property rather than a finite-order coincidence. We further show that this structural identity propagates through the planar Bogdanov–Takens normal form algorithm of Kuznetsov (2005). Although the pure coefficients vanish identically, the corresponding normal form coefficients are generated by mixed quadratic interactions. This leads to an explicit characterization of the degenerate Bogdanov–Takens locus and reduces the search for higher-order Bogdanov–Takens degeneracies to a single explicit algebraic condition.

For the planar Liénard differential system $\dot{x}=F(x)-y$, $\dot{y}=x$, where $F(x)$ is a piecewise linear function, Tonnelier (SIAM J. Appl. Math., 2002) conjectured that the maximum number of limit cycles of the system is $n$ when $F(x)$ has $n$ fold points and no jump points, and $2n$ when $F(x)$ has $n$ jump points and no fold points. This conjecture was confirmed by Llibre et al. (J. Nonlinear Sci., 2015) (resp. Chen et al. (J. London Math. Soc., 2026a)) when $F(x)$ has one fold point and no jump points (resp. two fold points). More recently, Chen et al. (J. London Math. Soc., 2026b) proved that the conjecture is correct when $F(x)$ has no fold points and one jump point. All other cases remain open. Here we verify that the lower bound for the maximum number of limit cycles of the system can be $n$ when $F(x)$ has only $n$ fold points, and $2n$ when $F(x)$ has only $n$ jump points, thereby confirming the lower bound part of Tonnelier's conjecture. Moreover, when $F(x)$ has $m$ jump points and $n-m$ fold points, $0\le m\le n$, we also show that the system can have $n+m=(n-m)+2m$ limit cycles. In addition, a complete classification of the {dynamics} near infinity for this class of systems is provided.

Let $M$ be a closed connected Riemannian manifold. A continuous semimartingale segment can be closed by a Borel family of paths of uniformly bounded length, producing a singular homology class $H_t\in H_1(M;\R)$. For every linear choice of smooth closed representatives of $H^1(M;\R)$, the corresponding Stratonovich homology differs from $H_t$ by a uniformly bounded term, almost surely and uniformly in time. The classes $H_t$ are additive under time shift up to a uniformly bounded error. For Brownian motion this comparison yields the Gaussian central limit theorem with covariance given by the normalized Hodge inner product, a functional central limit theorem for the polygonal interpolation of the closed homology classes, and the almost sure limit $H_t/t\to0$. For elliptic diffusions, the large-deviation principle of Galkin–Mariani passes to $H_T/T$ with the same rate function. The construction uses Schwartzman's closing procedure and is independent, at these asymptotic scales, of the chosen bounded closing family.