math.DS — Pith

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math.DS 2026-05-13 2 theorems

Timed dengue controls suppress transmission risk in seasonal models

Optimal Scheduling of Dengue Vector Control

Miami temperature simulations show that adjoint-optimized schedules of larvicide, adulticide and breeding-site reduction lower the time-var

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Dengue transmission is shaped by the population dynamics of the Aedes aegypti mosquito, making vector control a central strategy for disease mitigation. The impact of interventions such as larvicide, adulticide, and breeding-site reduction depends critically on their timing under fluctuating environmental conditions. We build on a high-fidelity, non-Markovian mechanistic model of the Aedes life cycle that captures stage-structured, temperature-dependent developmental delays, and mortality, and extend it to incorporate multiple vector control measures. Rather than using continuous abstract control amplitudes as in standard optimal control formulations, we introduce intervention-specific temporal profiles that better reflect operational practice. We then develop an adjoint-based gradient descent framework to compute the optimal timing of a sequence of interventions by minimizing the time-dependent dengue reproduction number, R0. Numerical simulations based on seasonal temperature data from Miami, Florida, show that appropriately timed combinations of interventions can substantially suppress transmission risk, with outcomes strongly influenced by seasonal temperature variation and intervention duration. We further propose embedding the resulting optimization framework within a Model Predictive Control architecture, yielding a closed-loop approach for real-time, surveillance-driven vector management under environmental and operational uncertainty.

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math.DS 2026-05-13 Recognition

Entropy monotonicity holds for power-law unimodal maps

Topological Entropy for Power-Law Unimodal Maps

Kneading sequences increase with a for any critical exponent r > 1, preserving the quadratic-family structure

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In this paper we prove that the monotonicity of kneading sequences and topological entropy, a fundamental structural property of the quadratic family, extends to the class of power-law unimodal maps $f_a(x)=a-|x|^r$ for arbitrary critical exponent $r>1$. This generalization is nontrivial: the absence of polynomial structure and the presence of non-integer criticality preclude the direct use of classical arguments. Our approach adapts and extends the Milnor-Thurston framework by introducing a Thurston-type operator associated with the critical orbit and establishing a determinant identity that relates its linearization to the parameter derivative of the orbit. The main difficulty proving positivity of this determinant in the absence of algebraic structure - is resolved via a contraction argument on an associated Torelli space endowed with the Teichm\"uller metric, extending Thurston's pullback construction beyond the polynomial setting, that is to critical powers $r=2^\nu/k$, $\nu\geq 1$, $k$ odd, and finally use continuity in $r$. As a consequence, we show that the kneading sequence varies monotonically with the parameter, and hence that the topological entropy is an increasing function of $a$. Our results show that the combinatorial organization of parameter space familiar from the quadratic family persists for unimodal maps with arbitrary power-law criticality, indicating that monotonicity of entropy is a robust phenomenon beyond polynomial dynamics.

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math.DS 2026-05-13 Recognition

Conformable Laplace transform solves diffusion equations

On solution of Diffusion Equation using Conformable Laplace Transform

Inversion and convolution theorems allow closed-form solutions for initial-boundary value problems.

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The inversion theorem and convolution theorem of the conformable fractional Laplace transforms are developed. All the elementary properties of the classical Laplace transform are extended to the conformable fractional transform, and using these properties, we found analytical solutions to the initial-boundary value problems of the diffusion equation.

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math.DS 2026-05-13 Recognition

SL(d,R) cocycles over chaotic bases either split or support positive-entropy measures

Cocycles with Quasi-Conformality II: Ergodic measures with positive entropy

A multiple covering principle yields approximations that stably preserve positive-entropy measures on bounded fiber orbits for non-isometric

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As the second part of a series on linear cocycles over chaotic systems, this paper establishes a "multiple covering principle" that robustly yields positive-entropy ergodic measures supported on fiberwise uniformly bounded orbits. Using this mechanism, we prove that any continuous $\mathrm{SL}(d,\mathbb{R})$ cocycle over a positive-entropy subshift of finite type either admits a dominated splitting or can be $C^0$-approximated by one that $C^\alpha$-stably supports such measures ($\alpha>0$). Additionally, for non-isometric cocycles, we show that the topological entropy of these bounded orbits is strictly less than that of the base subshift.

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math.DS 2026-05-13 2 theorems

Weakening edges can raise connectivity and induce synchronization

Spectral Sensitivity of Directed Weighted Networks: Why Weakening Edges May Trigger Synchronization

A perturbation formula shows how selected reductions in coupling strength increase kappa and improve coherence in nonlinear systems on digr

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Synchronization in dynamical systems on directed weighted networks is often associated with stronger coupling and denser interactions. This paper shows that the opposite can also occur: weakening selected edges may increase the generalized algebraic connectivity, denoted by $\kappa$, and in some nonlinear systems this spectral improvement is accompanied by a transition from nonsynchronization to synchronization. To explain this effect, we develop a perturbation-based spectral sensitivity framework for directed weighted networks. We derive an explicit first-order formula for the response of $\kappa$ to edge-weight perturbations and show that it decomposes into a directed cut-energy term and a stationary redistribution term. This decomposition clarifies how asymmetric flow structure and invariant-mass redistribution jointly determine the synchronization role of each edge. Based on this theory, we design sensitivity-guided algorithms for edge weakening, edge deletion, negative-edge insertion, and edge strengthening. Experiments on synthetic and real networks show that these methods identify critical edges whose modification yields substantial gains in $\kappa$. Simulations of first- and second-order nonlinear consensus dynamics further show markedly faster convergence and, in some cases, a transition from incoherence to synchronization. The results provide a local spectral mechanism by which reducing or reallocating coupling can enhance synchronization-related performance.

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math.DS 2026-05-13 1 theorem

Degree gap forces dynamical Mordell-Lang for curves

Local height arguments toward the dynamical Mordell-Lang conjecture

When k exceeds twice the max multiplicity at periodic points of the map at infinity, orbits intersect curves finitely and all periodic sets,

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We consider regular endomorphisms of the complex affine space with a degree gap $k$. They are endomorphisms $f$ of $\mathbb{A}_{\mathbb{C}}^{N}$ of the form $f(x_1,\dots,x_N)=(f_1(x_1,\dots,x_N)+g_1(x_1,\dots,x_N),\dots,f_N(x_1,\dots,x_N)+g_N(x_1,\dots,x_N))$, in which $f_1,\dots,f_N$ are homogeneous polynomials of degree $d$ with no nonzero common zeros and $g_1,\dots,g_N$ are polynomials of degree $\leq d-k$. Such an endomorphism extends to an endomorphism of $\mathbb{P}_{\mathbb{C}}^{N}$. Let $H_{\infty}=\mathbb{P}_{\mathbb{C}}^{N}\setminus\mathbb{A}_{\mathbb{C}}^{N}$ be the infinity hyperplane and we denote $f_{\infty}$ as the induced endomorphism of $H_{\infty}$. Suppose that $k$ is twice greater than the multiplicities of $f_{\infty}$ at the periodic closed points, i.e. $k>2\max\limits_{P\in\mathrm{Per}(f_\infty)}e_{f_{\infty}}(P)$. Then we prove that $f$ satisfies the dynamical Mordell-Lang conjecture for curves. As a by-product of our proof, we show that in this case every periodic curve of $f$ is a "vertical line", i.e. a straight line passing through the origin. There are many examples which satisfy our condition $k>2\max\limits_{P\in\mathrm{Per}(f_\infty)}e_{f_{\infty}}(P)$. Indeed, we prove that for every $d\geq2$, a general endomorphism $f_{\infty}$ of $H_{\infty}\cong\mathbb{P}_{\mathbb{C}}^{N-1}$ of degree $d$ satisfies $\max\limits_{P\in H_{\infty}(\mathbb{C})}e_{f_{\infty}}(P)\leq(N-1)!\cdot2^{N-1}$. So if we take $k=(N-1)!\cdot2^N+1$, then $f$ will satisfy our condition if $f_{\infty}$ is general (of an arbitrary degree $d\geq k$). Moreover, we provide examples to illustrate that this condition is optimal to force every periodic curve to be a vertical line, in the sense that one cannot change "$>$" into "$\geq$".

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math.DS 2026-05-13 2 theorems

One futile cycle network combines bistability with fixed final product

Bistability, Absolute Concentration Robustness, and Hysteresis in Dual-Site Futile Cycles with Bifunctional Enzymes

Two stable states share identical final modification levels despite different intermediates, a feature absent from the other three networks.

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Bifunctional enzymes, which catalyze both the forward and reverse steps of a substrate modification reaction, arise naturally in bacterial two-component signaling systems and metabolic regulation. Beyond their well-known role in conferring absolute concentration robustness (ACR) on substrate species, bifunctional enzymes profoundly shape the dynamical landscape of the networks in which they appear. We study a class of dual-site futile cycles in which the reverse modification steps are carried out by bifunctional enzyme-substrate compounds, and provide a complete mathematical analysis of all four such networks, characterizing the existence, number, and stability of steady states, as well as the bifurcation structure as total substrate is varied. All four networks admit boundary steady states, in contrast to the non-bifunctional case. The networks differ in the number and stability of boundary steady states, in the maximum number of positive steady states (ranging from two to four), and in whether bistability is present. In two networks, a transcritical bifurcation connects the boundary and positive steady state branches; in one case this is a backward bifurcation, producing hysteresis. Perhaps the most striking phenomenon occurs in one of the four networks, which simultaneously exhibits bistability and ACR in the final modification state, where the system can settle into either of two stable steady states with different intermediate concentrations yet identical final product concentration.

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math.DS 2026-05-12 1 theorem

Non-Abelian dynamics on Veech surfaces lack zero Lyapunov exponents

Infinitesimal random dynamics of certain Veech groups on SU(2)-character varieties

Surfaces with degenerate Abelian spectra show fully non-degenerate behavior in the corresponding SU(2) setting.

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Almost 20 years ago, the first and fourth authors found examples of SL(2,R)-invariant subbundles of Hodge bundles over Teichm\"uller curves having maximally degenerate Lyapunov spectrum. For these same surfaces, we show that a natural non-Abelian analogue has no zero Lyapunov exponents.

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math.DS 2026-05-12 3 theorems

Exact Hausdorff dimensions computed for self-affine attractors

Hausdorff Dimension of a Class of Self-Affine Sets

When one map eventually becomes a similarity and linear parts commute with a derived operator the open set condition gives precise values.

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In this paper, exact Hausdorff dimension formulas for a class of self-affine attractors generated by affine Iterated Function Systems are derived. We consider systems containing an affine map whose $n$-th iterate is a similarity contraction, alongside standard similarities whose linear parts commute with the symmetric operator $A^\top A$, where $A$ is the linear part of the affine map. We prove that the attractor of such a system exists uniquely, and, under the Open Set Condition, we compute its exact Hausdorff dimension. We extend this framework to systems where all map compositions of some fixed length are similarities, and to systems where overlaps are exact homothetic copies of the attractor. We unify these approaches to establish dimension formulas for hybrid systems that combine multiple eventually contractive affine maps with universally aligned similarities. Finally, we conclude with a topological classification of these systems in the plane. For a two-map system comprising an affine map whose second iterate is a similarity with contraction ratio $c$, alongside an $f$-aligned similarity with ratio $r$, we prove that the precise parameter balance $c + r = 1$ acts as a strict topological bottleneck uniquely guaranteeing both the open set condition and the connectedness of the attractor.

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math.DS 2026-05-12 2 theorems

Nonlocal equations form semi-dynamical systems with attractors

Long-time behaviors and attractors for time-nonlocal generalized Rayleigh-Stokes equations

Restricting to a compact-convergence subspace recovers the semigroup property and produces attracting sets under dissipativity.

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Fractional systems generally can not generate a standard semi-dynamical systems, as their solution trajectories do not possess the semigroup property. In this paper, we consider an autonomous semi-dynamical system driven by semilinear nonlocal evolution equations, these type equations are used to generalize the Rayleigh-Stokes problem for a non-Newtonain fluid to a generalized second grade fluid. We first investigates the global well-posedness of solutions consisting of global Lipschitz condition by a weighted space $\mathcal C$. Utilizing the subset space $\mathcal C_\rho$ of $\mathcal C$ with the topology convergence on compact subset, we construct a semi-dynamical system that satisfies the semi-group structure. It also is shown that this semi-dynamical system has an attracting set in $\mathcal{C}_\rho$ when the vector field function satisfies a dissipativity condition as well as a local Lipschitz condition. With the compactness, we also get the existence of attractors.

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math.DS 2026-05-12 2 theorems

Rescaling limits of rational maps form degree-bounded trees

Generalized rescaling limits of a sequence of rational maps

For quadratic maps the structure yields a uniform bound on cycles with small multipliers.

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We consider a sequence of complex rational maps (f_n) of a fixed degree d at least 2. Building on the seminal work of Kiwi, we introduce the notion of generalized rescaling limits. These are rational maps possibly defined over a non-Archimedean field obtained by renormalizing at some scale a fixed iterate of the sequence (f_n). We explain that the set of all generalized rescaling limits is naturally organized as a tree, and bound the size of this tree in term of the degree d. We apply our theory to quadratic rational maps. Using Kiwi's classification, we describe all possible trees in this case, and prove a uniform bound on the number of cycles with small multipliers.

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math.DS 2026-05-12 Recognition

Rational maps realize any d-basin layout for d at least 3

Simultaneous Approximation by Attracting Basins

Open sets sharing a boundary are approximated by attracting basins whose Julia set matches the boundary.

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We show that any $d\geq3$ pairwise-disjoint open sets $A_1$, ..., $A_d\subset\widehat{\mathbb{C}}$ sharing a common boundary $J$ can be simultaneously approximated by the $d$ attracting basins $\mathcal{A}_1$, ..., $\mathcal{A}_d$ of a rational map $r$ having Fatou set $\mathcal{F}(r)=\mathcal{A}_1\sqcup...\sqcup\mathcal{A}_d$ and so that the Julia set $\mathcal{J}(r)$ approximates $J$.

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math.DS 2026-05-11 2 theorems

Symbolic model connects geodesics to continued fractions

An elegant model of the geodesic flow on the modular surface

The coding of paths on the modular surface mirrors number expansions, bridging geometry and arithmetic.

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Caroline Series' [{\em The modular surface and continued fractions}, J. Lond. Math. Soc. (2), {\bf 31}, no.~1, (1985), 69--80] gives a clear framework linking, in a deceptively simple way, the dynamics of the geodesic flow on the modular surface with the dynamics of the regular continued fraction, through a well-chosen symbolic coding. It has been called {\em required reading} for those interested in the symbolic dynamics of geodesic flows, and has had consequences in symbolic dynamics, ergodic theory, hyperbolic geometry, and continued fraction theory. In this overview, we give an indication of why this is so, sketch some of the history related to the paper, and also point to some later works.

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math.DS 2026-05-11 2 theorems

Hopf points in four-wing chaos admit centers of higher cyclicity

Cyclicity of centers on center manifolds in a 3D chaotic system with a four-wing attractor

Center-manifold focal-value calculations raise the known lower bound on small limit cycles that emerge from equilibria.

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In this work, we investigate the conditions that guarantee the existence of centers on the center manifold, arising from Hopf points, in the new three-dimensional quadratic chaotic system introduced by B. Khaled et al. in 2024 in the Int. J. Data Netw. Sci. For some of the Hopf points of the system, we solve the center-focus problem on the center manifold, analyzing both its isochronicity and cyclicity. Our results significantly improve the previously known lower bound on the number of limit cycles bifurcating from Hopf points in this system, as established by B. M. Mohammed in 2025 in the Int. J. Bifurc. Chaos Appl. Sci. Eng.

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math.DS 2026-05-11 2 theorems

Nonlinear Fourier-Fejér discretisation converges to self-consistent fixed points

Fixed-point approximation for self-consistent transfer operators with Newton's method

The method yields exponential iteration and quadratic Newton convergence for mean-field coupled systems.

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Self consistent transfer operators arise naturally in the study of mean-field coupled dynamical systems and are closely related to kinetic PDEs such as the Vlasov equation. Despite substantial progress on existence and uniqueness of fixed points for self-consistent transfer operators, the development of fast, reliable, and provably accurate numerical methods remains largely unresolved. In this work, we construct a nonlinear Fourier-Fej\'er discretisation and establish convergence of the resulting finite-dimensional fixed point to that of the original self-consistent transfer operator. Further, using the nonlinear Fourier-Fej\'er discretisation, we prove exponential convergence of a sequential iteration scheme and develop a Newton framework with quadratic convergence. We present numerical examples demonstrating the efficiency and flexibility of the above methods.

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math.DS 2026-05-11 Recognition

Greedy and lazy maps each have unique ACIM equivalent to Lebesgue on I_Q

Invariant measure for double base expansions

Exactness implies that almost every point admits a continuum of Q-expansions when q0 + q1 exceeds q0 q1.

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Given a pair $Q=(q_0,q_1)\in(1,\infty)^2$ with $q_0+q_1\ge q_0q_1$, a sequence $(c_i)\in\set{0,1}^\infty$ is called a $Q$-expansion of $x$ if \begin{equation*} x=\sum_{i=1}^{\infty}\frac{c_i}{q_{c_1}\cdots q_{c_i}}. \end{equation*} We primarily study the dynamical properties of the greedy and lazy maps, which are the piecewise-linear maps on the interval $I_Q=[0,\,1/(q_1-1)]$ defined by the corresponding algorithms for $Q$-expansions. We show that the greedy and lazy maps each of which has a unique absolutely continuous invariant probability measure, equivalent to the Lebesgue measure on the intervals \begin{equation*} \left[0,\frac{q_0}{q_1}\right)\qtq{and}\left(\frac{q_1}{q_0(q_1-1)}-1,\frac{1}{q_1-1}\right], \end{equation*} respectively. Furthermore, the corresponding dynamical systems are exact on $I_Q$. As a dynamical consequence, under the stronger condition $q_0+q_1>q_0q_1$ the set of points having unique $Q$-expansions has Lebesgue measure zero, and almost every $x\in I_{Q}$ admits a continuum of $Q$-expansions.

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math.DS 2026-05-11 2 theorems

Mucube straight-line flow periodic via SL(2,Z) subgroup

Straight-line trajectories on the Mucube

Complete characterization uses genus-one quotient, yields Veech group, and proves density of periodic and ergodic directions.

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The dynamics of straight line flows on compact half-translation surfaces (surfaces formed by gluing Euclidean polygons edge-to-edge via translations possibly composed with rotation by $\pi$) has been widely studied due to their connections to polygonal billiards and Teichm\"uller theory. However, much less is known when the underlying surface is non-compact or infinite type. In this paper, we consider the straight line flow of the Mucube -- an infinite $\mathbb{Z}^3$-periodic half-translation square-tiled surface -- first written about by Coxeter and Petrie and more recently studied by Athreya--Lee and Guti\'errez-Romo--Lee--S\'anchez. We give a geometric description of the flow's periodic and drift orbits in terms of the Mucube's rigid symmetries, and we give a complete characterization of the set of directions in which the straight line flow is periodic on the Mucube -- first in terms of a genus one quotient and second in terms of an infinitely generated subgroup of $\mathrm{SL}_2(\mathbb{Z})$. We use the latter characterization to obtain the Veech group (i.e. group of derivatives of affine diffeomorphisms) of the Mucube. Finally, we prove density of the sets of periodic and ergodic directions.

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math.DS 2026-05-11 2 theorems

Switching control makes gene densities forget their initial state

Predictive-Switching Control of Stochastic Gene Regulatory Networks: A Contractive PIDE Framework

L1-contractivity of the closed-loop PIDE ensures the probability distribution evolves independently of starting conditions.

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This paper develops a predictive switching control algorithm for stochastic gene regulatory networks described by a Partial Integro-Differential Equation (PIDE) model, which enables direct shape control of the probability density function. Control inputs are selected from a finite candidate set to minimize a prescribed cost functional. A hybrid framework is proposed for scalability in higher-dimensional systems, using neural networks to approximate the control policy. A central theoretical contribution is a contraction-based analysis of the closed-loop PIDE dynamics. The paper establishes $L^ 1$-contractivity under the proposed control scheme, yielding formal stability guarantees and showing that the evolution of the probability density becomes progressively independent of the initial condition. Moreover, under strictly positive leakage terms, exponential convergence is obtained. The effectiveness and flexibility of the approach, together with the theoretical contractivity results, are illustrated through numerical simulations on three representative examples of increasing dimensionality.

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math.DS 2026-05-11 2 theorems

Proof validates transverse heteroclinic orbit in Shimizu-Morioka system

Computer-Assisted Proofs in Dynamical Systems: A Case Study of a Heteroclinic Orbit in the Shimizu--Morioka System

Radii polynomial method confirms connecting orbit between equilibria using parameterization and boundary-value validation.

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The radii polynomial approach is an a posteriori validation method based on the contraction of a quasi-Newton operator. We apply this strategy to give a computer-assisted proof of a transverse heteroclinic orbit in the Shimizu--Morioka system, validating the equilibria and eigenpairs, the local invariant manifolds via the parameterization method, and the connecting orbit via a boundary-value problem. For each subproblem we present a four-step procedure: $(i)$ zero-finding formulation, $(ii)$ approximate zero, $(iii)$ approximate inverse, and $(iv)$ bound estimates. This highlights the unifying structure behind the a posteriori validation method. Alongside the analysis, we include code snippets implemented in Julia using the RadiiPolynomial library.

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math.DS 2026-05-11 Recognition

SSM reduction enables efficient nonlinear analysis of rotating structures

Non-intrusive spectral submanifold model reduction for geometrically nonlinear rotating structures with Coriolis and centrifugal forces

Non-intrusive models anchored at centrifugal equilibrium accurately predict backbone and forced responses while accounting for Coriolis and

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Rotating structures are widely observed in engineering applications such as turbomachinary and wind turbine. These rotating structures, particularly for blades made by lightweight materials, can undergo large deformation in operations and display complex nonlinear dynamics under the coupling interaction of geometric nonlinearity, Coriolis effect and centrifugal force. Finite element (FE) methods provide a powerful and accurate modeling approach for capturing the complex nonlinear dynamics for realistic rotating structures, yet its high-dimensionality causes significant challenge to efficient prediction for the nonlinear vibration. Here, we present a non-intrusive spectral submanifold (SSM) model reduction for these FE models of rotating structures. We use COMSOL to establish FE models and simulate these FE models to verify the accuracy of SSM-based reduction. We first compute nontrivial static equilibrium induced by the centrifugal force and then construct non-intrusively SSM based reduced-order model (ROM) anchored at the equilibrium. These SSM-based ROMs enable efficient and accurate extraction of backbone and forced response curves. We use a suite of examples with increasing complexity to demonstrate the effectiveness of the SSM reduction, including a rotating beam, a twisted plate, a rotor with two disks, and an internally resonant fan with three blades. The obtained results also highlight the significant effects of Coriolis force on the nonlinear vibration of rotating structures.

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math.DS 2026-05-11 2 theorems

Minimal wave speed read from infinity in reaction-diffusion system

Minimal speed of unbounded traveling wave solutions for a 1D reaction-diffusion equation and their relationship with the dynamics at infinity

Poincaré compactification classifies all trajectories and yields explicit minimal speed for asymptotically linear reaction terms.

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This paper presents results on the unboundedness and minimal speed of traveling wave solutions for a one-dimensional spatial reaction-diffusion equation with an asymptotically linear reaction term and a saturation parameter. By applying a Poincar\'e-type compactification, we reveal the full dynamics (including infinity) of the two-dimensional system of ordinary differential equations satisfied by traveling wave solutions. This yields essential information characterizing traveling wave solutions: the classification of trajectories in the phase plane, the positivity and unboundedness of front-type and sign-changing profiles, and the explicit form of the minimal speed. This paper examines a special equation with an asymptotically linear reaction term. While, our results differ from those of conventional linear determinacy. We claim that the minimal speed is derived from information at infinity within the traveling wave system.

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math.DS 2026-05-08 2 theorems

Flowers limit expanding map subsets to linear complexity

Expanding Maps on Flowers, Interval Exchange Transformations, and Ergodic Optimization

Any set inside a flower has at most linear word complexity and connects to interval exchange transformations, with numerics suggesting they

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In this paper, we discuss expanding maps on a class of invariant sets called flowers. We show that any set contained in a flower has at most linear complexity, and we present a relationship between flowers and a special class of interval exchange transformations. This extends work of Bullett and Sentenac, who showed that any Sturmian system may be embedded into the circle as a doubling-invariant subset that is contained in a half circle. Flowers were first introduced in the context of ergodic optimization, as candidate sets for supporting maximizing measures. We discuss the relationship to ergodic optimization, and present numerical results that support the conjecture that trigonometric polynomials are maximized on flowers.

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math.DS 2026-05-08 Recognition

Rational eigenfunctions yield closed-form quadratic ODE solutions

Analytical solutions for some quadratic ODEs found via linear rational eigenfunctions and the rational eigenfunction variety

Imposing a linear rational form reduces the eigenfunction equation to algebra and locates exact solutions for specific families of systems.

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Many important systems across biology, engineering, physics, and economics are characterized by polynomial ordinary differential equations (ODEs), yet analytical solutions are rare. We develop a framework for identifying and solving a broad class of two-dimensional quadratic ODEs using linear rational Koopman eigenfunctions. By imposing a linear rational form on the eigenfunctions, we convert the Koopman eigenfunction PDE into a large algebraic system of polynomials. We then study the solutions of this polynomial system that satisfy the ODE restrictions; we call the solution set the rational eigenfunction variety of an ODE system. The nonlinear algebra method uses formal algebraic geometry theory to analyze and solve systems otherwise intractable and to discover relationships between ODE and eigenfunction parameters that must hold to extract eigenfunctions. We identify families of quadratic ODEs that can be solved analytically, characterize their eigenfunction parameters, and use the resulting eigenfunctions to produce closed-form analytical solutions.

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math.DS 2026-05-08

Tiny analytic perturbations restore integrability to any order

Integrable perturbations of polynomial Hamiltonian systems

For non-resonant equilibria, a function vanishing to order M+1 makes the Hamiltonian completely integrable everywhere.

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We consider a Hamiltonian system on the symplectic space $({\mathbb{R}}^{2n}, dy\wedge dx)$ with a real-analytic Hamiltonian $H : {\mathbb{R}}^{2n}\to {\mathbb{R}}$. We assume that the system has a non-degenerate equilibrium position at the origin. Under some nonresonance assumptions we prove the following. For any positive integer $M$ there exists a real-analytic function $F:{\mathbb{R}}^{2n}\to{\mathbb{R}}$ such that (1) $F = O\big( (|x|+|y|)^{M+1} \big)$ at the origin, (2) the system with Hamiltonian $H+F$ is completely integrable in ${\mathbb{R}}^{2n}$.

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math.DS 2026-05-08

Conjugacy class restriction preserves orbital growth asymptotics

Orbital Counting in Conjugacy Classes

For cocompact and convex cocompact actions on negatively curved spaces, counts inside one fixed class grow at the same exponential rate as 1

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In this article we consider a restricted orbital counting problem for the action of certain discrete groups on suitable spaces. In particular, we present asymptotics for counting those points in an orbit restricted to a single conjugacy class. A classical example would be cocompact actions of a discrete group acting isometrically on a simply connected manifold with pinched negative curvature. More generally, we obtain results for convex cocompact actions on $CAT(-1)$ spaces.

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math.DS 2026-05-08

Two-layer KANs recover symbolic Koopman dictionaries after training

Deep-Koopman-KANDy: Dictionary Discovery for Deep-Koopman Operators with Kolmogorov-Arnold Networks for Dynamics

The method exposes compositional structure in learned observables, recovering exact terms on Lorenz and Fourier bases on maps without precho

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Symbolic library -- or Koopman dictionary -- selection is a fundamental challenge in data-driven dynamical systems. Extended Dynamic Mode Decomposition (EDMD), Sparse Identification of Nonlinear Dynamics (SINDy), and Kolmogorov--Arnold Networks for Dynamics (KANDy) all require the practitioner to commit to a function library at training time; Deep-Koopman Operators avoid this commitment but produce uninterpretable latent observables. We propose Deep-Koopman-KANDy, a structured approach to post-hoc symbolic dictionary readout that combines Deep-Koopman modeling with Kolmogorov-Arnold Networks for Dynamics (KANDy). The encoder and decoder of a Deep-Koopman Operator are replaced with two-layer Kolmogorov--Arnold Networks (KANs), and a level-set construction together with a chain-rule gradient identity exposes the compositional structure of the learned observables in a basis chosen \emph{after} training. We evaluate the method on the Lorenz system, the Chirikov standard map, the Ikeda map, and the Arnold cat map. On Lorenz it recovers the target dictionary $\{x,y,z,xy,xz\}$ with perfect recall and Jaccard score $0.79\pm0.06$; on the standard map it recovers a low-order Fourier basis matching the analytical structure; on Ikeda -- which has no sparse polynomial representation -- a misspecified polynomial readout still recovers the correct foliation coordinate $g\approx x^2+y^2$ together with a nontrivial outer function; and on the Arnold cat map -- used as a negative control because finite-dimensional Koopman closure is provably impossible -- the method fails to find a sparse closure, as expected.

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math.DS 2026-05-08

Martingale framework produces maximal large deviation estimates

Martingale Methods for Maximal Large Deviations and Young Towers

For invertible systems with correlation decay, it yields bounds and Young structures for diffeomorphisms and slowly mixing billiards.

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We develop a martingale approximation framework yielding quantitative maximal large deviations estimates for invertible dynamical systems. From suitable decay of correlations, we deduce these estimates and, as an application, we obtain Young structures with matching recurrence tails for partially hyperbolic diffeomorphisms with mostly expanding central direction. In a second application, we prove maximal large deviation estimates for systems modelled by Young towers with subexponential contraction and expansion. Many examples of slowly mixing billiards are covered by this result.

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math.DS 2026-05-08

Tangency multiplicity fixes counts of cycles emerging from grazing loops

Bifurcations of grazing loops of arbitrary tangent multiplicity in piecewise-smooth systems

Higher-order tangencies in piecewise-smooth systems produce more crossing limit cycles and sliding loops once a functional perturbation and

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In piecewise-smooth differential systems, a hyperbolic limit cycle of a subsystem loses its structural stability if it grazes the switching manifold at a tangent point. Such a cycle is called a grazing loop and in this paper we investigate its bifurcations for arbitrary tangent multiplicity. For the low-multiplicity tangency, the recurrences are comprehensively captured by a functional perturbation with two parameters in previous publications, where the parameters characterize the recurrences near the tangent point and the limit cycle respectively. However, for high-multiplicity tangency, these parameters fail to capture the recurrences and thus, Poincare return maps can not be defined as usual. To address these challenges, we construct a functional perturbation with functions to clarify the recurrences and simultaneously, propose a localization method to make these two recurrences equivalent. We finally establish a quantitative relationship between the multiplicity of tangency and the numbers of crossing limit cycles, sliding loops bifurcating from the grazing loop and the number of tangent points on these sliding loops.

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math.DS 2026-05-08

Blaschke products realize every post-critically finite multimodal circle map

Blaschke-type models for multimodal circle maps

The finite family supplies a unique algebraic model up to rotation for all maps meeting the dynamical conditions.

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For each integer $m \geq 1$, we construct a finite-dimensional family of rational maps, given by Blaschke-type products, whose restriction to the unit circle consists of $2m$-multimodal maps. We show that every post-critically finite $2m$-multimodal circle map satisfying natural dynamical conditions is topologically conjugate to a map in this family. Moreover, we prove that this realization is unique up to rotation: two maps in the family that are topologically conjugate on the circle differ by a rigid rotation. In particular, the family provides a canonical model realizing all post-critically finite combinatorics in this class. The proofs combine a detailed description of the critical geometry of these Blaschke-type maps with a Thurston-type fixed point argument for a pull-back operator on the parameter space.

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math.DS 2026-05-08 Recognition

Exact symplectic map found for vibro-impact oscillator with friction

Mixed Global Dynamics of the Forced Vibro-Impact Oscillator with Coulomb Friction and its Symplectic Structure, KAM Tori, and Persistence

On the maximal non-sticking set the time-T stroboscopic map preserves symplectic form, yielding KAM tori near elliptic orbits and mixed co-4

abstract click to expand

The forced vibro-impact oscillator with Amonton-Coulomb friction and elastic walls was shown by Gendelman et al. (2019) to exhibit a coexistence of Hamiltonian stability islands and dissipative attractors in a single phase space. We provide a complete mathematical analysis of this phenomenon. We prove global well-posedness of the associated Filippov flow and construct a global lift to a piecewise smooth Hamiltonian system on a covering manifold. On the maximal forward-invariant non-sticking set, we show that the time-$T$ stroboscopic map is exact symplectic, within the formalism of symplectic dynamics. We derive a closed-form existence equation for symmetric $T$-periodic orbits and establish a parameter-dependent saddle-center bifurcation at $f_{\rm sc}(F,\omega,R)$, correcting a universality claim in prior work. Using Moser's twist theorem, we prove the existence of invariant Cantor families (KAM tori) near elliptic non-sticking periodic orbits, while a Melnikov analysis yields hyperbolic dynamics conjugate to a Bernoulli shift near the associated saddle. We further show that any positive restitution defect or viscous damping destroys the conservative structure: elliptic periodic orbits persist but become asymptotically stable, replacing Hamiltonian islands by a single attracting basin. The approach extends to multi-particle systems with elastic collisions, where a symplectic structure and higher-dimensional KAM tori are obtained. A computer-assisted proof verifies the existence and ellipticity of a non-sticking periodic orbit at a specific parameter point.

0

math.DS 2026-05-08

Friction oscillator map is exact symplectic on non-sticking set

Mixed Global Dynamics of the Forced Vibro-Impact Oscillator with Coulomb Friction and its Symplectic Structure, KAM Tori, and Persistence

The time-T map preserves symplectic structure there, supporting KAM tori near elliptic orbits; any damping replaces islands with attracting盆

abstract click to expand

The forced vibro-impact oscillator with Amonton-Coulomb friction and elastic walls was shown by Gendelman et al. (2019) to exhibit a coexistence of Hamiltonian stability islands and dissipative attractors in a single phase space. We provide a complete mathematical analysis of this phenomenon. We prove global well-posedness of the associated Filippov flow and construct a global lift to a piecewise smooth Hamiltonian system on a covering manifold. On the maximal forward-invariant non-sticking set, we show that the time-$T$ stroboscopic map is exact symplectic, within the formalism of symplectic dynamics. We derive a closed-form existence equation for symmetric $T$-periodic orbits and establish a parameter-dependent saddle-center bifurcation at $f_{\rm sc}(F,\omega,R)$, correcting a universality claim in prior work. Using Moser's twist theorem, we prove the existence of invariant Cantor families (KAM tori) near elliptic non-sticking periodic orbits, while a Melnikov analysis yields hyperbolic dynamics conjugate to a Bernoulli shift near the associated saddle. We further show that any positive restitution defect or viscous damping destroys the conservative structure: elliptic periodic orbits persist but become asymptotically stable, replacing Hamiltonian islands by a single attracting basin. The approach extends to multi-particle systems with elastic collisions, where a symplectic structure and higher-dimensional KAM tori are obtained. A computer-assisted proof verifies the existence and ellipticity of a non-sticking periodic orbit at a specific parameter point.

0

math.DS 2026-05-07

Power-mean spectra for p-adic Schneider map given by polylogarithms

Multifractal analysis of power means for the Schneider map on pmathbb{Z}_p

Thermodynamic formalism produces explicit Hausdorff dimensions of level sets, unlike the real continued-fraction case.

abstract click to expand

We study the asymptotic power means of the coefficients associated with the Schneider continued fraction map on $p\mathbb{Z}_p$. Using tools from thermodynamic formalism, we compute the Hausdorff dimension of the corresponding level sets and obtain explicit formulas for the associated multifractal spectra. The locally constant nature of the geometric potential enables a precise description in terms of polylogarithm functions, in sharp contrast with the classical real setting.

0

math.DS 2026-05-07

r-continued-fraction maps beat Mersenne Twister in randomness tests

Producing Quality Pseudorandomness with a Generalized Gauss Continued-Fraction Map

Generalized Gauss maps produce sequences that pass Dieharder, PractRand and TestU01 at higher quality than standard generators.

abstract click to expand

Well-known chaotic maps, such as the logistic and tent maps, have been used to generate cryptographically secure pseudorandomness, yet we know of no efforts which attempt to use the Gauss continued-fraction map, a known chaotic map, as a starting point for producing quality pseudorandom output. In this paper, we consider the family of $r$-continued-fraction maps, which generalize the Gauss map, and use them to generate pseudorandom output which outperforms many standard generators, such as the Mersenne Twister, in statistical quality, as ascertained by use of the Dieharder, PractRand, and TestU01 suites. In this way, we demonstrate the potential viability of these maps as a starting point for novel generators, and provide practical motivation for further study of the properties of both the exact and finite-precision $r$-continued fraction maps.

0

math.DS 2026-05-07

Measure-theoretic dynamics detailed for Baker domain boundaries

Boundaries of Baker domains of entire functions. A finer approach

Strengthened results on radial extensions of inner functions improve topology and dynamics for functions like z + e to the minus z

abstract click to expand

We consider transcendental entire functions having doubly parabolic Baker domains, such that the Denjoy-Wolff point of the associated inner function is not a singularity. We describe in a very precise way the dynamics on the boundary from a measure-theoretical point of view. Applications of such results lead to a better understanding of the topology and the dynamics on the boundaries. In particular, we improve some of the results in [N. Fagella and A. Jov\'e, A model for boundary dynamics of Baker domains], for the Baker domain of $z+e^{-z}$. In fact, our conclusions are obtained by applying new results established here on the dynamics of the radial extension of one component doubly parabolic inner functions, which strengthen those of [O. Ivrii and M. Urba\'nski, Inner functions, composition operators, symbolic dynamics and thermodynamic formalism].

0

math.DS 2026-05-07

Most homogeneous self-similar measures admit a minimal weighted IFS

On the minimal generating weighted IFS of self-similar measure

Generic parameters on the line yield a smallest generating system even when images overlap, via exponential-polynomial factorization.

abstract click to expand

We concern the structrue of generating weighted IFSs of a self-similar measure on the real line. We provide various sufficient conditions for the existence of a minimal generating weighted IFS of a self-similar measure on the real line. Under the homogeneity, we show that `most' self-similar measures on the real line have a minimal generating weighted IFS, without separation conditions. The ingredients of our proofs are based on the zero distribution and factorization theory of exponential polynomials, logarithmic commensurability (with a dynamical system argument), and results on the structure of generating IFSs of a self-similar sets.

0

math.DS 2026-05-07

Bounds set on crossing limit cycles for most piecewise isochronous centers

Crossing limit cycles of discontinuous piecewise differential systems with Pleshkan's isochronous centers

Twelve classes get explicit upper limits while three remain open, with examples proving three cycles possible in all fifteen.

abstract click to expand

In recent decades, piecewise linear differential systems have attracted considerable attention due to their ability to describe a wide range of phenomena. A central problem, as in the theory of general planar differential systems, is to determine the existence and the maximal number of crossing limit cycles. However, deriving sharp upper bounds for this quantity remains a highly challenging problem. In this work we study crossing limit cycles in planar discontinuous piecewise differential systems separated by a straight line, where each subsystem is either a linear center or a cubic isochronous center with homogeneous nonlinearities. Within this setting, we consider all possible combinations arising from these families, leading to fifteen distinct classes of piecewise systems. Using the existence of first integrals, we reduce the detection of crossing limit cycles to algebraic closing conditions on the discontinuity set, which allows for a systematic and unified analysis across all configurations. As a consequence, we establish explicit upper bounds for the number of crossing limit cycles in all cases except for three configurations that remain open. In addition, we construct examples exhibiting three crossing limit cycles in every class, providing a nontrivial uniform lower bound. Our results extend and complement earlier work in the literature by including previously unstudied configurations and improving some known bounds, thereby providing a comprehensive description of the number of crossing limit cycles within this class of systems

0

math.DS 2026-05-07

Residual derivatives fix first two Cantor-Bendixson levels

Residual stratification and the Cantor-Bendixson structures of dual algebraic coframes

In dual algebraic coframes with order-compatible topologies, the initial levels of the Cantor-Bendixson structure are determined exactly by残

abstract click to expand

We introduce a notion of residual derivative for elements of a preordered set, a construction that generalizes both the Frattini subgroup in algebra and the Cantor-Bendixson derivative in T1 topological spaces. For dual algebraic coframes with topologies compatible with order, we establish a partial correspondence between the Cantor-Bendixson structure of the lattice and the residual derivatives of its elements. Within this framework, we provide a complete characterization of the first two Cantor-Bendixson levels in terms of the lattice's residual structure. This provides a unified lens through which to study the Cantor-Bendixson structures of topological spaces across domains ranging from algebra to functional analysis and dynamics, facilitating the transfer of analytic techniques between them.

0

math.DS 2026-05-07

Stability criterion holds for all fractional orders in two-delay equations

Stability and Bifurcation Analysis of Fractional Delay Differential Equation with a Delay-dependent Coefficient

A general result for the linearized fractional system with positive delays remains valid independently of alpha and the first delay value.

abstract click to expand

This paper investigates the stability of different regions in the $(k,\gamma)$-plane for a class of fractional delay differential equations given by \begin{equation} D^{\alpha} x(t) = -\gamma x(t) + g\big(x(t - \tau_1)\big) - e^{-\gamma \tau_2}\, g\big(x(t - \tau_1 - \tau_2)\big), \qquad 0 < \alpha \le 1, \end{equation} where $k = g'(0)$. The primary focus is on the stability of the trivial equilibrium of the corresponding linearized system. A detailed stability and bifurcation analysis is carried out for the particular case $\tau_1 = 0$ and $\tau_2 \ge 0$. Furthermore, a general result is established for the case $\tau_1 > 0$, $\tau_2 \ge 0$, which holds for all values of $\alpha$ and $\tau_1$. In addition, illustrative examples are provided in the form of stability diagrams in the $(\tau_1,\tau_2)$-plane for fixed values of $\alpha$, $k$, and $\gamma$. These diagrams are generated using appropriate numerical methods to visualize the stability regions and to support the theoretical results.

0

math.DS 2026-05-07

Asymmetric crease coupling creates sequential origami deployment fronts

Programming sequential deployment of origami via kinematic transition fronts

Nonlinear recurrence relations from degree-4 vertices admit heteroclinic orbits that propagate folding in domino fashion.

abstract click to expand

Propagating transition fronts, in which local interactions sequentially trigger state changes, are widely observed across natural, biological, and engineered systems. While such propagation has been engineered using energy-driven instabilities, front propagation governed purely by geometric constraints remains underexplored and lacks a general design framework. In particular, how to program sequential deployment in origami through such kinematic propagation remains an open challenge. Here, we develop a systematic design framework for kinematic transition fronts based on their correspondence with heteroclinic orbits in discrete dynamical systems. Focusing on strips of developable and flat-foldable degree-4 origami vertices, we show that asymmetric coupling between adjacent creases produces nonlinear recurrence relations whose composition generically gives rise to heteroclinic orbits connecting developed and flat-folded states, enabling domino-like sequential deployment. We further show that macroscopic shape can be programmed independently of propagation behavior by exploiting invariances in the recurrence relation, and illustrate the approach through a representative thick-panel origami prototype. These results enable programmable sequential deployment in origami via transition fronts, while also establishing a general framework for kinematic transition fronts in geometrically constrained systems.

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math.DS 2026-05-06

Periodic billiard orbits persist along deformation paths

Persistence of periodic billiard orbits under domain deformation

A combinatorial criterion on the orbit guarantees continuous paths of polygons where the orbit type stays the same for every shape.

abstract click to expand

We prove that if a polygon admits a periodic billiard orbit satisfying a certain combinatorial criterion, then there are paths of polygons in parameter space for which every polygon in the path admits a periodic billiard orbit of the same type.

0

math.DS 2026-05-06

Neural network outputs symbolic governing equations

Symbolic Regression via Neural Networks

The model recovers interpretable expressions for classical dynamical systems directly from data.

abstract click to expand

Identifying governing equations for a dynamical system is a topic of critical interest across an array of disciplines, from mathematics to engineering to biology. Machine learning -- specifically deep learning -- techniques have shown their capabilities in approximating dynamics from data, but a shortcoming of traditional deep learning is that there is little insight into the underlying mapping beyond its numerical output for a given input. This limits their utility in analysis beyond simple prediction. Simultaneously, a number of strategies exist which identify models based on a fixed dictionary of basis functions, but most either require some intuition or insight about the system, or are susceptible to overfitting or a lack of parsimony. Here we present a novel approach that combines the flexibility and accuracy of deep learning approaches with the utility of symbolic solutions: a deep neural network that generates a symbolic expression for the governing equations. We first describe the architecture for our model, then show the accuracy of our algorithm across a range of classical dynamical systems.

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math.DS 2026-05-06 3 theorems

Power system DOA boundaries equal unions of stable manifolds

Calculating Domain of Attraction Boundary of Power Systems Based on the Gentlest Ascent Dynamics

Gentlest ascent dynamics locates index-1 saddles and orbits whose manifolds form the exact separatrix between stable and unstable post-fault

abstract click to expand

The power system, a fundamental public utility, is increasingly important due to growing global electricity demand. Recent large-scale blackouts (e.g., Iberian Peninsula, UK) have raised concerns about transient stability under impact faults. Transient stability is determined by post-disturbance synchronizing capability of synchronous generators, formulated as identifying the domain of attraction (DOA) boundary of the asymptotically stable equilibrium. Using a benchmark model of synchronous-generator-dominated power systems, this report employs a gentlest ascent dynamics (GAD) method for 1-saddle points, an adjoint operator method for periodic orbits, and stable manifold algorithms to compute the DOA boundary. These algorithms transform DOA boundary determination into constructing unstable critical elements (saddle points and periodic orbits) and their stable manifolds. Theoretically, under certain assumptions we prove that the DOA boundary is the closure of the union of stable manifolds of index-1 critical elements, and establish a stability theory for a perturbed GAD system. Numerical experiments on two-machine and three-machine systems (with only saddle points or with periodic orbits) validate the effectiveness and accuracy. Results show the algorithms accurately capture the geometric structure of the DOA boundary, providing a new numerical tool for transient stability analysis.

0

math.DS 2026-05-06 3 theorems

General dichotomies allow smooth linearization of nonautonomous systems

Smooth linearization of nonautonomous dynamics under general dichotomic behaviour

Time reparametrization converts μ-dichotomies to exponential ones, extending linearization to polynomial and logarithmic growth rates.

abstract click to expand

The main purpose of this paper is to formulate new conditions for smooth linearization of nonautonomous systems with discrete and continuous time. Our results assume that the linear part admits a very general form of dichotomy known as $\mu$-dichotomy and that the associated $\mu$-dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. We observe that our notion of $\mu$-dichotomy encompasses the classical notions of exponential, polynomial and logarithmic dichotomies as very particular cases. In particular, our result is in sharp contrast to most of the previous results in the literature which assumed that the linear part admits an exponential dichotomy. Our techniques exploit the relationship between $\mu$-dichotomy and exponential dichotomy via a suitable reparametrization of time.

0

math.DS 2026-05-06 3 theorems

Generic nine-point blowups make cubics nonlinearizable

Nonlinearizable embeddings of elliptic curves in rational surfaces

For any smooth cubic, a dense set of point choices yields a strict transform whose normal bundle has infinite order, settling Ogus's 1975

abstract click to expand

We show that for any smooth cubic in $\mathbb{P}^2$, there exists a dense $G_\delta$ set of configurations of 9 distinct points such that blowing up $\mathbb{P}^2$ at these 9 points, the strict transform of the cubic is not linearizable and has nontorsion normal bundle. This answers a problem raised by Ogus in 1975.

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math.DS 2026-05-06

Actions force random compacts to be finite or full almost surely

Invariant random compacts

Sufficient conditions and natural examples establish IC-rigidity; the Chacon system is only weakly rigid and implies multiplicative largness

abstract click to expand

For a compact metric space $X$ with a group $G$ acting on it continuously, an invariant random compact is a Borel probability measure on the space of nonempty compact subsets of $X$ that is invariant under the action of $G$. The action is IC-rigid if, with respect to every invariant random compact, every compact set is almost surely either finite or $X$. We give sufficient conditions for an action to be IC-rigid, and show there are natural examples of such actions. We further consider a notion of weak IC-rigidity, and prove that the Chacon system is weakly IC-rigid but not IC-rigid. As an application, we prove results concerning multiplicative largeness of dilations of sets on the circle.

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math.DS 2026-05-06

Reentrant fields on graphs form stable delayed reaction-diffusion systems

Reentrant value fields as delayed coupled reaction-diffusion systems on finite graphs

Symbolic and geometric fields coupled by delays admit compact attractors and global stability independent of delay size when coupling obeys

abstract click to expand

This article develops a field theory of synthetic cognition in which a symbolic field $H_L$ and a geometric field $X_R$, each a section of a vertex bundle over a finite graph, are coupled through a bipartite Hilbert-Schmidt operator with propagation delays. The central object is a retarded functional differential equation (RFDE) on the history space: the reaction-diffusion equation is the operative equation of the theory. Nine synthetic design blueprints specify admissibility conditions for each architectural component; each condition carries a dynamical consequence. The main formal results are: (1) well-posedness of the full deterministic RFDE under constant input $u^*$, (2) existence of a compact global attractor from compact viability and eventual compactness of solution segments, (3) delay-independent global stability of the principal components $(H_L,X_R,P)$ in the closed stability regime with fixed interfield coupling operators satisfying $C_{\mathcal{K}}^2<\mu_L\mu_R$, (4) $\mathrm{SE}(d)$-invariance of the scalar geometric feature dynamics, and (5) an $O(1/\kappa_Y)$ fast relaxation estimate for the valuative variable. Joint non-emptiness of all admissible classes is assumed. The well-posedness and attractor results allow Lipschitz state-dependent attention operators. The stability theorem is stated for the fixed-coupling principal subsystem, with the extra small-gain terms for state-dependent coupling identified explicitly.

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math.DS 2026-05-06 2 theorems

Delayed reaction-diffusion on graphs models synthetic cognition

Reentrant value fields as delayed coupled reaction-diffusion systems on finite graphs

The RFDE framework proves well-posedness, compact attractors, and stability of principal fields independent of delays under a coupling bound

abstract click to expand

This article develops a field theory of synthetic cognition in which a symbolic field $H_L$ and a geometric field $X_R$, each a section of a vertex bundle over a finite graph, are coupled through a bipartite Hilbert-Schmidt operator with propagation delays. The central object is a retarded functional differential equation (RFDE) on the history space: the reaction-diffusion equation is the operative equation of the theory. Nine synthetic design blueprints specify admissibility conditions for each architectural component; each condition carries a dynamical consequence. The main formal results are: (1) well-posedness of the full deterministic RFDE under constant input $u^*$, (2) existence of a compact global attractor from compact viability and eventual compactness of solution segments, (3) delay-independent global stability of the principal components $(H_L,X_R,P)$ in the closed stability regime with fixed interfield coupling operators satisfying $C_{\mathcal{K}}^2<\mu_L\mu_R$, (4) $\mathrm{SE}(d)$-invariance of the scalar geometric feature dynamics, and (5) an $O(1/\kappa_Y)$ fast relaxation estimate for the valuative variable. Joint non-emptiness of all admissible classes is assumed. The well-posedness and attractor results allow Lipschitz state-dependent attention operators. The stability theorem is stated for the fixed-coupling principal subsystem, with the extra small-gain terms for state-dependent coupling identified explicitly.

0

math.DS 2026-05-06 4 theorems

Generalized Gause system nonintegrable outside parameter regions

Liouvillian and Analytic Integrability of a Generalized Gause System

Nonintegrability of the predator-prey model implies the same for its reduced Abel equation and rules out Liouvillian closed-form solutions.

abstract click to expand

In this work, we identify the regions of the parameter space in which a predator-prey system, derived from the classical Gause model with a generalized Holling response function and logistic prey growth in the absence of predators, fails to be Liouvillian integrable. Although the model parameters have biological meaning only when restricted to appropriate real domains, our analysis is carried out in the complex setting, which provides a unified algebraic framework; the resulting nonintegrability conditions remain valid in the biologically relevant regime. As a consequence, we establish the nonintegrability of an Abel differential equation of the second kind with polynomial coefficients obtained from the system. Finally, we analyze the existence of a local analytic first integral in neighborhoods of the equilibrium points.

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math.DS 2026-05-06

Cusped singularities organize mixed-mode oscillations

Cusped singularities organize mixed-mode oscillations in mutually inhibitory slow-fast systems

They generate small-amplitude cycles near singular Hopf points that combine with large excursions in inhibitory slow-fast systems.

abstract click to expand

Mutual inhibition is a common motif in neural systems. Here, we establish that cusped singularities - folded singularities located at cusp points of critical manifolds - provide a universal organizing mechanism for mixed-mode oscillations (MMOs) in coupled slow-fast systems with mutual inhibition. We show that the geometric setup of these systems generically satisfies the conditions required by established geometric singular perturbation theory and blow-up methods, guaranteeing that such cusped singularities yield small-amplitude oscillations (SAOs). MMOs appear from the SAOs combined with an appropriate return mechanism. Further, we show that the geometric presence of a cusped singularity is strictly related to occurrence of a nearby singular Hopf bifurcation. We demonstrate the efficacy of this framework in two distinct neuronal models: the Curtu rate model of mutually inhibitory neural populations and coupled Morris-Lecar neurons with synaptic inhibition. In both cases, pushing the full system equilibrium near the cusped singularity triggers SAOs as the system passes near the cusp and approaches a full-system saddle-focus related to the singular Hopf bifurcation. Large-amplitude oscillations appear as the system spirals away from the saddle-focus, leading to MMOs, which may exhibit distinctive alternating patterns, in contrast to standard saddle-node induced MMOs. Our results establish cusped singularities as a generic, biologically relevant mechanism for complex oscillatory dynamics in inhibitory neural networks as well as for other inhibitory slow-fast systems.

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math.DS 2026-05-06

Coupled saddle-node systems unfold in four parameters

Local interaction of two systems with saddle-node bifurcations: mutualistic and mixed cases

Simultaneous saddle-nodes in two systems lead to homoclinic and SNIC bifurcations requiring four parameters to unfold fully.

abstract click to expand

The saddle-node bifurcation is the simplest example of a generic bifurcation in smooth ordinary differential equations, and is associated with the creation or destruction of a pair of equilibria. In this paper we examine the unfolding of the dynamics that occur when two generically coupled systems have simultaneous saddle-node bifurcations. We note that four parameters are required to generically unfold the interactions, and the dynamics are surprisingly complicated relative to the simplicity of a single saddle-node bifurcation. In the unfolding, in addition to saddle-node, Hopf and codimension-two local bifurcations, we also find a variety of global bifurcations, including homoclinic, SNIC, SNICeroclinic and non-central SNIC bifurcations. The latter two are codimension-two bifurcations that occur at the termination of a curve of SNIC bifurcations. A further contribution of this work is the development of numerical continuation techniques for the tracking of these codimension-two bifurcations through parameter space.

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math.DS 2026-05-05

Hamiltonian normal forms are explicit functionals of the original system

Combinatorics of Hamiltonian Normal Forms

The normal form near a singular point is obtained by direct substitution rather than iterative normalization.

abstract click to expand

We discuss algebraic and combinatorial aspects of the Hamiltonian normal form theory. The main objective is to describe the normal form near a singular point purely in terms of the original Hamiltonian, avoiding the normalization procedure. In the case of one degree of freedom we compute the normal form as an explicit nonlinear functional, applied to the original Hamiltonian. We present analogous results in arbitrary dimension. The corresponding formulas are more complicated but still explicit.

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math.DS 2026-05-05

Ergodic measures on Deroin-Tholozan varieties are finite orbits or Liouville

Invariant measures for Deroin-Tholozan representations

Mapping class group invariants on the compact components are either counting measures on finite orbits or match the Goldman Liouville form.

abstract click to expand

We classify mapping class group invariant probability measures on the character varieties of Deroin-Tholozan representations, namely the compact components of relative $\mathrm{PSL}_2\mathbb{R}$-character varieties. We prove that an ergodic measure is either the counting measure on a finite orbit or agrees with the Liouville measure induced by the Goldman symplectic form. Our approach is based on measure disintegration along transverse Lagrangian tori fibrations.

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math.DS 2026-05-05 3 theorems

Hyperbolic group metrics correspond to reparameterized flows with integer periodic orbits

A geometric correspondence for reparameterizations of geodesic flows

The correspondence yields the first continuous reparameterizations of geodesic flows on negatively curved manifolds where every periodic orb

abstract click to expand

For any non-elementary, torsion-free hyperbolic group, we provide a correspondence between the left-invariant Gromov-hyperbolic metrics on the group that are quasi-isometric to a word metric, and continuous reparameterizations of the associated Mineyev's flow space. From this correspondence, we produce the first examples of continuous reparameterizations of geodesic flows on negatively curved manifolds with all periodic orbits having integer lengths. For surface and free groups, this also yields isometric actions on Gromov-hyperbolic spaces on which loxodromic elements are precisely the non-simple elements. Key ingredients in our proof are an analysis of the geometry of Mineyev's flow space (such as the metric-Anosov property recently proven by Dilsavor), and the density of Green metrics in the moduli space of (symmetric) metrics on the group. We further establish continuity of the Bowen--Margulis--Sullivan geodesic current map on the moduli space of metrics, as well as a Bowen-type description of these currents as limits of sums of appropriately normalized atomic geodesic currents. For surface groups, we apply this continuity result to show that the Bowen--Margulis--Sullivan map restricts to a topological embedding on Hitchin components (up to contragradient involution) when equipped with their Hilbert lengths.

0

math.DS 2026-05-05 3 theorems

Isospectral graphons arise from different geometries

Graphons, Geometry, and Dynamics: Forward and Inverse Perspectives

Heat-kernel constructions from Neumann and Dirichlet drums show spectrum does not fix combinatorial class or always control Kuramoto stabl

abstract click to expand

In this work, we explore the interplay between graph limit theory, the geometry of underlying probability spaces, spectral theory, and network dynamical systems. We investigate two primary questions concerning forward and inverse perspectives: first, whether a graphon retains information about the geometry of the space on which it is defined, and second, whether spectral properties can distinguish graphons that originate from different geometric spaces. To address these questions, we differentiate between combinatorial equivalence and geometric structure, highlighting how these concepts are captured simultaneously by the class of pure graphons. Furthermore, we construct explicit examples of isospectral graphons -- graphons whose integral operators share the same spectrum -- that differ in their underlying geometry. By utilizing the heat kernels of Neumann- and Dirichlet-isospectral drums, we demonstrate that these graphons are not combinatorially equivalent. Finally, we establish new connections between the geometric aspects of graph limit theory and dynamical systems by analyzing a continuum Kuramoto model with graphon-defined interactions. We demonstrate that while isospectrality implies identical stability properties in certain cases, this correspondence breaks down when the differing boundary conditions of our specific Neumann and Dirichlet constructions are considered.

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math.DS 2026-05-05 3 theorems

Green functions exist for random loxodromic automorphism products

Random dynamics of plane polynomial automorphisms

Non-elementary groups of plane polynomial automorphisms yield compactly supported stationary measures and stiffness in the non-dissipative (

abstract click to expand

Let $\mu$ be a finitely supported probability measure on the group of automorphisms of $\mathbb{A}^2_\mathbb{C}$. If the group generated by the support of $\mu$ is non-elementary and contains only loxodromic elements, we show the existence of dynamical Green functions associated to random products. We derive consequences for $\mu$-stationary measures: they are compactly supported, and we can apply Roda's theorem to show stiffness when the action is non-dissipative.

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math.DS 2026-05-05 4 theorems

Neural approximation yields periodic paths along brain heteroclinic cycles

Modeling sequential cognitive states via population level cortical dynamics

Target dynamics containing sequential cognitive states are realized as periodic trajectories in high-dimensional neural fields that preserve

abstract click to expand

In this work, we present a mathematical model for cyclic and sequential patterns of brain activity, combining heteroclinic dynamics with discrete neural-field models. We first show that spatial-discrete neural-field equations with biologically realistic equilibria cannot support heteroclinic cycles. On the other hand, heterocline dynamics often arise in Lotka-Volterra-type systems, but these equations do not directly correspond to neuronal processes. To address this, we use a version of the Universal Approximation Theorem to approximate any target dynamics by a neural network interpretable as a high-dimensional Amari-type neural-field system. When the target dynamics contains a heteroclinic cycle, the approximating vector field generates a periodic trajectory that closely follows the heteroclinic connection. As a case study, we consider the cognitive processes underlying focused-attention meditation. We show how the model reproduces sequential transitions among cognitive states and we conclude providing a neural interpretation of the approximating dynamics.

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math.DS 2026-05-05 3 theorems

Maximal d-1 factor is topological characteristic factor of order d

On higher order regionally proximal relations and topological characteristic factors for group actions

Holds modulo almost one-to-one factors for cubic configurations in general group actions and for arithmetic progressions in finitely gen. ab

abstract click to expand

We study several aspects of higher-order regionally proximal relations for group actions. First, we develop an algebraic approach to study higher-order regionally proximal relations. To this end, we introduce a new topology on a subgroup of the universal minimal system, which can be seen as a higher-order analogue of the classical $\tau$-topology. Using this topology, we obtain an algebraic characterization of the relation $\mathbf{RP}^{[d]}$ for abelian actions. Then, we study higher-order regionally proximal relations via recurrence sets, extending results of Huang, Shao, and Ye for $\mathbb{Z}$-actions to more general group actions under suitable assumptions. We then study topological characteristic factors and prove, modulo almost one-to-one factors, that the maximal factor of order $d-1$ is the topological characteristic factor of order d for cubic configurations for arbitrary group actions, and for arithmetic progressions for finitely generated abelian group actions. As a consequence, we show that $\mathbf{RP}^{[d]}$ and $\mathbf{AP}^{[d]}$ coincide on minimal points for finitely generated abelian group actions, and we apply this to obtain results on independence along arithmetic progressions.

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math.DS 2026-05-05 2 theorems

Lyapunov exponent stays continuous for Gevrey cocycles when s+η is between 1 and 2

Continuity of Lyapunov Exponent for Quasi-Periodic Gevrey Cocycles

Quasi-periodic systems with limited smoothness keep their average expansion rate stable under small changes precisely in that parameter band

abstract click to expand

It is shown that for the quasi-periodic cocycles in Gevrey space $G^{s}$ and subexponential Brjuno class frequency $\Omega(\eta)$, the Lyapunov exponent is continuous provided that $1<s+\eta<2$.

0

math.DS 2026-05-04 3 theorems

Data deforms unknown vector field to single saddle-node

Realizing Saddle-Node Bifurcations from Finite Data

In dimension 6 or higher, an isolating block with matching Conley index yields a canonical model that leaves the dynamics unchanged outside

abstract click to expand

Given a finite set of data generated by an unknown ordinary differential equation it is impossible to exactly determine the associated vector field, and hence, bifurcation theory tells us that it is impossible, in general, to correctly characterize the underlying dynamics. In this paper, we bypass the effort of obtaining an analytic approximation of the vector field, and we adopt an approach based on Occam's razor: identify the simplest robust characterization of the dynamics that is compatible with the given data. Our fundamental assumption is that the data allows for the construction of an isolating block over a parameter space whose homological Conley index is consistent with a saddle-node bifurcation. Our main result establishes that, for phase spaces of dimension greater than or equal to 6, the original vector field can be smoothly deformed into a canonical model exhibiting exactly one structurally stable saddle-node bifurcation. Crucially, this deformation leaves the vector field unaltered outside the isolating block, ensuring strict compatibility with the observed data.

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math.DS 2026-05-04

EML operator yields single-block overshoot modules for biology models

Non-Monotone Response Modules and Cascades from the EML Operator for Reduced Models of Biological Dynamics

Replaces paired saturating blocks with one activation-suppression unit in reduced nonlinear ODEs

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Standard saturating response functions, such as the Hill function, are monotone and therefore cannot represent recruitment-induced overshoot or adaptive transients with a single block. Reproducing such non-monotone responses from saturating primitives requires at least a difference of two blocks with opposing amplitudes, doubling the static-block parameter count. Here, building on a recent mathematical result that a single binary operator, EML, generates all standard elementary functions, we use EML as a structured grammar for reduced nonlinear ODEs. This yields an activation-suppression module that captures overshoot directly. We validate the framework in three settings. First, on PKA-R relocalization data, the EML grammar discovers a reduced surrogate consistent with established mechanistic biology. Second, on Rho-GTPase recruitment data, an exhaustive search over EML expression trees selects the same compositional form across all four perturbation-response traces. Third, a 50-state simulated network is compressed by an EML cascade acting as a fixed temporal basis. Thus we demonstrate the power and potential of EML for reduced models of biological dynamics.

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math.DS 2026-05-04

Graph cycles decide when translated self-similar sets stay self-similar

For equal-ratio attractors, the union with ordered shifts is self-similar exactly when the translation graph contains a cycle.

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In this paper, we explore the self-similarity of unions of self-similar sets and their translations. For $N \in \mathbb{N}$ and $0< \beta < 1/(N+1)$, let $\Gamma$ be the self-similar set generated by the IFS \[ \Big\{ \phi_i(x)=\beta x + i \frac{1-\beta}{N}: i=0,1,\ldots, N \Big\}. \] We provide a complete characterization of translation vectors $\boldsymbol{t} =(t_0,t_1, \ldots, t_m) \in \mathbb{R}^{m+1}$ with $0=t_0 < t_1 < \cdots < t_m$ for which the union $\bigcup_{j=0}^m (\Gamma+t_j)$ is a self-similar set, by determining the existence of cycles in associated directed graphs. This extends the result of [Derong Kong, Wenxia Li, Zhiqiang Wang, Yuanyuan Yao, Yunxiu Zhang. On the union of homogeneous symmetric Cantor set with its translations. Math. Z., 2024]. Additionally, we present two types of self-similar sets for which the union with their translations cannot be self-similar.

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math.DS 2026-05-04

Rank-one matrices split epidemic parameters into four stable regions

From the Volterra type Lyapunov functions of Rahman-Zou towards a competitive exclusion partition property for rank one models

Concave incidence plus a Perron-Volterra Lyapunov function certifies unique equilibria (disease-free, single-strain, or coexistence) in each

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This paper presents a Perron-Volterra framework that unifies explicit Lyapunov constructions for multi-strain epidemic models with rank-one next-generation matrices. At each boundary equilibrium on a siphon face, the Lyapunov function consists of a Volterra entropy on resident variables plus a Perron-weighted linear functional on invaders, derived from the left Perron eigenvector of the transversal Jacobian. A balance identity cancels coupling terms, reducing global stability to recursive computation of invasion numbers on the siphon lattice. For two-strain models with concave, increasing incidence, we prove the competitive exclusion partition property (CEPP): the parameter space splits into four open regions, each possessing a unique globally asymptotically stable equilibrium (disease-free, single-strain, or coexistence) certified by an explicit Lyapunov function. The same mechanism extends to an arbitrary number of singleton strains and to models with one scalar strain and one irreducible rank-one block. We implement the algorithmic approach in the Mathematica package EpidCRN, which constructs candidate Lyapunov functions, verifies the balance identity, and partitions the parameter space recursively. For two rank-one matrix blocks, the standard ansatz fails; we characterize the obstruction and propose an augmented cross-equilibrium Lyapunov function. A local Lyapunov theorem for siphon faces is also provided. The framework offers a systematic stability analysis of rank-one models.

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math.DS 2026-05-04

Optimizing periodic orbits stay typical after joint small changes to maps and potentials

Joint typical periodic optimization: systems with stable hyperbolicity

Axiomatic extension shows open dense locking sets for Axiom A diffeomorphisms, hyperbolic rationals, quadratics, and one-dimensional maps.

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The framework of joint typical periodic optimization, in which both the dynamical system and the potential function are allowed to vary simultaneously, was introduced in [HHJL25], in a direction motivated by the work of Yang, Hunt & Ott [YHO00]. For certain classes of hyperbolic systems, it was shown there that optimizing periodic orbits persist under simultaneous perturbation, yielding joint locking sets that contain open dense subsets of the relevant product spaces. In the present article we broaden the scope of this theory, by developing an axiomatic joint perturbation framework that accommodates a wider class of stably hyperbolic systems, and by establishing new joint typical periodic optimization results for several natural and important families: Axiom A diffeomorphisms with the no-cycle property, hyperbolic rational maps on the Riemann sphere, real quadratic polynomials, and $C^r$ maps in one dimension.

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math.DS 2026-05-04

Carrying simplex conjugates competitive Carathéodory ODEs to lower dimension

Time-periodic carrying simplex for a competitive system of Carath\'eodory ODEs

The conjugacy holds for time-periodic Kolmogorov systems whose growth rates meet only Carathéodory conditions and supplies a numerical route

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We consider time-periodic competitive systems of ordinary differential equations of Kolmogorov type. However, compared with standard assumptions, we relax the regularity of the time-dependent per-capita growth rates by imposing much weaker regularity, namely Carath\'eodory conditions. An important tool in investigating such systems is the concept of carrying simplex, that is, of an unordered invariant manifold of codimension one that attracts all nonzero orbits. We define the carrying simplex via the compact attractor of compact sets of an extended flow, and that attractor can be obtained as the limit of the actions of the solution operator on some set. Compared with previous papers, our approach has more dynamical flavour, and, further, provides a method of numerical approximation of the carrying simplex. Another feature of our paper is that we prove that the system restricted to the extended carrying simplex is topologically conjugate to a system of one dimension less. This property, appearing in the path-breaking paper by Morris W. Hirsch, has been almost universally neglected in the later papers.

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math.DS 2026-05-04

Shadowing ensures stability for local iterated function systems

Stability Theory for Local Iterated Function Systems

Concordant shadowing yields attractor semicontinuity and code space persistence, giving combinatorial and topological stability under the开放集

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We develop a stability theory for contractive local IFSs on compact metric spaces. Unlike the classical global setting, local systems may exhibit a richer symbolic and geometric structure, including code spaces that are not of finite type and attractors with endpoints, leading to new mechanisms of instability. We first prove that concordant shadowing implies upper semicontinuity of the local attractor and persistence of the code space, yielding a criterion for combinatorial stability under perturbations. Under the open set condition, we establish a strong form of topological stability for combinatorially stable contractive local systems, and prove the converse implication on compact manifolds of dimension at least three. In particular, we show that contractive graph-directed IFSs are topologically stable. We also construct contractive local IFSs derived from beta-transformations that are combinatorially unstable. These results show that stability in the local setting is governed by the interplay between contraction and the combinatorial rigidity of the code space. Applications to graph-directed IFSs and pseudogroup actions are also given.

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math.DS 2026-05-04

Cat-map entropy stays fixed on circulant graphs

Coupled Arnol'd cat maps on circulant graphs

Simulations show Kolmogorov-Sinai entropy independent of connectivity thanks to translational symmetry of the graph.

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This paper investigates the chaotic properties of Arnol'd cat maps (ACMs) coupled on the nodes of a circulant graph. By demanding that the system's evolution matrix be symplectic, we determine the coupling matrix, which is naturally interpreted as the adjacency matrix of a circulant graph. Specifically, the study analyses the system's Lyapunov spectra and Kolmogorov-Sinai (K-S) entropy. Numerical simulations yield the counterintuitive result that the entropy production does not increase as the connectivity of the graph increases, due to the translational symmetry of the circulant graph. Moreover, we analyse the spectra of the periods of the evolution matrix on a finite toroidal phase space of the dynamical system.

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math.DS 2026-05-04

Gluing two polynomials yields rational maps with finite curve attractor

On the Global Curve Attractor for polynomial gluing

Intersection numbers with separating arcs decay under pullback, forcing all non-peripheral curves into a finite set of homotopy classes.

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Pilgrim's Finite Global Attractor Conjecture has been verified for polynomials [1], but remains open for general rational maps. In this paper, we prove the conjecture for a family of rational maps obtained by gluing two PCF polynomials along the boundaries of their finite superattracting basins. Adapting the idea of [17], we show that a suitably defined intersection number with a finite family of separating arcs eventually decays under pullback, yielding a finite collection of homotopy classes that attracts all non-peripheral curves under iteration.

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math.DS 2026-05-04

Cosine memristor adds Neimark-Sacker bifurcations and multi-chimera states to Chialvo map

Dynamical analysis of r-Chialvo neuron map with cosine memristive

The discrete model exhibits multistable limit cycles, period-five and chaotic attractors plus network patterns absent from prior Chialvo研究.

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In this work, we construct a novel two-dimensional discrete neuron map by incorporating a cosine-based memristor into the reduced Chialvo neuron map to examine the dynamical analysis of electromagnetic modulation. The nonlinear current-voltage characteristics of the memristor enrich the neuron map's behavior, leading to diverse firing regimes, stability behaviors, and chaotic attractors. This study begins to establish the equilibrium points using both analytical and numerical methods. Additionally, we determine the conditions on parameters under which the proposed map exhibits a Neimark-Sacker bifurcation. Further, the numerical study reveals the antimonotonicity structure through the forward and backward bifurcation diagrams. The model exhibits a wide range of codimension-one and codimension-two bifurcation patterns, including Neimark-Sacker, period-doubling, saddle-node, generalized period-doubling, cusp-point, fold-flip, and various resonance structures (1:1, 1:2, 1:3, and 1:4). We also observe that the coexistence of multistable attractors including a stable limit cycle, a period-five attractor, and a chaotic attractor, along with their respective basins of attraction. Furthermore, we extend this analysis to the network of neurons under the ring-star configuration and discuss several spatiotemporal patterns. This network investigation reveals complex collective patterns, including imperfect synchronization, clustered patterns, and multi-chimera state phenomena, which have not been previously observed in existing Chialvo-based studies. These results highlight the potential of the discrete memristor-based neuron map for advancing theoretical neurodynamics and offer a robust framework for investigating low-dimensional yet dynamically rich neuron systems.

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math.DS 2026-05-04

Hyperbolic minima force complete devil's staircases in twist maps

On Aubry's completeness conjecture

Rotation numbers versus cohomology classes become purely singularly continuous when minimal configurations are uniformly hyperbolic.

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In this paper, we prove Aubry's completeness stating conjecture that for a twist map the graph of rotation numbers as a function of the cohomology classes is a purely singularly continuous function (called complete devil's staircase by Aubry) when the set of all minimal configurations is uniformly hyperbolic. Such a phenomenon is crucial for characterizing the chain of atoms being an insulator for the Frenkel-Kontorova model, and can be considered as the analogue of the phase locking phenomenon in critical circle maps as well as the fractional quantum Hall effect. In contrast, in the presence of a positive measure set of KAM tori, we prove that the devil's staircase is incomplete.

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math.DS 2026-05-01

Stability of interval translation maps equals no critical connections plus matching

Characterisation of Stability for Interval Translation Maps

The equivalence supplies the first concrete test for robustness in these non-bijective generalizations of interval exchanges.

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An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ of an interval $I$ into $r \ge 2$ subintervals. In contrast to classical interval exchange transformations (IETs), we do not require that the images of these subintervals are disjoint; in particular, ITMs are not assumed to be bijective. Thus, ITMs provide a natural non-invertible generalisation of IETs. In this paper, we formulate an appropriate notion of stability for general interval translation mappings and prove a characterisation of stability in terms of two dynamically natural properties called the Absence of Critical Connections and Matching. This result can be viewed as the foundational step towards the stability theory of general ITMs.

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math.DS 2026-05-01

Finite type maps dominate interval translations for any r

Topological Prevalence of Finite Type Interval Translation Maps

Their attractors collapse to finite unions of intervals, so generic maps behave like exchanges on the surviving core.

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An interval translation map (ITM) is a map $T \colon I \to I$ defined as a piecewise translation on a finite partition of an interval $I$ into $r \ge 2$ subintervals. Unlike classical interval exchange transformations (IETs), the images of these subintervals are allowed to overlap, making ITMs a natural generalisation of IETs. An ITM $T$ is said to be \textit{of finite type} if its attractor $\bigcap_{n\ge 0} T^n(I)$ is a finite union of intervals; in this case, restricted to this invariant set, $T$ is bijective and hence behaves like an IET. Otherwise, $T$ is of infinite type. In this paper, for every $r \ge 2$, we prove that the set of finite type ITMs contains an open and dense subset in the space of all possible ITMs on $r$ subintervals. This confirms a topological version of a long-standing conjecture due to Boshernitzan and Kornfeld.

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math.DS 2026-05-01

Transversality controls first returns in interval translation maps

Transversality for Interval Translation Maps

A perturbation result adjusts returns to subintervals while the overall dynamics stays fixed, serving as a tool for stability and related 0-

abstract click to expand

An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ of an interval $I$ into $r \ge 2$ subintervals. In contrast to classical interval exchange transformations (IETs), we do not require that the images of these subintervals are disjoint; in particular, ITMs are not assumed to be bijective. Thus, ITMs provide a natural non-invertible generalisation of IETs. In this paper, we prove a transversality theorem for a family of dynamically defined vector subspaces that encode the dynamics of a given ITM. As a consequence, we establish a perturbation result that gives a precise control of the first return dynamics to subintervals in $I$, while preserving the remaining global dynamics of the system. Beyond their independent interest, these results are a key technical ingredient in the proof of the Characterisation of Stability of ITMs in arXiv:2605.00190, and in the establishment of the topological version of the Boshernitzan--Kornfeld Conjecture in arXiv:2605.00186.

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math.DS 2026-05-01

Transcendental shift-like maps always have non-empty Julia sets

Two remarks on transcendental shift-like maps on mathbb{C}^N

An explicit example also shows escaping wandering domains, unlike the polynomial case.

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In \cite{Bedford}, the dynamics of a particular polynomial diffeomorphism of $\mathbb{C}^N$, called a polynomial shift-like map of type $\nu$, has been studied as a higher dimensional analog of H\'enon maps. In this note, we prove that the Julia set of their transcendental counterpart is non-empty. In addition, an example of a transcendental shift-like map with an escaping wandering domain has been provided which, in fact, showcases a contrast with the dynamics of a polynomial shift map.

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math.DS 2026-05-01

Endogenous measures exist on finite sigma-algebra systems

Endogenous Measures and Refinement Dynamics on Finite {σ}-Algebra Systems

They remain invariant under admissible refinements and induce dynamics on the lattice of algebras.

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We study systems of {\sigma}-algebras ordered by refinement and introduce the notion of an endogenous probability measure, invariant under admissible refinement transformations. We prove existence and structural properties of such measures on finite systems and show how refinement operators induce a natural dynamical structure on the lattice of {\sigma}-algebras.

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math.DS 2026-05-01

Linear response yields explicit optimal drift perturbations on the torus

Optimal response for stochastic differential equations in mathbb{T}^d with perturbations on the drift term

A first-order formula for stationary densities gives unique maximizers via Fourier series that work in any dimension.

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We study stochastic differential equations on the $d$-dimensional flat torus $\mathbb{T}^d$ with drift and perturbation coefficients in $L^{\infty}(\mathbb{T}^d;\mathbb{R}^d)$ and additive non-degenerate noise. For the associated transfer operators, we analyse the dependence of the stationary measure and of the expectation of a given observable on small perturbations of the drift. In this framework, we prove a linear response formula for the invariant density and for the expectation of a given observable. We then address an optimal response problem, namely the determination of admissible perturbations that maximise the first-order variation of a prescribed observable. We establish existence of optimal perturbations and, in a Hilbert space framework, prove uniqueness and provide an explicit characterisation of the optimiser. This yields a practical Fourier-based numerical method, which we implement in several numerical examples, including both low and high-dimensional settings.

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math.DS 2026-04-30

Singular 3D flows have finitely many maximal entropy measures

Thermodynamics formalism for singular flows

C^∞ flows with positive topological entropy retain only a finite number of ergodic measures of maximal entropy even with zero-velocity sing

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We establish that $C^\infty$ three-dimensional flows with positive topological entropy admit only finitely many ergodic measures of maximal entropy, even when singularities (zero-velocity points) are present. Furthermore, every ergodic measure of maximal entropy is rapid mixing for such flows within a $C^\infty$ open and dense subset. To prove this, we develop a novel symbolic coding system for flows with singularities, which serves as a fundamental tool in this work. We also define the strong positive recurrence (SPR) property for singular flows and verify that SPR flows can be coded by suspension flows of SPR symbolic systems. This framework extends to other singular flows, including star flows, and to equilibrium states.

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math.DS 2026-04-30

Polynomial ODEs with provable bounds discovered from data

Data-driven discovery of polynomial ODEs with provably bounded solutions

The approach simultaneously identifies the dynamics and a polynomial Lyapunov function to ensure trajectories stay within compact sets.

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We introduce SILAS, a data-driven framework for discovering polynomial ordinary differential equations (ODEs) with provably bounded trajectories. Boundedness is certified by compact absorbing sets defined via polynomial Lyapunov functions. We jointly identify the ODE vector field and the Lyapunov function using a well-posed nonconvex optimization problem built using polynomial optimization tools. We solve this problem using an alternating block-coordinate optimization scheme with convex subproblems, whose feasibility is ensured by a novel model-agnostic initialization that identifies a candidate Lyapunov function from data. Our methods extend prior approaches for quadratic ODEs with absorbing ellipsoids to a significantly broader class of ODEs and absorbing sets. A suite of over 100 examples demonstrates that SILAS can recover accurate and provably bounded ODE models for a broad range of nonlinear dynamical systems.

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math.DS 2026-04-30

Four-symbol coding gives bijection between words and pedal triangles

Primitive Two-Dimensional Words and Iterated Pedal Triangles via Symbolic Coding

The construction accounts for why the counts of primitive 2-by-n binary words match the triangles first similar to themselves at the nth ped

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The notion of a two-dimensional word arises naturally in the study of combinatorics on words, while the iterative construction of pedal triangles results in a rich dynamical system in the study of geometry. At first, these two classes of objects seem to be unrelated. However, it is known that for all $n \geq 1$, the number of primitive two-dimensional words of dimension $2 \times n$ over a binary alphabet agrees with the number of triangles whose first similar pedal triangle is their $n$th pedal triangle. We construct a finite four-symbol coding of the sorted pedal map and use the resulting branch itineraries to give a bijection between these two classes.

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math.DS 2026-04-30

Logistic reformulation cuts Lipschitz constants in gene network models

Beyond Linear Additive and Hill Functions: A General Logistic Reformulation of Delay-Coupled Gene Regulatory Networks with Equilibrium Analysis, Hopf Bifurcation, and Lipschitz Stability

Matching half-max slopes and basal rates keeps equilibria and delay-induced Hopf points while allowing larger integration steps.

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Hill functions, dominant in gene regulatory network modeling, carry fundamental limitations: at non-integer cooperativity exponents, routine when fitting dose-response data, derivatives diverge at the origin, complex arithmetic corrupts ODE trajectories, and zero output at zero activation traps models in off-states. This paper employs logistic-based models that are globally $C^\infty$, real-valued, and strictly positive at zero concentration, resolving all three pathologies while preserving sigmoidal dynamics. Using the delay-coupled two-gene mutual-activation and self-repression network of Vinoth et al.\ as a concrete model, we analyze two reformulations: linear additive activation with logistic self-repression, and a fully sigmoidal form with both terms logistic. A closed-form matching relation $\lambda = n/\theta$ follows from equating slopes at half-maximal points. Closed-form parameters of the weighted logistic formulation are derived by matching basal rates and local slopes to the Hill-linear hybrid model. The unique biologically feasible equilibrium is computed in each case; it is lower in the weighted logistic case, the reduction arising from saturation of the bounded activation term. In the delay-free case ($\tau=0$), local asymptotic stability holds in both formulations since the Jacobian trace is strictly negative for all positive parameters; stability persists for $\tau\in(0,\tau_c)$ and is lost via Hopf bifurcation at the critical delay $\tau_c$. Numerical solution of the full transcendental system locates $\tau_c$, with higher-order bifurcations characterised numerically in each case. Replacing linear additive with weighted logistic activation substantially reduces both the global Lipschitz constant of the right-hand side and that of its Jacobian, permitting larger integration steps.

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math.DS 2026-04-30

Invariant sets lift uniquely via normal cones to boundary ODEs

Invariant Sets and Boundary Systems of Nonautonomous Differential Inclusions

This turns analysis of uncertainty propagation and control dynamics into a finite-dimensional deterministic ODE problem with unique minimal

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In this paper we propose a finite-dimensional and deterministic approach to the study of invariant sets of certain nonautonomous differential inclusions naturally arising in the context of random and control dynamical systems, as well as in systems modeling the dynamical propagation of uncertainty. In particular, to any such differential inclusion, we associate a finite-dimensional and deterministic system of nonautonomous ordinary differential equations, which we call the boundary system, due to its following characteristic property: invariant sets of the differential inclusion lift in a unique way to backward invariant unit normal cones of the associated boundary system, and these are even invariant if the boundary is smooth. We further illustrate the strength of this approach in the study of minimal attractors of nonautonomous linear differential inclusions. Under the natural assumption of exponential stability for the unperturbed problem, we establish existence and uniqueness of a minimal attractor for the differential inclusion with fiberwise strictly convex, closed, and continuously differentiable boundaries. We also show that the unit normal bundle is in fact the pullback attractor for the skew-product flow associated to the boundary system which extends to the global attractor when the underlying admits a compact base.

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