Dynamical Systems
See recent articles
Showing new listings for Friday, 21 August 2026
- [1] arXiv:2608.19289 [pdf, html, other]
-
Title: Existence of periodic solutions for Hamiltonian inclusion systems using Clarke dualitySubjects: Dynamical Systems (math.DS)
We prove the existence of periodic solutions for Hamiltonian differential inclusions under growth conditions involving a G-function.
- [2] arXiv:2608.19360 [pdf, html, other]
-
Title: Hodge Coercivity and Global Dynamics in Two-Field Edge-Cochain Systems with MHD-Type CancellationSubjects: Dynamical Systems (math.DS); Social and Information Networks (cs.SI)
A finite-dimensional two-field system for divergence-free edge cochains is introduced. Its MHD-type designation refers only to a quadratic exchange pattern and exact total-energy cancellation; it is not a physical MHD discretization. A general cancellation class is separated from a corrected explicit realization: the anticommutator $D(a)J+JD(a)$ is skew-symmetric for diagonal $D(a)$ and skew-symmetric $J$, and its projected bilinear map has the required trilinear antisymmetry. The central result is a Hodge coercivity criterion: the full divergence-free space admits the Poincaré-type estimate needed for dissipativity if and only if its harmonic $1$-cochain space is trivial. Under this condition, global existence, an exact energy identity, an absorbing ball, and a compact global attractor follow. When harmonic modes are present, a harmonic-decoupled interaction class yields invariant harmonic affine fibres and fibre-wise attractors. Deterministic disk, annular, and two-hole examples illustrate the spectral criterion, energy law, and distinction between general harmonic exchange and harmonic-fibre invariance.
- [3] arXiv:2608.19393 [pdf, other]
-
Title: Outer Contact BilliardsComments: 30 pages, 13 figuresSubjects: Dynamical Systems (math.DS); Symplectic Geometry (math.SG)
We introduce outer contact billiards as an odd dimensional counterpart to outer symplectic billiards. Until now, outer billiards have only been considered in even dimensional symplectic vector spaces. By projectivizing outer symplectic billiards, we obtain outer contact billiards, where the affine midpoint condition descends to its projective analog, namely harmonic conjugation.
We prove that outer contact billiards generate contactomorphisms. For quadratic surfaces in $\mathbb{RP}^3$, we show that the correspondence is completely integrable: its domain is foliated by invariant quadrics and, on each leaf, the dynamics is determined by the iteration of an explicit linear transformation. We also establish two rigidity results for periodic trajectories: outer contact billiards admit no 3-periodic orbits and, among quadratic tables, only one admits 4-periodic orbits. - [4] arXiv:2608.19424 [pdf, html, other]
-
Title: A rock-paper-scissors Mandelbrot setSubjects: Dynamical Systems (math.DS); Mathematical Software (cs.MS); Complex Variables (math.CV); Rings and Algebras (math.RA)
The titular object of this paper is an analogue of the Mandelbrot set over the 3-dimensional real algebra whose multiplication is the bilinear extension of the rock-paper-scissors operation. In order to study the dynamics of the mappings $x\mapsto x^2+c$ in this setting, a notion of holomorphy for functions on general finite-dimensional real algebras is introduced. Under a mild assumption it is shown that for such algebras holomorphy is always equivalent to solving a finite system of linear first-order PDEs generalizing the Cauchy-Riemann equations. Code is provided for generating animations of these fractals and for exploring them in a video game format.
- [5] arXiv:2608.19457 [pdf, html, other]
-
Title: Equidistribution and thermodynamics at infinityComments: 45 pages, no figuresSubjects: Dynamical Systems (math.DS)
We prove level-2 large deviation upper bounds for potentials on countable Markov shifts and for suspension semi-flows over countable Markov shifts. For strongly positive recurrent potentials, we establish equidistribution of weighted empirical measures toward the corresponding equilibrium state. We then apply these results to interval maps, obtaining, in particular, equidistribution and large-deviation estimates for measures supported on boundary points. For the Gauss map, this yields equidistribution results on rational numbers, including a theorem of David and Shapira.
- [6] arXiv:2608.19542 [pdf, html, other]
-
Title: The number of limit cycles of piecewise linear Liénard systemsSubjects: Dynamical Systems (math.DS)
For the planar Liénard differential system $\dot{x}=F(x)-y$, $\dot{y}=x$, where $F(x)$ is a piecewise linear function, Tonnelier (SIAM J. Appl. Math., 2002) conjectured that the maximum number of limit cycles of the system is $n$ when $F(x)$ has $n$ fold points and no jump points, and $2n$ when $F(x)$ has $n$ jump points and no fold points. This conjecture was confirmed by Llibre et al. (J. Nonlinear Sci., 2015) (resp. Chen et al. (J. London Math. Soc., 2026a)) when $F(x)$ has one fold point and no jump points (resp. two fold points). More recently, Chen et al. (J. London Math. Soc., 2026b) proved that the conjecture is correct when $F(x)$ has no fold points and one jump point. All other cases remain open.
Here we verify that the lower bound for the maximum number of limit cycles of the system can be $n$ when $F(x)$ has only $n$ fold points, and $2n$ when $F(x)$ has only $n$ jump points, thereby confirming the lower bound part of Tonnelier's conjecture. Moreover, when $F(x)$ has $m$ jump points and $n-m$ fold points, $0\le m\le n$, we also show that the system can have $n+m=(n-m)+2m$ limit cycles. In addition, a complete classification of the {dynamics} near infinity for this class of systems is provided. - [7] arXiv:2608.19563 [pdf, html, other]
-
Title: An exact structural identity and its normal form consequences in a modified Leslie--Gower modelComments: 16 pages, 1 figureSubjects: Dynamical Systems (math.DS)
Zhao and Zhao (2026) established a codimension-four Bogdanov--Takens singularity in a modified Leslie--Gower predator--prey model through a recursive normal form computation. We show that the nonlinear coefficients governing the restriction of the reduced system to the singularity's distinguished eigendirection are not independent. Instead, a single exact rational identity, already present in the transformed equations, generates the entire family of pure Taylor coefficients and proves that their simultaneous vanishing is an exact all orders property rather than a finite-order coincidence. We further show that this structural identity propagates through the planar Bogdanov--Takens normal form algorithm of Kuznetsov (2005). Although the pure coefficients vanish identically, the corresponding normal form coefficients are generated by mixed quadratic interactions. This leads to an explicit characterization of the degenerate Bogdanov--Takens locus and reduces the search for higher-order Bogdanov--Takens degeneracies to a single explicit algebraic condition.
- [8] arXiv:2608.19697 [pdf, html, other]
-
Title: Open cover specification property and ChaosSubjects: Dynamical Systems (math.DS)
This paper investigates the open cover specification property and its role in the study of topological dynamical systems. We show that this property is a topological invariant and a natural extension of the classical topological specification property from uniform spaces to general topological spaces, and that it is preserved under infinite products with compositions. We further prove that on regular spaces, the strong open cover specification property implies Devaney chaos, and that on locally compact, first countable Hausdorff spaces, it guarantees both distributional chaos and Li Yorke chaos. These results demonstrate the significance of the open cover specification property in linking topological structure with dynamical complexity.
- [9] arXiv:2608.19785 [pdf, html, other]
-
Title: Learning piecewise-smooth dynamical systemsSubjects: Dynamical Systems (math.DS); Machine Learning (cs.LG); Numerical Analysis (math.NA)
Discovering dynamical systems from trajectory data is a central problem in applied mathematics and engineering. Whilst recent advances in machine learning have led to strong progress in data-driven system identification, much less attention has been given to systems with discontinuous dynamics. These systems are nevertheless highly relevant in applications, including climate dynamics and mechanical systems with friction. In this work, we consider the problem of identifying piecewise-smooth dynamical systems directly from trajectory data. Compared with the smooth setting, this requires recovering the governing equations and detecting the switching hyperplanes that separate different dynamical regimes and characterising their behaviour, such as sliding motion. We present a modular framework for discovering such systems by first estimating switching hyperplanes from data and then learning smooth dynamics within each region using geometry-constrained neural networks. The geometry-learning phase is studied from a statistical perspective, analysing the identifiability of the discontinuities and the robustness of the procedure. We also introduce a novel neural network architecture with a prescribed discontinuity set, and provide a theoretical analysis of its approximation properties. The approach is tested on low-dimensional benchmark problems, including dry-friction oscillators and the PP04 climate model for the ice ages.
- [10] arXiv:2608.19958 [pdf, html, other]
-
Title: The Symmetry and Linear Stability of Convex 1+5 Coorbital Central Configurations with Homogeneous PotentialComments: 14 pages, 1 figures, 28 conferencesSubjects: Dynamical Systems (math.DS)
For the planar Newtonian 1+N-body problem when the N masses tend to zero, the corresponding relative equilibria become coorbital around the dominant mass. In this work, we focus on convex central configurations in the planar 1+N coorbital problem. For the 1+5 coorbital problem with the homogeneous potential, we prove that any convex coorbital central configuration with symmetric masses must have an axis of symmetry. Furthermore, under explicit restrictions on the angular variables in a homogeneous potential, we prove the linear stability of both convex symmetric 1+5 and convex 1+N coorbital central configurations.
- [11] arXiv:2608.20004 [pdf, html, other]
-
Title: Sets of nice recurrence are partition regularComments: 6 pagesSubjects: Dynamical Systems (math.DS); Combinatorics (math.CO)
A set of positive integers $R$ is called a set of nice recurrence if for any measure preserving system $(X,\mu,T)$, for all measurable $A\subseteq X$, and each $\varepsilon>0$, there exists $n\in R$ such that $\mu(A\cap T^{-n}A)\geqslant \mu(A)^2 - \varepsilon$. Answering a long-standing question of Bergelson, we show that sets of nice recurrence have the following Ramsey property: any finite colouring of a set of nice recurrence admits a monochromatic set of nice recurrence.
- [12] arXiv:2608.20008 [pdf, html, other]
-
Title: An arithmetic approach to parabolic multiplicity in complex dynamicsComments: 26 pagesSubjects: Dynamical Systems (math.DS)
When $\omega$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z) = \omega z (1 -z)$ and the entire map $F(z) = \omega z \mathrm{e}^{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $F^{\circ n}(z) = z \bigl( 1 +c z^n +\mathcal{O}(z^{n+1}) \bigr)$ with $c \neq 0$. The classical proof of this fact is transcendental. We present an arithmetic proof which may be extracted from [Towards global models near homoclinic tangencies of dissipative diffeomorphisms; H. Broer, C. Simó, J.C. Tatjer] in the transcendental case and requires working in $\mathbb{Z}/(n -1) \mathbb{Z}$, and which is new in the polynomial case and requires working in the $p$-adic field $\mathbb{Q}_p$ for a suitable prime $p$ such that the order of $2$ in $(\mathbb{Z}/p \mathbb{Z})^\times$ is exactly $n$.
- [13] arXiv:2608.20165 [pdf, html, other]
-
Title: Orbit equivalence and total weak mixing of free group actionsComments: 6 pagesSubjects: Dynamical Systems (math.DS); Group Theory (math.GR); Logic (math.LO)
We prove that the orbit equivalence class of every free ergodic probability-measure-preserving (pmp) action of a free group contains a totally weak mixing action. Equivalently, every ergodic treeable pmp equivalence relation of cost $n\in\mathbf{N}\cup\{\infty\}$ is generated by a free totally weak mixing action of $\mathbf{F}_n$. This answers a question of Miller and Tserunyan.
The proof goes by considering a Polish space of edge slidings along a fixed mixing transformation and proving that for every $w\not=e\in\mathbf{F}_n$ the set of edge slidings that produce an action with $w$ weakly mixing forms a comeager set.
New submissions (showing 13 of 13 entries)
- [14] arXiv:2608.18485 (cross-list from math.LO) [pdf, html, other]
-
Title: Finite-Index Lifting of Strong Topological Rokhlin Property and Descriptive ComplexitySubjects: Logic (math.LO); Dynamical Systems (math.DS)
We give a finite symbolic reformulation of the strong topological Rokhlin property in terms of globally realizable tuples. We prove that the strong topological Rokhlin property passes from a finite-index subgroup to a finitely generated overgroup. We also study the descriptive complexity of the class of countable groups having the strong topological Rokhlin property. In the standard compact space of countable groups, this class belongs to $\mathbf{\Pi}^0_4$ and is $\mathbf{\Sigma}^0_2$-hard. We also isolate a barrier to Borel rank four: if the class is not $\mathbf{\Sigma}^0_3$, then there is a non-finitely-presented group with the strong topological Rokhlin property.
- [15] arXiv:2608.19499 (cross-list from math.FA) [pdf, html, other]
-
Title: Resolving the generalized hyperbolicity conjecture for shadowingComments: 16 pagesSubjects: Functional Analysis (math.FA); Dynamical Systems (math.DS)
It is known that generalized hyperbolicity implies the shadowing property for invertible bounded linear operators on a Banach space. Whether the converse holds has been a central open problem in linear dynamics and has been conjectured to have a positive answer. We show that this conjecture fails on general Banach spaces by constructing a counterexample, whereas it holds on separable Hilbert spaces. The distinction is explained by the gap that may occur on Banach spaces between surjectivity and right invertibility, a gap that disappears on Hilbert spaces. The main ingredients of the proofs are a recent spectral characterization of shadowing in terms of the surjective spectrum and a new characterization of generalized hyperbolicity in terms of right resolvent functions near the unit circle.
- [16] arXiv:2608.19579 (cross-list from cs.AI) [pdf, html, other]
-
Title: Enforcing LLM Safety through DMD-based Classification of Prompt-Response Embedding DynamicsSubjects: Artificial Intelligence (cs.AI); Dynamical Systems (math.DS)
Large Language Models (LLMs) are increasingly deployed in high-stakes applications, yet their tendency to generate toxic, harmful, or policy-violating content poses significant risks. Detecting these unsafe outputs efficiently in a black-box manner remains an open challenge. In this paper, we extend a recently proposed dynamical systems framework designed for hallucination detection to LLM safety classification. By projecting both prompts and responses into high-dimensional embedding spaces and fitting separate Koopman-based predictive models for safe and unsafe regimes, we classify new outputs using a new differential residual score that compares prediction errors of the safe and unsafe regimes. A key contribution is the incorporation of the prompt and response embedding dynamics, yielding fitted Koopman operators that capture crucial interaction patterns. We evaluate our black-box method across three safety benchmarks using three embedding models. Our results show that incorporating prompt embeddings yields consistent improvements, particularly for interaction-dependent violations when paired with causal decoders (e.g., in Llama-3), while response-only violations benefit more from dense semantic embedding representations. These findings opens the door for using dynamical systems to analyze AI systems rather than the dominant paradigm of using AI to model dynamical systems.
- [17] arXiv:2608.19814 (cross-list from physics.chem-ph) [pdf, html, other]
-
Title: Rare Fluctuations from Normally Hyperbolic Invariant ManifoldsComments: 37 pages, 2 figuresSubjects: Chemical Physics (physics.chem-ph); Dynamical Systems (math.DS)
Freidlin--Wentzell theory converts weak-noise large deviations into a Hamiltonian variational problem. We study how a $k$-dimensional normally hyperbolic invariant manifold (NHIM) $N$ of the deterministic dynamics appears in this Hamiltonian system. Its zero-momentum copy $N_0=N\times\{0\}$ is invariant, but the Hamiltonian dynamics has $2k$ center directions near $N_0$: $k$ tangent to $N$ and $k$ conjugate covector directions. We construct the resulting local symplectic geometry and show that fluctuation extremals approaching $N_0$ backward in time at the strong normal rate form an $n$-dimensional exact Lagrangian invariant manifold carrying a single-valued action. A counterexample shows that a corresponding zero-energy section need not be normally hyperbolic within $H_{\mathrm{FW}}^{-1}(0)$. We then consider reaction dynamics. If a parameter moves a deterministic trajectory toward a codimension-one reactivity boundary, the minimum Freidlin--Wentzell action required to reach the boundary is quadratic in the distance from the threshold parameter. Its coefficient is determined by the relative motion of trajectory and boundary and by how effectively the available noise acts transversely. This deterministic boundary is distinct from a noise-dependent stochastic transition state or a committor surface. In a solvent--solute model, varying solvent mass moves the phase-space reactivity boundary while leaving the potential-energy surface fixed, changing the rare-event cost without changing the potential-energy barrier.
- [18] arXiv:2608.20010 (cross-list from math.CO) [pdf, html, other]
-
Title: Graphs with connectivity $3/4 - \varepsilon$ are globally synchronizingComments: 17 pagesSubjects: Combinatorics (math.CO); Dynamical Systems (math.DS)
We study synchronization in the Kuramoto model on finite graphs. We prove that there is an absolute constant $\eta>0$ such that every finite simple graph $G$ on $n$ vertices with minimum degree at least $(3/4-\eta)n$ has no local minima of the Kuramoto energy other than the fully synchronized states. This strictly improves the previous $3/4$ upper bound and refutes a conjecture of Bandeira, Kireeva, Maillard, and Rödder.
- [19] arXiv:2608.20017 (cross-list from nlin.CD) [pdf, html, other]
-
Title: Understanding the superiority of multi-model ensemble forecasts through reservoir computingDaniel Estevez Moya, Francesco Martinuzzi, Edmilson Roque dos Santos, Erick Alejandro Madrigal Solis, Ernesto Estevez Rams, Holger KantzSubjects: Chaotic Dynamics (nlin.CD); Dynamical Systems (math.DS)
Weather forecasting and climate projection frequently use multi-model ensembles (MMEs) to improve short-term forecasts by averaging across models. However, this practice is often not well justified or validated. Using reservoir computing (RC) as a computationally efficient alternative to large-scale physical models, we assess the validity of the MME approach for chaotic time series. By training multiple randomly constructed RCs on the same dataset, we create a multi-model ensemble in which each model has its own unique error. These model errors lead to very different forecasting performances, with forecast error distributions that exhibit heavy tails. The arithmetic mean across forecasts from multiple models for the same target is usually closer to the ground truth than most individual forecasts, and further improvement is achieved by weighted arithmetic means where the weights are constructed based on each model's test-set performance. We show that iterated forecasts over many time steps deviate from the ground truth along the unstable manifold of the target point, in both directions, so that, if forecast errors were independent and had zero mean, the arithmetic mean forecast should approach the true target like $1/\sqrt{\nens}$ where $\nens$ is the size of the multi-model ensemble. We observe deviations from this behavior, which we attribute to the tails of the error distribution of random RCs.
- [20] arXiv:2608.20043 (cross-list from eess.SY) [pdf, other]
-
Title: Wave-Based Bilateral Teleoperation between Nonlinear Manipulators with Direct Contact Force FeedbackComments: 65th IEEE Conference on Decision and Control (CDC), Honolulu, HI, USA, Dec. 2026Subjects: Systems and Control (eess.SY); Robotics (cs.RO); Dynamical Systems (math.DS); Optimization and Control (math.OC)
We study bilateral teleoperation between nonlinear, multi-DOF robotic manipulators in the presence of constant communication delays. Unlike classical wave-transformation architectures that transmit a coordinating force, we consider the case where the environmental force is reflected to the master side to enhance teleoperation transparency. Since direct contact force feedback might destabilize the closed-loop system, we first develop a passivity-shortage characterization for the Euler--Lagrange remote system using a linear matrix inequality (LMI) approach. An upper strictly passive communication law is then employed to compensate for the computed passivity shortage so that the closed-loop stability under delays as well as position and force synchronization are preserved under appropriate conditions. Simulations with nonlinear 2-DOF robotic manipulators in different settings illustrate our approach.
- [21] arXiv:2608.20143 (cross-list from math.GT) [pdf, html, other]
-
Title: The entropy spectrum of hyperbolic surfacesComments: 62 pages, 14 figuresSubjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Dynamical Systems (math.DS)
This article introduces and studies the entropy spectrum of a hyperbolic surface, that is the set of entropies of its subsurfaces. The main results are that the entropy spectrum is a reverse well-ordered multiset, with finite multiplicities, and that there is a quantifiable gap around the value $1$. This gap comes from a counting result on the number of non-filling geodesics which in turn comes from explicit estimates on the number of curves on surfaces with boundary in terms of geometric data. The geometric data includes lengths of boundary geodesics and so-called boundary width, which measures maximal distance to the boundary but can also be interpreted in terms of the topology of the surface and the systole.
- [22] arXiv:2608.20191 (cross-list from math.CO) [pdf, html, other]
-
Title: Spectrum of the refined Diophantine exponentSubjects: Combinatorics (math.CO); Formal Languages and Automata Theory (cs.FL); Dynamical Systems (math.DS); Number Theory (math.NT)
The refined Diophantine exponent, recently introduced by the author, is a quantity that measures the periodicity of an infinite word. In this article, we study this exponent from combinatorial and topological viewpoints. First, we show that, over a ternary alphabet, the spectrum of the refined Diophantine exponent is $[1,\infty]$. Second, we show that this exponent has topological properties similar to those of the set of Liouville numbers. Finally, we provide concrete examples with the Champernowne, Rudin--Shapiro, and Thue--Morse words, words coming from coding a rotation by intervals, and bracket words.
Cross submissions (showing 9 of 9 entries)
- [23] arXiv:2411.11290 (replaced) [pdf, html, other]
-
Title: Chebyshev's method for exponential mapsComments: 28 pages, 10 figuresSubjects: Dynamical Systems (math.DS)
It is proved that the Chebyshev's method applied to an entire function $f$ is a rational map if and only if $f(z) = p(z) e^{q(z)}$, for some polynomials $p$ and $q$. These are referred to as rational Chebyshev maps, and their fixed points are discussed in this article. It is seen that $\infty$ is a parabolic fixed point with multiplicity one bigger than the degree of $q$. Considering $q(z)=p(z)^n+c$, where $p$ is a linear polynomial, $n \in \mathbb{N}$ and $c$ is a non-zero constant, we show that the Chebyshev's method applied to $ pe^q$ is affine conjugate to that applied to $z e^{z^n}$. We denote this by $C_n$. All the finite extraneous fixed points of $C_n$ are shown to be repelling. The Julia set $\mathcal{J}(C_n)$ of $C_n$ is found to be preserved under rotations of order $n$ about the origin. For each $n$, the immediate basin of $0$ is proved to be simply connected. For all $n \leq 16$, we prove that $\mathcal{J}(C_n)$ is connected. For $n$ even, the non-existence of Herman ring and Siegel disk of $C_n$ is proved. Under some additional hypothesis, the same is also proved for odd $n$. The Newton's method applied to $ze^{z^n}$ is found to be conjugate to a polynomial, and its dynamics is also completely determined.
- [24] arXiv:2505.13064 (replaced) [pdf, html, other]
-
Title: Symmetric Lyapunov Subcenter Manifolds for Periodic Regulation of Mechanical SystemsComments: 21 pages, 27 figures, submitted to AutomaticaSubjects: Dynamical Systems (math.DS); Robotics (cs.RO)
Multi-body mechanical systems have rich internal dynamics, whose solutions can be exploited as energy-efficient control targets. Yet, solutions non-trivially depend on system parameters, obscuring feasible properties for use as target trajectories. For periodic regulation tasks in robotics applications, we investigate properties of nonlinear oscillations collected in Lyapunov subcenter manifolds (LSMs) of conservative mechanical systems (CMs). Using a time-symmetry of CMs, it is shown that mild non-resonance conditions guarantee that LSMs exclusively consist of oscillations between two points of zero velocity. The existence of a unique generator is proven, which is a connected, 1D manifold that collects these points of zero velocity for a given LSM. Furthermore, it is shown that an additional spatial symmetry provides LSMs with yet stronger properties of Rosenberg manifolds. Here all oscillations pass through a unique equilibrium configuration, which can be favorable for control applications. These theoretical results are numerically confirmed on two mechanical systems: a double pendulum and a 5-link pendulum.
- [25] arXiv:2506.19989 (replaced) [pdf, html, other]
-
Title: An Ergodic Spectral Decomposition Theorem for Singular Star FlowsComments: 114 pages, 2 figuresSubjects: Dynamical Systems (math.DS)
For Axiom A diffeomorphisms and flows, Smale's Spectral Decomposition Theorem asserts that the non-wandering set decomposes into finitely many isolated hyperbolic basic sets, each given by a homoclinic class. For singular star flows, which may be viewed as "Axiom A flows with singularities", the corresponding spectral decomposition remains open and is known as the Spectral Decomposition Conjecture.
We provide a positive answer to an ergodic formulation of this conjecture: $C^1$-open and dense among singular star flows with positive topological entropy, there is a unique measure of maximal entropy. More generally, we prove the uniqueness of equilibrium states for Hölder continuous potentials under a mild and natural pressure gap condition. We further establish that $C^1$-open and dense star flows are almost expansive and that the topological pressure of continuous potentials varies continuously with respect to the vector field in the $C^1$ topology.
Our approach combines ergodic and geometric arguments adapted to the multi-singular setting. In this context, classical hyperbolic tools such as uniform local product structure or invariant splittings on the tangent bundle are no longer available. To overcome this, we develop new mechanisms to control the geometry of orbit segments and to produce transversal intersections on large subsets uniformly detected by good invariant measures. These ingredients allow us to extend classical arguments to the multi-singular setting through structural properties of equilibrium states combined with refined shadowing and specification at the level of invariant measures. - [26] arXiv:2507.11958 (replaced) [pdf, html, other]
-
Title: Interacting Hosts with Microbiome Exchange: An Extension of Metacommunity Theory for Discrete Interactions between HostsComments: 57 pages; revised versionSubjects: Dynamical Systems (math.DS); Statistical Mechanics (cond-mat.stat-mech); Social and Information Networks (cs.SI); Adaptation and Self-Organizing Systems (nlin.AO); Populations and Evolution (q-bio.PE)
Microbiomes, which are collections of interacting microbes in an environment, often substantially impact the environmental patches or living hosts that they occupy. In microbiome models, it is important to consider both the local dynamics within an environment and exchanges of microbiomes between environments. One way to incorporate these and other interactions across multiple scales is to employ metacommunity theory. Metacommunity models commonly assume continuous microbiome dispersal between the environments in which local microbiome dynamics occur. Under this assumption, a single parameter between each pair of environments controls the dispersal rate between those environments. This metacommunity framework is well-suited to abiotic environmental patches, but it fails to capture an essential aspect of the microbiomes of living hosts. Living hosts generally do not interact continuously with each other. Instead, living hosts interact with each other in discrete time intervals. In this paper, we develop a modeling framework that encodes such discrete interactions and uses two parameters to separately control the interaction frequencies between hosts and the amount of microbiome exchange during each interaction. We derive analytical approximations of models in our framework in three parameter regions and prove that they are accurate in these regions. We compare these approximations to numerical simulations for an illustrative model, and we demonstrate that both parameters in our modeling framework are necessary to determine microbiome dynamics. Key features of the dynamics, such as microbiome convergence across hosts, depend sensitively on the interplay between interaction frequency and strength.
- [27] arXiv:2608.13302 (replaced) [pdf, html, other]
-
Title: Input-to-state stability of chemical reaction networks with application to molecular computationComments: 28 pages, 3 figures; Revised version; an illustrative example has been added to the Appendix. The main theoretical results remain unchangedSubjects: Dynamical Systems (math.DS)
In biological reaction systems, reaction rates may vary over time due to environmental fluctuations, regulation, or coupling with other reaction modules. Input-to-state stability (ISS) provides a useful tool for analyzing the robustness of time-varying chemical reaction networks (CRNs). Existing ISS results for CRNs typically rely on restrictive structural assumptions, such as zero deficiency, a single linkage class, or weak reversibility. This paper makes two main contributions. First, we establish ISS for a broader class of weakly reversible CRNs, allowing nonzero deficiency and multiple linkage classes. Second, we extend the analysis to certain CRNs that are not weakly reversible by using network transformation techniques (linear conjugacy and reconstruction). Together, these results enlarge the class of CRNs for which robustness under time-varying reaction-rate inputs can be certified. Since CRNs are a standard framework for biomolecular computation, our results further enable the stability analysis of parallel molecular computing systems, an important problem in biomolecular computation where multiple CRN-based computing modules operate simultaneously and perturb one another through time-varying effective reaction rates.
- [28] arXiv:2401.01129 (replaced) [pdf, html, other]
-
Title: Variational Lifting and Optimal Gauges on Riemannian Homogeneous SpacesComments: 22 pages, 2 figuresSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY); Differential Geometry (math.DG); Dynamical Systems (math.DS)
Building on standard Euler-Poincaré reduction on Lie groups, we develop a variational lifting framework for mechanical systems on Riemannian homogeneous spaces $H=G/K$. A mechanical action on $H$ is lifted to $G$ through a functional whose kinetic energy depends only on the horizontal component of the velocity, and we prove that its critical points project precisely onto those of the original action.
The lifted functional possesses a natural gauge invariance under $H^1$ curves with values in the isotropy subgroup $K$. Consequently, every lifted critical curve decomposes into a smooth horizontal representative and an arbitrary vertical gauge, and the associated Euler-Poincaré equations with symmetry breaking are obtained in reduced form. We then introduce a second variational problem that selects a distinguished lift by minimizing the vertical kinetic energy.
The optimal gauge is a length-minimizing geodesic on $K$, yielding an explicit decomposition of the total energy into projected and vertical contributions and a geometric interpretation in terms of holonomy for closed projected curves. The framework is illustrated first for pure quantum states on $\mathbb{CP}^{n}\cong SU(n+1)/S(U(1)\times U(n))$, where the gauge freedom corresponds to the isotropy of unitary lifts, and then for $S^2\cong SO(3)/SO(2)$ through an optimal camera-orientation problem for an axisymmetric satellite subject to an undesirable pointing region. - [29] arXiv:2503.05606 (replaced) [pdf, html, other]
-
Title: Control analysis and synthesis for general control-affine systemsComments: The last version corresponds to the peer-reviewed article published in the SIAM Journal on Control and OptimizationSubjects: Optimization and Control (math.OC); Classical Analysis and ODEs (math.CA); Dynamical Systems (math.DS)
We study controllability and constructive synthesis for control-affine systems. We introduce trajectory-dependent Gramian maps that extend the linear time-varying Gramian and yield explicit fixed-point synthesis maps. On feasible coercivity classes (uniform eigenvalue lower bounds), the Gramian map is Lipschitz, and under a comparison estimate criterion, synthesis iterates exhibit decay, and the Banach fixed-point theorem gives a unique fixed point that steers the system and satisfies an energy identity. When, in addition, an orthogonality condition holds, this fixed point coincides with the unique global minimum-energy control on the feasible set; if the coercivity bound holds uniformly for all bounded controls, the same conclusion holds on the full bounded-control space. We provide structural conditions on the input matrix that ensure the nonemptiness of the feasible class (and, in fully actuated regimes, equality with the full space) and sufficient conditions for underactuated systems via bounded-amplitude reference controls. Case studies on Hopfield network dynamics illustrate refined estimates that enlarge reachable targets. A trajectory-freezing and compactness step extends the synthesis to general nonlinear control-affine systems. The results yield verifiable controllability criteria with explicit, numerically implementable controllers.
- [30] arXiv:2608.00197 (replaced) [pdf, other]
-
Title: Inertial manifolds for the nonlocal parabolic problemSubjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS); Functional Analysis (math.FA)
This paper provides an abstract framework for studying inertial manifolds associated with a class of nonlocal parabolic problems. In particular, by suitably modifying the nonlocal term outside the absorbing ball and changing the scale of time, we derive a corresponding spectral gap condition. As applications, we establish the existence of inertial manifolds for two classes of two-dimensional modified nonlocal parabolic equations on a square domain, whose diffusion coefficients depend on the $L^2$-norm of the solution and of its gradient, respectively.