Dynamical Systems

New submissions

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New submissions for Wed, 15 May 24

[1]  arXiv:2405.08236 [pdf, other]

Title: Existence of attracting invariant 2-curves in fibred quadratic dynamics

Authors: Igsyl Domínguez

Subjects: Dynamical Systems (math.DS)

We present a construction of new invariant sets for fibred polynomial dynamics with base an irrational rotation over the unit circle, called multi-curves. Furthermore, the local dynamical theory for attracting invariant curves is extended to these objects.

[2]  arXiv:2405.08444 [pdf, ps, other]

Title: Multi-dimensional piecewise contractions are asymptotically periodic

Authors: Jose Pedro Gaivao, Benito Pires

Comments: 23 pages

Subjects: Dynamical Systems (math.DS)

Piecewise contractions (PCs) are piecewise smooth maps that decrease distance between pair of points in the same domain of continuity. The dynamics of a variety of systems is described by PCs. During the last decade, a lot of effort has been devoted to proving that in parametrized families of one-dimensional PCs, the $\omega$-limit set of a typical PC consists of finitely many periodic orbits while there exist atypical PCs with Cantor $\omega$-limit sets. In this article, we extend these results to the multi-dimensional case. More precisely, we provide criteria to show that an arbitrary family $\{f_{\mu}\}_{\mu\in U}$ of locally bi-Lipschitz piecewise contractions $f_\mu:X\to X$ defined on a compact metric space $X$ is asymptotically periodic for Lebesgue almost every parameter $\mu$ running over an open subset $U$ of the $M$-dimensional Euclidean space $\mathbb{R}^M$. As a corollary of our results, we prove that piecewise affine contractions of $\mathbb{R}^d$ defined in generic polyhedral partitions are asymptotically periodic.

[3]  arXiv:2405.08532 [pdf, other]

Title: A dynamical view of Tijdeman's solution of the chairman assignment problem

Authors: Valérie Berthé (IRIF (UMR\_8243)), Olivier Carton (IRIF (UMR\_8243)), Nicolas Chevallier (IRIMAS), Wolfgang Steiner (IRIF (UMR\_8243)), Reem Yassawi

Subjects: Dynamical Systems (math.DS)

In 1980, R. Tijdeman provided an on-line algorithm that generates sequences over a finite alphabet with minimal discrepancy, that is, such that the occurrence of each letter optimally tracks its frequency. In this article, we define discrete dynamical systems generating these sequences. The dynamical systems are defined as exchanges of polytopal pieces, yielding cut and project schemes, and they code tilings of the line whose sets of vertices form model sets. We prove that these sequences of low discrepancy are natural codings of toral translations with respect to polytopal atoms, and that they generate a minimal and uniquely ergodic subshift with purely discrete spectrum. Finally, we show that the factor complexity of these sequences is of polynomial growth order $n^{d-1}$, where $d$ is the cardinality of the alphabet.

[4]  arXiv:2405.08537 [pdf, other]

Title: GS-PINN: Greedy Sampling for Parameter Estimation in Partial Differential Equations

Authors: Ali Forootani, Harshit Kapadia, Sridhar Chellappa, Pawan Goyal, Peter Benner

Subjects: Dynamical Systems (math.DS)

Partial differential equation parameter estimation is a mathematical and computational process used to estimate the unknown parameters in a partial differential equation model from observational data. This paper employs a greedy sampling approach based on the Discrete Empirical Interpolation Method to identify the most informative samples in a dataset associated with a partial differential equation to estimate its parameters. Greedy samples are used to train a physics-informed neural network architecture which maps the nonlinear relation between spatio-temporal data and the measured values. To prove the impact of greedy samples on the training of the physics-informed neural network for parameter estimation of a partial differential equation, their performance is compared with random samples taken from the given dataset. Our simulation results show that for all considered partial differential equations, greedy samples outperform random samples, i.e., we can estimate parameters with a significantly lower number of samples while simultaneously reducing the relative estimation error. A Python package is also prepared to support different phases of the proposed algorithm, including data prepossessing, greedy sampling, neural network training, and comparison.

[5]  arXiv:2405.08592 [pdf, ps, other]

Title: Horocycle flows on abelian covers of surfaces of negative curvature

Authors: Roberto Castorrini, Davide Ravotti

Comments: 40 pages

Subjects: Dynamical Systems (math.DS)

We consider the unit speed parametrization of the horocycle flow on infinite Abelian covers of compact surfaces of negative curvature. We prove an asymptotic result for the ergodic integrals of sufficiently regular functions. In the case of constant curvature, where the unit speed and the uniformly contracting parametrizations of horocycles coincide, we recover a result by Ledrappier and Sarig. Our method, which does not use symbolic dynamics, is based on a general Fourier decomposition for Abelian covers and on the study of spectral theory of weighted (and twisted) transfer operators for the geodesic flow acting on appropriate anisotropic Banach spaces.

[6]  arXiv:2405.08613 [pdf, other]

Title: GN-SINDy: Greedy Sampling Neural Network in Sparse Identification of Nonlinear Partial Differential Equations

Authors: Ali Forootani, Peter Benner

Subjects: Dynamical Systems (math.DS); Machine Learning (cs.LG)

The sparse identification of nonlinear dynamical systems (SINDy) is a data-driven technique employed for uncovering and representing the fundamental dynamics of intricate systems based on observational data. However, a primary obstacle in the discovery of models for nonlinear partial differential equations (PDEs) lies in addressing the challenges posed by the curse of dimensionality and large datasets. Consequently, the strategic selection of the most informative samples within a given dataset plays a crucial role in reducing computational costs and enhancing the effectiveness of SINDy-based algorithms. To this aim, we employ a greedy sampling approach to the snapshot matrix of a PDE to obtain its valuable samples, which are suitable to train a deep neural network (DNN) in a SINDy framework. SINDy based algorithms often consist of a data collection unit, constructing a dictionary of basis functions, computing the time derivative, and solving a sparse identification problem which ends to regularised least squares minimization. In this paper, we extend the results of a SINDy based deep learning model discovery (DeePyMoD) approach by integrating greedy sampling technique in its data collection unit and new sparsity promoting algorithms in the least squares minimization unit. In this regard we introduce the greedy sampling neural network in sparse identification of nonlinear partial differential equations (GN-SINDy) which blends a greedy sampling method, the DNN, and the SINDy algorithm. In the implementation phase, to show the effectiveness of GN-SINDy, we compare its results with DeePyMoD by using a Python package that is prepared for this purpose on numerous PDE discovery

[7]  arXiv:2405.08634 [pdf, other]

Title: Stabilization of Integral Delay Equations by solving Fredholm equations

Authors: Jean Auriol (L2S)

Comments: IEEE Control Systems Letters, In press

Subjects: Dynamical Systems (math.DS)

In this paper, we design a stabilizing state-feedback control law for a system represented by a general class of integral delay equations subject to a pointwise and distributed input delay. The proposed controller is defined in terms of integrals of the state and input history over a fixed-length time window. We show that the closed-loop stability is guaranteed, provided the controller integral kernels are solutions to a set of Fredholm equations. The existence of solutions is guaranteed under an appropriate spectral controllability assumption, resulting in an implementable stabilizing control law. The proposed methodology appears simpler and more general compared to existing results in the literature. In particular, under additional regularity assumptions, the proposed approach can be expanded to address the degenerate case where only a distributed control term is present.

[8]  arXiv:2405.08736 [pdf, other]

Title: Polytropic Dynamical Systems with Time Singularity

Authors: Oday Hazaimah

Comments: 18 pages, 7 figures

Subjects: Dynamical Systems (math.DS)

In this paper we consider a class of second order singular homogeneous differential equations called the Lane-Emden-type with time singularity in the drift coefficient. Lane-Emden equations are singular initial value problems that model phenomena in astrophysics such as stellar structure and are governed by polytropics with applications in isothermal gas spheres. A hybrid method is proposed to approximate the solution of this type of dynamic equations.

[9]  arXiv:2405.08771 [pdf, other]

Title: Multi-objective SINDy for parameterized model discovery from single transient trajectory data

Authors: Javier A. Lemus, Benjamin Herrmann

Subjects: Dynamical Systems (math.DS)

The sparse identification of nonlinear dynamics (SINDy) has been established as an effective technique to produce interpretable models of dynamical systems from time-resolved state data via sparse regression. However, to model parameterized systems, SINDy requires data from transient trajectories for various parameter values over the range of interest, which are typically difficult to acquire experimentally. In this work, we extend SINDy to be able to leverage data on fixed points and/or limit cycles to reduce the number of transient trajectories needed for successful system identification. To achieve this, we incorporate the data on these attractors at various parameter values as constraints in the optimization problem. First, we show that enforcing these as hard constraints leads to an ill-conditioned regression problem due to the large number of constraints. Instead, we implement soft constraints by modifying the cost function to be minimized. This leads to the formulation of a multi-objective sparse regression problem where we simultaneously seek to minimize the error of the fit to the transients trajectories and to the data on attractors, while penalizing the number of terms in the model. Our extension, demonstrated on several numerical examples, is more robust to noisy measurements and requires substantially less training data than the original SINDy method to correctly identify a parameterized dynamical system.

[10]  arXiv:2405.08791 [pdf, other]

Title: On the basin of attraction of a critical three-cycle of a model for the secant map

Authors: Ernest Fontic, Antonio Garijo, Xavier Jarque

Subjects: Dynamical Systems (math.DS)

We consider the secant method $S_p$ applied to a real polynomial $p$ of degree $d+1$ as a discrete dynamical system on $\mathbb R^2$. If the polynomial $p$ has a local extremum at a point $\alpha$ then the discrete dynamical system generated by the iterates of the secant map exhibits a critical periodic orbit of period 3 or three-cycle at the point $(\alpha,\alpha)$. We propose a simple model map $T_{a,d}$ having a unique fixed point at the origin which encodes the dynamical behaviour of $S_p^3$ at the critical three-cycle. The main goal of the paper is to describe the geometry and topology of the basin of attraction of the origin of $T_{a,d}$ as well as its boundary. Our results concern global, rather than local, dynamical behaviour. They include that the boundary of the basin of attraction is the stable manifold of a fixed point or contains the stable manifold of a two-cycle, depending on the values of the parameters of $d$ (even or odd) and $a\in \mathbb R$ (positive or negative).

[11]  arXiv:2405.08811 [pdf, other]

Title: Slow-growing counterexamples to the strong Eremenko Conjecture

Authors: Andrew P. Brown

Comments: 26 pages, 2 figures

Subjects: Dynamical Systems (math.DS); Complex Variables (math.CV)

Let $f\colon\mathbb{C} \to\mathbb{C}$ be a transcendental entire function. In 1989, Eremenko asked the following question concerning the set $I(f)$ of points that tend to infinity under iteration: can every point of $I(f)$ be joined to $\infty$ by a curve in $I(f)$? This is known as the strong Eremenko conjecture and was disproved in 2011 by Rottenfu{\ss}er, R\"uckert, Rempe and Schleicher. The function has relatively small infinite order: it can be chosen such that $\log \log \,\lvert f(z)\rvert = (\log \lvert z\rvert)^{1+o(1)}$ as $f(z)\to \infty$. Moreover, $f$ belongs to the \emph{Eremenko--Lyubich class $\mathcal{B}$}. Rottenfu{\ss}er et al also show that the strong Eremenko conjecture does hold for any $f\in\mathcal{B}$ of finite order. We consider how slow a counterexample $f\in\mathcal{B}$ can grow. Suppose that $\Theta\colon [t_0,\infty)\to [0,\infty)$ satisfies $\Theta(t) \to 0$ and \[ (\log t)^{-\beta \Theta(\log t)}/\Theta(t) \to 0 \quad\text{ as $t\to \infty$} \] for some $0<\beta<1$, along with a certain regularity assumption. Then there exists a counterexample $f\in\mathcal{B}$ as above such that \[ \log \log \vert f(z)\vert = O ( (\log \vert z \vert)^{1 + \Theta( \log \vert z \vert )}) \] as $\vert f(z)\vert \to\infty$. The hypotheses are satisfied, in particular, for $\Theta(t) = 1/(\log \log t)^{\alpha}$, for any $\alpha>0$.

[12]  arXiv:2405.08812 [pdf, other]

Title: Chaotic dynamics at the boundary of a basin of attraction via non-transversal intersections for a non-global smooth diffeomorphism

Authors: Ernest Fontich, Antonio Garijo, Xavier Jarque

Subjects: Dynamical Systems (math.DS)

In this paper we give analytic proofs of the existence of transversal homoclinic points for a family of non-globally smooth diffeomorphisms having the origin as a fixed point which come out as a truncated map governing the local dynamics near a critical period three cycle associated to the Secant map. Using Moser's version of Birkhoff-Smale's Theorem, we prove that the boundary of the basin of attraction of the origin contains a Cantor-like invariant subset such that the restricted dynamics to it is conjugate to the full shift of $N$-symbols for any integer $N\ge 2$ or infinity.

Cross-lists for Wed, 15 May 24

[13]  arXiv:nlin/0408039 (cross-list from nlin.AO) [pdf, ps, other]

Title: Stability and Diversity in Collective Adaptation

Authors: Yuzuru Sato, Eizo Akiyama, James P. Crutchfield

Comments: 22 pages, 23 figures; updated references, corrected typos, changed content

Subjects: Adaptation and Self-Organizing Systems (nlin.AO); Machine Learning (cs.LG); Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD); Machine Learning (stat.ML)

We derive a class of macroscopic differential equations that describe collective adaptation, starting from a discrete-time stochastic microscopic model. The behavior of each agent is a dynamic balance between adaptation that locally achieves the best action and memory loss that leads to randomized behavior. We show that, although individual agents interact with their environment and other agents in a purely self-interested way, macroscopic behavior can be interpreted as game dynamics. Application to several familiar, explicit game interactions shows that the adaptation dynamics exhibits a diversity of collective behaviors. The simplicity of the assumptions underlying the macroscopic equations suggests that these behaviors should be expected broadly in collective adaptation. We also analyze the adaptation dynamics from an information-theoretic viewpoint and discuss self-organization induced by information flux between agents, giving a novel view of collective adaptation.

[14]  arXiv:2405.08735 (cross-list from q-bio.PE) [pdf, other]

Title: Competition in the nutrient-driven self-cycling fermentation process

Authors: Stacey R. Smith, Tyler Meadows, Gail S.K. Wolkowicz

Comments: 17 pages, 2 figures

Subjects: Populations and Evolution (q-bio.PE); Dynamical Systems (math.DS)

Self-cycling fermentation is an automated process used for culturing microorganisms. We consider a model of $n$ distinct species competing for a single non-reproducing nutrient in a self-cycling fermentor in which the nutrient level is used as the decanting condition. The model is formulated in terms of impulsive ordinary differential equations. We prove that two species are able to coexist in the fermentor under certain conditions. We also provide numerical simulations that suggest coexistence of three species is possible and that competitor-mediated coexistence can occur in this case. These results are in contrast to the chemostat, the continuous analogue, where multiple species cannot coexist on a single nonreproducing nutrient.

[15]  arXiv:2405.08778 (cross-list from math-ph) [pdf, other]

Title: Quantum Integrable Systems arising from Separation of Variables on S3

Authors: Sean Dawson, Holger Dullin

Subjects: Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA); Dynamical Systems (math.DS)

We study the family of quantum integrable systems that arise from separating the Schr\"odinger equation in all 6 separable orthogonal coordinates on the 3 sphere: ellipsoidal, prolate, oblate, Lam\'{e}, spherical and cylindrical. On the one hand each separating coordinate system gives rise to a quantum integrable system on S2 x S2, on the other hand it also leads to families of harmonic polynomials in R4. We show that separation in ellipsoidal coordinates yields a generalised Lam\'{e} equation - a Fuchsian ODE with 5 regular singular points. We seek polynomial solutions so that the eigenfunctions are analytic at all finite singularities. We classify eigenfunctions by their discrete symmetry and compute the joint spectrum for each symmetry class. The latter 5 separable coordinate systems are all degenerations of the ellipsoidal coordinates. We perform similar analyses on these systems and show how the ODEs degenerate in a fashion akin to their respective coordinates. For the prolate system we show that there exists a defect in the joint spectrum which prohibits a global assignment of quantum numbers: the system has quantum monodromy. This is a companion paper to our previous work where the respective classical systems were studied.

Replacements for Wed, 15 May 24

[16]  arXiv:2210.16908 (replaced) [pdf, other]

Title: Statistical properties for mixing Markov chains with applications to dynamical systems

Authors: Ao Cai, Pedro Duarte, Silvius Klein

Comments: 42 pages, 1 figure. Compared to the previous version we added several references and rephrased the main result in a more general setting

Subjects: Dynamical Systems (math.DS); Probability (math.PR)

[17]  arXiv:2211.11234 (replaced) [pdf, ps, other]

Title: The measure transfer for subshifts induced by a morphism of free monoids

Authors: Nicolas Bédaride, Arnaud Hilion, Martin Lustig

Comments: The second half of section 5, concerning the discussion around "recognizable for aperiodic points" (including Figure 1), is new. Small errors have been corrected, and parts of the introduction has been rewritten

Subjects: Dynamical Systems (math.DS)

[18]  arXiv:2212.07561 (replaced) [pdf, other]

Title: Monotonicity of the period map for the equation $-\varphi''+\varphi-\varphi^{k}=0$

Authors: Fábio Natali, Giovana Alves

Comments: The correction of a small imperfection in the proof of Lemma 3.3 has been made. Accepted for publication in Monatshefte f\"ur Mathematik (2024)

Subjects: Dynamical Systems (math.DS); Classical Analysis and ODEs (math.CA)

[19]  arXiv:2403.10383 (replaced) [pdf, other]

Title: A new canonical reduction of three-vortex motion and its application to vortex-dipole scattering

Authors: Atul Anurag, Roy H. Goodman, Ellison K. O'Grady

Comments: This is a major restructuring on the paper. Whereas the initial submission focused on the scattering problem, the revision focuses on the method of the reduction, its mathematical structure, and its advantages. It treats the scattering problem as an application of the method and adds a second application: the problem of three identical vortices. The title has been changed to reflect this

Subjects: Dynamical Systems (math.DS)

[20]  arXiv:2405.05834 (replaced) [pdf, other]

Title: The Riemann hypothesis and dynamics of Backtracking New Q-Newton's method

Authors: Thuan Quang Tran, Tuyen Trung Truong

Comments: 19 pages. Some typos are fixed, references are updated. Exposition is improved, Section 4 is expanded. Comments are welcome!

Subjects: Dynamical Systems (math.DS); Complex Variables (math.CV); Number Theory (math.NT); Optimization and Control (math.OC)

[21]  arXiv:2405.07645 (replaced) [pdf, ps, other]

Title: Ergodicity of skew-products over typical IETs

Authors: Fernando Argentieri, Przemysław Berk, Frank Trujillo

Subjects: Dynamical Systems (math.DS)

[22]  arXiv:2306.06045 (replaced) [pdf, ps, other]

Title: Global solution and blow-up for the SKT model in Population Dynamics

Authors: Ichraf Belkhamsa, Messaoud Souilah

Subjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS)

[23]  arXiv:2405.00610 (replaced) [pdf, ps, other]

Title: Growth in products of matrices: fastest, average, and generic

Authors: Vladimir Shpilrain

Comments: 10 pages. Comments are welcome

Subjects: Group Theory (math.GR); Cryptography and Security (cs.CR); Combinatorics (math.CO); Dynamical Systems (math.DS); Probability (math.PR)

• New submissions
• Cross-lists
• Replacements