Dynamical Systems

New submissions

• New submissions
• Cross-lists
• Replacements

New submissions for Mon, 29 Apr 24

[1]  arXiv:2404.17076 [pdf, ps, other]

Title: Thermodynamic formalism and hyperbolic Baker domains: Real-analyticity of the Hausdorff dimension

Authors: Adrián Esparza-Amador

Subjects: Dynamical Systems (math.DS)

We consider the family of entire maps given by $f_{\ell,c}(z)=\ell+c-(\ell-1)\log c-e^z$, where $c\in D(\ell,1)$ and $\ell\in\mathbb N$, $\ell\geq2$. By using the property of $f_{\ell,c}$ to be dynamically projected to an infinite cylinder $\mathbb C/2\pi I\mathbb Z$, where the thermodynamic formalism tools are well-defined, we prove as a main result on this work, the real-analyticity of the map $c\mapsto HD(\mathcal{J}_r(f_{\ell,c}))$, here $\mathcal{J}_r(f_{\ell,c})$ is the radial Julia set.

[2]  arXiv:2404.17321 [pdf, other]

Title: Fractional Order Sunflower Equation: Stability, Bifurcation and Chaos

Authors: Deepa Gupta, Sachin Bhalekar

Comments: 10 pages, 28 figures

Subjects: Dynamical Systems (math.DS)

The sunflower equation describes the motion of the tip of a plant due to the auxin transportation under the influence of gravity. This work proposes the fractional-order generalization to this delay differential equation. The equation contains two fractional orders and infinitely many equilibrium points. The coefficients in the linearized equation near the equilibrium points are delay-dependent. We provide a detailed stability analysis of each equilibrium point. We observed the following bifurcation phenomena: stable for all the delay values, a single stable region in the delayed interval, and a stability switch. We also observed a multi-scroll chaotic attractor for some values of the parameters.

[3]  arXiv:2404.17356 [pdf, other]

Title: Phase and amplitude responses for delay equations using harmonic balance

Authors: Rachel Nicks, Robert Allen, Stephen Coombes

Comments: 5 pages, 2 figures

Subjects: Dynamical Systems (math.DS)

Robust delay induced oscillations, common in nature, are often modeled by delay-differential equations (DDEs). Motivated by the success of phase-amplitude reductions for ordinary differential equations with limit cycle oscillations, there is now a growing interest in the development of analogous approaches for DDEs to understand their response to external forcing. When combined with Floquet theory, the fundamental quantities for this reduction are phase and amplitude response functions. Here, we develop a framework for their construction that utilises the method of harmonic balance.

[4]  arXiv:2404.17414 [pdf, ps, other]

Title: Multifractal analysis of the power-2-decaying Gauss-like expansion

Authors: Xue-Jiao Wang

Subjects: Dynamical Systems (math.DS)

Each real number $x\in[0,1]$ admits a unique power-2-decaying Gauss-like expansion (P2GLE for short) as $x=\sum_{i\in\mathbb{N}} 2^{-(d_1(x)+d_2(x)+\cdots+d_i(x))}$, where $d_i(x)\in\mathbb{N}$. For any $x\in(0,1]$, the Khintchine exponent $\gamma(x)$ is defined by $\gamma(x):=\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^nd_j(x)$ if the limit exists. We investigate the sizes of the level sets $E(\xi):=\{x\in(0,1]:\gamma(x)=\xi\}$ for $\xi\geq 1$. Utilizing the Ruelle operator theory, we obtain the Khintchine spectrum $\xi\mapsto\dim_H E(\xi)$, where $\dim_H$ denotes the Hausdorff dimension. We establish the remarkable fact that the Khintchine spectrum has exactly one inflection point, which was never proved for the corresponding spectrum in continued fractions. As a direct consequence, we also obtain the Lyapunov spectrum. Furthermore, we find the Hausdorff dimensions of the level sets $\{x\in(0,1]:\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}\log(d_j(x))=\xi\}$ and $\{x\in(0,1]:\lim_{n\to\infty}\frac{1}{n}\sum_{j=1}^{n}2^{d_j(x)}=\xi\}$.

[5]  arXiv:2404.17500 [pdf, ps, other]

Title: On the integrability of generalized $N$-center problems

Authors: Eddaly Guerra-Velasco, Boris Percino-Figueroa, Russell-Aarón Quiñones-Estrella

Comments: 15 pages

Subjects: Dynamical Systems (math.DS)

In this paper, we study the rational integrability of the $N$-center problem with rational weak and moderate forces. We show that the problem is not rationally integrable for all but a finite number of values $\alpha\in]0,2[$, where $\alpha$ is the order of the singularities. We identify the remaining cases and give the necessary conditions for integrability.

Cross-lists for Mon, 29 Apr 24

[6]  arXiv:2404.16861 (cross-list from nlin.AO) [pdf, other]

Title: Universal resonancelike emergence of chaos in complex networks of damped-driven nonlinear systems

Authors: Ricardo Chacón, Pedro J. Martínez

Comments: 11 pages (including supplemental material) with included figures

Subjects: Adaptation and Self-Organizing Systems (nlin.AO); Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD)

Characterizing the emergence of chaotic dynamics of complex networks is an essential task in nonlinear science with potential important applications in many fields such as neural control engineering, microgrid technologies, and ecological networks. Here, we solve a critical outstanding problem in this multidisciplinary research field: The emergence and persistence of spatio-temporal chaos in complex networks of damped-driven nonlinear oscillators in the significant weak-coupling regime, while they exhibit regular behavior when uncoupled. By developing a comprehensive theory with the aid of standard analytical methods, a hierarchy of lower-dimensional effective models, and extensive numerical simulations, we uncover and characterize the basic physical mechanisms concerning both heterogeneity-induced and impulse-induced emergence, enhancement, and suppression of chaos in starlike and scale-free networks of periodically driven, dissipative nonlinear oscillators.

[7]  arXiv:2404.17082 (cross-list from physics.soc-ph) [pdf, other]

Title: Evolutionary game dynamics with environmental feedback in a network with two communities

Authors: Katherine Betz, Feng Fu, Naoki Masuda

Comments: 8 figures, 2 tables

Subjects: Physics and Society (physics.soc-ph); Social and Information Networks (cs.SI); Dynamical Systems (math.DS); Populations and Evolution (q-bio.PE)

Recent developments of eco-evolutionary models have shown that evolving feedbacks between behavioral strategies and the environment of game interactions, leading to changes in the underlying payoff matrix, can impact the underlying population dynamics in various manners. We propose and analyze an eco-evolutionary game dynamics model on a network with two communities such that players interact with other players in the same community and those in the opposite community at different rates. In our model, we consider two-person matrix games with pairwise interactions occurring on individual edges and assume that the environmental state depends on edges rather than on nodes or being globally shared in the population. We analytically determine the equilibria and their stability under a symmetric population structure assumption, and we also numerically study the replicator dynamics of the general model. The model shows rich dynamical behavior, such as multiple transcritical bifurcations, multistability, and anti-synchronous oscillations. Our work offers insights into understanding how the presence of community structure impacts the eco-evolutionary dynamics within and between niches.

[8]  arXiv:2404.17220 (cross-list from math.AP) [pdf, other]

Title: Infinite dimensional Slow Manifolds for a Linear Fast-Reaction System

Authors: Christian Kuehn, Pascal Lehner, Jan-Eric Sulzbach

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

The aim of this expository paper is twofold. We start with a concise overview of the theory of invariant slow manifolds for fast-slow dynamical systems starting with the work by Tikhonov and Fenichel to the most recent works on infinite-dimensional fast-slow systems. The main part focuses on a class of linear fast-reaction PDE, which are particular forms of fast-reaction systems. The first result shows the convergence of solutions of the linear system to the limit system as the time-scale parameter $\varepsilon$ goes to zero. Moreover, from the explicit solutions the slow manifold is constructed and the convergence to the critical manifold is proven. The subsequent result, then, states a generalized version of the Fenichel-Tikhonov theorem for linear fast-reaction systems.

[9]  arXiv:2404.17506 (cross-list from math.AP) [pdf, other]

Title: Chemotaxis-inspired PDE model for airborne infectious disease transmission: analysis and simulations

Authors: Pierluigi Colli, Gabriela Marinoschi, Elisabetta Rocca, Alex Viguerie

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

Partial differential equation (PDE) models for infectious disease have received renewed interest in recent years. Most models of this type extend classical compartmental formulations with additional terms accounting for spatial dynamics, with Fickian diffusion being the most common such term. However, while diffusion may be appropriate for modeling vector-borne diseases, or diseases among plants or wildlife, the spatial propagation of airborne diseases in human populations is heavily dependent on human contact and mobility patterns, which are not necessarily well-described by diffusion. By including an additional chemotaxis-inspired term, in which the infection is propagated along the positive gradient of the susceptible population (from regions of low- to high-density of susceptibles), one may provide a more suitable description of these dynamics. This article introduces and analyzes a mathematical model of infectious disease incorporating a modified chemotaxis-type term. The model is analyzed mathematically and the well-posedness of the resulting PDE system is demonstrated. A series of numerical simulations are provided, demonstrating the ability of the model to naturally capture important phenomena not easily observed in standard diffusion models, including propagation over long spatial distances over short time scales and the emergence of localized infection hotspots

Replacements for Mon, 29 Apr 24

[10]  arXiv:2012.02303 (replaced) [pdf, other]

Title: Decentralized State-Dependent Markov Chain Synthesis with an Application to Swarm Guidance

Authors: Samet Uzun, Nazim Kemal Ure, Behcet Acikmese

Comments: arXiv admin note: text overlap with arXiv:2012.01928

Subjects: Optimization and Control (math.OC); Multiagent Systems (cs.MA); Dynamical Systems (math.DS); Probability (math.PR)

[11]  arXiv:2104.03943 (replaced) [src]

Title: Un exemple de somme de série de vecteurs propres à valeurs propres de module un, non récurrente

Authors: Pierre Mazet, Eric Saias

Comments: After submitting to arXiv, we found a trivial proof for the main result

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

[12]  arXiv:2302.04078 (replaced) [pdf, ps, other]

Title: Generalisations of Thompson's group V arising from purely infinite groupoids

Authors: Eusebio Gardella, Owen Tanner

Comments: 25 pages. v2: One of the conditions in Theorem A was removed, since it is not equivalent to the remaining ones. Otherwise only minor changes

Subjects: Group Theory (math.GR); Dynamical Systems (math.DS); Operator Algebras (math.OA)

[13]  arXiv:2403.08052 (replaced) [pdf, other]

Title: A Computational Method for $H_2$-optimal Estimator and State Feedback Controller Synthesis for PDEs

Authors: Sachin Shivakumar, Matthew Peet

Subjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP); Dynamical Systems (math.DS)

• New submissions
• Cross-lists
• Replacements