Dynamical Systems
- [1] arXiv:2405.09917 [pdf, ps, other]
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Title: Conundrums for continuous Lebesgue measure-preserving interval mapsSubjects: Dynamical Systems (math.DS)
We consider continuous maps of the interval which preserve the Lebesgue measure. Except for the identity map or $1 - \id$ all such maps have topological entropy at least $\log2/2$ and generically they have infinite topological entropy. In this article we show that thegeneric map has zero measure-theoretic entropy. This implies that there are dramatic differences in the topological versus metric behavior both for injectivity as well as for the structure of thelevel sets of generic maps.
- [2] arXiv:2405.09954 [pdf, ps, html, other]
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Title: Quantization of Cantor-Like Set on the Real Projective LineComments: 15 pages, 5 figuresSubjects: Dynamical Systems (math.DS)
In this article, an iterated function system (IFS) is considered on the real projective line $\mathbb{RP}^1$ so that the attractor is a Cantor-like set. Hausdorff dimension of this attractor is estimated. The existence of a probability measure associated with this IFS on $\mathbb{RP}^1$ is also demonstrated. It is shown that the $n$-th quantization error of order $r$ for the push-forward measure is a constant multiple of the $n$-th quantization error of order $r$ of the original measure. Finally, an upper bound for the $n$-th quantization error of order $2$ for this measure is provided.
- [3] arXiv:2405.10076 [pdf, ps, html, other]
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Title: Travelling Waves and Exponential Nonlinearities in the Zeldovich-Frank-Kamenetskii EquationSubjects: Dynamical Systems (math.DS)
We prove the existence of a family of travelling wave solutions in a variant of the $\textit{Zeldovich-Frank-Kamenetskii (ZFK) equation}$, a reaction-diffusion equation which models the propagation of planar laminar premixed flames in combustion theory. Our results are valid in an asymptotic regime which corresponds to a reaction with high activation energy, and provide a rigorous and geometrically informative counterpart to formal asymptotic results that have been obtained for similar problems using $\textit{high activation energy asymptotics}$. We also go beyond the existing results by (i) proving smoothness of the minimum wave speed function $\overline c(\epsilon)$, where $0< \epsilon \ll 1$ is the small parameter, and (ii) providing an asymptotic series for a flat slow manifold which plays a role in the construction of travelling wave solutions for non-minimal wave speeds $c > \overline c(\epsilon)$. The analysis is complicated by the presence of an exponential nonlinearity which leads to two different scaling regimes as $\epsilon \to 0$, which we refer to herein as the $\textit{convective-diffusive}$ and $\textit{diffusive-reactive}$ zones. The main idea of the proof is to use the geometric blow-up method to identify and characterise a $(c,\epsilon)$-family of heteroclinic orbits which traverse both of these regimes, and correspond to travelling waves in the original ZFK equation. More generally, our analysis contributes to a growing number of studies which demonstrate the utility of geometric blow-up approaches to the study dynamical systems with singular exponential nonlinearities.
- [4] arXiv:2405.10144 [pdf, ps, html, other]
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Title: Note on existence of physical measures for systems with mixed central behaviorComments: 14 pages, 1 figureSubjects: Dynamical Systems (math.DS)
We show the existence of physical measures for $C^{\infty}$ smooth instances of certain partially hyperbolic dynamics, both continuous and discrete, exhibiting mixed behavior (positive and negative Lyapunov exponents) along the central non-uniformly hyperbolic multidimensional invariant direction, as a consequence of assuming the existence of certain types of ''regular points'' on positive volume subsets. This includes the $C^3$ robust class of multidimensional non-hyperbolic attractors obtained by Viana, and the $C^1$ robust classes of $3$-sectionally hyperbolic wild strange attractors presented by Shilnikov and Turaev.
- [5] arXiv:2405.10179 [pdf, ps, html, other]
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Title: Conditions on the continuity of the Hausdorff measureComments: 25 pages, 3 figuresSubjects: Dynamical Systems (math.DS)
Let $b_k$ be strictly decreasing sequence of real numbers such that $b_0 = 1$ and $f_k$ be decreasing, linear functions such that $f_k(b_k) = 1$ and $f_k(b_{k-1}) = 0$, $k = 1, 2, \dots$. We define iterated function system (IFS) $S_n$ by limiting the collection of functions $f_k$ to first n, meaning $S_n = \{f_k \}_{k=1}^n$. Let $J_n$ denote the limit set of $S_n$. We show that if $S_n$ fulfills the following two conditions: (1)~$\lim\limits_{n \to \infty} \left(1-h_n\right) \ln{n} = 0 $ where $h_n$ is the Hausdorff dimension of $J_n$, and (2)~$\sup \limits_{k\in \mathbb{N}} \left \{\frac{b_k-b_{k+1}}{b_{k+1}} \right \} < \infty $, then $\lim\limits_{n\to \infty} H_{h_n}(J_n) = 1 = H_1(J)$, where $h_n$ is the Hausdorff dimension of $J_n$ and $H_{h_n}$ is the corresponding Hausdorff measure. We also show examples of families of IFSes fulfilling those properties.
New submissions for Friday, 17 May 2024 (showing 5 of 5 entries )
- [6] arXiv:2405.09684 (cross-list from math.AG) [pdf, ps, html, other]
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Title: Construction of a Basis of K\"{a}hler differentials for generic plane branchesComments: 52 pagesSubjects: Algebraic Geometry (math.AG); Complex Variables (math.CV); Dynamical Systems (math.DS)
Let ${\mathcal C}$ be a fixed equisingularity class of irreducible germs of complex analytic plane curves. We compute a basis of the ${\mathbb C}[[x]]$-module of Kähler differentials for generic $\Gamma \in {\mathcal C}$, algorithmically, and study its behaviour under blow-up.
As an application, we give an alternative proof for a formula of Genzmer, that provides the dimension of the generic component of the moduli of analytic classes in the equisingularity class of $\Gamma$. - [7] arXiv:2405.10209 (cross-list from math.GT) [pdf, ps, html, other]
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Title: Remarks on discrete subgroups with full limit sets in higher rank Lie groupsComments: Comments are welcome!Subjects: Geometric Topology (math.GT); Dynamical Systems (math.DS); Group Theory (math.GR)
We show that real semi-simple Lie groups of higher rank contain (infinitely generated) discrete subgroups with full limit sets in the corresponding Furstenberg boundaries. Additionally, we provide criteria under which discrete subgroups of $G = \operatorname{SL}(3,\mathbb{R})$ must have a full limit set in the Furstenberg boundary of $G$.
In the appendix, we show the the existence of Zariski-dense discrete subgroups $\Gamma$ of $\operatorname{SL}(n,\mathbb{R})$, where $n\ge 3$, such that the Jordan projection of some loxodromic element $\gamma \in\Gamma$ lies on the boundary of the limit cone of $\Gamma$. - [8] arXiv:2405.10236 (cross-list from math-ph) [pdf, ps, other]
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Title: A systematic path to non-Markovian dynamics II: Probabilistic response of nonlinear multidimensional systems to Gaussian colored noise excitationComments: 46 pages, 8 figures, 2 appendices, 103 references, Supplementary materialSubjects: Mathematical Physics (math-ph); Dynamical Systems (math.DS); Probability (math.PR)
The probabilistic characterization of non-Markovian responses to nonlinear dynamical systems under colored excitation is an important issue, arising in many applications. Extending the Fokker-Planck-Kolmogorov equation, governing the first-order response probability density function (pdf), to this case is a complicated task calling for special treatment. In this work, a new pdf-evolution equation is derived for the response of nonlinear dynamical systems under additive colored Gaussian noise. The derivation is based on the Stochastic Liouville equation (SLE), transformed, by means of an extended version of the Novikov-Furutsu theorem, to an exact yet non-closed equation, involving averages over the history of the functional derivatives of the non-Markovian response with respect to the excitation. The latter are calculated exactly by means of the state-transition matrix of variational, time-varying systems. Subsequently, an approximation scheme is implemented, relying on a decomposition of the state-transition matrix in its instantaneous mean value and its fluctuation around it. By a current-time approximation to the latter, we obtain our final equation, in which the effect of the instantaneous mean value of the response is maintained, rendering it nonlinear and non-local in time. Numerical results for the response pdf are provided for a bistable Duffing oscillator, under Gaussian excitation. The pdfs obtained from the solution of the novel equation and a simpler small correlation time (SCT) pdf-evolution equation are compared to Monde Carlo (MC) simulations. The novel equation outperforms the SCT equation as the excitation correlation time increases, keeping good agreement with the MC simulations.
- [9] arXiv:2405.10252 (cross-list from math.NT) [pdf, ps, html, other]
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Title: Bass Note Spectra of Binary FormsComments: 38 pagesSubjects: Number Theory (math.NT); Dynamical Systems (math.DS)
We show that the spectrum of every $\mathbb{R}-$isotropic homogeneous binary form $P$ of degree $n\geq3$ is an interval of the form $[0,M_P],$ where $M_P$ is some positive constant. This completes the discussion around a conjecture of Mordell from 1940 (disproved by Davenport) regarding the existence of spectral gaps for binary cubic forms and further settles Mahler's program for binary forms of every degree.
Cross submissions for Friday, 17 May 2024 (showing 4 of 4 entries )
- [10] arXiv:2109.10704 (replaced) [pdf, ps, html, other]
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Title: Proof of the $C^2$-stability conjecture for geodesic flows of closed surfacesComments: 34 pages, 6 figures; final version, as publishedJournal-ref: Duke Mathematical Journal 173 (2024), no. 2, 347-390Subjects: Dynamical Systems (math.DS); Differential Geometry (math.DG); Symplectic Geometry (math.SG)
We prove that a $C^2$-generic Riemannian metric on a closed surface has either an elliptic closed geodesic or an Anosov geodesic flow. As a consequence, we prove the $C^2$-stability conjecture for Riemannian geodesic flows of closed surfaces: a $C^2$-structurally stable Riemannian geodesic flow of a closed surface is Anosov. In order to prove these statements, we establish a general result that may be of independent interest and provides sufficient conditions for a Reeb flow of a closed 3-manifold to be Anosov.
- [11] arXiv:2210.13126 (replaced) [pdf, ps, html, other]
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Title: On Ruelle-Walters formula of random metric mean dimensionSubjects: Dynamical Systems (math.DS)
The present paper contributes to develop metric mean dimension theory of continuous random dynamical systems, which is driven by Tsukamoto's problem [\emph{Adv. Math.} \textbf{361} (2020), 106935, 53 pp.]: For Brody curves of complex dynamical systems, why is mean dimension connected to the certain integral?
For continuous random dynamical systems, we introduce the concept of metric mean dimension with potential, and inspired by the Ruelle and Walters' idea of topological pressure determining measure-theoretical entropy, we introduce the concept of measure-theoretical metric mean dimension of probability measures. With the help of separation theorem of convex sets, Stone vector lattice, outer measure theory, and Von Neumann's ergodic theorem, we establish a Ruelle-Walters formula of random metric mean dimension by using functional analysis techniques. This demonstrates the deeper ergodic theoretic phenomena hidden behind of the random metric mean dimension theory. - [12] arXiv:2302.08806 (replaced) [pdf, ps, html, other]
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Title: Periodic Normal Forms for Bifurcations of Limit Cycles in DDEsComments: 53 pages, 1 figure. arXiv admin note: text overlap with arXiv:2207.02480Subjects: Dynamical Systems (math.DS); Functional Analysis (math.FA)
A recent work by the authors on the existence of a periodic smooth finite-dimensional center manifold near a nonhyperbolic cycle in delay differential equations motivates the derivation of periodic normal forms. In this paper, we prove the existence of a special coordinate system on the center manifold that will allow us to describe the local dynamics on the center manifold near the cycle in terms of these periodic normal forms. To construct the linear part of this coordinate system, we prove the existence of time periodic smooth Jordan chains for the original and adjoint system. Moreover, we establish duality and spectral relations between both systems by using tools from the theory of delay equations and Volterra integral equations, dual perturbation theory, duality theory and evolution semigroups.
- [13] arXiv:2304.04202 (replaced) [pdf, ps, html, other]
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Title: Continuous eigenfunctions of the transfer operator for the Dyson modelSubjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph)
We prove that there exists a continuous eigenfunction for the transfer operator corresponding to pair potentials that satisfy a square summability condition, when the inverse temperature is subcritical. As a corollary we obtain a continuous eigenfunction for the classical Dyson model, with interactions $\J(k)=\beta \, k^{-\alpha}$, $k\ge1$, in the whole subcritical regime $\beta<\beta_c$ for which the parameter $\alpha$ is greater than $3/2$.
- [14] arXiv:2310.14066 (replaced) [pdf, ps, html, other]
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Title: Topological lower bounds for the R\"ossler SystemComments: 24 pages, 32 figuresSubjects: Dynamical Systems (math.DS); Mathematical Physics (math-ph)
The Rössler System is one of the best known chaotic dynamical systems, exhibiting a plethora of complex phenomena - and yet, only a few studies tackled its complexity analytically. Building on previous work by the author, in this paper we characterize the dynamical complexity for the Rössler System at parameter values at which the flow satisfies a certain heteroclinic condition. This will allow us to characterize the knot type of infinitely many periodic trajectories for the flow - and reduce the Rössler system to a simpler hyperbolic flow, capturing its essential dynamics.
- [15] arXiv:2401.10482 (replaced) [pdf, ps, html, other]
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Title: Periodic orbits of the Stark problemSubjects: Dynamical Systems (math.DS)
The Stark problem is Kepler problem with an external constant acceleration. In this paper, we study the periodic orbits for Stark problem for both planar case and spatial case. We have conducted a detailed analysis of the invariant tori and periodic orbits appearing in the Stark problem, providing a more refined characterization of the properties of the orbits. Interestingly, there exists a family of circular orbits in the spatial case, some of which are quite stable with $L$ being fixed.
- [16] arXiv:2403.13744 (replaced) [pdf, ps, html, other]
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Title: Mean value theorems in multiplicative systems and joint ergodicity of additive and multiplicative actionsComments: 49 pagesSubjects: Dynamical Systems (math.DS); Number Theory (math.NT)
In this paper we are concerned with the study of additive ergodic averages in multiplicative systems and the investigation of the "pretentious" dynamical behaviour of these systems. We prove a mean ergodic theorem (Theorem A) that generalises Halász's mean value theorem for finitely generated multiplicative functions taking values in the unit circle. In addition, we obtain two structural results concerning the "pretentious" dynamical behaviour of finitely generated multiplicative systems. Moreover, motivated by the independence principle between additive and multiplicative structures of the integers, we explore the joint ergodicity (as a natural notion of independence) of an additive and a finitely generated multiplicative action, both acting on the same probability space. In Theorem B, we show that such actions are jointly ergodic whenever no "local obstructions" arise, and we give a concrete description of these "local obstructions". As an application, we obtain some new combinatorial results regarding arithmetic configurations in large sets of integers including refinements of a special case of Szemerédi's theorem.
- [17] arXiv:2405.09479 (replaced) [pdf, ps, html, other]
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Title: Scenarios for the appearance of strange attractors in a model of three interacting microbubble contrast agentsJournal-ref: 2024. Chaos, Solitons & Fractals, 182, p.114785Subjects: Dynamical Systems (math.DS)
We study nonlinear dynamics in a model of three interacting encapsulated gas bubbles in a liquid. The model is a system of three coupled nonlinear oscillators with an external periodic force. Such bubbles have numerous applications, for instance, they are used as contrast agents in ultrasound visualization. Certain types of bubbles oscillations may be beneficial or undesirable depending on a given application and, hence, the dependence of the regimes of bubbles oscillations on the control parameters is worth studying. We demonstrate that there is a wide variety of types of dynamics in the model by constructing a chart of dynamical regimes in the control parameters space. Here we focus on hyperchaotic attractors characterized by three positive Lyapunov exponents and strange attractors with one or two positive Lyapunov exponents possessing an additional zero Lyapunov exponent, which have not been observed previously in the context of bubbles oscillations. We also believe that we provide a first example of a hyperchaotic attractor with additional zero Lyaponov exponent. Furthermore, the mechanisms of the onset of these types of attractors are still not well studied. We identify two-parametric regions in the control parameter space where these hyperchaotic and chaotic attractors appear and study one-parametric routes leading to them. We associate the appearance of hyperchaotic attractors with three positive Lyapunov exponents with the inclusion of a periodic orbit with a three-dimensional unstable manifold, while the onset of chaotic oscillations with an additional zero Lyapunov exponent is connected to the partial synchronization of bubbles oscillations. We propose several underlying bifurcation mechanisms that explain the emergence of these regimes. We believe that these bifurcation scenarios are universal and can be observed in other systems of coupled oscillators.
- [18] arXiv:2403.07572 (replaced) [pdf, ps, html, other]
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Title: On Weakly Contracting Dynamics for Convex OptimizationComments: 16 pages, 4 FiguresSubjects: Optimization and Control (math.OC); Dynamical Systems (math.DS)
We analyze the convergence behavior of \emph{globally weakly} and \emph{locally strongly contracting} dynamics. Such dynamics naturally arise in the context of convex optimization problems with a unique minimizer. We show that convergence to the equilibrium is \emph{linear-exponential}, in the sense that the distance between each solution and the equilibrium is upper bounded by a function that first decreases linearly and then exponentially. As we show, the linear-exponential dependency arises naturally in certain dynamics with saturations. Additionally, we provide a sufficient condition for local input-to-state stability. Finally, we illustrate our results on, and propose a conjecture for, continuous-time dynamical systems solving linear programs.