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Dynamical Systems

New submissions

[ total of 26 entries: 1-26 ]
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New submissions for Tue, 20 Mar 18

[1]  arXiv:1803.06369 [pdf, other]
Title: A Family of Minimal and Renormalizable Rectangle Exchange Maps
Comments: 33 pages, 12 figures
Subjects: Dynamical Systems (math.DS)

A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a PV number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.

[2]  arXiv:1803.06372 [pdf, other]
Title: Stochastic basins of attraction and generalized committor functions
Subjects: Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD)

We generalize the concept of basin of attraction of a stable state in order to facilitate the analysis of dynamical systems with noise and to assess stability properties of metastable states and long transients. To this end we examine the notions of mean sojourn times and absorption probabilities for Markov chains and study their convergence to the basin of attraction in the limiting cases. Since any dynamical system described by a transfer operator on a compact domain can be approximated by a Markov chain our approach is applicable to a large variety of problems.

[3]  arXiv:1803.06402 [pdf, ps, other]
Title: Shadowing for nonautonomous dynamics
Subjects: Dynamical Systems (math.DS)

We prove that whenever a sequence of invertible and bounded operators $(A_m)_{m\in \mathbb{Z}}$ acting on a Banach space $X$ admits an exponential dichotomy and a sequence of differentiable maps $f_m \colon X\to X$, $m\in \mathbb{Z}$, has bounded and H\"{o}lder derivatives, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in \mathbb{Z}$ has various shadowing properties. Hence, we extend recent results of Bernardes Jr. et al. in several directions. As a nontrivial application of our results, we give a new proof of the nonautonomous Grobman-Hartman theorem.

[4]  arXiv:1803.06537 [pdf, other]
Title: Renormalization of the Hutchinson Operator
Authors: Yann Demichel
Subjects: Dynamical Systems (math.DS)

One of the easiest and common ways of generating fractal sets in $\mathbb{R}^D$ is as attractors of affine iterated function systems (IFS). The classic theory of IFS's requires that they are made with contractive functions. In this paper, we relax this hypothesis considering a new operator $H_\rho$ obtained by renormalizing the usual Hutchinson operator $H$. Namely, the $H_\rho$-orbit of a given compact set $K_0$ is built from the original sequence $(H^n(K_0))_n$ by rescaling each set by its distance from $0$. We state several results for the convergence of these orbits and give a geometrical description of the corresponding limit sets. In particular, it provides a way to construct some eigensets for $H$. Our strategy to tackle the problem is to link these new sequences to some classic ones but it will depend on whether the IFS is strictly linear or not. We illustrate the different results with various detailed examples. Finally, we discuss some possible generalizations.

[5]  arXiv:1803.06572 [pdf, ps, other]
Title: Non-hyperbolic ergodic measures and horseshoes in partially homoclinic classes
Comments: 25 pages and 1 figure
Subjects: Dynamical Systems (math.DS)

We study a rich family of robustly non-hyperbolic transitive diffeomorphisms and we show that each ergodic measure is approached by hyperbolic sets in weak$*$-topology and in entropy. For hyperbolic ergodic measures, it is a classical result of A. Katok. The novelty here is to deal with non-hyperbolic ergodic measures.

[6]  arXiv:1803.06583 [pdf, ps, other]
Title: Circular orders, ultrahomogeneity and topological groups
Comments: 17 pages
Subjects: Dynamical Systems (math.DS)

We study topological groups $G$ for which the universal minimal $G$-system $M(G)$, or the universal irreducible affine $G$-system $IA(G)$ are tame. We call such groups intrinsically tame and convexly intrinsically tame. These notions are generalized versions of extreme amenability and amenability, respectively. When $M(G)$, as a $G$-system, admits a circular order we say that $G$ is intrinsically circularly ordered. This implies that $G$ is intrinsically tame.
We show that for every circularly ultrahomogeneous action $G \curvearrowright X$ on a circularly ordered set $X$ the topological group $G$, in its pointwise convergence topology, is intrinsically circularly ordered. This result is a "circular" analog of Pestov's result about the extremal amenability of ultrahomogeneous actions on linearly ordered sets by linear order preserving transformations.
In the case where $X$ is countable, the corresponding Polish group of circular automorphisms $G$ admits a concrete description. Using the Kechris-Pestov-Todorcevic construction we show that $M(G)$ is a circularly ordered compact space obtained by splitting the rational points on the circle. We show also that $G$ is Roelcke precompact, satisfies Kazhdan's property $T$ (using results of Evans-Tsankov) and has the automatic continuity property (using results of Rosendal-Solecki).

[7]  arXiv:1803.06665 [pdf, ps, other]
Title: Les exposants de Liapounoff du flot de Teichmüller
Comments: in French
Journal-ref: Ast\'erisque, Soci\'et\'e Math\'ematique de France, 2014, 361 (1060), pp.43-75
Subjects: Dynamical Systems (math.DS)

The purpose of the paper under review is to explain the main ideas and the main ingredients of the involved and delicate work of A. Eskin, M. Kontsevich and A. Zorich concerning the sum of the positive Lyapunov exponents of the so-called Kontsevich- Zorich cocycle acting on the first cohomology spaces of translation surfaces.

[8]  arXiv:1803.06796 [pdf, ps, other]
Title: Different Statistical Future of Dynamical Orbits over Expanding or Hyperbolic Systems (II): Nonempty Syndetic Center
Comments: The present paper is a continuation of arXiv:1701.01910 which consider dynamical orbits without syndetic center. Here sixteen different statistical behavior of dynamical orbits with nonempty syndetic center may appear but up to now only two cases are well-studied
Subjects: Dynamical Systems (math.DS)

In [30] different statistical behavior of dynamical orbits without syndetic center are considered. In present paper we continue this project and consider different statistical behavior of dynamical orbits with nonempty syndetic center: Two of sixteen cases appear (for which other fourteen cases are still unknown) in transive topologically expanding or hyperbolic systems and are discovered to have full topological entropy for which it is also true if combined with non-recurrence and multifractal analysis such as quasi-regular set, irregular set and level sets. In this process a strong entropy-dense property, called minimal-entropy-dense, is established. In particular, we show that points that are minimal (or called almost periodic), a classical and important concept in the study of dynamical systems, form a set with full topological entropy if the dynamical system satisfies shadowing or almost specification property.

[9]  arXiv:1803.06859 [pdf, ps, other]
Title: Approximation of non-archimedean Lyapunov exponents and applications over global fields
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)

Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-archimedean quantitative approximation allows us to show:
- a quantified version of Silverman's and Ingram's recent comparison between the critical height and any ample height on the moduli space $\mathcal{M}_d(\bar{\mathbb{Q}})$,
- two improvements of McMullen's finiteness of the mutiplier maps: reduction to multipliers of cycles of exact given period and an effective bound from below on the period,
- a characterization of non-affine isotrivial rational maps defined over the function field $\mathbb{C}(X)$ of a normal projective variety $X$ in terms of the growth of the degree of the multipliers.

[10]  arXiv:1803.06863 [pdf, ps, other]
Title: Approximation property on entropies for surface diffeomorphisms
Subjects: Dynamical Systems (math.DS)

In this paper, we prove that for any $C^1$ surface diffeomorphism $f$ with positive topological entropy, there exists a diffeomorphism $g$ arbitrarily close (in the $C^1$ topology) to $f$ exhibiting a horseshoe $\Lambda$, such that the topological entropy of $g$ restricted on $\Lambda$ can arbitrarily approximate the topological entropy of $f$. This extends the Theorem \cite[Theorem 1.1]{Gan} of Gan.

[11]  arXiv:1803.06865 [pdf, other]
Title: Spatio-temporal Poisson processes for visits to small sets
Subjects: Dynamical Systems (math.DS)

For many measure preserving dynamical systems $(\Omega,T,m)$ the successive hitting times to a small set is well approximated by a Poisson process on the real line. In this work we define a new process obtained from recording not only the successive times $n$ of visits to a set $A$, but also the position $T^n(x)$ in $A$ of the orbit, in the limit where $m(A)\to0$.
We obtain a convergence of this process, suitably normalized, to a Poisson point process in time and space under some decorrelation condition. We present several new applications to hyperbolic maps and SRB measures, including the case of a neighborhood of a periodic point, and some billiards such as Sinai billiards, Bunimovich stadium and diamond billiard.

[12]  arXiv:1803.06918 [pdf, other]
Title: Correcting Observation Model Error in Data Assimilation
Subjects: Dynamical Systems (math.DS); Data Analysis, Statistics and Probability (physics.data-an)

Standard methods of data assimilation assume prior knowledge of a model that describes the system dynamics and an observation function that maps the model state to a predicted output. An accurate mapping from model state to observation space is crucial in filtering schemes when adjusting the estimate of the system state during the filter's analysis step. However, in many applications the true observation function may be unknown and the available observation model may have significant errors, resulting in a suboptimal state estimate. We propose a method for observation model error correction within the filtering framework. The procedure involves an alternating minimization algorithm used to iteratively update a given observation function to increase consistency with the model and prior observations, using ideas from attractor reconstruction. The method is demonstrated on the Lorenz 1963 and Lorenz 1996 models, and on a single-column radiative transfer model with multicloud parameterization.

[13]  arXiv:1803.06996 [pdf, ps, other]
Title: Existence and Ergodic properties of equilibrium measures for maps associated with inducing schemes of hyperbolic type
Subjects: Dynamical Systems (math.DS)

We consider maps $f:X\to X$ admitting inducing schemes of hyperbolic type introduced in \cite{ind} as well as the induced maps $\tilde{f}:\tilde{X}\to \tilde{X}$ and the associated tower maps $\hat{f}:\hat{X} \to \hat {X}$. For a certain class of potential functions $\varphi$ on $X$, that includes all H\"older continuous functions, we establish thermodynamic formalism for the above three systems. We study relations among the corresponding equilibrium measures and their ergodic properties. We establish decay of correlations, the Central Limit Theorem (CLT), the Bernoulli property for the three systems with respect to their corresponding equilibrium measures. Finally, we prove analyticity of the pressure function for the three systems.

Cross-lists for Tue, 20 Mar 18

[14]  arXiv:1803.06439 (cross-list from math.SG) [pdf, other]
Title: Global surfaces of section for dynamically convex Reeb flows on lens spaces
Comments: 29 pages, 4 figures
Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS)

We show that a dynamically convex Reeb flow on a lens space $L(p, 1), p>1,$ admits a $p$-unknotted closed Reeb orbit $P$ which is the binding of a rational open book decomposition with disk-like pages. Each page is a rational global surface of section for the Reeb flow and the Conley-Zehnder index of the $p$-th iterate of $P$ is $3$. This result applies to the H\'enon-Heiles Hamiltonian whose energy level presents $\mathbb{Z}_3$-symmetric and for all energies $<1/6$ the flow restricted to the sphere-like component descends to a dynamically convex Reeb flow on $L(3,1)$. Due to a $\mathbb{Z}_4$-symmetry the result also applies to Hill's lunar problem.

[15]  arXiv:1803.06511 (cross-list from nlin.SI) [pdf, ps, other]
Title: On discretization of the Euler top
Authors: A.V. Tsiganov
Comments: 12 pages, 2 figures, LaTeX with AMS fonts
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Dynamical Systems (math.DS)

Application of the intersection theory to construction of n-point finite-difference equations associated with classical integrable systems is discussed. As an example, we present a few new discretizations of motion of the Euler top sharing the integrals of motion with the continuous time system and the Poisson bracket up to the integer scaling factor.

[16]  arXiv:1803.06717 (cross-list from math.AP) [pdf, other]
Title: High frequency limits for invariant Ruelle densities
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Dynamical Systems (math.DS); Spectral Theory (math.SP)

We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank one. More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective products of resonant and co-resonant states converge weakly to the Liouville measure. We prove this result by establishing an explicit quantum-classical correspondence between eigenspaces of the scalar Laplacian and the resonant states of the first band of Ruelle resonances which also leads to a new description of Patterson-Sullivan distributions.

[17]  arXiv:1803.06920 (cross-list from nlin.CD) [pdf, other]
Title: On the Keldysh Problem of Flutter Suppression
Subjects: Chaotic Dynamics (nlin.CD); Dynamical Systems (math.DS)

This work is devoted to the Keldysh model of flutter suppression and rigorous approaches to its analysis. To solve the stabilization problem in the Keldysh model we use an analog of direct Lyapunov method for differential inclusions. The results obtained here are compared with the results of Keldysh obtained by the method of harmonic balance (describing function method), which is an approximate method for analyzing the existence of periodic solutions. The limitations of the use of describing function method for the study of systems with dry friction and stationary segment are demonstrated.

[18]  arXiv:1803.07065 (cross-list from physics.app-ph) [pdf]
Title: Sensorless Resonance Tracking of Resonant Electromagnetic Actuator through Back-EMF Estimation for Mobile Devices
Authors: Youngjun Cho
Subjects: Applied Physics (physics.app-ph); Systems and Control (cs.SY); Signal Processing (eess.SP); Dynamical Systems (math.DS)

Resonant electromagnetic actuators have been broadly used as vibration motors for mobile devices given their ability of generating relatively fast, strong, and controllable vibration force at a given resonant frequency. Mechanism of the actuators that is based on mechanical resonance, however, limits their use to a situation where their resonant frequencies are known and unshifted. In reality, there are many factors that alter the resonant frequency: for example, manufacturing tolerances, worn mechanical components such as a spring, nonlinearity in association with different input voltage levels. Here, we describe a sensorless resonance tracking method that actuates the motor and automatically detects its unknown damped natural frequency through the estimation of back electromotive force (EMF) and inner mass movements. We demonstrate the tracking performance of the proposed method through a series of experiments. This approach has the potential to control residual vibrations and then improve vibrotactile feedback, which can potentially be used for human-computer interaction, cognitive and affective neuroscience research.

Replacements for Tue, 20 Mar 18

[19]  arXiv:1501.01368 (replaced) [pdf, ps, other]
Title: On parameter loci of the Hénon family
Comments: 70 pages, 38 figures, 3 tables
Subjects: Dynamical Systems (math.DS)
[20]  arXiv:1603.08360 (replaced) [pdf, other]
Title: Lyapunov spectrum of Markov and Euclid trees
Comments: Slightly improved version with more details added to the proof of Theorem 3
Journal-ref: Nonlinearity, Volume 30, Number 12 (2017), 4428-53
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)
[21]  arXiv:1611.06470 (replaced) [pdf, ps, other]
Title: Bounded orbits of Diagonalizable Flows on finite volume quotients of products of $SL_2(\mathbb{R})$
Comments: Added Tue Ly as author. Other minor changes
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)
[22]  arXiv:1704.05130 (replaced) [pdf, ps, other]
Title: Rotation number of interval contracted rotations
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)
[23]  arXiv:1802.07654 (replaced) [pdf, ps, other]
Title: Intersecting limit sets of Kleinian subgroups and Susskind's conjecture
Subjects: Dynamical Systems (math.DS)
[24]  arXiv:1803.00402 (replaced) [pdf, ps, other]
Title: On Periodic Solutions to Lagrangian System With Constraints
Authors: Oleg Zubelevich
Subjects: Dynamical Systems (math.DS)
[25]  arXiv:1711.00778 (replaced) [pdf, other]
Title: Asymptotic behaviour of a network of oscillators coupled to thermostats of finite energy
Authors: Andrey V. Dymov
Comments: 22 pages. In comparison with the previous version where a chain of oscillators was considered, the result is generalized to the case when the oscillators form arbitrary network
Subjects: Mathematical Physics (math-ph); Dynamical Systems (math.DS)
[26]  arXiv:1801.01165 (replaced) [pdf, other]
Title: Automorphism groups and Ramsey properties of sparse graphs
Comments: 40 pages, 3 figures, new proof of Theorem 3.19 and other minor revisions
Subjects: Group Theory (math.GR); Discrete Mathematics (cs.DM); Combinatorics (math.CO); Dynamical Systems (math.DS); Logic (math.LO)
[ total of 26 entries: 1-26 ]
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