ISSN:

1078-0947

eISSN:

1553-5231

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## Discrete and Continuous Dynamical Systems

November 2019 , Volume 39 , Issue 11

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*+*[Abstract](2173)

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**Abstract:**

For the system of polyconvex adiabatic thermoelasticity, we define a notion of dissipative measure-valued solution, which can be considered as the limit of a viscosity approximation. We embed the system into a symmetrizable hyperbolic one in order to derive the relative entropy. Exploiting the weak-stability properties of the transport and stretching identities, we base our analysis in the original variables, instead of the symmetric ones (in which the entropy is convex) and we prove measure-valued weak versus strong uniqueness using the averaged relative entropy inequality.

*+*[Abstract](1868)

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**Abstract:**

We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term, in which a vector field is replaced by a Probability Vector Field, that is a probability distribution on the tangent bundle; on the other side, a source term. Such new formulation allows to write in a unified way both classical transport and diffusion with finite speed, together with creation of mass.

The main result of this article shows that, by introducing a suitable Wasserstein-like functional, one can ensure existence of solutions to Measure Differential Equations with sources under Lipschitz conditions. We also prove a uniqueness result under the following additional hypothesis: the measure dynamics needs to be compatible with dynamics of measures that are sums of Dirac masses.

*+*[Abstract](2236)

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**Abstract:**

In this paper, we devote to the study of the Bowen's entropy for fixed-point free flows and show that the Bowen entropy of the whole compact space is equal to the topological entropy. To obtain this result, we establish the Brin-Katok's local entropy formula for fixed-point free flows in ergodic case.

*+*[Abstract](2196)

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**Abstract:**

We study the structure of the escape orbits for a certain class of interval maps. This structure is encoded in the escape transition matrix

*+*[Abstract](2473)

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**Abstract:**

The material conservation of vorticity in fluid flows confined to a thin layer on the surface of a large rotating sphere, is a central result of geophysical fluid dynamics. In this paper we revisit the conservation of vorticity in the context of global scale flows on a rotating sphere. Starting from the vorticity equation instead of the Euler equation, we examine the kinematical and dynamical assumptions that are necessary to arrive at this result. We argue that, in contrast to the planar case, a two-dimensional velocity field does not lead to a single component vorticity equation on the sphere. The shallow fluid approximation is then used to argue that only one component of the vorticity equation is significant for global scale flows. Spherical coordinates are employed throughout, and no planar approximation is used.

*+*[Abstract](2057)

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**Abstract:**

We develop a Thermodynamic Formalism for bounded continuous potentials defined on the sequence space

We then apply our results to the construction of invariant measures of time-homogeneous Markov chains taking values on a general Borel standard space and obtain exponential asymptotic stability for a class of Markov operators. We also construct conformal measures for an infinite collection of interacting random paths which are associated to a potential depending on infinitely many coordinates. Under an additional differentiability hypothesis, we show how this process is related after a proper scaling limit to a certain infinite-dimensional diffusion.

*+*[Abstract](1704)

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**Abstract:**

We consider a mass-critical system of nonlinear Schrödinger equations

where

*+*[Abstract](1706)

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**Abstract:**

Global feedback stabilizability results are derived for nonautonomous coupled systems arising from the linearization around a given time-dependent trajectory of FitzHugh-CNagumo type systems. The feedback is explicit and is based on suitable oblique (nonorthogonal) projections in Hilbert spaces. The actuators are, typically, a finite number of indicator functions and act only in the parabolic equation. Subsequently, local feedback stabilizability to time-dependent trajectories results are derived for nonlinear coupled parabolic-ODE systems of the FitzHugh-CNagumo type. Simulations are presented showing the stabilizing performance of the feedback control.

*+*[Abstract](1431)

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**Abstract:**

We obtain spectral estimates for the iterations of Ruelle operators

*+*[Abstract](1578)

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**Abstract:**

For ${\mathbb{Z}}_2-$symmetric analytic deformation of the circle (with certain Fourier decaying rate), the necessary condition for the corresponding billiard map to keep the coexistence of period 2, 3 caustics is that the deformation has to be an isometric transformation.

*+*[Abstract](1748)

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**Abstract:**

In this paper, we provide a technical result on the existence of Gibbs $ cu $-states for diffeomorphisms with dominated splittings. More precisely, for given $ C^2 $ diffeomorphim $ f $ with dominated splitting $ T_{\Lambda}M = E\oplus F $ on an attractor $ \Lambda $, by considering some suitable random perturbation of $ f $, we show that for any zero noise limit of ergodic stationary measures, if it has positive integrable Lyapunov exponents along invariant sub-bundle $ E $, then its ergodic components contain Gibbs $ cu $-states associated to $ E $. With this technique, we show the existence of SRB measures and physical measures for some systems exhibiting dominated splittings and weak hyperbolicity.

*+*[Abstract](2112)

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**Abstract:**

In this paper, we focus on the exponential stability of stochastic differential equations driven by fractional Brownian motion (fBm) with Hurst parameter $ H\in(1/2, 1) $. Based on the generalized Itô formula and representation of the fBm, some sufficient conditions for exponential stability of a class of SDEs with additive fractional noise are given. Besides, we present a criterion on the exponential stability for the fractional Ornstein-Uhlenbeck process with Markov switching. A numerical example is provided to illustrate our results.

*+*[Abstract](1636)

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**Abstract:**

In this paper, we prove the propagation of the Gevrey regularity of solutions to the three-dimensional incompressible ideal magnetohydrodynamics (MHD) equations. We also obtain an uniform estimate of Gevrey radius for the solution of MHD equation.

*+*[Abstract](1460)

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**Abstract:**

For a positive integer $ M $ and a real base $ q\in(1, M+1] $, let $ {\mathcal{U}}_q $ denote the set of numbers having a unique expansion in base $ q $ over the alphabet $ \{0, 1, \dots, M\} $, and let $ \mathbf{U}_q $ denote the corresponding set of sequences in $ \{0, 1, \dots, M\}^ {\mathbb{N}} $. Komornik et al. [ *Adv. Math.* 305 (2017), 165–196] showed recently that the Hausdorff dimension of $ {\mathcal{U}}_q $ is given by $ h(\mathbf{U}_q)/\log q $, where $ h(\mathbf{U}_q) $ denotes the topological entropy of $ \mathbf{U}_q $. They furthermore showed that the function $ H: q\mapsto h(\mathbf{U}_q) $ is continuous, nondecreasing and locally constant almost everywhere. The plateaus of $ H $ were characterized by Alcaraz Barrera et al. [ *Trans. Amer. Math. Soc.*, **371** (2019), 3209–3258]. In this article we reinterpret the results of Alcaraz Barrera et al. by introducing a notion of composition of fundamental words, and use this to obtain new information about the structure of the function $ H $. This method furthermore leads to a more streamlined proof of their main theorem.

*+*[Abstract](2201)

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**Abstract:**

In this paper, we consider the following fractional Laplacian system with one critical exponent and one subcritical exponent

where

*+*[Abstract](1995)

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**Abstract:**

We consider

*+*[Abstract](1687)

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**Abstract:**

In this paper the geodesic flow on the 2-torus in a non-zero magnetic field is considered. Suppose that this flow admits an additional first integral

*+*[Abstract](1763)

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**Abstract:**

We establish almost sure invariance principles (ASIP), a strong form of approximation by Brownian motion, for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau-Manneville map. Quenched ASIP for random compositions of these maps is also obtained.

*+*[Abstract](2020)

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**Abstract:**

In this paper, we study the following nonlinear Schrödinger equation

where

*+*[Abstract](1877)

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**Abstract:**

We introduce the notion of relative topological entropy dimension to classify the different intermediate levels of relative complexity for factor maps. By considering the dimension or ''density" of special class of sequences along which the entropy is encountered, we provide equivalent definitions of relative entropy dimension. As applications, we investigate the corresponding localization theory and obtain a disjointness theorem involving relative entropy dimension.

*+*[Abstract](2068)

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**Abstract:**

We consider the focusing nonlinear Schrödinger equation

In this work we give an alternative proof of scattering for both NLS and gHartree equations in the radial setting in the inter-critical regime, following the approach of Dodson and Murphy [

*+*[Abstract](1714)

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**Abstract:**

In this paper, the global smooth solution of Cauchy's problem of incompressible, resistive, viscous Hall-magnetohydrodynamics (Hall-MHD) is studied. By exploring the nonlinear structure of Hall-MHD equations, a class of large initial data is constructed, which can be arbitrarily large in

*+*[Abstract](1728)

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**Abstract:**

We investigate the system consisting of the the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting in the context of closed

*+*[Abstract](2017)

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**Abstract:**

This paper concerns the Cauchy problem of the two-dimensional density-dependent Boussinesq equations on the whole space

*+*[Abstract](1822)

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**Abstract:**

We consider a jumping problem for nonlocal singular problems. We apply a recent variational approach for nonlocal singular problem, together with a minimax method in the framework of nonsmooth critical point theory.

*+*[Abstract](1724)

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**Abstract:**

We consider radial solutions of a general elliptic equation involving a weighted Laplace operator. We establish the uniqueness of the radial bound state solutions to

2020
Impact Factor: 1.392

5 Year Impact Factor: 1.610

2020 CiteScore: 2.2

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