ISSN:

1078-0947

eISSN:

1553-5231

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## Discrete & Continuous Dynamical Systems - A

January 2019 , Volume 39 , Issue 1

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**Abstract:**

We study the neutral periodic points of Markov-Dyck shifts of finite strongly connected directed graphs. Under certain hypothesis on the structure of the graphs $G$ we show, that the topological conjugacy of their Markov-Dyck shifts implies the isomorphism of the graphs.

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**Abstract:**

This paper deals with a predator-prey model with Beddington-DeAngelis functional response, in which a protection zone is created for the prey species. Whether the combination of the protection zone and the Beddington-DeAngelis functional response can yield new results or not is of interest. The result reveals that they jointly produce a new critical value, which is smaller than that determined by either the protection zone or the functional response singly. As a result, rather different stationary patterns can be found and the combined effects are very prominent. Then the effect of the parameter $k$ in the Beddington-DeAngelis functional response is studied. The result deduces that as $k$ is large enough, there exists a unique positive stationary solution and it is linearly stable except a special case. Actually, we can obtain that the positive stationary solution is globally asymptotically stable.

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**Abstract:**

We investigate a nonlocal reaction-diffusion-advection model which describes the growth of a single phytoplankton species in a water column with crowding effect. The longtime dynamical behavior of this model and the asymptotic profiles of its positive steady states for small crowding effect and large advection rate are established. The results show that there is a critical death rate such that the phytoplankton species survives if and only if its death rate is less than the critical death rate. In contrast to the model without crowding effect, our results show that the density of the phytoplankton species will have a finite limit rather than go to infinity when the death rate disappears. Furthermore, for large sinking rate, the phytoplankton species concentrates at the bottom of the water column with a finite population density. For large buoyant rate, the phytoplankton species concentrates at the surface of the water column with a finite population density.

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We study the phase portraits on the Poincaré disc for all the linear type centers of polynomial Hamiltonian systems of degree

*+*[Abstract](99)

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**Abstract:**

This paper discusses some properties of the topological entropy of the systems generated by polynomials of degree

*+*[Abstract](104)

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**Abstract:**

In his classical work on synchronization, Kuramoto derived the formula for the critical value of the coupling strength corresponding to the transition to synchrony in large ensembles of all-to-all coupled phase oscillators with randomly distributed intrinsic frequencies. We extend this result to a large class of coupled systems on convergent families of deterministic and random graphs. Specifically, we identify the critical values of the coupling strength (transition points), between which the incoherent state is linearly stable and is unstable otherwise. We show that the transition points depend on the largest positive or/and smallest negative eigenvalue(s) of the kernel operator defined by the graph limit. This reveals the precise mechanism, by which the network topology controls transition to synchrony in the Kuramoto model on graphs. To illustrate the analysis with concrete examples, we derive the transition point formula for the coupled systems on Erdős-Rényi, small-world, and

In the second part of this work [

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**Abstract:**

We study the convexity preserving property for a class of time-dependent Hamilton-Jacobi equations in a complete geodesic space. Assuming that the Hamiltonian is nondecreasing, we show that in a Busemann space the unique metric viscosity solution preserves the geodesic convexity of the initial value at any time. We provide two approaches and also discuss several generalizations for more general geodesic spaces including the lattice grid.

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**Abstract:**

In this paper, we investigate the asymptotic behavior of the solutions of the two-dimensional stochastic Navier-Stokes equations via the stationary Wong-Zakai approximations given by the Wiener shift. We prove the existence and uniqueness of tempered pullback attractors for the random equations of the Wong-Zakai approximations with a Lipschitz continuous diffusion term. Under certain conditions, we also prove the convergence of solutions and random attractors of the approximate equations when the step size of approximations approaches zero.

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**Abstract:**

We consider the asymptotic behavior in time of solutions to the nonlinear Schrödinger equation with fourth order anisotropic dispersion (4NLS) which describes the propagation of ultrashort laser pulses in a medium with anomalous time-dispersion in the presence of fourth-order time-dispersion. We prove existence of a solution to (4NLS) which scatters to a solution of the linearized equation of (4NLS) as

*+*[Abstract](127)

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**Abstract:**

We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and present the study of its numerical properties. By following [

*+*[Abstract](114)

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**Abstract:**

In this paper we propose a new sufficient condition for disjointness with all minimal systems.

Using proposed approach we construct a transitive dynamical system

*+*[Abstract](124)

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**Abstract:**

We consider the Cauchy problem for a model of non-linear acoustic, named the Kuznetsov equation, describing a sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the inviscid case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a

*+*[Abstract](99)

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**Abstract:**

In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincaré equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincaré equations.

Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.

*+*[Abstract](91)

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**Abstract:**

The existence and uniqueness/multiplicity of phase locked solution for continuum Kuramoto model was studied in [

*+*[Abstract](116)

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This paper is concerned with the qualitative properties of the solutions of mixed integro-differential equation

with

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**Abstract:**

We study in this article a stochastic 3D globally modified Allen-Cahn-Navier-Stokes model in a bounded domain. We prove the existence and uniqueness of a strong solutions. The proof relies on a Galerkin approximation, as well as some compactness results. Furthermore, we discuss the relation between the stochastic 3D globally modified Allen-Cahn-Navier-Stokes equations and the stochastic 3D Allen-Cahn-Navier-Stokes equations, by proving a convergence theorem. More precisely, as a parameter $N$ tends to infinity, a subsequence of solutions of the stochastic 3D globally modified Allen-Cahn-Navier-Stokes equations converges to a weak martingale solution of the stochastic 3D Allen-Cahn-Navier-Stokes equations.

*+*[Abstract](108)

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**Abstract:**

In this paper, we consider a simple superlinear Duffing equation

with impulses, where

*+*[Abstract](124)

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**Abstract:**

In this paper, we consider the existence of non-trivial solutions for the following equation

where

We show that problem (1) admits at least a solution provided that in the case

*+*[Abstract](109)

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**Abstract:**

Let

where

and

in a dynamical system with the pseudo-orbit tracing property, where

*+*[Abstract](111)

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**Abstract:**

This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as

*+*[Abstract](142)

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**Abstract:**

We establish a symmetry result for a non-autonomous overdetermined problem associated to a sublinear fractional equation. To this purpose we prove, in particular, that the solution of the corresponding Dirichlet problem is monotonically increasing with respect to the domain. We also obtain a strong minimum principle and a boundary-point lemma for linear fractional equations that may have an independent interest.

*+*[Abstract](134)

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**Abstract:**

We prove that for the geodesic flow of a rank 1 Riemannian surface which is expansive but not Anosov the Hausdorff dimension of the set of vectors with only zero Lyapunov exponents is large.

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We consider the mass-subcritical NLS in dimensions

*+*[Abstract](99)

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In this work we prove the lower bound for the number of

*+*[Abstract](106)

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In this paper, we prove that the solution of the Landau-Lifshitz flow

*+*[Abstract](148)

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**Abstract:**

In this paper, we study a fully non-local reaction-diffusion equation which is non-local both in time and space. We apply subordination principles to construct the fundamental solutions of this problem, which we use to find a representation of the mild solutions. Moreover, using techniques of Harmonic Analysis and Fourier Multipliers, we obtain the temporal decay rates for the mild solutions.

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