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Discrete & Continuous Dynamical Systems - A

January 2019 , Volume 39 , Issue 1

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Markov-Dyck shifts, neutral periodic points and topological conjugacy
Wolfgang Krieger and Kengo Matsumoto
2019, 39(1): 1-18 doi: 10.3934/dcds.2019001 +[Abstract](228) +[HTML](82) +[PDF](222.69KB)
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.

Combined effects of the spatial heterogeneity and the functional response
Yu-Xia Wang and Wan-Tong Li
2019, 39(1): 19-39 doi: 10.3934/dcds.2019002 +[Abstract](216) +[HTML](91) +[PDF](416.29KB)
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.

Single phytoplankton species growth with light and crowding effect in a water column
Danfeng Pang, Hua Nie and Jianhua Wu
2019, 39(1): 41-74 doi: 10.3934/dcds.2019003 +[Abstract](218) +[HTML](73) +[PDF](941.69KB)
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.

Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form $ H = H_1(x)+H_2(y)$
Jaume Llibre, Y. Paulina Martínez and Claudio Vidal
2019, 39(1): 75-113 doi: 10.3934/dcds.2019004 +[Abstract](211) +[HTML](75) +[PDF](2063.3KB)
Abstract:

We study the phase portraits on the Poincaré disc for all the linear type centers of polynomial Hamiltonian systems of degree \begin{document} $5$ \end{document} with Hamiltonian function \begin{document} $H(x,y) = H_1(x)+H_2(y)$ \end{document}, where \begin{document} $H_1(x) = \frac{1}{2} x^2+\frac{a_3}{3}x^3+ \frac{a_4}{4}x^4+ \frac{a_5}{5}x^5$ \end{document} and \begin{document} $H_2(y) = \frac{1}{2} y^2+ \frac{b_3}{3}y^3+ \frac{b_4}{4}y^4+ \frac{b_5}{5}y^5$ \end{document} as function of the six real parameters \begin{document} $a_3, a_4, a_5, b_3, b_4$ \end{document} and \begin{document} $b_5$ \end{document} with \begin{document} $a_5 b_5≠ 0$ \end{document}. We characterize the type and multiplicity of the roots of the polynomials \begin{document} $\hat{p}(y) = 1+b_3y + b_4 y^2+b_5y^3$ \end{document} and \begin{document} $\hat{q}(x) = 1+a_3x+a_4x^2+a_5x^3$ \end{document} and we prove that the finite equilibria are saddles, centers, cusps or the union of two hyperbolic sectors. For the infinite equilibria we found that there only exist two nodes on the Poincaré disc with opposite stability. We also characterize the separatrices of the equilibria and analyze the possible connections between them. As a complement we use the energy level to complete the global phase portrait.

Core entropy of polynomials with a critical point of maximal order
Domingo González and Gamaliel Blé
2019, 39(1): 115-130 doi: 10.3934/dcds.2019005 +[Abstract](99) +[HTML](54) +[PDF](491.3KB)
Abstract:

This paper discusses some properties of the topological entropy of the systems generated by polynomials of degree \begin{document}$ d$\end{document} with two critical points. A partial order in the parameter space is defined. The monotonicity of the topological entropy of postcritically finite polynomials of degree \begin{document}$ d$\end{document} acting on Hubbard tree is generalized.

The mean field analysis of the Kuramoto model on graphs Ⅰ. The mean field equation and transition point formulas
Hayato Chiba and Georgi S. Medvedev
2019, 39(1): 131-155 doi: 10.3934/dcds.2019006 +[Abstract](104) +[HTML](49) +[PDF](533.64KB)
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 \begin{document}$ k$\end{document}-nearest-neighbor families of graphs. As a result of independent interest, we provide a rigorous justification for the mean field limit for the Kuramoto model on graphs. The latter is used in the derivation of the transition point formulas.

In the second part of this work [8], we study the bifurcation corresponding to the onset of synchronization in the Kuramoto model on convergent graph sequences.

Convexity preserving properties for Hamilton-Jacobi equations in geodesic spaces
Qing Liu and Atsushi Nakayasu
2019, 39(1): 157-183 doi: 10.3934/dcds.2019007 +[Abstract](90) +[HTML](53) +[PDF](539.71KB)
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.

Asymptotic behavior of random Navier-Stokes equations driven by Wong-Zakai approximations
Anhui Gu, Kening Lu and Bixiang Wang
2019, 39(1): 185-218 doi: 10.3934/dcds.2019008 +[Abstract](123) +[HTML](72) +[PDF](650.61KB)
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.

Asymptotic behavior in time of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion
Jean-Claude Saut and Jun-Ichi Segata
2019, 39(1): 219-239 doi: 10.3934/dcds.2019009 +[Abstract](105) +[HTML](61) +[PDF](472.99KB)
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 \begin{document}$t\to∞$\end{document}.

Adaptive isogeometric methods with hierarchical splines: An overview
Cesare Bracco, Annalisa Buffa, Carlotta Giannelli and Rafael Vázquez
2019, 39(1): 241-262 doi: 10.3934/dcds.2019010 +[Abstract](127) +[HTML](55) +[PDF](10124.61KB)
Abstract:

We consider an adaptive isogeometric method (AIGM) based on (truncated) hierarchical B-splines and present the study of its numerical properties. By following [10,12,11], optimal convergence rates of the AIGM can be proved when suitable approximation classes are considered. This is in line with the theory of adaptive methods developed for finite elements, recently well reviewed in [43]. The important output of our analysis is the definition of classes of admissibility for meshes underlying hierarchical splines and the design of an optimal adaptive strategy based on these classes of meshes. The adaptivity analysis is validated on a selection of numerical tests. We also compare the results obtained with suitably graded meshes related to different classes of admissibility for 2D and 3D configurations.

Double minimality, entropy and disjointness with all minimal systems
Piotr Oprocha
2019, 39(1): 263-275 doi: 10.3934/dcds.2019011 +[Abstract](114) +[HTML](78) +[PDF](414.55KB)
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 \begin{document}$(X,T)$ \end{document} disjoint with every minimal system and such that the set of transfer times \begin{document}$N(x,U)$ \end{document} is not an \begin{document}$\text{IP}^*$ \end{document}-set for some nonempty open set \begin{document}$U\subset X$ \end{document} and every \begin{document}$x∈ X$ \end{document}. This example shows that the new condition sharpens sufficient conditions for disjointness below previous bounds. In particular our approach does not rely on distality of points or sets.

Cauchy problem for the Kuznetsov equation
Adrien Dekkers and Anna Rozanova-Pierrat
2019, 39(1): 277-307 doi: 10.3934/dcds.2019012 +[Abstract](124) +[HTML](60) +[PDF](601.33KB)
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 \begin{document}$L^2$\end{document}-stability estimate, obtained between a regular and a less regular solutions.

The variational discretization of the constrained higher-order Lagrange-Poincaré equations
Anthony Bloch, Leonardo Colombo and Fernando Jiménez
2019, 39(1): 309-344 doi: 10.3934/dcds.2019013 +[Abstract](99) +[HTML](52) +[PDF](670.09KB)
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.

Convergence and stability of generalized gradient systems by Łojasiewicz inequality with application in continuum Kuramoto model
Zhuchun Li, Yi Liu and Xiaoping Xue
2019, 39(1): 345-367 doi: 10.3934/dcds.2019014 +[Abstract](91) +[HTML](84) +[PDF](508.35KB)
Abstract:

The existence and uniqueness/multiplicity of phase locked solution for continuum Kuramoto model was studied in [12,29]. However, its asymptotic behavior is still unknown. In this paper we concern the asymptotic property of classic solutions to continuum Kuramoto model. In particular, we prove the convergence towards a phase locked state and its stability, provided suitable initial data and coupling strength. The main strategy is the quasi-gradient flow approach based on Łojasiewicz inequality. For this aim, we establish a Łojasiewicz type inequality in infinite dimensions for continuum Kuramoto model which is a nonlocal integro-differential equation. General theorems for convergence and stability of (generalized) quasi-gradient system in an abstract setting are also provided based on Łojasiewicz inequality.

Qualitative properties of positive solutions for mixed integro-differential equations
Patricio Felmer and Ying Wang
2019, 39(1): 369-393 doi: 10.3934/dcds.2019015 +[Abstract](116) +[HTML](89) +[PDF](564.83KB)
Abstract:

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

with \begin{document}$N≥ 1$\end{document}, \begin{document}$M≥ 1$\end{document} and \begin{document}$α∈ (0,1)$\end{document}. We study decay and symmetry properties of the solutions to this equation. Difficulties arise due to the mixed character of the integro-differential operators. Here, a crucial role is played by a version of the Hopf's Lemma we prove in our setting. In studying the decay, we construct appropriate super and sub solutions and we use the moving planes method to prove the symmetry properties.

On the convergence of a stochastic 3D globally modified two-phase flow model
Theodore Tachim Medjo
2019, 39(1): 395-430 doi: 10.3934/dcds.2019016 +[Abstract](99) +[HTML](50) +[PDF](527.08KB)
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.

An application of Moser's twist theorem to superlinear impulsive differential equations
Yanmin Niu and Xiong Li
2019, 39(1): 431-445 doi: 10.3934/dcds.2019017 +[Abstract](108) +[HTML](47) +[PDF](408.24KB)
Abstract:

In this paper, we consider a simple superlinear Duffing equation

with impulses, where \begin{document} $p(t+1) = p(t)$ \end{document} is an integrable function in \begin{document} $\mathbb{R}$ \end{document}. In order to apply Moser's twist theorem, we need to ensure that the corresponding Poincaré map of (0.1) is quite close to a standard twist map but it is not usually achieved due to the existence of impulses. Two types of impulsive functions which overcome this problem with different effects in the Poincaré map are provided here. In both cases, there are large invariant curves diffeomorphism to circles surrounding the origin and going to the infinity, which confine the solutions in its interior and therefore lead to the boundedness of all solutions. Furthermore, it turns out that the solutions starting at \begin{document} $t = 0$ \end{document} on the invariant curves are quasiperiodic.

Normalized solutions of higher-order Schrödinger equations
Aliang Xia and Jianfu Yang
2019, 39(1): 447-462 doi: 10.3934/dcds.2019018 +[Abstract](124) +[HTML](53) +[PDF](422.37KB)
Abstract:

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

where \begin{document} $\mathcal{H}_{0J}$ \end{document} is the higher-order Schrödinger operator with \begin{document} $J∈\mathbb{N}$ \end{document}, \begin{document} $2<p<\frac{4J+6}{3}$ \end{document}, and \begin{document} $λ∈\mathbb{R}$ \end{document} is a parameter. Let \begin{document} $E(u)$ \end{document} be the corresponding variational functional of problem (1). We look for solutions of equation (1) by finding minimizers of the minimization problem

We show that problem (1) admits at least a solution provided that in the case \begin{document} $J$ \end{document} being odd, \begin{document} $2<p<3$ \end{document} and \begin{document} $ρ>0$ \end{document} small or \begin{document} $2+J<p<\frac{4J+6}{3}$ \end{document} and \begin{document} $ρ>0$ \end{document} large; and for the case \begin{document} $J$ \end{document} being even, \begin{document} $3<p<\frac{4J+6}{3}$ \end{document} and \begin{document} $ρ>0$ \end{document} small.

The conditional variational principle for maps with the pseudo-orbit tracing property
Zheng Yin and Ercai Chen
2019, 39(1): 463-481 doi: 10.3934/dcds.2019019 +[Abstract](109) +[HTML](61) +[PDF](437.25KB)
Abstract:

Let \begin{document} $(X,d,f)$ \end{document} be a topological dynamical system, where \begin{document} $(X,d)$ \end{document} is a compact metric space and \begin{document} $f:X \to X$ \end{document} is a continuous map. We define \begin{document} $n$ \end{document}-ordered empirical measure of \begin{document} $x \in X$ \end{document} by

where \begin{document} $δ_y$ \end{document} is the Dirac mass at \begin{document} $y$ \end{document}. Denote by \begin{document} $V(x)$ \end{document} the set of limit measures of the sequence of measures \begin{document} $\mathscr{E}_n(x)$ \end{document}. In this paper, we obtain conditional variational principles for the topological entropy of

and

in a dynamical system with the pseudo-orbit tracing property, where \begin{document} $I$ \end{document} is a certain subset of \begin{document} $\mathscr M_{\rm inv}(X,f)$ \end{document}.

Uniqueness of limit cycles for quadratic vector fields
José Luis Bravo, Manuel Fernández, Ignacio Ojeda and Fernando Sánchez
2019, 39(1): 483-502 doi: 10.3934/dcds.2019020 +[Abstract](111) +[HTML](58) +[PDF](441.33KB)
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 \begin{document}$x' = a_1 x-y-a_3x^2+(2 a_2+a_5)xy + a_6 y^2$\end{document}, \begin{document}$y' = x+a_1 y + a_2x^2+(2 a_3+a_4)xy -a_2y^2$\end{document}. In particular, we study the semi-varieties defined in terms of the parameters \begin{document}$a_1, a_2, ..., a_6$\end{document} where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry.

Non-local sublinear problems: Existence, comparison, and radial symmetry
Antonio Greco and Vincenzino Mascia
2019, 39(1): 503-519 doi: 10.3934/dcds.2019021 +[Abstract](142) +[HTML](55) +[PDF](449.9KB)
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.

Non-hyperbolic behavior of geodesic flows of rank 1 surfaces
Katrin Gelfert
2019, 39(1): 521-551 doi: 10.3934/dcds.2019022 +[Abstract](134) +[HTML](59) +[PDF](765.38KB)
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.

The radial mass-subcritical NLS in negative order Sobolev spaces
Rowan Killip, Satoshi Masaki, Jason Murphy and Monica Visan
2019, 39(1): 553-583 doi: 10.3934/dcds.2019023 +[Abstract](108) +[HTML](54) +[PDF](606.75KB)
Abstract:

We consider the mass-subcritical NLS in dimensions \begin{document}$d≥ 3$\end{document} with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.

Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems
Paolo Gidoni and Alessandro Margheri
2019, 39(1): 585-605 doi: 10.3934/dcds.2019024 +[Abstract](99) +[HTML](50) +[PDF](3710.76KB)
Abstract:

In this work we prove the lower bound for the number of \begin{document}$T$\end{document}-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, \begin{document}$T$\end{document}-periodic in time, with \begin{document}$T$\end{document}-Maslov indices \begin{document}$i_0, i_∞$\end{document} at the origin and at infinity, has at least \begin{document}$|i_∞-i_0|$\end{document} periodic solutions, and an additional one if \begin{document}$i_0$\end{document} is even. Our argument combines the Poincaré-Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.

Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces
Ze Li and Lifeng Zhao
2019, 39(1): 607-638 doi: 10.3934/dcds.2019025 +[Abstract](106) +[HTML](76) +[PDF](629.43KB)
Abstract:

In this paper, we prove that the solution of the Landau-Lifshitz flow \begin{document}$u(t, x)$\end{document} from \begin{document}$\mathbb{H}^2$\end{document} to \begin{document}$\mathbb{H}^2$\end{document} converges to some harmonic map as \begin{document}$t\to∞$\end{document}. The main idea is to construct Tao's caloric gauge in the case where nontrivial harmonic maps exist and use it to prove the convergence to harmonic maps. On one side, since in our case the stationary solutions are asymptotically stable under the heat flow, the caloric gauge of Tao provides a natural geometric linearization. On the other side, although there exist infinite numbers of harmonic maps from \begin{document}$\Bbb H^2$\end{document} to \begin{document}$\Bbb H^2$\end{document}, the heat flow initiated from \begin{document}$u(t, x)$\end{document} for any given \begin{document}$t>0$\end{document} converges to the same harmonic map as the heat flow initiated from \begin{document}$u(0, x)$\end{document}. The two observations enable us to construct Tao's caloric gauge to reduce the convergence to harmonic maps for the Landau-Lifshitz flow to the decay of the corresponding heat tension field. This idea also works for dispersive geometric flows, see our succeeding works on wave maps for instance.

Fundamental solutions and decay of fully non-local problems
Juan C. Pozo and Vicente Vergara
2019, 39(1): 639-666 doi: 10.3934/dcds.2019026 +[Abstract](148) +[HTML](60) +[PDF](534.23KB)
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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