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Discrete & Continuous Dynamical Systems - A

June 2018 , Volume 38 , Issue 6

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Ergodic theorems for nonconventional arrays and an extension of the Szemerédi theorem
Yuri Kifer
2018, 38(6): 2687-2716 doi: 10.3934/dcds.2018113 +[Abstract](20) +[HTML](8) +[PDF](561.84KB)

The paper is primarily concerned with the asymptotic behavior as \begin{document}$N\to∞$\end{document} of averages of nonconventional arrays having the form \begin{document}${N^{ - 1}}\sum\limits_{n = 1}^N {\prod\limits_{j = 1}^\ell {{T^{{P_j}(n,N)}}} } {f_j}$\end{document} where \begin{document}$f_j$\end{document}'s are bounded measurable functions, \begin{document}$T$\end{document} is an invertible measure preserving transformation and \begin{document}$P_j$\end{document}'s are polynomials of \begin{document}$n$\end{document} and \begin{document}$N$\end{document} taking on integer values on integers. It turns out that when \begin{document}$T$\end{document} is weakly mixing and \begin{document}$P_j(n, N) = p_jn+q_jN$\end{document} are linear or, more generally, have the form \begin{document}$P_j(n, N) = P_j(n)+Q_j(N)$\end{document} for some integer valued polynomials \begin{document}$P_j$\end{document} and \begin{document}$Q_j$\end{document} then the above averages converge in \begin{document}$L^2$\end{document} but for general polynomials \begin{document}$P_j$\end{document} of both \begin{document}$n$\end{document} and \begin{document}$N$\end{document} the \begin{document}$L^2$\end{document} convergence can be ensured even in the "conventional" case \begin{document}$\ell = 1$\end{document} only when \begin{document}$T$\end{document} is strongly mixing while for \begin{document}$\ell>1$\end{document} strong \begin{document}$2\ell$\end{document}-mixing should be assumed. Studying also weakly mixing and compact extensions and relying on Furstenberg's structure theorem we derive an extension of Szemerédi's theorem saying that for any subset of integers \begin{document}$\Lambda $\end{document} with positive upper density there exists a subset \begin{document}${\cal N}_\Lambda $\end{document} of positive integers having uniformly bounded gaps such that for \begin{document}$N∈{\cal N}_\Lambda $\end{document} and at least \begin{document}$\varepsilon N, \, \varepsilon >0$\end{document} of \begin{document}$n$\end{document}'s all numbers \begin{document}$p_jn+q_jN, \, j = 1, ..., \ell, $\end{document} belong to \begin{document}$\Lambda $\end{document}. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemerédi theorem.

Partially hyperbolic sets with a dynamically minimal lamination
Luiz Felipe Nobili França
2018, 38(6): 2717-2729 doi: 10.3934/dcds.2018114 +[Abstract](14) +[HTML](7) +[PDF](384.91KB)

We study partially hyperbolic sets of \begin{document}$C^1$\end{document}-diffeomorphisms. For these sets there are defined the strong stable and strong unstable laminations.A lamination is called dynamically minimal when the orbit of each leaf intersects the set densely.

We prove that partially hyperbolic sets having a dynamically minimal lamination have empty interior. We also study the Lebesgue measure and the spectral decomposition of these sets. These results can be applied to \begin{document}$C^1$\end{document}-generic/robustly transitive attractors with one-dimensional center bundle.

Asymptotic properties of various stochastic cucker-smale dynamics
Laure Pédèches
2018, 38(6): 2731-2762 doi: 10.3934/dcds.2018115 +[Abstract](11) +[HTML](6) +[PDF](986.9KB)

Starting from the stochastic Cucker-Smale model introduced in [14], we look into its asymptotic behaviours for different kinds of interaction. First in term of ergodicity, when $t$ goes to infinity, seeking invariant probability measures and using Lyapunov functionals. Second, when the number $N$ of particles becomes large, leading to results about propagation of chaos.

Remarks on the critical coupling strength for the Cucker-Smale model with unit speed
Seung-Yeal Ha, Dongnam Ko and Yinglong Zhang
2018, 38(6): 2763-2793 doi: 10.3934/dcds.2018116 +[Abstract](18) +[HTML](9) +[PDF](871.21KB)

We present a non-trivial lower bound for the critical coupling strength to the Cucker-Smale model with unit speed constraint and short-range communication weight from the viewpoint of a mono-cluster(global) flocking. For a long-range communication weight, the critical coupling strength is zero in the sense that the mono-cluster flocking emerges from any initial configurations for any positive coupling strengths, whereas for a short-range communication weight, a mono-cluster flocking can emerge from an initial configuration only for a sufficiently large coupling strength. Our main interest lies on the condition of non-flocking. We provide a positive lower bound for the critical coupling strength. We also present numerical simulations for the upper and lower bounds for the critical coupling strength depending on initial configurations and compare them with analytical results.

Synchronization of positive solutions for coupled Schrödinger equations
Chuangye Liu and Zhi-Qiang Wang
2018, 38(6): 2795-2808 doi: 10.3934/dcds.2018118 +[Abstract](17) +[HTML](14) +[PDF](441.85KB)

In this paper, we analyze synchronized positive solutions for a coupled nonlinear Schrödinger equation

where \begin{document}$ 2< p<\frac{n}{n-2}, $\end{document} if \begin{document}$ n\ge 3$\end{document} and \begin{document}$ 2< p<+∞ $\end{document}, if \begin{document}$ n = 1, 2, $\end{document} and \begin{document}$μ_1, μ_2, β>0 $\end{document} are positive constants. Our goal is two fold. On one hand we study the question under what conditions the ground states are nontrivial synchronized positive solutions, giving precise conditions in terms of the size of the coupling constant. On the other hand, we examine the questions on whether all positive solutions are synchronized solutions. We have a complete answer for the case \begin{document}$ n = 1 $\end{document} by proving that positivity implies synchronization. The latter result enables us to obtain the exact number of positive solutions even though no uniqueness result holds in the case, and this is quite different from the case \begin{document}$ p = 2 $\end{document} for which uniqueness of positive solutions was known ([19]).

Ruelle's inequality in negative curvature
Felipe Riquelme
2018, 38(6): 2809-2825 doi: 10.3934/dcds.2018119 +[Abstract](12) +[HTML](6) +[PDF](390.47KB)

In this paper we study different notions of entropy for measure-preserving dynamical systems defined on noncompact spaces. We see that some classical results for compact spaces remain partially valid in this setting. We define a new kind of entropy for dynamical systems defined on noncompact Riemannian manifolds, which satisfies similar properties to the classical ones. As an application, we prove Ruelle's inequality and Pesin's entropy formula for the geodesic flow in manifolds with pinched negative sectional curvature.

Introduction to tropical series and wave dynamic on them
Nikita Kalinin and Mikhail Shkolnikov
2018, 38(6): 2827-2849 doi: 10.3934/dcds.2018120 +[Abstract](14) +[HTML](9) +[PDF](516.7KB)

The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves.

Reducibility of three dimensional skew symmetric system with Liouvillean basic frequencies
Dongfeng Zhang, Junxiang Xu and Xindong Xu
2018, 38(6): 2851-2877 doi: 10.3934/dcds.2018123 +[Abstract](14) +[HTML](9) +[PDF](518.58KB)

In this paper we consider the system \begin{document}$\dot{x} = (A(\epsilon)+ \epsilon^{m} P(t;\epsilon)) x, x∈\mathbb{R}^{3}, $\end{document} where \begin{document}$\epsilon$\end{document} is a small parameter, \begin{document}$A, P$\end{document} are all \begin{document}$3×3$\end{document} skew symmetric matrices, \begin{document}$A$\end{document} is a constant matrix with eigenvalues \begin{document}$± i\bar{λ}(\epsilon)$\end{document} and 0, where \begin{document}$\bar{λ}(\epsilon) = λ+a_{m_{0}}\epsilon^{m_{0}} + O(\epsilon^{m_{0}+1}) (m_{0}< m),$\end{document} \begin{document}$a_{m_{0}}≠ 0,$\end{document} \begin{document}$P$\end{document} is a quasi-periodic matrix with basic frequencies \begin{document}$ω = (1,α)$\end{document} with \begin{document}$α$\end{document} being irrational. First, it is proved that for most of sufficiently small parameters, this system can be reduced to a rotation system. Furthermore, if the basic frequencies satisfy that \begin{document}$ 0≤β(α) < r,$\end{document} where \begin{document}$β(α)$\end{document} measures how Liouvillean \begin{document}$α$\end{document} is, \begin{document}$r$\end{document} is the initial analytic radius, it is proved that for most of sufficiently small parameters, this system can be reduced to constant system by means of a quasi-periodic change of variables.

Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces
Qunyi Bie, Haibo Cui, Qiru Wang and Zheng-An Yao
2018, 38(6): 2879-2910 doi: 10.3934/dcds.2018124 +[Abstract](18) +[HTML](9) +[PDF](629.27KB)

The present paper is devoted to the compressible nematic liquid crystal flow in the whole space \begin{document}$ \mathbb{R}^N\,(N≥ 2)$\end{document}. Here we concentrate on the incompressible limit in the \begin{document}$ L^p$\end{document} type critical Besov spaces setting. We first establish the existence of global solutions in the framework of \begin{document}$ L^p$\end{document} type critical spaces provided that the initial data are close to some equilibrium states. Based on the global existence, we then consider the incompressible limit problem in the ill prepared data case. We justify the low Mach number convergence to the incompressible flow of liquid crystals in proper function spaces. In addition, the accurate converge rates are obtained.

Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system
Min Ding and Hairong Yuan
2018, 38(6): 2911-2943 doi: 10.3934/dcds.2018125 +[Abstract](14) +[HTML](9) +[PDF](582.8KB)

We study supersonic flow past a convex corner which is surrounded by quiescent gas. When the pressure of the upstream supersonic flow is larger than that of the quiescent gas, there appears a strong rarefaction wave to rarefy the supersonic gas. Meanwhile, a transonic characteristic discontinuity appears to separate the supersonic flow behind the rarefaction wave from the static gas. In this paper, we employ a wave front tracking method to establish structural stability of such a flow pattern under non-smooth perturbations of the upcoming supersonic flow. It is an initial-value/free-boundary problem for the two-dimensional steady non-isentropic compressible Euler system. The main ingredients are careful analysis of wave interactions and construction of suitable Glimm functional, to overcome the difficulty that the strong rarefaction wave has a large total variation.

Isolated singularities for elliptic equations with hardy operator and source nonlinearity
Huyuan Chen and Feng Zhou
2018, 38(6): 2945-2964 doi: 10.3934/dcds.2018126 +[Abstract](13) +[HTML](9) +[PDF](470.64KB)

In this paper, we concern the isolated singular solutions for semi-linear elliptic equations involving Hardy-Leray potential

We classify the isolated singularities and obtain the existence and stability of positive solutions of (1). Our results are based on the study of nonhomogeneous Hardy problem in a new distributional sense.

Lozi-like maps
Michał Misiurewicz and Sonja Štimac
2018, 38(6): 2965-2985 doi: 10.3934/dcds.2018127 +[Abstract](10) +[HTML](7) +[PDF](583.08KB)

We define a broad class of piecewise smooth plane homeomorphisms which have properties similar to the properties of Lozi maps, including the existence of a hyperbolic attractor. We call those maps Lozi-like. For those maps one can apply our previous results on kneading theory for Lozi maps. We show a strong numerical evidence that there exist Lozi-like maps that have kneading sequences different than those of Lozi maps.

Propagation of monostable traveling fronts in discrete periodic media with delay
Shi-Liang Wu and Cheng-Hsiung Hsu
2018, 38(6): 2987-3022 doi: 10.3934/dcds.2018128 +[Abstract](12) +[HTML](5) +[PDF](606.44KB)

This paper is devoted to study the front propagation for a class of discrete periodic monostable equations with delay and nonlocal interaction. We first establish the existence of rightward and leftward spreading speeds and prove their coincidence with the minimal wave speeds of the pulsating traveling fronts in the right and left directions, respectively. The dependency of the speeds of propagation on the heterogeneity of the medium and the delay term is also investigated. We find that the periodicity of the medium increases the invasion speed, in comparison with a homogeneous medium; while the delay decreases the invasion speed. Further, we prove the uniqueness of all noncritical pulsating traveling fronts. Finally, we show that all noncritical pulsating traveling fronts are globally exponentially stable, as long as the initial perturbations around them are uniformly bounded in a weight space.

High energy solutions of the Choquard equation
Daomin Cao and Hang Li
2018, 38(6): 3023-3032 doi: 10.3934/dcds.2018129 +[Abstract](15) +[HTML](6) +[PDF](369.22KB)

In this paper we are concerned with the existence of positive high energy solutions of the Choquard equation. Under certain assumptions, the ground state of Choquard equation does not exist. However, by global compactness analysis, we prove that there exists a positive high energy solution.

A singular cahn-hilliard-oono phase-field system with hereditary memory
Monica Conti, Stefania Gatti and Alain Miranville
2018, 38(6): 3033-3054 doi: 10.3934/dcds.2018132 +[Abstract](9) +[HTML](9) +[PDF](447.78KB)

We consider a phase-field system modeling phase transition phenomena, where the Cahn-Hilliard-Oono equation for the order parameter is coupled with the Coleman-Gurtin heat law for the temperature. The former suitably describes both local and nonlocal (long-ranged) interactions in the material undergoing phase-separation, while the latter takes into account thermal memory effects. We study the well-posedness and longtime behavior of the corresponding dynamical system in the history space setting, for a class of physically relevant and singular potentials. Besides, we investigate the regularization properties of the solutions and, for sufficiently smooth data, we establish the strict separation property from the pure phases.

Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction
Gilbert Peralta and Karl Kunisch
2018, 38(6): 3055-3083 doi: 10.3934/dcds.2018133 +[Abstract](15) +[HTML](12) +[PDF](523.25KB)

A coupled parabolic-hyperbolic system of partial differential equations modeling the interaction of a structure submerged in a fluid is studied. The system being considered incorporates delays in the interaction on the interface between the fluid and the solid. We study the stability properties of the interaction model under suitable assumptions between the competing strengths of the delays and the feedback controls.

Liouville theorems for periodic two-component shallow water systems
Qiaoyi Hu, Zhixin Wu and Yumei Sun
2018, 38(6): 3085-3097 doi: 10.3934/dcds.2018134 +[Abstract](20) +[HTML](11) +[PDF](393.31KB)

We establish Liouville-type theorems for periodic two-component shallow water systems, including a two-component Camassa-Holm equation (2CH) and a two-component Degasperis-Procesi (2DP) equation. More presicely, we prove that the only global, strong, spatially periodic solutions to the equations, vanishing at some point \begin{document}$(t_0, x_0)$\end{document}, are the identically zero solutions. Also, we derive new local-in-space blow-up criteria for the dispersive 2CH and 2DP.

Exit time asymptotics for small noise stochastic delay differential equations
David Lipshutz
2018, 38(6): 3099-3138 doi: 10.3934/dcds.2018135 +[Abstract](16) +[HTML](8) +[PDF](672.4KB)

Dynamical system models with delayed dynamics and small noise arise in a variety of applications in science and engineering. In many applications, stable equilibrium or periodic behavior is critical to a well functioning system. Sufficient conditions for the stability of equilibrium points or periodic orbits of certain deterministic dynamical systems with delayed dynamics are known and it is of interest to understand the sample path behavior of such systems under the addition of small noise. We consider a small noise stochastic delay differential equation (SDDE). We obtain asymptotic estimates, as the noise vanishes, on the time it takes a solution of the stochastic equation to exit a bounded domain that is attracted to a stable equilibrium point or periodic orbit of the corresponding deterministic equation. To obtain these asymptotics, we prove a sample path large deviation principle (LDP) for the SDDE that is uniform over initial conditions in bounded sets. The proof of the uniform sample path LDP uses a variational representation for exponential functionals of strong solutions of the SDDE. We anticipate that the overall approach may be useful in proving uniform sample path LDPs for other infinite-dimensional small noise stochastic equations.

Sign-changing multi-bump solutions for Kirchhoff-type equations in $\mathbb{R}^3$
Yinbin Deng and Wei Shuai
2018, 38(6): 3139-3168 doi: 10.3934/dcds.2018137 +[Abstract](25) +[HTML](12) +[PDF](610.26KB)

We are interested in the existence of sign-changing multi-bump solutions for the following Kirchhoff equation

where \begin{document}$λ$\end{document}>0 is a parameter and the potential \begin{document}$V(x)$\end{document} is a nonnegative continuous function with a potential well \begin{document}$Ω: = int(V^{-1}(0))$\end{document} which possesses \begin{document}$k$\end{document} disjoint bounded components \begin{document}$Ω_1,Ω_2,···,Ω_k$\end{document}. Under some conditions imposed on \begin{document}$f(u)$\end{document}, multiple sign-changing multi-bump solutions are obtained. Moreover, the concentration behavior of these solutions as \begin{document}$λ→ +∞$\end{document} are also studied.

Normality and uniqueness of Lagrange multipliers
Karla L. Cortez and Javier F. Rosenblueth
2018, 38(6): 3169-3188 doi: 10.3934/dcds.2018138 +[Abstract](28) +[HTML](20) +[PDF](151.64KB)

In this paper we study, for certain problems in the calculus of variations and optimal control, two different questions related to uniqueness of multipliers appearing in first order necessary conditions. One deals with conditions under which a given multiplier associated with an extremal of a fixed function is unique, a property which, in nonlinear programming, is known to be equivalent to the strict Mangasarian-Fromovitz constraint qualification. We show that, for isoperimetric problems in the calculus of variations, a similar characterization holds, but not in optimal control where the corresponding condition is only sufficient for the uniqueness of the multiplier. The other question is related to the set of multipliers associated with all functions for which a solution to the constrained problem is given. We prove that, for both types of problems, this set is a singleton if and only if a strong normality assumption holds.

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