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Discrete and Continuous Dynamical Systems

July 2018 , Volume 38 , Issue 7

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Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families
Jun Hu, Oleg Muzician and Yingqing Xiao
2018, 38(7): 3189-3221 doi: 10.3934/dcds.2018139 +[Abstract](5557) +[HTML](327) +[PDF](2822.6KB)

In [6], regularly ramified rational maps are constructed and Julia sets of these maps in some one-parameter families are explored through computer-generated pictures. It is observed that they have classifications similar to the Julia sets of maps in the families \begin{document}$ f_n^{c}(z) = z^n+\frac{c}{z^n}$\end{document}, where \begin{document}$ n≥ 2$\end{document} and \begin{document}$ c$\end{document} is a complex number. A new type of Julia set is also presented, which has not appeared in the literature. We call such a Julia set an exploded McMullen necklace. We prove in this paper: if a map \begin{document}$ f$\end{document} in the one-parameter families given in [6] has a superattracting fixed point of order greater than 2, then its Julia set \begin{document}$ J(f)$\end{document} is either connected, a Cantor set, or a McMullen necklace (either exploded or not); if such a map \begin{document}$ f$\end{document} has a superattracting fixed point of order equal to 2, then \begin{document}$ J(f)$\end{document} is either connected or a Cantor set.

Dicritical nilpotent holomorphic foliations
Percy Fernández-Sánchez, Jorge Mozo-Fernández and Hernán Neciosup
2018, 38(7): 3223-3237 doi: 10.3934/dcds.2018140 +[Abstract](4280) +[HTML](277) +[PDF](408.85KB)

We study in this paper several properties concerning singularities of foliations in \begin{document}$ {\left( {{\mathbb{C}}^{3}}\rm{,}\bf{0} \right)}$\end{document} that are pull-back of dicritical foliations in \begin{document}$ {\left( {{\mathbb{C}}^{2}}\rm{,}\bf{0} \right)}$\end{document}. Particularly, we will investigate the existence of first integrals (holomorphic and meromorphic) and the dicriticalness of such a foliation. In the study of meromorphic first integrals we follow the same method used by R. Meziani and P. Sad in dimension two. While the foliations we study are pull-back of foliations in \begin{document}$ {\left( {{\mathbb{C}}^{2}}\rm{,}\bf{0} \right)}$\end{document}, the adaptations are not straightforward.

Low Mach number limit for a model of accretion disk
Donatella Donatelli, Bernard Ducomet and Šárka Nečasová
2018, 38(7): 3239-3268 doi: 10.3934/dcds.2018141 +[Abstract](4459) +[HTML](280) +[PDF](635.4KB)

We study an hydrodynamical model describing the motion of thick astrophysical disks relying on compressible Navier-Stokes-Fourier-Poisson system. We also suppose that the medium is electrically charged and we include energy exchanges through radiative transfer. Supposing that the system is rotating, we study the singular limit of the system when the Mach number, the Alfven number and Froude number go to zero and we prove convergence to a 3D incompressible MHD system with radiation with two stationary linear transport equations for transport of radiation intensity.

The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity
Filomena Feo, Pablo Raúl Stinga and Bruno Volzone
2018, 38(7): 3269-3298 doi: 10.3934/dcds.2018142 +[Abstract](4803) +[HTML](243) +[PDF](578.21KB)

For \begin{document}$0<s<1$\end{document}, we consider the Dirichlet problem for the fractional nonlocal Ornstein-Uhlenbeck equation

where \begin{document}$Ω$\end{document} is a possibly unbounded open subset of \begin{document}$\mathbb{R}^n$\end{document}, \begin{document}$n≥2$\end{document}. The appropriate functional settings for this nonlocal equation and its corresponding extension problem are developed. We apply Gaussian symmetrization techniques to derive a concentration comparison estimate for solutions. As consequences, novel \begin{document}$L^p$\end{document} and \begin{document}$L^p(\log L)^α$\end{document} regularity estimates in terms of the datum \begin{document}$f$\end{document} are obtained by comparing \begin{document}$u$\end{document} with half-space solutions.

Derivation of a non-autonomous linear Boltzmann equation from a heterogeneous Rayleigh gas
Karsten Matthies and George Stone
2018, 38(7): 3299-3355 doi: 10.3934/dcds.2018143 +[Abstract](4568) +[HTML](237) +[PDF](651.78KB)

A linear Boltzmann equation with non-autonomous collision operator is rigorously derived in the Boltzmann-Grad limit for the deterministic dynamics of a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with heterogeneously distributed background particles, which do not interact among each other. The validity of the linear Boltzmann equation holds for arbitrary long times under moderate assumptions on spatial continuity and higher moments of the initial distributions of the tagged particle and the heterogeneous, non-equilibrium distribution of the background. The empiric particle dynamics are compared to the Boltzmann dynamics using evolution systems for Kolmogorov equations of associated probability measures on collision histories.

Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind
Hung Le
2018, 38(7): 3357-3385 doi: 10.3934/dcds.2018144 +[Abstract](4417) +[HTML](214) +[PDF](551.38KB)

In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in Hölder spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.

Navier-Stokes-Oseen flows in the exterior of a rotating and translating obstacle
Trinh Viet Duoc
2018, 38(7): 3387-3405 doi: 10.3934/dcds.2018145 +[Abstract](5370) +[HTML](246) +[PDF](448.07KB)

In this paper, we investigate Navier-Stokes-Oseen equation describing flows of incompressible viscous fluid passing a translating and rotating obstacle. The existence, uniqueness, and polynomial stability of bounded and almost periodic weak mild solutions to Navier-Stokes-Oseen equation in the solenoidal Lorentz space \begin{document}$ L^{3}_{σ, w} $\end{document} are shown. Moreover, we also prove the unique existence of time-local mild solutions to this equation in the solenoidal Lorentz spaces \begin{document}$ L^{3,q}_{σ} $\end{document}.

On the fundamental solution and a variational formulation for a degenerate diffusion of Kolmogorov type
Manh Hong Duong and Hoang Minh Tran
2018, 38(7): 3407-3438 doi: 10.3934/dcds.2018146 +[Abstract](4307) +[HTML](262) +[PDF](524.05KB)

In this paper, we construct the fundamental solution to a degenerate diffusion of Kolmogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. We establish the convergence of the scheme to the solution of the adjoint equation generalizing previously known results for the Fokker-Planck equation and the Kramers equation.

Non-floquet invariant tori in reversible systems
Xiaocai Wang
2018, 38(7): 3439-3457 doi: 10.3934/dcds.2018147 +[Abstract](4344) +[HTML](238) +[PDF](434.11KB)

In this paper we obtain a theorem about the persistence of non-floquet invariant tori of analytic reversible systems by an improved KAM iteration. This theorem can be applied to solve the persistence problem of completely hyperbolic-type degenerate invariant tori for a class of reversible system.

On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions
Maxim A. Olshanskii, Leo G. Rebholz and Abner J. Salgado
2018, 38(7): 3459-3477 doi: 10.3934/dcds.2018148 +[Abstract](5008) +[HTML](311) +[PDF](511.86KB)

We study well-posedness of a velocity-vorticity formulation of the Navier-Stokes equations, supplemented with no-slip velocity boundary conditions, a corresponding zero-normal condition for vorticity on the boundary, along with a natural vorticity boundary condition depending on a pressure functional. In the stationary case we prove existence and uniqueness of a suitable weak solution to the system under a small data condition. The topic of the paper is driven by recent developments of vorticity based numerical methods for the Navier-Stokes equations.

Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes
Bernold Fiedler and Carlos Rocha
2018, 38(7): 3479-3545 doi: 10.3934/dcds.2018149 +[Abstract](5547) +[HTML](286) +[PDF](3318.96KB)

Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic,

on the unit interval \begin{document}$0 < x <1$\end{document} with Neumann boundary conditions. Equilibria \begin{document}$v_t = 0$\end{document} are assumed to be hyperbolic.

Geometrically, we study the resulting Thom-Smale dynamic complex with cells defined by the fast unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a regular cell complex. In the first two papers we characterized 3-ball Sturm attractors \begin{document}$\mathcal{A}$\end{document} as 3-cell templates \begin{document}$\mathcal{C}$\end{document}. The characterization involves bipolar orientations and hemisphere decompositions which are closely related to the geometry of the fast unstable manifolds.

An equivalent combinatorial description was given in terms of the Sturm permutation, alias the meander properties of the shooting curve for the equilibrium ODE boundary value problem. It involves the relative positioning of extreme 2-dimensionally unstable equilibria at the Neumann boundaries \begin{document}$x = 0$\end{document} and \begin{document}$x = 1$\end{document}, respectively, and the overlapping reach of polar serpents in the shooting meander.

In the present paper we apply these descriptions to explicitly enumerate all 3-ball Sturm attractors \begin{document}$\mathcal{A}$\end{document} with at most 13 equilibria. We also give complete lists of all possibilities to obtain solid tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27 equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and dodecahedra, we indicate a reduction to mere planar considerations as discussed in our previous trilogy on planar Sturm attractors.

Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source
Chunhua Jin
2018, 38(7): 3547-3566 doi: 10.3934/dcds.2018150 +[Abstract](4668) +[HTML](287) +[PDF](504.46KB)

In this paper, we deal with a coupled chemotaxis-fluid model with logistic source \begin{document} $γ n-μ n^2$ \end{document}. We prove the existence of global classical solution for the chemotaxis-Stokes system in a bounded domain \begin{document} $Ω\subset \mathbb R^3$ \end{document} for any large initial data. On the basis of this, we further prove that if \begin{document} $γ>0$ \end{document}, the zero solution is not stable; if \begin{document} $γ = 0$ \end{document}, the zero solution is globally asymptotically stable; and if \begin{document}$ 0 < γ < 16μ^2$ \end{document}, the nontrivial steady state \begin{document} $\left(\fracγμ, \fracγμ, 0\right)$ \end{document} is globally asymptotically stable.

On critical Choquard equation with potential well
Zifei Shen, Fashun Gao and Minbo Yang
2018, 38(7): 3567-3593 doi: 10.3934/dcds.2018151 +[Abstract](5154) +[HTML](310) +[PDF](538.96KB)

In this paper we are interested in the following nonlinear Choquard equation

where \begin{document} $λ, β∈\mathbb{R}^+$ \end{document}, \begin{document} $0<μ<N, N≥4, 2_{μ}^{*} = (2N-μ)/(N-2)$ \end{document} is the upper critical exponent due to the Hardy-Littlewood-Sobolev inequality and the nonnegative potential function \begin{document} $V∈ \mathcal{C}(\mathbb{R}^N, \mathbb{R})$ \end{document} such that \begin{document} $Ω : = \mbox{int} V^{-1}(0)$ \end{document} is a nonempty bounded set with smooth boundary. If \begin{document} $β>0$ \end{document} is a constant such that the operator \begin{document} $-Δ +λ V(x)-β$ \end{document} is non-degenerate, we prove the existence of ground state solutions which localize near the potential well int \begin{document} $V^{-1}(0)$ \end{document} for \begin{document} $λ$ \end{document} large enough and also characterize the asymptotic behavior of the solutions as the parameter \begin{document} $λ$ \end{document} goes to infinity. Furthermore, for any \begin{document} $0<β<β_{1}$ \end{document}, we are able to prove the existence of multiple solutions by the Lusternik-Schnirelmann category theory, where \begin{document} $β_{1}$ \end{document} is the first eigenvalue of \begin{document} $-Δ$ \end{document} on \begin{document} $Ω$ \end{document} with Dirichlet boundary condition.

Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity
Hai-Yang Jin
2018, 38(7): 3595-3616 doi: 10.3934/dcds.2018155 +[Abstract](4832) +[HTML](270) +[PDF](561.79KB)

This paper is concerned with the following Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity

in a bounded smooth domain \begin{document}$Ω\subset \mathbb{R}^2$\end{document} with homogeneous Neumann boundary conditions, where \begin{document}$a≥0$\end{document} and \begin{document}$b>0$\end{document} are constants, and the functions \begin{document}$d(c)$\end{document} and \begin{document}$χ(c)$\end{document} satisfy the following assumptions:

\begin{document}$(d(c), χ (c))∈ [C^2([0, ∞))]^2$\end{document} with \begin{document}$d(c), χ(c)>0$\end{document} for all \begin{document}$c≥0$\end{document}, \begin{document}$d'(c)<0$\end{document} and \begin{document}$\lim\limits_{c\to∞}d(c) = 0$\end{document}.

\begin{document}$\lim\limits_{c\to∞} \frac{χ (c)}{d(c)}$\end{document} and \begin{document}$\lim\limits_{c\to∞}\frac{d'(c)}{d(c)}$\end{document} exist.

The difficulty in analysis of system (*) is the possible degeneracy of diffusion due to the condition \begin{document}$\lim\limits_{c\to∞}d(c) = 0$\end{document}. In this paper, we will use function \begin{document}$d(c)$\end{document} as weight function and employ the method of energy estimate to establish the global existence of classical solutions of (*) with uniform-in-time bound. Furthermore, by constructing a Lyapunov functional, we show that the global classical solution \begin{document}$(n, c, u)$\end{document} will converge to the constant state \begin{document}$(\frac{a}{b}, \frac{a}{b}, 0)$\end{document} if \begin{document}$b>\frac{K_0}{16}$\end{document} with \begin{document}$K_0 = \max\limits_{0≤c ≤∞}\frac{|χ(c)|^2}{d(c)}$\end{document}.

Global dynamics in a two-species chemotaxis-competition system with two signals
Xinyu Tu, Chunlai Mu, Pan Zheng and Ke Lin
2018, 38(7): 3617-3636 doi: 10.3934/dcds.2018156 +[Abstract](4981) +[HTML](392) +[PDF](523.92KB)

In this paper, we consider a chemotaxis-competition system of parabolic-elliptic-parabolic-elliptic type

with homogeneous Neumann boundary conditions in an arbitrary smooth bounded domain \begin{document}$Ω\subset R^n$\end{document}, \begin{document}$n≥2$\end{document}, where \begin{document}$χ_{i}$\end{document}, \begin{document}$μ_{i}$\end{document} and \begin{document}$a_{i}$\end{document} \begin{document}$(i = 1, 2)$\end{document} are positive constants. It is shown that for any positive parameters \begin{document}$χ_{i}$\end{document}, \begin{document}$μ_{i}$\end{document}, \begin{document}$a_{i}$\end{document} \begin{document}$(i = 1, 2)$\end{document} and any suitably regular initial data \begin{document}$(u_{0}, w_{0})$\end{document}, this system possesses a global bounded classical solution provided that \begin{document}$\frac{χ_{i}}{μ_{i}}$\end{document} are small. Moreover, when \begin{document}$a_{1}, a_{2}∈ (0, 1)$\end{document} and the parameters \begin{document}$μ_{1}$\end{document} and \begin{document}$μ_{2}$\end{document} are sufficiently large, it is proved that the global solution \begin{document}$(u, v, w, z)$\end{document} of this system exponentially approaches to the steady state \begin{document}$\left(\frac{1-a_{1}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{2}}{1-a_{1}a_{2}}, \frac{1-a_{1}}{1-a_{1}a_{2}}\right)$\end{document} in the norm of \begin{document}$L^{∞}(Ω)$\end{document} as \begin{document}$t\to ∞$\end{document}; If \begin{document}$a_{1}≥1>a_{2}>0$\end{document} and \begin{document}$μ_{2}$\end{document} is sufficiently large, the solution of the system converges to the constant stationary solution \begin{document}$\left(0, 1, 1, 0\right)$\end{document} as time tends to infinity, and the convergence rates can be calculated accurately.

Homoclinic and stable periodic solutions for differential delay equations from physiology
Vera Ignatenko
2018, 38(7): 3637-3661 doi: 10.3934/dcds.2018157 +[Abstract](4359) +[HTML](229) +[PDF](730.1KB)

A one-parameter family of Mackey-Glass type differential delay equations is considered. The existence of a homoclinic solution for suitable parameter value is proved. As a consequence, one obtains stable periodic solutions for nearby parameter values. An example of a nonlinear functions is given, for which all sufficient conditions of our theoretical results can be verified numerically. Numerically computed solutions are shown.

Regular measurable dynamics for reaction-diffusion equations on narrow domains with rough noise
Fuzhi Li, Yangrong Li and Renhai Wang
2018, 38(7): 3663-3685 doi: 10.3934/dcds.2018158 +[Abstract](3885) +[HTML](219) +[PDF](491.0KB)

This paper is concerned with the regular random dynamics for the reaction-diffusion equation defined on a thin domain and perturbed by rough noise, where the usual Winner process is replaced by a general stochastic process satisfied the basic convergence. A bi-spatial attractor is obtained when the non-initial space is $p$-times Lebesgue space or Sobolev space. The measurability of the solution operator is proved, which leads to the measurability of the attractor in both state spaces. Finally, the upper semi-continuity of attractors under the $p$-norm is established when the narrow domain degenerates onto a lower dimensional set. Both methods of symbolical truncation and spectral decomposition provide all required uniform estimates in both Lebesgue and Sobolev spaces.

Scattering and inverse scattering for nonlinear quantum walks
Masaya Maeda, Hironobu Sasaki, Etsuo Segawa, Akito Suzuki and Kanako Suzuki
2018, 38(7): 3687-3703 doi: 10.3934/dcds.2018159 +[Abstract](5056) +[HTML](244) +[PDF](413.32KB)

We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak nonlinear regime. The proof is based on space-time estimate of (linear) QWs such as dispersive estimates and Strichartz estimate. Such argument is standard in the study of nonlinear Schrödinger equations and discrete nonlinear Schrödinger equations but it seems to be the first time to be applied to QWs.

Corrigendum to: On specification and measure expansiveness
Welington Cordeiro, Manfred Denker and Xuan Zhang
2018, 38(7): 3705-3706 doi: 10.3934/dcds.2018160 +[Abstract](3963) +[HTML](230) +[PDF](199.7KB)

We correct a flaw in the proof of Theorem D in [1].

2020 Impact Factor: 1.392
5 Year Impact Factor: 1.610
2021 CiteScore: 2.4




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