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Discrete & Continuous Dynamical Systems - B

2011 , Volume 16 , Issue 3

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Navier--Stokes equations on the $\beta$-plane
Mustafa A. H. Al-Jaboori and D. Wirosoetisno
2011, 16(3): 687-701 doi: 10.3934/dcdsb.2011.16.687 +[Abstract](455) +[PDF](448.8KB)
Abstract:
We show that, given a sufficiently regular forcing, the solution of the two-dimensional Navier--Stokes equations on the periodic $\beta$-plane (i.e. with the Coriolis force varying as $f_0+\beta y$) will become nearly zonal: with the vorticity $\omega(x,y,t)=\bar\omega(y,t)+\tilde\omega(x,y,t),$ one has $|\tilde\omega|_{H^s}^2 \le \beta^{-1} M_s(\cdots)$ as $t\to\infty$. We use this show that, for sufficiently large $\beta$, the global attractor of this system reduces to a point.
Almost periodic and asymptotically almost periodic solutions of Liénard equations
Tomás Caraballo and David Cheban
2011, 16(3): 703-717 doi: 10.3934/dcdsb.2011.16.703 +[Abstract](409) +[PDF](427.6KB)
Abstract:
The aim of this paper is to study the almost periodic and asymptotically almost periodic solutions on $(0,+\infty)$ of the Liénard equation

$ x''+f(x)x'+g(x)=F(t), $

where $F: T\to R$ ($ T= R_+$ or $R$) is an almost periodic or asymptotically almost periodic function and $g:(a,b)\to R$ is a strictly decreasing function. We study also this problem for the vectorial Liénard equation.
   We analyze this problem in the framework of general non-autonomous dynamical systems (cocycles). We apply the general results obtained in our early papers [3, 7] to prove the existence of almost periodic (almost automorphic, recurrent, pseudo recurrent) and asymptotically almost periodic (asymptotically almost automorphic, asymptotically recurrent, asymptotically pseudo recurrent) solutions of Liénard equations (both scalar and vectorial).

Dynamical behavior of a ratio dependent predator-prey system with distributed delay
Canan Çelik
2011, 16(3): 719-738 doi: 10.3934/dcdsb.2011.16.719 +[Abstract](427) +[PDF](405.9KB)
Abstract:
In this paper, we consider a predator-prey system with distributed time delay where the predator dynamics is logistic with the carrying capacity proportional to prey population. In [1] and [2], we studied the impact of the discrete time delay on the stability of the model, however in this paper, we investigate the effect of the distributed delay for the same model. By choosing the delay time $\tau $ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay time $\tau $ passes some critical values. Using normal form theory and central manifold argument, we establish the direction and the stability of Hopf bifurcation. Some numerical simulations for justifying the theoretical analysis are also presented.
Existence of radial stationary solutions for a system in combustion theory
Jérôme Coville and Juan Dávila
2011, 16(3): 739-766 doi: 10.3934/dcdsb.2011.16.739 +[Abstract](391) +[PDF](506.4KB)
Abstract:
In this paper, we construct radially symmetric solutions of a nonlinear non-cooperative elliptic system derived from a model for flame balls with radiation losses. This model is based on a one step kinetic reaction and our system is obtained by approximating the standard Arrehnius law by an ignition nonlinearity, and by simplifying the term that models radiation. We prove the existence of 2 solutions using degree theory.
Shape minimization of the dissipated energy in dyadic trees
Xavier Dubois de La Sablonière, Benjamin Mauroy and Yannick Privat
2011, 16(3): 767-799 doi: 10.3934/dcdsb.2011.16.767 +[Abstract](445) +[PDF](1298.0KB)
Abstract:
In this paper, we study the role of boundary conditions on the optimal shape of a dyadic tree in which flows a Newtonian fluid. Our optimization problem consists in finding the shape of the tree that minimizes the viscous energy dissipated by the fluid with a constrained volume, under the assumption that the total flow of the fluid is conserved throughout the structure. These hypotheses model situations where a fluid is transported from a source towards a 3D domain into which the transport network also spans. Such situations could be encountered in organs like for instance the lungs and the vascular networks.
   Two fluid regimes are studied: (i) low flow regime (Poiseuille) in trees with an arbitrary number of generations using a matricial approach and (ii) non linear flow regime (Navier-Stokes, moderate regime with a Reynolds number $100$) in trees of two generations using shape derivatives in an augmented Lagrangian algorithm coupled with a 2D/3D finite elements code to solve Navier-Stokes equations. It relies on the study of a finite dimensional optimization problem in the case (i) and on a standard shape optimization problem in the case (ii). We show that the behaviours of both regimes are very similar and that the optimal shape is highly dependent on the boundary conditions of the fluid applied at the leaves of the tree.
Thermalization time in a model of neutron star
Bernard Ducomet and Šárka Nečasová
2011, 16(3): 801-818 doi: 10.3934/dcdsb.2011.16.801 +[Abstract](377) +[PDF](440.1KB)
Abstract:
We consider an initial boundary value problem for the equation describing heat conduction in a spherical model of neutron star considered by Lattimer et al. We estimate the asymptotic decay of the solution, which provides a plausible estimate for a "thermalization time" for the system.
Front propagation in diffusion-aggregation models with bi-stable reaction
Mikhail Kuzmin and Stefano Ruggerini
2011, 16(3): 819-833 doi: 10.3934/dcdsb.2011.16.819 +[Abstract](459) +[PDF](410.2KB)
Abstract:
In this paper, necessary and sufficient conditions are given for the existence of travelling wave solutions of the reaction-diffusion-aggregation equation

$v_\tau=(D(v)v_x)_{x}+f(v), $

where the diffusivity $D$ changes sign twice in the interval $(0,1)$ (from positive to negative and again to positive) and the reaction $f$ is bi-stable. We show that classical travelling waves with decreasing profile do exist for a single admissible value of their speed of propagation which can be either positive or negative, according to the behavior of $f$ and $D$. An example is given, illustrating the employed techniques. The results are then generalized to a diffusivity $D$ with $2n$ sign changes.

Robustness of signaling gradient in drosophila wing imaginal disc
Jinzhi Lei, Frederic Y. M. Wan, Arthur D. Lander and Qing Nie
2011, 16(3): 835-866 doi: 10.3934/dcdsb.2011.16.835 +[Abstract](496) +[PDF](860.6KB)
Abstract:
Quasi-stable gradients of signaling protein molecules (known as morphogens or ligands) bound to cell receptors are known to be responsible for differential cell signaling and gene expressions. From these follow different stable cell fates and visually patterned tissues in biological development. Recent studies have shown that the relevant basic biological processes yield gradients that are sensitive to small changes in system characteristics (such as expression level of morphogens or receptors) or environmental conditions (such as temperature changes). Additional biological activities must play an important role in the high level of robustness observed in embryonic patterning for example. It is natural to attribute observed robustness to various type of feedback control mechanisms. However, our own simulation studies have shown that feedback control is neither necessary nor sufficient for robustness of the morphogen decapentaplegic (Dpp) gradient in wing imaginal disc of Drosophilas. Furthermore, robustness can be achieved by substantial binding of the signaling morphogen Dpp with nonsignaling cell surface bound molecules (such as heparan sulfate proteoglygans) and degrading the resulting complexes at a sufficiently rapid rate. The present work provides a theoretical basis for the results of our numerical simulation studies.
Long time behavior of some epidemic models
Fang Li and Nung Kwan Yip
2011, 16(3): 867-881 doi: 10.3934/dcdsb.2011.16.867 +[Abstract](432) +[PDF](336.1KB)
Abstract:
In this paper, we prove two results concerning the long time behavior of two systems of reaction diffusion equations motivated by the S-I-R model in epidemic modeling. The results generalize and simplify previous approaches. In particular, we consider the presence of directed diffusions between the two species. The new system contains an ill-posed region for arbitrary parameters. Our result is established under the assumption of small initial data.
Phase transitions of a phase field model
Honghu Liu
2011, 16(3): 883-894 doi: 10.3934/dcdsb.2011.16.883 +[Abstract](525) +[PDF](462.5KB)
Abstract:
We consider a phase field model for the mixture of two viscous incompressible uids with the same density. The model leads to a coupled Navier-Stokes/Cahn-Hilliard system. We explore the dynamics of the system near the critical point via a dynamic phase transition theory developed recently by Ma and Wang [7, 8]. Our analysis shows qualitatively the same phase transition result as the purely dissipative Cahn-Hilliard equation, which implies that the hydrodynamics does not play a role in the phase transition process of binary systems. This is different from the sharp interface situation, where numerical studies (see e.g. [3, 6]) suggest quite different behaviors between these two models.
Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology
Judith R. Miller and Huihui Zeng
2011, 16(3): 895-925 doi: 10.3934/dcdsb.2011.16.895 +[Abstract](497) +[PDF](416.9KB)
Abstract:
We use spectral methods to prove a general stability theorem for traveling wave solutions to the systems of integrodifference equations arising in spatial population biology. We show that non-minimum-speed waves are exponentially asymptotically stable to small perturbations in appropriately weighted $L^\infty$ spaces, under assumptions which apply to examples including a Laplace or Gaussian dispersal kernel a monotone (or non-monotone) growth function behaving qualitatively like the Beverton-Holt function (or Ricker function with overcompensation), and a constant probability $p\in [0,1)$ (or $p=0$) of remaining sedentary for a single population; as well as to a system of two populations exhibiting non-cooperation (in particular, Hassell and Comins' model [6]) with $p=0$ and Laplace or Gaussian dispersal kernels which can be different for the two populations.
The logistic map of matrices
Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras and Minvydas Ragulskis
2011, 16(3): 927-944 doi: 10.3934/dcdsb.2011.16.927 +[Abstract](524) +[PDF](1018.6KB)
Abstract:
The standard iterative logistic map is extended by replacing the scalar variable by a square matrix of variables. Dynamical properties of such an iterative map are explored in detail when the order of matrices is 2. It is shown that the evolution of the logistic map depends not only on the control parameter but also on the eigenvalues of the matrix of initial conditions. Several computational examples are used to demonstrate the convergence to periodic attractors and the sensitivity of chaotic processes to initials conditions.
Tikhonov's theorem and quasi-steady state
Lena Noethen and Sebastian Walcher
2011, 16(3): 945-961 doi: 10.3934/dcdsb.2011.16.945 +[Abstract](386) +[PDF](362.4KB)
Abstract:
There exists a systematic approach to asymptotic properties for quasi-steady state phenomena via the classical theory of Tikhonov and Fenichel. This observation allows, on the one hand, to settle convergence issues, which are far from trivial in asymptotic expansions. On the other hand, even if one takes convergence for granted, the approach yields a natural way to compute a reduced system on the slow manifold, with a reduced equation that is frequently simpler than the one obtained by the ad hoc approach. In particular, the reduced system is always rational. The paper includes a discussion of necessary and sufficient conditions for applicability of Tikhonov's and Fenichel's theorems, computational issues and a direct determination of the reduced system. The results are applied to several relevant examples.
The flashing ratchet and unidirectional transport of matter
Dmitry Vorotnikov
2011, 16(3): 963-971 doi: 10.3934/dcdsb.2011.16.963 +[Abstract](335) +[PDF](309.4KB)
Abstract:
We study the flashing ratchet model of a Brownian motor, which consists in cyclical switching between the Fokker-Planck equation with an asymmetric ratchet-like potential and the pure diffusion equation. We show that the motor indeed performs unidirectional transport of mass, for proper parameters of the model, by analyzing the attractor of the problem and the stationary vector of a related Markov chain.
Existence of traveling wavefront for discrete bistable competition model
Chin-Chin Wu
2011, 16(3): 973-984 doi: 10.3934/dcdsb.2011.16.973 +[Abstract](376) +[PDF](333.4KB)
Abstract:
We study traveling wavefront solutions for a two-component competition system on a one-dimensional lattice. We combine the monotonic iteration method with a truncation to obtain the existence of the traveling wavefront solution.
Attractors for autonomous and nonautonomous 3D Navier-Stokes-Voight equations
Gaocheng Yue and Chengkui Zhong
2011, 16(3): 985-1002 doi: 10.3934/dcdsb.2011.16.985 +[Abstract](453) +[PDF](457.0KB)
Abstract:
In this paper we study the long time behavior of the three dimensional Navier-Stokes-Voight model of viscoelastic incompressible fluid for the autonomous and nonautonomous cases. A useful decomposition method is introduced to overcome the difficulties in proving the asymptotical regularity of the 3D Navier-Stokes-Voight equations. For the autonomous case, we prove the existence of global attractor when the external forcing belongs to $V'.$ For the nonautonomous case, we only assume that $f(x,t)$ is translation bounded instead of translation compact, where $f=Pg$ and $P$ is the Helmholz-Leray orthogonal projection. By means of this useful decomposition methods, we prove the asymptotic regularity of solutions of 3D Navier-Stokes-Voight equations and also obtain the existence of the uniform attractor. Finally, we describe the structure of the uniform attractor and its regularity.
Influence of neurobiological mechanisms on speeds of traveling wave fronts in mathematical neuroscience
Linghai Zhang, Ping-Shi Wu and Melissa Anne Stoner
2011, 16(3): 1003-1037 doi: 10.3934/dcdsb.2011.16.1003 +[Abstract](383) +[PDF](1670.2KB)
Abstract:
We study speeds of traveling wave fronts of the following integral differential equation

$ \frac{\partial u}{\partial t}+f(u)\hspace{6cm} $

$=(\alpha-au)\int^{\infty}_0\xi(c)[\int_R K(x-y) H(u(y,t-\frac{1}{c}|x-y|)-\theta)dy]dc $

$ +(\beta-bu)\int^{\infty}_0\eta(\tau)[\int_RW(x-y) H(u(y,t-\tau)-\Theta)dy]d\tau. $

This model equation is motivated by previous models which arise from synaptically coupled neuronal networks. In this equation, $f(u)$ is a smooth function of $u$, usually representing sodium current in the neuronal networks. Typical examples include $f(u)=u$ and $f(u)=u(u-1)(Du-1)$, where $D>1$ is a constant. The transmission speed distribution $\xi$ and the feedback delay distribution $\eta$ are probability density functions. The kernel functions $K$ and $W$ represent synaptic couplings between neurons in the neuronal networks. The function $H$ stands for the Heaviside step function: $H(u-\theta)=0$ for all $u<\theta$, $H(0)=\frac{1}{2}$ and $H(u-\theta)=1$ for all $u>\theta$. Here $H$ represents the gain function. The parameters $a \geq 0$, $b \geq 0$, $ \alpha \geq 0$, $\beta \geq 0$, $\theta > 0$ and $\Theta > 0$ represent biological mechanisms in the neuronal networks.
We will use mathematical analysis to investigate the influence of neurobiological mechanisms on the speeds of the traveling wave fronts. We will derive new estimates for the wave speeds. These results are quite different from the results obtained before, complementing the estimates obtained in many previous papers [11], [14], [15], and [16].
We will also use MATLAB to perform numerical simulations to investigate how the neurobiological mechanisms $a$, $b$, $\alpha$, $\beta$, $\theta$ and $\Theta$ influence the wave speeds.

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