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Discrete & Continuous Dynamical Systems - B

2013 , Volume 18 , Issue 1

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Second order corrector in the homogenization of a conductive-radiative heat transfer problem
Grégoire Allaire and  Zakaria Habibi
2013, 18(1): 1-36 doi: 10.3934/dcdsb.2013.18.1 +[Abstract](102) +[PDF](1223.1KB)
This paper focuses on the contribution of the so-called second order corrector in periodic homogenization applied to a conductive-radiative heat transfer problem. More precisely, heat is diffusing in a periodically perforated domain with a non-local boundary condition modelling the radiative transfer in each hole. If the source term is a periodically oscillating function (which is the case in our application to nuclear reactor physics), a strong gradient of the temperature takes place in each periodicity cell, corresponding to a large heat flux between the sources and the perforations. This effect cannot be taken into account by the homogenized model, neither by the first order corrector. We show that this local gradient effect can be reproduced if the second order corrector is added to the reconstructed solution.
The basic reproduction number of discrete SIR and SEIS models with periodic parameters
Hui Cao and  Yicang Zhou
2013, 18(1): 37-56 doi: 10.3934/dcdsb.2013.18.37 +[Abstract](138) +[PDF](436.9KB)
Seasonal fluctuations have been observed in many infectious diseases. Discrete epidemic models with periodic epidemiological parameters are formulated and studied to take into account seasonal variations of infectious diseases. The definition and the formula of the basic reproduction number $R_0$ are given by following the framework in [1,2,3,4,5]. Threshold results for a general model are obtained which show that the magnitude of $R_0$ determines whether the disease will go extinct (when $R_0<1$) or not (when $R_0>1$) in the population. Applications of these general results to discrete periodic SIR and SEIS models are demonstrated. The disease persistence and the existence of the positive periodic solution are established. Numerical explorations of the model properties are also presented via several examples including the calculations of the basic reproduction number, conditions for the disease extinction or persistence, and the existence of periodic solutions as well as its stability.
Weak KAM theory for nonregular commuting Hamiltonians
Andrea Davini and  Maxime Zavidovique
2013, 18(1): 57-94 doi: 10.3934/dcdsb.2013.18.57 +[Abstract](35) +[PDF](573.7KB)
In this paper we consider the notion of commutation for a pair of continuous and convex Hamiltonians, given in terms of commutation of their Lax--Oleinik semigroups. This is equivalent to the solvability of an associated multi--time Hamilton--Jacobi equation. We examine the weak KAM theoretic aspects of the commutation property and show that the two Hamiltonians have the same weak KAM solutions and the same Aubry set, thus generalizing a result recently obtained by the second author for Tonelli Hamiltonians. We make a further step by proving that the Hamiltonians admit a common critical subsolution, strict outside their Aubry set. This subsolution can be taken of class $C^{1,1}$ in the Tonelli case. To prove our main results in full generality, it is crucial to establish suitable differentiability properties of the critical subsolutions on the Aubry set. These latter results are new in the purely continuous case and of independent interest.
Slow passage through multiple bifurcation points
Younghae Do and  Juan M. Lopez
2013, 18(1): 95-107 doi: 10.3934/dcdsb.2013.18.95 +[Abstract](55) +[PDF](546.7KB)
The slow passage problem, the slow variation of a control parameter, is explored in a model problem that posses several co-existing equilibria (fixed points, limit cycles and 2-tori), and these are either created or destroyed or change their stability as control parameters are varied through Hopf, Neimark-Sacker and torus break-up bifurcations. The slow passage through the Hopf bifurcation behaves as determined in previous studies (the delay in the observation of oscillations depends only on how far from critical the ramped parameter is at the start of the ramp--a memory effect), and that through the Neimark-Sacker bifurcation also behaves similarly. We show that the range of the ramped parameter over which a Hopf oscillation can be observed (limited by the subsequent onset of torus oscillations from the Neimark-Sacker bifurcation) is twice that predicted from a static-parameter bifurcation analysis, and this is a memory-less result independent of the initial value of the ramped parameter. These delay and memory effects are independent of the ramp rate, for small enough ramp rates. The slow passage through the torus break-up bifurcation is qualitatively different. It does not depend on the initial value of the ramped parameter, but instead is found to depend, on average, on the square-root of the ramp rate. This is typical of transient behavior. We show that this transient behavior is due to the stable and unstable manifolds of the saddle limit cycles forming a very narrow escape tunnel for trajectories originating near the unstable 2-torus no matter how slow a ramp speed is used. The type of bifurcation sequence in the model problem studied (Hopf, Neimark-Sacker, torus break-up) is typical of those for the transition to spatio-temporal chaos in hydrodynamic problems, and in those physical problems the transition can occur over a very small range of the control parameter, and so the inevitable slow drift of the parameter in an experiment may lead to observations where the slow passage results reported here need to be taken into account.
Stability results for a size-structured population model with delayed birth process
Xianlong Fu and  Dongmei Zhu
2013, 18(1): 109-131 doi: 10.3934/dcdsb.2013.18.109 +[Abstract](43) +[PDF](525.2KB)
In this paper, we discuss the qualitative behavior of an age/size-structured population equation with delay in the birth process. The linearization about stationary solutions is analyzed by semigroup and spectral methods. In particular, the spectrally determined growth property of the linearized semigroup is derived from its long-term regularity. These analytical results allow us to derive linearized stability and instability results under some conditions. The principal stability criterions are given in terms of a modified net reproduction rate. Finally, two examples are presented and simulated to illustrated the obtained conclusions.
Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria
Fengjie Geng , Junfang Zhao , Deming Zhu and  Weipeng Zhang
2013, 18(1): 133-145 doi: 10.3934/dcdsb.2013.18.133 +[Abstract](28) +[PDF](436.5KB)
In this paper, using the local moving frame approach, we investigate bifurcations of nongeneric heteroclinic loop with a nonhyperbolic equilibrium $p_1$ and a hyperbolic saddle $p_2$, where $p_1$ is assumed to undergo a transcritical bifurcation. Firstly, we establish the persistence of a nongeneric heteroclinic loop, the existence of a homoclinic loop and a periodic orbit when the transcritical bifurcation does not occur. Secondly, bifurcations of a nongeneric heteroclinic loop accompanied with a transcritical bifurcation are discussed. We obtain the existence of heteroclinic orbits, a homoclinic loop, a heteroclinic loop and a periodic orbit. Some bifurcation patterns different from the case of the generic heteroclinic loop accompanied with transcritical bifurcation are revealed. The results achieved here can be extended to higher dimensional systems.
Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation
Tzy-Wei Hwang and  Feng-Bin Wang
2013, 18(1): 147-161 doi: 10.3934/dcdsb.2013.18.147 +[Abstract](43) +[PDF](419.0KB)
Dengue fever is a virus-caused disease in the world. Since the high infection rate of dengue fever and high death rate of its severe form dengue hemorrhagic fever, the control of the spread of the disease is an important issue in the public health. In an effort to understand the dynamics of the spread of the disease, Esteva and Vargas [2] proposed a SIR v.s. SI epidemiological model without crowding effect and spatial heterogeneity. They found a threshold parameter $R_0,$ if $R_0<1,$ then the disease will die out; if $R_0>1,$ then the disease will always exist.
    To investigate how the spatial heterogeneity and crowding effect influence the dynamics of the spread of the disease, we modify the autonomous system provided in [2] to obtain a reaction-diffusion system. We first define the basic reproduction number in an abstract way and then employ the comparison theorem and the theory of uniform persistence to study the global dynamics of the modified system. Basically, we show that the basic reproduction number is a threshold parameter that predicts whether the disease will die out or persist. Further, we demonstrate the basic reproduction number in an explicit way and construct suitable Lyapunov functionals to determine the global stability for the special case where coefficients are all constant.
Exact travelling wave solutions and their dynamical behavior for a class coupled nonlinear wave equations
Jibin Li and  Fengjuan Chen
2013, 18(1): 163-172 doi: 10.3934/dcdsb.2013.18.163 +[Abstract](25) +[PDF](162.8KB)
For the system of KP like equation coupled to a Schrödinger equation, a corresponding four-dimensional travelling wave systems and a two-order linear non-autonomous system are studied by using Congrove's results and dynamical system method. For the four-dimensional travelling wave systems, exact explicit homoclinic orbit families, periodic and quasi-periodic wave solution families are obtained. The existence of homoclinic manifolds to four kinds of equilibria including a hyperbolic equilibrium, a center-saddle and an equilibrium with zero pair of eigenvalues is revealed. For the two-order linear non-autonomous system, the dynamical behavior of the bounded solutions is discussed.
A note on the global stability of an SEIR epidemic model with constant latency time and infectious period
Yoshiaki Muroya , Yoichi Enatsu and  Huaixing Li
2013, 18(1): 173-183 doi: 10.3934/dcdsb.2013.18.173 +[Abstract](29) +[PDF](359.3KB)
In this note, under the condition for the permanence used by [Beretta and Breda, An SEIR epidemic model with constant latency time and infectious period, Math. Biosci. Eng. 8 (2011) 931-952], applying modified monotone sequences, we establish the global asymptotic stability of the endemic equilibrium of this SEIR epidemic model, without any other additional conditions on the global stability.
Direct exponential ordering for neutral compartmental systems with non-autonomous $\mathbf{D}$-operator
Rafael Obaya and  Víctor M. Villarragut
2013, 18(1): 185-207 doi: 10.3934/dcdsb.2013.18.185 +[Abstract](26) +[PDF](474.1KB)
We study closed compartmental systems described by neutral functional differential equations with non-autonomous stable $D$-operator which are monotone for the direct exponential ordering. Under some appropriate conditions on the induced semiflow including uniform stability for the exponential order and the differentiability of the $D$-operator along the base flow, we establish the 1-covering property of omega-limit sets, in order to describe the long-term behavior of the trajectories.
Time dependent perturbation in a non-autonomous non-classical parabolic equation
Felipe Rivero
2013, 18(1): 209-221 doi: 10.3934/dcdsb.2013.18.209 +[Abstract](26) +[PDF](420.8KB)
n this paper we study the existence and characterization of a pullback attractor for a non-autonomous non-classical parabolic equation of the form \begin{equation}\label{EQnoncla} \left\{ \begin{split} &u_t-\gamma(t)\Delta u_t-\Delta u=f(u) \mbox{ in }\Omega,\\ &u=0 \mbox{ on }\partial\Omega \end{split} \right. (1) \end{equation} in a sufficiently smooth bounded domain $\Omega\subset\mathbb R^n$ with $f$ and $\gamma$ satisfying some suitable natural conditions. We prove the well posedness of this model and the existence of a pullback attractor. We show that this pullback attractor is characterized as the union of unstable sets of the associated equilibria and that this characterization is stable under time dependent perturbation of the nonlinearity.
$\omega$-limit sets for porous medium equation with initial data in some weighted spaces
Liangwei Wang , Jingxue Yin and  Chunhua Jin
2013, 18(1): 223-236 doi: 10.3934/dcdsb.2013.18.223 +[Abstract](39) +[PDF](397.2KB)
We discuss the $\omega$-limit set for the Cauchy problem of the porous medium equation with initial data in some weighted spaces. Exactly, we show that there exists some relationship between the $\omega$-limit set of the rescaled initial data and the $\omega$-limit set of the spatially rescaled version of solutions. We also give some applications of such a relationship.
On the multiple spike solutions for singularly perturbed elliptic systems
Weichung Wang , Tsung-Fang Wu and  Chien-Hsiang Liu
2013, 18(1): 237-258 doi: 10.3934/dcdsb.2013.18.237 +[Abstract](110) +[PDF](776.3KB)
We study the multiplicity of positive solutions for the two coupled nonlinear Schrödinger equations in bounded domains in this paper. By using Nehari manifold and Lusternik-Schnirelmann category, we prove the existence of multiple positive solutions for two coupled nonlinear Schrödinger equations in bounded domains. We also propose a numerical scheme that leads to various new numerical predictions regarding the solution characteristics.
Global dynamics and bifurcations in a four-dimensional replicator system
Yuanshi Wang , Hong Wu and  Shigui Ruan
2013, 18(1): 259-271 doi: 10.3934/dcdsb.2013.18.259 +[Abstract](27) +[PDF](557.6KB)
In this paper, the four-dimensional cyclic replicator system $\dot{u}_i = {u}_i [-(Bu)_i + \sum_{j=1}^{4} u_j (Bu)_j ],1\le i \le 4$, with $b_1 = b_3$ is considered, in which the first row of the matrix $B$ is $(0~ b_1~ b_2~ b_3)$ and the other rows of $B$ are cyclic permutations of the first row. Our aim is to study the global dynamics and bifurcations in the system, and to show how and when all but one species go to extinction. By reducing the four-dimensional system to a three-dimensional one, we show that there is no periodic orbit in the system. For the case $b_1 b_2 < 0$, we give complete analysis on the global dynamics. For the case $b_1 b_2 \ge 0$, we extend some results obtained by Diekmann and van Gils (2009). By combining our work with that in Diekmann and van Gils (2009), we present the dynamics and bifurcations of the system on the whole $(b_1, b_2)$-plane. The analysis leads to explanations for the phenomena that in some semelparous species, all but one brood go extinct.
On the spectrum of the superposition of separated potentials.
J. Douglas Wright
2013, 18(1): 273-281 doi: 10.3934/dcdsb.2013.18.273 +[Abstract](28) +[PDF](331.8KB)
Suppose that $V(x)$ is an exponentially localized potential and $L$ is a constant coefficient differential operator. A method for computing the spectrum of $L+V(x-x_1) + ... + V(x-x_N)$ given that one knows the spectrum of $L+V(x)$ is described. The method is functional theoretic in nature and does not rely heavily on any special structure of $L$ or $V$ apart from the exponential localization. The result is aimed at applications involving the existence and stability of multi-pulses in partial differential equations.

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