# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

January 2011 , Volume 15 , Issue 1

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2011, 15(1): 1-14 doi: 10.3934/dcdsb.2011.15.1 +[Abstract](1209) +[PDF](266.9KB)
Abstract:
A bacillary dysentery model with seasonal fluctuation is formulated and studied. The basic reproductive number $\mathcal {R}_0$ is introduced to investigate the disease dynamics in seasonal fluctuation environments. It is shown that there exists only the disease-free periodic solution which is globally asymptotically stable if $\mathcal {R}_0<1$, and there exists a positive periodic solution if $\mathcal {R}_0>1$. $\mathcal {R}_0$ is a threshold parameter, its magnitude determines the extinction or the persistence of the disease. Parameters in the model are estimated on the basis of bacillary dysentery epidemic data. Numerical simulations have been carried out to describe the transmission process of bacillary dysentery in China.
2011, 15(1): 15-43 doi: 10.3934/dcdsb.2011.15.15 +[Abstract](1419) +[PDF](368.4KB)
Abstract:
We consider a system of partial differential equations which describes anti-plane shear in the context of a strain-gradient theory of plasticity proposed by Gurtin in [6]. The problem couples a fully nonlinear degenerate parabolic system and an elliptic equation. It features two types of degeneracies: the first one is caused by the nonlinear structure, the second one by the dependence of the principal part on twice the curl of a planar vector field. Furthermore, the elliptic equation depends on the divergence of such vector field - which is not controlled by twice the curl - and the boundary conditions suggested in [6] are of mixed type.
To overcome the latter complications we use a suitable, time-dependent representation of a divergence-free vector field which plays the role of the elastic stress. To handle the nonlinearities, by a suitable reformulation of the problem we transform the original system into one satisfying a monotonicity property which is more "robust" than the gradient flow structure inherited as an intrinsic feature of the mechanical model. These two insights make it possible to prove existence and uniqueness of a solution to the original system.
2011, 15(1): 45-60 doi: 10.3934/dcdsb.2011.15.45 +[Abstract](1097) +[PDF](303.5KB)
Abstract:
We investigate the permeation flow of cholesteric liquid crystal polymers (CLCPs) subject to a small amplitude oscillatory shear using a tensor theory developed by the authors [8]. We model the material system by the Stokes hydrodynamic equations coupled with the orientational dynamics. At low frequencies, the steady permeation modes are recovered and the director rotates in phase with the applied shear. At high frequencies, the out of phase component dominates the dynamics. The asymptotic formulas for the loss modulus ($G''$) and storage modulus ($G^{'}$) are obtained at both low and high frequencies. In the low frequency limit, both the loss modulus and the storage modulus are shown to exhibit a classical frequency $\omega$ dependence ($G^{''} \propto \omega$, $G^{'} \propto \omega^2$ ) with the proportionality of order $O(Er)$ and $O(q)$, respectively, where $\frac{2\pi}{q}$ defines the pitch of the chiral liquid crystal and $Er$ is the Ericksen number of the liquid crystal polymer system. The magnitudes of dimensionless complex flow rate and complex viscosity are calculated. They are shown to have two Newtonian plateaus at low and high frequencies while a power-law response at intermediate frequencies.
2011, 15(1): 61-74 doi: 10.3934/dcdsb.2011.15.61 +[Abstract](1544) +[PDF](194.0KB)
Abstract:
In this paper, we establish the global asymptotic stability of equilibria for an SIR model of infectious diseases with distributed time delays governed by a wide class of nonlinear incidence rates. We obtain the global properties of the model by proving the permanence and constructing a suitable Lyapunov functional. Under some suitable assumptions on the nonlinear term in the incidence rate, the global dynamics of the model is completely determined by the basic reproduction number $R_0$ and the distributed delays do not influence the global dynamics of the model.
2011, 15(1): 75-92 doi: 10.3934/dcdsb.2011.15.75 +[Abstract](1448) +[PDF](784.3KB)
Abstract:
We study numerically the solutions of the steady advection-diffu-sion problem in bounded domains with prescribed boundary conditions when the Péclet number Pe is large. We approximate the solution at high, but finite Péclet numbers by the solution to a certain asymptotic problem in the limit Pe $\to \infty$. The asymptotic problem is a system of coupled 1-dimensional heat equations on the graph of streamline-separatrices of the cellular flow, that was developed in [21]. This asymptotic model is implemented numerically using a finite volume scheme with exponential grids. We conclude that the asymptotic model provides for a good approximation of the solutions of the steady advection-diffusion problem at large Péclet numbers, and even when Pe is not too large.
2011, 15(1): 93-112 doi: 10.3934/dcdsb.2011.15.93 +[Abstract](1505) +[PDF](1841.6KB)
Abstract:
The main purpose of this paper is to explore the dynamics of an epidemic model with a general nonlinear incidence $\beta SI^p/(1+\alpha I^q)$. The existence and stability of multiple endemic equilibria of the epidemic model are analyzed. Local bifurcation theory is applied to explore the rich dynamical behavior of the model. Normal forms of the model are derived for different types of bifurcations, including Hopf and Bogdanov-Takens bifurcations. Concretely speaking, the first Lyapunov coefficient is computed to determine various types of Hopf bifurcations. Next, with the help of the Bogdanov-Takens normal form, a family of homoclinic orbits is arising when a Hopf and a saddle-node bifurcation merge. Finally, some numerical results and simulations are presented to illustrate these theoretical results.
2011, 15(1): 113-135 doi: 10.3934/dcdsb.2011.15.113 +[Abstract](1073) +[PDF](3081.3KB)
Abstract:
We investigate a stage-structured model of an iteroparous population with two age classes. The population is assumed to exhibit Allee effects through reproduction. The asymptotic dynamics of the model depend on the maximal reproductive number of the population. The population may persist if the maximal reproductive number is greater than one. There exists a population threshold in terms of the unstable interior equilibrium. The host population will become extinct if its initial distribution lies below the threshold and the host population can persist indefinitely if its initial distribution lies above the threshold. In addition, if the unstable equilibrium is a saddle point and the system has no $2$-cycles, then the stable manifold of the saddle point provides the Allee threshold for the host. Based on this host population system, we construct a host-parasitoid model to study the impact of Allee effects upon the population interaction. The parasitoid population may drive the host to below the Allee threshold so that both populations become extinct. On the other hand, under some conditions on the parameters, the host-parasitoid system may possess an interior equilibrium and the populations may coexist as an interior equilibrium.
2011, 15(1): 137-170 doi: 10.3934/dcdsb.2011.15.137 +[Abstract](1415) +[PDF](295.4KB)
Abstract:
Rayleigh's problem of an infinite flat plate set into uniform motion impulsively in its own plane is studied by using the BKW model, the linearized Boltzmann equation and the full Boltzmann equation, respectively. The purpose is to study the gas motion under the diffuse reflection boundary condition. For a small impulsive velocity (small Mach number) and short time, the flow behaves like a free molecule flow. Our analysis is based on certain pointwise estimates for the solution of the problem and flow velocity.
2011, 15(1): 171-196 doi: 10.3934/dcdsb.2011.15.171 +[Abstract](1451) +[PDF](312.8KB)
Abstract:
Using a new method of monotone iteration of a pair of smooth lower- and upper-solutions, the traveling wave solutions of the classical Lotka-Volterra system are shown to exist for a family of wave speeds. Such constructed upper and lower solution pair enables us to derive the explicit value of the minimal (critical) wave speed as well as the asymptotic decay/growth rates of the wave solutions at infinities. Furthermore, the traveling wave corresponding to each wave speed is unique up to a translation of the origin. The stability of the traveling wave solutions with non-critical wave speed is also studied by spectral analysis of a linearized operator in exponentially weighted Banach spaces.
2011, 15(1): 197-215 doi: 10.3934/dcdsb.2011.15.197 +[Abstract](1254) +[PDF](288.1KB)
Abstract:
This article deals with the tracking control of periodic references in single-input single-output bilinear systems using a stable inversion-based approach. Assuming solvability of the exact tracking problem and asymptotic stability of the nominal error system, the study focuses on the output behavior when the control scheme uses a periodic approximation of the nominal feedforward input signal $u_d$. The investigation shows that this results in a periodic, asymptotically stable output; moreover, a sequence of periodic control inputs $u_n$ uniformly convergent to $u_d$ produce a sequence of output responses that, in turn, converge uniformly to the output reference. It is also shown that, for a special class of bilinear systems, the internal dynamics equation can be approximately solved by an iterative procedure that provides of closed-form analytic expressions uniformly convergent to its exact solution. Then, robustness in the face of bounded parametric disturbances/uncertainties is achievable through dynamic compensation. The theoretical analysis is applied to nonminimum phase switched power converters.
2011, 15(1): 217-230 doi: 10.3934/dcdsb.2011.15.217 +[Abstract](1337) +[PDF](3367.7KB)
Abstract:
This paper continues the analysis on a bimolecular autocatalytic reaction-diffusion model with saturation law. An improved result of steady state bifurcation is derived and the effect of various parameters on spatiotemporal patterns is discussed. Our analysis provides a better understanding on the rich spatiotemporal patterns. Some numerical simulations are performed to support the theoretical conclusions.
2011, 15(1): 231-254 doi: 10.3934/dcdsb.2011.15.231 +[Abstract](1412) +[PDF](478.7KB)
Abstract:
A semi-analytical procedure for studying stability and bifurcations of limit cycles in higher-dimensional nonlinear autonomous dynamical systems is developed. This procedure is based mainly on the incremental harmonic balance (IHB) method. It is composed of three key steps, namely, the determination of limit cycles by IHB method, the calculation of transition matrix by precise integration (PI) algorithm and the discrimination of limit cycle stability by Floquet theory. As an application, the procedure is used to investigate the dynamics of the limit cycle of a three-dimensional nonlinear autonomous system. The symmetry-breaking bifurcation, the first and the second period-doubling bifurcations of the limit cycle are identified. The critical parameter values corresponding to these bifurcations are calculated. The phase portraits and bifurcation points agree well with those of direct numerical integrations by using Runge-Kutta method.
2011, 15(1): 255-271 doi: 10.3934/dcdsb.2011.15.255 +[Abstract](1374) +[PDF](219.9KB)
Abstract:
The existence and multiplicity of homoclinic orbits for a class of the second order Hamiltonian systems $\ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0, \ \forall t \in \mathbb{R}$, are obtained via the concentration-compactness principle and the fountain theorem respectively, where $W(t, x)$ is superquadratic and need not satisfy the (AR) condition with respect to the second variable $x\in\mathbb{R}^{N}$.
2011, 15(1): 273-291 doi: 10.3934/dcdsb.2011.15.273 +[Abstract](1197) +[PDF](579.4KB)
Abstract:
In this paper, a predator-prey model with stage structure for the predator and a spatio-temporal delay describing the gestation period of the predator under homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states is established. Sufficient conditions are derived for the global attractiveness of the positive steady state and the global stability of the semi-trivial steady state of the proposed problem by using the method of upper-lower solutions and its associated monotone iteration scheme. Numerical simulations are carried out to illustrate the main results.
2011, 15(1): 293-308 doi: 10.3934/dcdsb.2011.15.293 +[Abstract](1335) +[PDF](214.4KB)
Abstract:
In this paper we study a delayed free boundary problem for the growth of tumors. The establishment of the model is based on the diffusion of nutrient and mass conservation for the two process proliferation and apoptosis(cell death due to aging). It is assumed the process of proliferation is delayed compared to apoptosis. By $L^p$ theory of parabolic equations and the Banach fixed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the asymptotic behavior of the solution, and prove that in the case $c$ is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to a dormant state as $t\rightarrow\infty.$
2011, 15(1): 309-323 doi: 10.3934/dcdsb.2011.15.309 +[Abstract](1478) +[PDF](237.2KB)
Abstract:
This article is concerned with the existence of traveling wave solutions, including standing waves, to some models based on configurational forces, describing respectively the diffusionless phase transitions of solid materials, e.g., Steel, and phase transitions due to interface motion by interface diffusion, e.g., Sintering. These models were proposed by Alber and Zhu in [3]. We consider both the order-parameter-conserved case and the non-conserved one, under suitable assumptions. Also we compare our results with the corresponding ones for the Allen-Cahn and the Cahn-Hilliard equations coupled with linear elasticity, which are models for diffusion-dominated phase transitions in elastic solids.

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