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1531-3492

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## Discrete & Continuous Dynamical Systems - B

March 2009 , Volume 11 , Issue 2

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2009, 11(2): 205-231
doi: 10.3934/dcdsb.2009.11.205

*+*[Abstract](2047)*+*[PDF](323.4KB)**Abstract:**

In this paper we prove the existence of a global attractor with respect to the weak topology of a suitable Banach space for a parabolic scalar differential equation describing a non-Newtonian flow. More precisely, we study a model proposed by Hébraud and Lequeux for concentrated suspensions.

2009, 11(2): 233-262
doi: 10.3934/dcdsb.2009.11.233

*+*[Abstract](2198)*+*[PDF](778.8KB)**Abstract:**

Designing trajectories for a submerged rigid body motivates this paper. Two approaches are addressed: the time optimal approach and the motion planning approach using a concatenation of kinematic motions. We focus on the structure of singular extremals and their relation to the existence of rank-one kinematic reductions; thereby linking the optimization problem to the inherent geometric framework. Using these kinematic reductions, we provide a solution to the motion planning problem in the under-actuated scenario, or equivalently, in the case of actuator failures. We finish the paper comparing a time optimal trajectory to one formed by a concatenation of pure motions.

2009, 11(2): 263-282
doi: 10.3934/dcdsb.2009.11.263

*+*[Abstract](1753)*+*[PDF](286.4KB)**Abstract:**

In this paper we construct and study discretizations of an extension of the Zakharov system occurring in plasma physics. This system is intermediate between Euler-Maxwell and Zakharov systems. The usual Zakharov system can be recovered by taking a singular limit. We introduce two numerical schemes that take into account this singular limit process and that are asymptotic preserving. We prove some stability and convergence results and we perform some numerical tests showing that the range of validity of the extended system is wider than that of the usual Zakharov system.

2009, 11(2): 283-314
doi: 10.3934/dcdsb.2009.11.283

*+*[Abstract](1888)*+*[PDF](409.1KB)**Abstract:**

In this paper we study an optimal boundary control problem for the 3D steady-state Navier-Stokes equation in a cylindrically perforated domain $\Omega_{\epsilon}$. The control is the boundary velocity field supported on the 'vertical' sides of thin cylinders. We minimize the vorticity of viscous flow through thick perforated domain. We show that an optimal solution to some limit problem in a non-perforated domain can be used as basis for the construction of suboptimal controls for the original control problem. It is worth noticing that the limit problem may take the form of either a variational calculation problem or an optimal control problem for Brinkman's law with another cost functional, depending on the cross-size of thin cylinders.

2009, 11(2): 315-345
doi: 10.3934/dcdsb.2009.11.315

*+*[Abstract](2307)*+*[PDF](402.6KB)**Abstract:**

In this paper, we investigate the mixed generalized Laguerre-Fourier spectral method and its applications to exterior problems of partial differential equations of fourth order. Some basic results on the mixed generalized Laguerre-Fourier orthogonal approximation are established, which play important roles in designing and analyzing various spectral methods for exterior problems of fourth order. As an important application, a mixed spectral scheme is proposed for the stream function form of the Navier-Stokes equations outside a disc. The numerical solution fulfills the compressibility automatically and keeps the same conservation property as the exact solution. The stability and convergence of proposed scheme are proved. Numerical results demonstrate its spectral accuracy in space, and coincide with the analysis very well.

2009, 11(2): 347-352
doi: 10.3934/dcdsb.2009.11.347

*+*[Abstract](2586)*+*[PDF](121.8KB)**Abstract:**

For three-dimensional competitive Lotka-Volterra systems, Zeeman identified 33 stable equivalence classes. Among these, only classes 26-31 may have limit cycles. We construct two limit cycles without a heteroclinic cycle (classes 30 and 31 in Zeeman's classification). Our construction together with Hofbauer and So [9] and Lu and Luo [10] gives a complete answer to Hofbauer's and So's problem [9] concerning two limit cycles for three-dimensional competitive Lotka-Volterra systems.

2009, 11(2): 353-364
doi: 10.3934/dcdsb.2009.11.353

*+*[Abstract](2179)*+*[PDF](202.3KB)**Abstract:**

We present two a priori $L^p$-stability estimates to the discrete velocity Boltzmann models. In a close-to-global Maxwellian regime, we derive a local-in-time $L^2$-stability estimate using a macro-micro decomposition and dispersion estimates for smooth perturbations, and as a direct application, we establish that classical solutions in Kawashima's framework [22, 24] are uniformly $L^2$-stable. In a close-to-vacuum regime, we also obtain a local-in-time $L^p$-stability estimates for classical solutions near vacuum.

2009, 11(2): 365-385
doi: 10.3934/dcdsb.2009.11.365

*+*[Abstract](2111)*+*[PDF](259.0KB)**Abstract:**

We describe a Lohner-type algorithm for the computation of rigorous upper bounds for reachable set for control systems, solutions of ordinary differential inclusions and perturbations of ODEs.

2009, 11(2): 387-419
doi: 10.3934/dcdsb.2009.11.387

*+*[Abstract](2215)*+*[PDF](330.8KB)**Abstract:**

The semi and fully discrete finite element methods are proposed for investigating vibration analysis of elastic plate-plate structures. In the space directions, the longitudinal displacements on plates are discretized by conforming linear elements, and the corresponding transverse displacements are discretized by the Morley element, leading to a semi-discrete finite element method for the problem under consideration. Applying the second order central difference to discretize the time derivative, a fully discrete scheme is obtained, and two approaches for choosing the initial functions are also introduced. The error analysis in the energy norm for the semi and fully discrete methods are established, and some numerical examples are included to validate the theoretical analysis.

2009, 11(2): 421-442
doi: 10.3934/dcdsb.2009.11.421

*+*[Abstract](1989)*+*[PDF](269.8KB)**Abstract:**

We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in a one dimensional random drift which is a mean zero stationary ergodic process with mixing property and local Lipschitz continuity. To prove the variational principle, we use the path integral representation of solutions, hitting time and large deviation estimates of the associated stochastic flows. The variational principle allows us to derive upper and lower bounds of the front speeds which decay according to a power law in the limit of large root mean square amplitude of the drift. This scaling law is different from that of the effective diffusion (homogenization) approximation which is valid for front speeds in incompressible periodic advection.

2009, 11(2): 443-457
doi: 10.3934/dcdsb.2009.11.443

*+*[Abstract](2651)*+*[PDF](241.7KB)**Abstract:**

We consider the growth of an epitaxial thin film on a continuously supplied substrate using both the Burton-Cabrara-Frank (BCF) mean-field model and kinetic Monte-Carlo (KMC) simulation. Of particular interest are effects due to the finite size of the deposition zone, which is modeled by imposing an up- and downwind adatom density equal to the adatom density on an infinite terrace in equilibrium with a step. For the BCF model, we find this scenario admits a steady-state pattern with a specific number of steps separated by alternating widths. The specific spacing between the steps depends sensitively on the processing speed and on whether the number of steps is odd or even, with the range of velocities admitting an odd number of steps typically much narrower. These predictions are only partially confirmed by KMC simulations, however, with particularly poor agreement for an odd number of steps. To investigate further, we consider alternative KMC simulations with the interactions between random walkers on the terraces neglected so as to conform more closely with the mean field model. The latter simulations also more readily allow one to disable the step detachment mechanism, in which case they agree well with the predictions of the BCF model.

2009, 11(2): 459-477
doi: 10.3934/dcdsb.2009.11.459

*+*[Abstract](2153)*+*[PDF](457.6KB)**Abstract:**

In this paper, self-adaptive proportional control method in economic chaotic system is discussed. It is not necessarily required for the fixed point having stable manifold in the method we used. One can stabilize chaos via time-dependent adjustments of control parameters; also can suppress chaos by adjusting external control signals. Two kinds of chaos about the output systems in duopoly are stabilized in a neighborhood of an unstable fixed point by using the chaos controlling method. The results show that performances of the system are improved by controlling chaos. Furthermore, their applications in practice are presented. The results also show that players can control chaos by adjusting their planned output or variable cost per unit according to the sign of marginal profit.

2009, 11(2): 479-496
doi: 10.3934/dcdsb.2009.11.479

*+*[Abstract](2141)*+*[PDF](236.7KB)**Abstract:**

In this paper, a new transmission model of human malaria in a partially immune population is formulated. We establish the basic reproduction number $\tilde{R}_0$ for the model. The existence and local stability of the equilibria are studied. Our results suggest that, if the disease-induced death rate is large enough, there may be endemic equilibrium when $\tilde{R}_0 < 1$ and the model undergoes a backward bifurcation and saddle-node bifurcation, which implies that bringing the basic reproduction number below 1 is not enough to eradicate malaria. Explicit subthreshold conditions in terms of parameters are obtained beyond the basic reproduction number which provides further guidelines for accessing control of the spread of malaria.

2009, 11(2): 497-517
doi: 10.3934/dcdsb.2009.11.497

*+*[Abstract](1830)*+*[PDF](260.0KB)**Abstract:**

We provide a comprehensive study on the planar (2D) orientational distributions of nematic polymers under an imposed shear flow of arbitrary strength. We extend previous analysis for persistence of equilibria in steady shear and for transitions to unsteady limit cycles, from closure models [21] to the Doi-Hess 2D kinetic equation. A variation on the Boltzmann distribution analysis of Constantin

*et al.*[3, 4, 5] and others [8, 22, 23] for potential flow is developed to solve for all persistent steady equilibria, and characterize parameter boundaries where steady states cease to exist, which predicts the transition to tumbling limit cycles.

2009, 11(2): 519-540
doi: 10.3934/dcdsb.2009.11.519

*+*[Abstract](2350)*+*[PDF](364.6KB)**Abstract:**

The goal of this paper is to examine the evaluation of interfacial stresses using a standard, finite difference based, immersed boundary method (IMBM). This calculation is not trivial for two fundamental reasons. First, the immersed boundary is represented by a localized boundary force which is distributed to the underlying fluid grid by a discretized delta function. Second, this discretized delta function is used to impose a spatially averaged no-slip condition at the immersed boundary. These approximations can cause errors in interpolating stresses near the immersed boundary.

To identify suitable methods for evaluating stresses, we investigate three model flow problems at very low Reynolds numbers. We compare the results of the immersed boundary calculations to those achieved by the boundary element method (BEM). The stress on an immersed boundary may be calculated either by direct evaluation of the fluid stress (FS) tensor or, for the stress jump, by direct evaluation of the locally distributed boundary force (wall stress or WS). Our first model problem is Poiseuille channel flow. Using an analytical solution of the immersed boundary formulation in this simple case, we demonstrate that FS calculations should be evaluated at a distance of approximately one grid spacing inward from the immersed boundary. For a curved immersed boundary we present a procedure for selecting representative interfacial fluid stresses using the concepts from the Poiseuille flow test problem. For the final two model problems, steady state flow over a bump in a channel and unsteady peristaltic pumping, we present an 'exclusion filtering' technique for accurately measuring stresses. Using this technique, these studies show that the immersed boundary method can provide reliable approximations to interfacial stresses.

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