# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

November 2006 , Volume 6 , Issue 6

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2006, 6(6): 1199-1212 doi: 10.3934/dcdsb.2006.6.1199 +[Abstract](521) +[PDF](249.8KB)
Abstract:
We investigate the existence, uniqueness and exponential stability of non-constant stationary solutions of stochastic semilinear evolution equations. Our main result shows, in particular, that noise can have a stabilization effect on deterministic equations. Moreover, we do not require any commutative condition on the noise terms.
2006, 6(6): 1213-1238 doi: 10.3934/dcdsb.2006.6.1213 +[Abstract](622) +[PDF](390.0KB)
Abstract:
In this work we present some results for the inverse problem of the identification of a single rigid body immersed in a fluid governed by the stationary Boussinesq equations. First, we establish a uniqueness result. Then, we show the way the observation depends on perturbations of the rigid body and we deduce some consequences. Finally, we present a new method for the partial identification of the body assuming that it can be deformed only through fields that, in some sense, are finite dimensional. In the proofs, we use various techniques, related to Carleman estimates, differentiation with respect to domains, data assimilation and controllability of PDEs.
2006, 6(6): 1239-1260 doi: 10.3934/dcdsb.2006.6.1239 +[Abstract](597) +[PDF](357.5KB)
Abstract:
Bacteria are able to respond to environmental signals by changing their rules of movement. When we take into account chemical signals in the environment, this behaviour is often called chemotaxis. At the individual-level, chemotaxis consists of several steps. First, the cell detects the extracellular signal using receptors on its membrane. Then, the cell processes the signal information through the intracellular signal transduction network, and finally it responds by altering its motile behaviour accordingly. At the population level, chemotaxis can lead to aggregation of bacteria, travelling waves or pattern formation, and the important task is to explain the population-level behaviour in terms of individual-based models. It has been previously shown that the transport equation framework [12, 13] is suitable for connecting different levels of modelling of bacterial chemotaxis. In this paper, we couple the transport equation for bacteria with the (parabolic/elliptic) equation for the extracellular signals. We prove global existence of solutions for the general hyperbolic chemotaxis models of cells which process the information about the extracellular signal through the intracellular biochemical network and interact by altering the extracellular signal as well. Working in one spatial dimension with multi-dimensional internal dynamics, conditions for global existence in terms of the properties of the signal transduction model are given.
2006, 6(6): 1261-1300 doi: 10.3934/dcdsb.2006.6.1261 +[Abstract](716) +[PDF](479.5KB)
Abstract:
In this paper we develop several numerical algorithms for the computation of invariant manifolds in quasi-periodically forced systems. The invariant manifolds we consider are invariant tori and the asymptotic invariant manifolds (whiskers) to these tori.
The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34].
The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties.
The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.
2006, 6(6): 1301-1320 doi: 10.3934/dcdsb.2006.6.1301 +[Abstract](811) +[PDF](1950.3KB)
Abstract:
The dynamics of a prey-predator model with impulsive state feedback control is studied by an autonomous system with impulses. The dynamical behavior of this system is discussed by means of both theoretical and numerical ways. The sufficient conditions of existence and stability of the semi-trivial periodic solution, positive period-one, and positive period-two solutions are obtained by using Lambert W function and the analogue of the Poincaré criterion. The bifurcation analysis shows that solutions appear via a cascade of period-doubling in some interval of parameters. The bifurcation diagrams, the Lyapunov exponents, and the phase portraits are given in two examples. The discussion of prey (pest) control strategy shows that the impulsive state feedback control is effective.
2006, 6(6): 1321-1338 doi: 10.3934/dcdsb.2006.6.1321 +[Abstract](628) +[PDF](268.7KB)
Abstract:
The main concern of this paper is to study the dynamic of a ratio dependent predator-prey system with diffusion and delay. Concretely, we study the dissipativeness and persistence of the system. We show that there are no non trivial steady states solutions for certain parameter's configuration; and discuss the existence of attracting periodic solutions.
2006, 6(6): 1339-1356 doi: 10.3934/dcdsb.2006.6.1339 +[Abstract](670) +[PDF](288.2KB)
Abstract:
In this paper we study a class of inequality problems for static frictional contact between a piezoelastic body and a foundation. The constitutive law is assumed to be electrostatic and involves a nonlinear elasticity operator. The contact is described by Clarke subdifferential relations of nonmonotone and multivalued character in the normal and tangential directions on the boundary. We derive a variational formulation which is a coupled system of a hemivariational inequality and an elliptic equation. The existence of solutions to the model is a consequence of a more general result obtained from the theory of pseudomonotone mappings. Conditions under which a solution of the system is unique are also presented.
2006, 6(6): 1357-1380 doi: 10.3934/dcdsb.2006.6.1357 +[Abstract](545) +[PDF](3872.7KB)
Abstract:
A two-dimensional thermal convection problem in a circular annulus subject to a constant inward radial gravity and heated from the inside is considered. A branch of spatio-temporal symmetric periodic orbits that are known only numerically shows a multi-critical codimension-two point with a triple +1-Floquet multiplier. The weakly nonlinear analysis of the dynamics near such point is performed by deriving a system of amplitude equations using a perturbation technique, which is an extension of the Lindstedt-Poincaré method, and solvability conditions. The results obtained using the amplitude equation are compared with those from the original system of partial differential equations showing a very good agreement.
2006, 6(6): 1381-1402 doi: 10.3934/dcdsb.2006.6.1381 +[Abstract](619) +[PDF](2323.7KB)
Abstract:
Dual-Petrov-Galerkin approximations to linear third-order equations and the Korteweg-de Vries equation on semi-infinite intervals are considered. It is shown that by choosing appropriate trial and test basis functions the Dual-Petrov-Galerkin method using Laguerre functions leads to strongly coercive linear systems which are easily invertible and enjoy optimal convergence rates. A novel multi-domain composite Legendre-Laguerre dual-Petrov-Galerkin method is also proposed and implemented. Numerical results illustrating the superior accuracy and effectiveness of the proposed dual-Petrov-Galerkin methods are presented.
2006, 6(6): 1403-1416 doi: 10.3934/dcdsb.2006.6.1403 +[Abstract](556) +[PDF](303.0KB)
Abstract:
The slow-manifold for the Lorenz-Krishnamurthy model has been studied. By minimizing the evolution rate we find that the analytical functions for the fast variables are devoid of high frequency oscillations. However upon solving this model with initial values of the fast variables obtained from the analytical functions, the LK model exhibits high frequency oscillations. Upon using the time derivatives of the analytic functions for computing the evolution of fast variables, we find a slow-manifold in the neighbourhood of the LK model.
Minimization of evolution rate does not guarantee the invariance of the manifold. Using a locally linear approximate reduction scheme, the invariance can be maintained. However, the solutions so obtained do develop high frequency oscillations. The onset of these high frequency oscillations is delayed vis-a-vis other previous studies. These methods have potential to be used in improving the predictions of weather systems.
2006, 6(6): 1417-1430 doi: 10.3934/dcdsb.2006.6.1417 +[Abstract](684) +[PDF](282.7KB)
Abstract:
We study a mathematical model for the interaction of HIV infection and CD4$^+$ T cells. Local and global analysis is carried out. Let $N$ be the number of HIV virus produced per actively infected T cell. After identifying a critical number $N_{crit}$, we show that if $N\le N_{crit},$ then the uninfected steady state $P_{0}$ is the only equilibrium in the feasible region, and $P_{0}$ is globally asymptotically stable. Therefore, no HIV infection persists. If $N>N_{crit},$ then the infected steady state $P^$* emerges as the unique equilibrium in the interior of the feasible region, $P_{0}$ becomes unstable and the system is uniformly persistent. Therefore, HIV infection persists. In this case, $P^$* can be either stable or unstable. We show that $P^$* is stable only for $r$ (the proliferation rate of T cells) small or large and unstable for some intermediate values of $r.$ In the latter case, numerical simulations indicate a stable periodic solution exists.
2006, 6(6): 1431-1444 doi: 10.3934/dcdsb.2006.6.1431 +[Abstract](598) +[PDF](282.8KB)
Abstract:
The dynamics of a supply chain have been modelled by several authors, yet no attempt has ever been made for finding the vendor's optimal production policy when facing such dynamics. In this paper, we model the dynamics of a supply chain as an infinite-horizon time-delayed optimal control problem. By approximating the time interval $[ 0,\infty )$ by $0,T_f$, we obtain an approximated problem $P(T_f)$ which can be easily solved by the control parametrization method. Moreover, we can show that the objective function of the approximated problem converges to that of the original problem as $T_f \to \infty$. Lastly, we also extend our method to solving a stochastic problem where the demand is a stochastic process with white noise input. Several examples for both the deterministic and the stochastic problems are solved to illustrate the efficiency of our method. In these examples, some important results relating the production rate to the demand are developed.

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