ISSN:

1531-3492

eISSN:

1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

2002 , Volume 2 , Issue 2

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2002, 2(2): 153-167
doi: 10.3934/dcdsb.2002.2.153

*+*[Abstract](187)*+*[PDF](483.7KB)**Abstract:**

Directed motion or ratchet-like behavior in many molecular scale systems is a consequence of diffusion mediated transport. The Brownian motor serves as a paradigm. The Parrondo Paradox is a pair of coin toss games, each of which is fair, or even losing, but become winning with a schedule of playing them in alternation. It has been proposed as a discrete analog of the Brownian motor. We examine the relationship between these two systems. We discover a class of Parrondo games with unusual ratchet-like behavior and for which diffusion plays a fundamentally different role than it does in the Brownian motor. Detailed balance is an important feature in these considerations.

The Brownian motor depends on details of the potential landscape in the system but the Parrondo game is decided on the potential difference alone. There are winning Parrondo games whose Brownian motor analogs move in the opposite direction. A general framework is discussed in section 7. The original Parrondo game, here in section 7.2, is completely determined by detailed balance.

2002, 2(2): 169-184
doi: 10.3934/dcdsb.2002.2.169

*+*[Abstract](184)*+*[PDF](601.2KB)**Abstract:**

This paper develops the observation control method for refining the Kalman-Bucy estimates, which is based on impulsive modeling of the transition matrix in an observation equation, thus engaging discrete-continuous observations. The impulse observation control generates on-line computable jumps of the estimate variance from its current position towards zero and, as a result, enables us to instantaneously obtain the estimate, whose variance is closer to zero. The filtering equations over impulse-controlled observations are obtained in the Kalman-Bucy filtering problem. The method for feedback design of control of the estimate variance is developed. First, the pure impulse control is used, and, next, the combination of the impulse and continuous control components is employed. The considered examples allow us to compare the properties of these control and filtering methodologies.

2002, 2(2): 185-204
doi: 10.3934/dcdsb.2002.2.185

*+*[Abstract](415)*+*[PDF](226.2KB)**Abstract:**

Flow based upscaling of absolute permeability has become an important step in practical simulations of flow through heterogeneous formations. The central idea is to compute upscaled, grid-block permeability from fine scale solutions of the flow equation. Such solutions can be either local in each grid-block or global in the whole domain. It is well-known that the grid-block permeability may be strongly influenced by the boundary conditions imposed on the flow equations and the size of the grid-blocks. We show that the upscaling errors due to both effects manifest as the resonance between the small physical scales of the media and the artificial size of the grid blocks. To obtain precise error estimates, we study the scale-up of single phase steady flows through media with periodic small scale heterogeneity. As demonstrated by our numerical experiments, these estimates are also useful for understanding the upscaling of general random media. It is further shown that the oversampling technique introduced in our previous work can be used to reduce the resonance error and obtain boundary-condition independent, grid-block permeability. Some misunderstandings in scale up studies are also clarified in this work.

2002, 2(2): 205-219
doi: 10.3934/dcdsb.2002.2.205

*+*[Abstract](174)*+*[PDF](178.3KB)**Abstract:**

The ocean thermohaline circulation, also called meridional overturning circulation, is caused by water density contrasts. This circulation has large capacity of carrying heat around the globe and it thus affects the energy budget and further affects the climate. We consider a thermohaline circulation model in the meridional plane under external wind forcing. We show that, when there is no wind forcing, the stream function and the density fluctuation (under appropriate metrics) tend to zero exponentially fast as time goes to infinity. With rapidly oscillating wind forcing, we obtain an averaging principle for the thermohaline circulation model. This averaging principle provides convergence results and comparison estimates between the original thermohaline circulation and the averaged thermohaline circulation, where the wind forcing is replaced by its time average. This establishes the validity for using the averaged thermohaline circulation model for numerical simulations at long time scales.

2002, 2(2): 221-242
doi: 10.3934/dcdsb.2002.2.221

*+*[Abstract](190)*+*[PDF](549.1KB)**Abstract:**

In this paper we illustrate a novel method for studying the role of complex dynamics in practical nonlinear systems of a certain form: Hamiltonian systems with a homoclinic connexion, subject to forcing and damping. We derive a set of optimal forcing functions which are better than any comparable waveform at inducing complex dynamics in the system in question via a break-up of the homoclinic orbit. These forcing functions are then used to investigate a practical problem relating to complex dynamics in a nonlinear system: how to achieve in-band disruption of a common nonlinear circuit, the phase-locked loop. This problem is chosen both for its intrinsic interest and as a motivational example of how such optimal forcing functions can be used to understand better complex dynamics in practical nonlinear systems. Numerical and experimental results are reported for a prototypical circuit which validate our approach. The importance and potential benefits of such an approach are discussed.

2002, 2(2): 243-256
doi: 10.3934/dcdsb.2002.2.243

*+*[Abstract](171)*+*[PDF](187.4KB)**Abstract:**

In previous article [18], we introduced a system of equations to model the superconductivity phenomena. We investigated its connection to Ginzburg-Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the dynamic behavior of solutions to the system and prove existence of global attractors and estimate their Hausdorff dimensions.

2002, 2(2): 257-264
doi: 10.3934/dcdsb.2002.2.257

*+*[Abstract](210)*+*[PDF](111.4KB)**Abstract:**

We formulate and analyze a model for an infectious disease which does not cause death but for which infectives remain infective for life. We derive the basic reproductive number $R_0$ and show that there is a unique globally asymptotically stable equilibrium, namely the disease - free equilibrium if $R_0 < 1$ and the endemic equilibrium if $R_0 > 1$. However, the relation between the basic reproductive number, the mean age at infection, and the mean life span depends on the distribution of life spans and may be quite different from that for exponentially distributed life spans or very short infective periods.

2002, 2(2): 265-278
doi: 10.3934/dcdsb.2002.2.265

*+*[Abstract](183)*+*[PDF](152.0KB)**Abstract:**

A sufficient condition is established for globally asymptotic stability of the positive equilibrium of a regulated logistic growth model with a delay in the state feedback. The result improves some existing criteria for this model. It is in a form that is related to the number $3/2$ and the coupling strength, and thus, is comparable to the well-known $3/2$ condition for the uncontrolled delayed logistic equation. The comparison seems to suggest that the mechanism of the control in this model might be inappropriate and new mechanism should be introduced.

2002, 2(2): 279-294
doi: 10.3934/dcdsb.2002.2.279

*+*[Abstract](245)*+*[PDF](194.9KB)**Abstract:**

In this paper we study the asymptotic behavior of a semidiscrete numerical approximation for the heat equation, $u_t = \Delta u$, in a bounded smooth domain with a nonlinear flux boundary condition, $(\partial u)/(\partial\eta)= u^p$. We focus in the behavior of blowing up solutions. We prove that every numerical solution blows up in finite time if and only if $p > 1$ and that the numerical blow-up time converges to the continuous one as the mesh parameter goes to zero. Also we show that the blow-up rate for the numerical scheme is different from the continuous one. Nevertheless we find that the blow-up set for the numerical approximations is contained in a small neighborhood of the blow-up set of the continuous problem when the mesh parameter is small enough.

2002, 2(2): 295-307
doi: 10.3934/dcdsb.2002.2.295

*+*[Abstract](334)*+*[PDF](156.0KB)**Abstract:**

The aim of this paper is to solve a tracking problem in a particular second order control system that requires indirect control. A complete knowledge of the plant parameters is assumed. The calculation of the indirect tracking depends on the solution of an inverse problem given by an ordinary differential equation. In spite of the instability of the generic solution of the differential equation, the existence of a bounded, periodic solution for the tracking of a periodic signal is proved. Finally, the periodic solution is approximated by the harmonic balance method, and the original tracking problem is solved.

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