ISSN:

1531-3492

eISSN:

1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

2007 , Volume 8 , Issue 4

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2007, 8(4): 735-772
doi: 10.3934/dcdsb.2007.8.735

*+*[Abstract](455)*+*[PDF](384.6KB)**Abstract:**

Partially ionized plasmas corresponding to different ionization degrees are derived and connected one with each other by the diffusion approximation methodology. These plasmas are the following electrical discharges: a thermal arc discharge, glow discharges in local thermodynamic equilibrium -LTE- and in non-LTE, and a non-LTE glow discharge interacting with an electron beam (or flow).

2007, 8(4): 773-800
doi: 10.3934/dcdsb.2007.8.773

*+*[Abstract](479)*+*[PDF](841.1KB)**Abstract:**

We revisit existence and stability of two-pulse solutions in the fifth-order Korteweg–de Vries (KdV) equation with two new results. First, we modify the Petviashvili method of successive iterations for numerical (spectral) approximations of pulses and prove convergence of iterations in a neighborhood of two-pulse solutions. Second, we prove structural stability of embedded eigenvalues of negative Krein signature in a linearized KdV equation. Combined with stability analysis in Pontryagin spaces, this result completes the proof of spectral stability of the corresponding two-pulse solutions. Eigenvalues of the linearized problem are approximated numerically in exponentially weighted spaces where embedded eigenvalues are isolated from the continuous spectrum. Approximations of eigenvalues and full numerical simulations of the fifth-order KdV equation confirm stability of two-pulse solutions associated with the minima of the effective interaction potential and instability of two-pulse solutions associated with the maxima points.

2007, 8(4): 801-831
doi: 10.3934/dcdsb.2007.8.801

*+*[Abstract](477)*+*[PDF](708.2KB)**Abstract:**

A system of impulsive ordinary differential equations is used to model the growth of microorganisms in a self-cycling fermentor. The microorganisms are being used to remove a non-reproducing contaminant that is limiting to growth at both high and low concentrations. Hence it is the concentration of the contaminant that triggers the emptying and refilling process. This model predicts that either the process fails or the process cycles indefinitely with one impulse per cycle. Success or failure can depend on the choice of microorganisms, the initial concentration of the microorganisms and contaminant, as well as the choice for the emptying/refilling fraction. Either there is no choice of this fraction that works or there is an interval of possible choices with an optimal choice within the interval. If more than one strain is available, it does not seem to be the strains that have the highest specific growth rate over the largest range of the concentrations of the contaminant, but rather the ones that have the highest specific growth rate over very low concentrations of the contaminant, just above the threshold that initiates recycling that appear to be the most efficient, i.e., processing the highest volume of medium over a specified time period.

2007, 8(4): 833-859
doi: 10.3934/dcdsb.2007.8.833

*+*[Abstract](524)*+*[PDF](344.8KB)**Abstract:**

In this paper, we consider multiscale approaches for solving parabolic equations with heterogeneous coefficients. Our interest stems from porous media applications and we assume that there is no scale separation with respect to spatial variables. To compute the solution of these multiscale problems on a coarse grid, we define global fields such that the solution smoothly depends on these fields. We present various finite element discretization techniques and provide analyses of these methods. A few representative numerical examples are presented using heterogeneous fields with strong non-local features. These numerical results demonstrate that the solution can be captured more accurately on the coarse grid when some type of limited global information is used.

2007, 8(4): 861-877
doi: 10.3934/dcdsb.2007.8.861

*+*[Abstract](468)*+*[PDF](225.8KB)**Abstract:**

A logarithmic fourth-order parabolic equation in one space dimension with periodic boundary conditions is analyzed. Using a new semi-discrete approximation in time, a first-order entropy–entropy dissipation inequality is proved. Passing to the limit of vanishing time discretization parameter, some regularity results are deduced. Moreover, it is shown that the solution is strictly positive for large time if it does so initially.

2007, 8(4): 879-900
doi: 10.3934/dcdsb.2007.8.879

*+*[Abstract](423)*+*[PDF](315.2KB)**Abstract:**

We shall be concerned with the mathematical analysis of a deterministic model describing the spread of a contamination which structures a population on different and continuous levels, each level representing a degree of contamination. Our approach is essentially devoted to describe a population when exposed to pollution or affected by any non environmentally-friendly source.

Mathematically, the problem consists of an advection-reaction partial differential equation with variable speed, coupled by mean of its boundary condition to an ordinary differential equation. Using a method of characteristics, we prove the global existence, uniqueness and nonnegativity of the mild solution to this system, and also the global boundedness of the total population when subjected to controlled growth dynamics such as so-called logistic behaviors.

2007, 8(4): 901-923
doi: 10.3934/dcdsb.2007.8.901

*+*[Abstract](444)*+*[PDF](277.2KB)**Abstract:**

This paper is devoted to the study of the existence of a capacity solution to the thermistor problem assuming that the thermal conductivity vanishes at points where the temperature is null and the electric conductivity is not bounded below by a positive constant value. This is a situation of practical interest including the case of metallic conduction under the Wiedemann-Franz law.

2007, 8(4): 925-941
doi: 10.3934/dcdsb.2007.8.925

*+*[Abstract](550)*+*[PDF](231.9KB)**Abstract:**

This work concerns with the existence of the time optimal controls for some linear evolution equations without the a priori assumption on the existence of admissible controls. Both global and local existence results are presented. Some necessary conditions, sufficient conditions, and necessary and sufficient conditions for the existence of time optimal controls are derived by establishing the relationship between controllability and time optimal control problems.

2007, 8(4): 943-970
doi: 10.3934/dcdsb.2007.8.943

*+*[Abstract](445)*+*[PDF](653.7KB)**Abstract:**

In this paper we consider the equation $\ddot x+x=\sin(\sqrt{2}t)+s(x)\,$ where $s(x)$ is a piece-wise linear map given by min$\{5x,1\}$ if $x\ge0$ and by max$\{-1, 5x\}$ if $x<0$. The existence of chaotic behaviour in the Smale sense inside the instability area is proven. In particular transversal homoclinic fixed point is found. The results follow from the application of topological degree theory the computer-assisted verification of a set of inequalities. Usually such proofs can not be verified by hands due to vast amount of computations, but the simplicity of our system leads to a small set of inequalities that can be verified by hand.

2007, 8(4): 971-1005
doi: 10.3934/dcdsb.2007.8.971

*+*[Abstract](381)*+*[PDF](7946.0KB)**Abstract:**

We numerically analyse different kinds of one-dimensional and two-dimensional attractors for the limit return map associated to the unfolding of homoclinic tangencies for a large class of three-dimensional dissipative diffeomorphisms. Besides describing the topological properties of these attractors, we often numerically compute their Lyapunov exponents in order to clarify where two-dimensional strange attractors can show up in the parameter space. Hence, we are specially interested in the case in which the unstable manifold of the periodic saddle taking part in the homoclinic tangency has dimension two.

2007, 8(4): 1007-1020
doi: 10.3934/dcdsb.2007.8.1007

*+*[Abstract](504)*+*[PDF](335.0KB)**Abstract:**

An algorithm for the construction of hash function based on optical time average moirè experimental technique is proposed in this paper. Algebraic structures of grayscale color functions and time average operators are constructed. Properties of time average operators and effects of digital image representation are explored. The fact that the inverse problem of identification of the original grayscale color function from its time averaged image is an ill-posed problem helps to construct an efficient algorithm for the construction of a new class of one-way collision free hash functions.

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