# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

2006 , Volume 6 , Issue 5

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2006, 6(5): 957-978 doi: 10.3934/dcdsb.2006.6.957 +[Abstract](194) +[PDF](271.6KB)
Abstract:
In this paper, we present the a posteriori error analysis for the finite element approximation of American option valuation problems. We introduce an efficient and reliable error estimator both for the semi discrete and fully discrete backward Euler scheme.
2006, 6(5): 979-1000 doi: 10.3934/dcdsb.2006.6.979 +[Abstract](134) +[PDF](1325.3KB)
Abstract:
We consider a $1$-dimensional Lagrangian averaged model for an inviscid compressible fluid. As previously introduced in the literature, such equations are designed to model the effect of fluctuations upon the mean flow in compressible fluids. This paper presents a traveling wave analysis and a numerical study for such a model. The discussion is centered around two issues. One relates to the intriguing wave motions supported by this model. The other is the appropriateness of using Lagrangian-averaged models for compressible flow to approximate shock wave solutions of the compressible Euler equations.
2006, 6(5): 1001-1026 doi: 10.3934/dcdsb.2006.6.1001 +[Abstract](218) +[PDF](353.2KB)
Abstract:
This paper deals with a thermomechanical model describing phase transitions with thermal memory in terms of balance and equilibrium equations for entropy and microforces, respectively. After a presentation and discussion of the model, the large time behaviour of the solutions to the related integro-differential system of PDE's is investigated.
2006, 6(5): 1027-1050 doi: 10.3934/dcdsb.2006.6.1027 +[Abstract](226) +[PDF](357.7KB)
Abstract:
In this paper, we prove new functional inequalities of Poincaré type on the one-dimensional torus $S^1$ and explore their implications for the long-time asymptotics of periodic solutions of nonlinear singular or degenerate parabolic equations of second and fourth order. We generically prove a global algebraic decay of an entropy functional, faster than exponential for short times, and an asymptotically exponential convergence of positive solutions towards their average. The asymptotically exponential regime is valid for a larger range of parameters for all relevant cases of application: porous medium/fast diffusion, thin film and logarithmic fourth order nonlinear diffusion equations. The techniques are inspired by direct entropy-entropy production methods and based on appropriate Poincaré type inequalities.
2006, 6(5): 1051-1076 doi: 10.3934/dcdsb.2006.6.1051 +[Abstract](190) +[PDF](406.4KB)
Abstract:
We consider an integrodifferential reaction-diffusion system which finds application in population dynamics. The memory kernels accounting for delay effects can be of both weak and strong type. Rescaling the kernels with a time relaxation $\varepsilon>0$, we show that the original model gives raise to a one-parameter family of dynamical systems in a suitable phase-space, We prove that this family is characterized by a corresponding family of exponential attractors which is stable as the delay effects vanish, i.e., when $\varepsilon$ goes to $0$.
2006, 6(5): 1077-1096 doi: 10.3934/dcdsb.2006.6.1077 +[Abstract](169) +[PDF](314.3KB)
Abstract:
Semi-discrete in time approximations of the velocity tracking problem are studied based on a pseudo-compressibility approach. Two different methods are used for the analysis of the corresponding optimality system. The first one, the classical penalty formulation, leads to estimates of order $k + \varepsilon$, under suitable regularity assumptions. The estimate is based on previously derived results for the solution of the unsteady Navier-Stokes problem by penalty methods (see e.g. Jie Shen [26]) and the Brezzi-Rappaz-Raviart theory (see e.g. [12]). The second one, based on the artificially compressible optimality system, leads to an improved estimate of the form $k + \varepsilon k$ for the linearized system.
2006, 6(5): 1097-1112 doi: 10.3934/dcdsb.2006.6.1097 +[Abstract](205) +[PDF](267.4KB)
Abstract:
In this paper, we analyze the damped Duffing equation by means of qualitative theory of planar systems. Under certain parametric choices, the global structure in the Poincaré phase plane of an equivalent two-dimensional autonomous system is plotted. Exact solutions are obtained by using the Lie symmetry and the coordinate transformation method, respectively. Applications of the second approach to some nonlinear evolution equations such as the two-dimensional dissipative Klein-Gordon equation are illustrated.
2006, 6(5): 1113-1119 doi: 10.3934/dcdsb.2006.6.1113 +[Abstract](172) +[PDF](188.9KB)
Abstract:
Analyzing the asymptotic properties of solutions to a class of second-order differential equations, we give criteria for the existence of edge waves for variable seabed profiles in the presence of longshore currents.
2006, 6(5): 1121-1139 doi: 10.3934/dcdsb.2006.6.1121 +[Abstract](198) +[PDF](293.4KB)
Abstract:
In this paper we study the dynamics of a Burgers' type equation (1). First, we use a new method called attractor bifurcation introduced by Ma and Wang in [4, 6] to study the bifurcation of Burgers' type equation out of the trivial solution. For Dirichlet boundary condition, we get pitchfork attrac- tor bifurcation as the parameter $\lambda$ crosses the first eigenvalue. For periodic boundary condition, we get bifurcated $S^{1}$ attractor consisting of steady states. Second, we study the long time behavior of the equation. We show that there exists a global attractor whose dimension is at least of the order of $\sqrt{\lambda}$. Thus it provides another example of extended system (see (2)) whose global attractor has a Hausdorff/fractal dimension that scales at least linearly in the system size while the long time dynamics is non-chaotic.
2006, 6(5): 1141-1156 doi: 10.3934/dcdsb.2006.6.1141 +[Abstract](231) +[PDF](381.8KB)
Abstract:
We investigate global behavior of

$x_{n+1} = T(x_{n}),\quad n=0,1,2,...$ (E)

where $T:\mathcal{ R}\rightarrow \mathcal{ R}$ is a competitive (monotone with respect to the south-east ordering) map on a set $\mathcal{R}\subset \mathbb{R}^2$ with nonempty interior. We assume the existence of a unique fixed point $\overline{e}$ in the interior of $\mathcal{ R}$. We give very general conditions which are easily verifiable for (E) to exhibit either competitive-exclusion or competitive-coexistence. More specifically, we obtain sufficient conditions for the interior fixed point $\overline{ e}$ to be a global attractor when $\mathcal{ R}$ is a rectangular region. We also show that when $T$ is strongly monotone in $\mathcal{ R}^{\circ}$ (interior of $\mathcal{ R}$), $\mathcal{ R}$ is convex, the unique interior equilibrium $\overline{ e}$ is a saddle, and a technical condition is satisfied, the corresponding global stable and unstable manifolds are the graphs of monotonic functions, and the global stable manifold splits the domain into two connected regions, which under additional conditions on $\mathcal{R}$ and on $T$ are shown to be basins of attraction of fixed points on the boundary of $\mathcal{R}$. Applications of the main results to specific difference equations are given.

2006, 6(5): 1157-1174 doi: 10.3934/dcdsb.2006.6.1157 +[Abstract](184) +[PDF](368.1KB)
Abstract:
In this paper we generalize analytic studies the problems related to suppression of chaos and non--feedback controlling chaotic motion. We develop an analytic method of the investigation of qualitative changes in chaotic dynamical systems under certain external periodic perturbations. It is proven that for polymodal maps one can stabilize chosen in advance periodic orbits. As an example, the quadratic family of maps is considered.
Also we demonstrate that for a piecewise linear family of maps and for a two-dimensional map having a hyperbolic attractor there are feedback-free perturbations which lead to the suppression of chaos and stabilization of certain periodic orbits.
2006, 6(5): 1175-1189 doi: 10.3934/dcdsb.2006.6.1175 +[Abstract](205) +[PDF](286.5KB)
Abstract:
This paper analyses front propagation of the equation

$\upsilon_{\tau}=[D(\upsilon)\upsilon_{x}]_{x}+f(\upsilon) \tau\ge 0, x\in R,$

where $f$ is a monostable (i.e. Fisher-type) nonlinear reaction term and $D(\upsilon)$ changes its sign once, from positive to negative values, in the interval $\upsilon \in [0, 1]$ where the process is studied. This model equation accounts for simultaneous diffusive and aggregative behaviors of a population dynamic depending on the population density $\upsilon$ at time $\tau$ and position $x$. The existence of infinitely many traveling wave solutions is proven. These fronts are parameterized by their wave speed and monotonically connect the stationary states $\upsilon \equiv 0$ and $\upsilon \equiv 1$. In the degenerate case, i.e. when $D(0) = 0$ and/or $D(1) = 0$, sharp profiles appear, corresponding to the minimum wave speed. They also have new behaviors, in addition to those already observed in diffusive models, since they can be right compactly supported, left compactly supported, or both. The dynamics can exhibit, respectively, the phenomena of finite speed of propagation, finite speed of saturation, or both.

2006, 6(5): 1191-1198 doi: 10.3934/dcdsb.2006.6.1191 +[Abstract](259) +[PDF](195.0KB)
Abstract:
In this paper we study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive integral evolution equation

$u'(t)= Au(t)+f(t,u(t),\int_0^tk(t,s)u(s)ds), t>0, t\ne t_i,$
$u(0)= u_0,$
$\Delta u(t_i)= I_i(u(t_i)). i= 1,2,....,p.$

in a Banach space X, where A is the infinitesimal generator of a strongly continuous semigroup,$\Delta u(t_i)=u(t^+_i)-u(t^-_i)$ and $I's$ are some operator. We apply the semigroup theory to study the existence and uniqueness of the mild solutions, and then show that the mild solution give rise to classical solution if $f$ is continuously differentiable.

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