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 1937-1632

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Volume 1, 2008

Discrete & Continuous Dynamical Systems - S

2008 , Volume 1 , Issue 2

Guest Editors
Boris Belinskiy, Kunquan Lan, Xin Lu, Alain Miranville, and R. Shivaji

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A new finite element scheme for Landau-Lifchitz equations
François Alouges
2008, 1(2): 187-196 doi: 10.3934/dcdss.2008.1.187 +[Abstract](74) +[PDF](185.3KB)
Abstract:
In this paper we describe a new implicit finite element scheme for the discretization of Landau-Lifchitz equations. A proof of convergence of the numerical solution to a (weak) solution of the original equations is given and numerical tests showing the applicability of the method are also provided.
Delayed feedback control near Hopf bifurcation
Fatihcan M. Atay
2008, 1(2): 197-205 doi: 10.3934/dcdss.2008.1.197 +[Abstract](47) +[PDF](144.7KB)
Abstract:
The stability of functional differential equations under delayed feedback is investigated near a Hopf bifurcation. Necessary and sufficient conditions are derived for the stability of the equilibrium solution using averaging theory. The results are used to compare delayed versus undelayed feedback, as well as discrete versus distributed delays. Conditions are obtained for which delayed feedback with partial state information can yield stability where undelayed feedback is ineffective. Furthermore, it is shown that if the feedback is stabilizing (respectively, destabilizing), then a discrete delay is locally the most stabilizing (resp., destabilizing) one among delay distributions having the same mean. The result also holds globally if one considers delays that are symmetrically distributed about their mean.
Strong stability of PDE semigroups via a generator resolvent criterion
George Avalos
2008, 1(2): 207-218 doi: 10.3934/dcdss.2008.1.207 +[Abstract](34) +[PDF](200.7KB)
Abstract:
In the context of a coupled partial differential equation (PDE) model, we provide a rather general procedure by which one may invoke a recently derived operator theoretic result in [21], so as to obtain strong stability of those dissipative $C_{0}$-semigroups which model PDEs in Hilbert space. In particular, the procedure is applied here to a PDE which models structural acoustic interactions; it is wellknown that for this interactive PDE the classical stability tools--i.e., the Nagy-Foias decomposition--is inapplicable. The novelty of adopting the present strong stability technique is that one does not need to have an explicit representation of the resolvent.
Delay equations and nonuniform exponential stability
Luis Barreira and  Claudia Valls
2008, 1(2): 219-223 doi: 10.3934/dcdss.2008.1.219 +[Abstract](42) +[PDF](110.5KB)
Abstract:
For nonautonomous linear delay equations $v'=L(t)v_t$ admitting a nonuniform exponential contraction, we establish the nonuniform exponential stability of the equation $v'=L(t) v_t +f(t,v_t)$ for a large class of nonlinear perturbations.
A priori estimate for the Nirenberg problem
Wenxiong Chen and  Congming Li
2008, 1(2): 225-233 doi: 10.3934/dcdss.2008.1.225 +[Abstract](47) +[PDF](165.8KB)
Abstract:
We establish a priori estimate for solutions to the prescribing Gaussian curvature equation

$ - \Delta u + 1 = K(x) e^{2u}, x \in S^2,$    (1)

for functions $K(x)$ which are allowed to change signs. In [16], Chang, Gursky and Yang obtained a priori estimate for the solution of (1) under the condition that the function K(x) be positive and bounded away from 0. This technical assumption was used to guarantee a uniform bound on the energy of the solutions. The main objective of our paper is to remove this well-known assumption. Using the method of moving planes in a local way, we are able to control the growth of the solutions in the region where K is negative and in the region where K is small and thus obtain a priori estimate on the solutions of (1) for general functions K with changing signs.

A semilinear $A$-spectrum
C. T. Cremins and  G. Infante
2008, 1(2): 235-242 doi: 10.3934/dcdss.2008.1.235 +[Abstract](31) +[PDF](134.6KB)
Abstract:
Using the theory of $A$-proper maps, we give a new definition of spectrum for a semilinear pair $(L,F)$. Properties of this spectrum are given.
Existence and uniqueness of a structural acoustic model involving a nonlinear shell
Inger Daniels and  Catherine Lebiedzik
2008, 1(2): 243-252 doi: 10.3934/dcdss.2008.1.243 +[Abstract](33) +[PDF](174.7KB)
Abstract:
We consider a structural acoustic model where the active wall is a nonlinear shell. We use a shell modeled with the intrinsic method of Michel Delfour and Jean-Paul Zolésio. We show the existence and uniqueness of solutions in the finite energy space as a consequence of a special trace estimate.
On a climate model with a dynamic nonlinear diffusive boundary condition
Jesús Ildefonso Díaz and  L. Tello
2008, 1(2): 253-262 doi: 10.3934/dcdss.2008.1.253 +[Abstract](34) +[PDF](161.6KB)
Abstract:
This work studies the sensitivity of a global climate model with deep ocean effect to the variations of a Solar parameter $Q$. The model incorporates a dynamic and diffusive boundary condition. We study the number of stationary solutions according to the positive parameter $Q$.
Emergence of travelling waves in smooth nerve fibres
S. Jacquir , S. Binczak , J. P. Gauthier and  J. M. Bilbault
2008, 1(2): 263-272 doi: 10.3934/dcdss.2008.1.263 +[Abstract](30) +[PDF](189.9KB)
Abstract:
An approximate analytical solution characterizing initial conditions leading to action potential firing in smooth nerve fibres is determined, using the bistable equation. In the first place, we present a non-trivial stationary solution wave. Then, we extract the main features of this solution to obtain a frontier condition between the initiation of the travelling waves and a decay to the resting state. This frontier corresponds to a separatrix in the projected dynamics diagram depending on the width and the amplitude of the stationary wave.
Invariant regions and global existence for a phase field model
A. Jiménez-Casas
2008, 1(2): 273-281 doi: 10.3934/dcdss.2008.1.273 +[Abstract](26) +[PDF](162.7KB)
Abstract:
In this paper we prove a global existence result for the solution of a phase-field model with initial data in high order Sobolev spaces using the invariant regions. This improves, in some sense, the result of [9].
Clamped elastic-ideally plastic beams and Prandtl-Ishlinskii hysteresis operators
Pavel Krejčí and  Jürgen Sprekels
2008, 1(2): 283-292 doi: 10.3934/dcdss.2008.1.283 +[Abstract](29) +[PDF](167.2KB)
Abstract:
We consider a model for one-dimensional transversal oscillations of an elastic-ideally plastic beam. It is based on the von Mises model of plasticity and leads after a dimensional reduction to a fourth-order partial differential equation with a hysteresis operator of Prandtl-Ishlinskii type whose weight function is given explicitly. In this paper, we study the case of clamped beams involving a kinematic hardening in the stress-strain relation. As main result, we prove the existence and uniqueness of a weak solution. The method of proof, based on spatially semidiscrete approximations, strongly relies on energy dissipation properties of one-dimensional hysteresis operators.
Computing long-lifetime science orbits around natural satellites
Martin Lara and  Sebastián Ferrer
2008, 1(2): 293-302 doi: 10.3934/dcdss.2008.1.293 +[Abstract](40) +[PDF](410.9KB)
Abstract:
Science missions around natural satellites require low eccentricity and high inclination orbits. These orbits are unstable because of the planetary perturbations, making control necessary to reach the required mission lifetime. Dynamical systems theory helps in improving lifetimes reducing fuel consumption. After a double averaging of the 3-DOF model, the initial conditions are chosen so that the orbit follows the stable-unstable manifold path of an equilibria of the 1-DOF reduced problem. Corresponding initial conditions in the non-averaged problem are easily computed from the explicit transformations provided by the Lie-Deprit perturbation method.
Dynamics of ratio-dependent Predator-Prey models with nonconstant harvesting
Benjamin Leard , Catherine Lewis and  Jorge Rebaza
2008, 1(2): 303-315 doi: 10.3934/dcdss.2008.1.303 +[Abstract](125) +[PDF](682.3KB)
Abstract:
The dynamics of constant harvesting of a single species has been studied extensively within the framework of ratio-dependent predator-prey models. In this work, we investigate the properties of a Michaelis-Menten ratio-dependent predator-prey model with two nonconstant harvesting functions depending on the prey population. Equilibria and periodic orbits are computed and their stability properties are analyzed. Several bifurcations are detected as well as connecting orbits, with an emphasis on analyzing the equilibrium points at which the species coexist. Smooth numerical continuation is performed that allows computation of branches of solutions.
Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping
Brenton LeMesurier
2008, 1(2): 317-327 doi: 10.3934/dcdss.2008.1.317 +[Abstract](36) +[PDF](491.3KB)
Abstract:
A mathematical model is introduced for weakly nonlinear wave phenomena in molecular systems like DNA and protein molecules that includes thermal effects: exchange of heat energy with the surrounding aqueous medium. The resulting equation is a stochastic discrete nonlinear Schrödinger equation with focusing cubic nonlinearity and "Thermal'' terms modeling heat input and loss: PDSDNLS.
    New numerical methods are introduced to handle the unusual combination of a conservative equation, stochastic, and fully nonlinear terms. Some analysis is given of accuracy needs, and the special issues of time step adjustment in stochastic realizations. Numerical studies are presented of the effects of thermalization on solitons, including damping induced self-trapping of wave energy, a discrete counterpart of single-point blowup.
Generalizations of logarithmic Sobolev inequalities
Jochen Merker
2008, 1(2): 329-338 doi: 10.3934/dcdss.2008.1.329 +[Abstract](53) +[PDF](145.5KB)
Abstract:
We generalize logarithmic Sobolev inequalities to logarithmic Gagliardo-Nirenberg inequalities, and apply these inequalities to prove ultracontractivity of the semigroup generated by the doubly nonlinear $p$-Laplacian

$\dot{u}=\Delta_p u^m.$

Our proof does not use Moser iteration, but shows that the time-dependent Lebesgue norm $\||u(t)|\|_{r(t)}$ stays bounded for a variable exponent $r(t)$ blowing up in arbitrary short time.

Stokes and Oseen flow with Coriolis force in the exterior domain
Šárka Nečasová
2008, 1(2): 339-351 doi: 10.3934/dcdss.2008.1.339 +[Abstract](41) +[PDF](180.3KB)
Abstract:
In the paper we will study the problem of steady viscous linear case with Coriolis force in the exterior domain.
Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise
Henri Schurz
2008, 1(2): 353-363 doi: 10.3934/dcdss.2008.1.353 +[Abstract](59) +[PDF](196.1KB)
Abstract:
One-dimensional wave equations with cubic power law perturbed by Q-regular additive space-time random noise are considered. These models describe the displacement of nonlinear strings excited by state-independent random external forces. The presented analysis is based on the representation of its solution in form of Fourier-series expansions along the eigenfunctions of Laplace operator with continuous, Markovian, unique Fourier coefficients (the so-called commutative case). We shall discuss existence and uniqueness of Fourier solutions using energy-type methods based on the construction of Lyapunov-functions. Appropriate truncations and finite-dimensional approximations are presented while exploiting the explicit knowledge on eigenfunctions of related second order differential operators. Moreover, some nonstandard partial-implicit difference methods for their numerical integration are suggested in order to control its energy functional in a dynamically consistent fashion. The generalized energy $\cE$ (sum of kinetic, potential and damping energy) is governed by the linear relation $\E [\varepsilon(t)] = \E [\varepsilon(0)] + b^2 trace (Q) t / 2$ in time $t \ge 0$, where $b$ is the scalar intensity of noise and $Q$ its covariance operator.

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