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2005

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Formal normal forms for holomorphic maps tangent to the identity
Marco Abate and Francesca Tovena
2005, 2005(Special): 1-10 doi: 10.3934/proc.2005.2005.1 +[Abstract](614) +[PDF](134.7KB)
Abstract:
We describe a procedure for constructing formal normal forms of holomorphic maps with a hypersurface of mixed points, and we apply it to obtain a complete list of formal normal forms for 2-dimensional holomorphic maps tangential to a curve of mixed points.
Multiscale methods for advection-diffusion problems
Assyr Abdulle
2005, 2005(Special): 11-21 doi: 10.3934/proc.2005.2005.11 +[Abstract](893) +[PDF](991.8KB)
Abstract:
The development of numerical methods for multiscale advection-diffusion problems presents a number of challenges. The one-scale structures may significantly in uence the coarser properties of the system, but are often impossible to solve in full details. The time integration of the evolution system is still due to the diffusion term and its stability properties have to be taken into account for its resolution. We discuss in this paper an algorithm, which combines Heterogeneous Multiscale Methods (HMM) with Orthogonal Runge-Kutta Chebyshev (ROCK) methods, for the efficient numerical resolution of multiscale advection-diffusion problems.
On a discrete version of the Korteweg-De Vries equation
M. Agrotis, S. Lafortune and P.G. Kevrekidis
2005, 2005(Special): 22-29 doi: 10.3934/proc.2005.2005.22 +[Abstract](627) +[PDF](214.3KB)
Abstract:
In this short communication, we consider a discrete example of how to perform multiple scale expansions and by starting from the discrete nonlinear Schröodinger equation (DNLS) as well as the Ablowitz-Ladik nonlinear Schrödinger equation (AL-NLS), we obtain the corresponding discrete versions of a Korteweg-de Vries (KdV) equation. We analyze in particular the equation obtained from the AL-NLS and discuss its integrability, as well as its connections with previously studied discrete versions of the KdV equation.
Doubly nonlinear evolution equations and Bean's critical-state model for type-II superconductivity
Goro Akagi
2005, 2005(Special): 30-39 doi: 10.3934/proc.2005.2005.30 +[Abstract](657) +[PDF](222.4KB)
Abstract:
This paper is intended as an investigation of the solvability of Cauchy problem for doubly nonlinear evolution equation of the form $dv(t)/dt + \partial \lambda^t(u(t)) \in 3 f(t)$, $v(t) \in \partial \psi(u(t))$, 0 < $t$ < $T$, where $\partial \lambda^t$ and $\partial \psi$ are subdifferential operators, and @'t depends on t explicitly. Our method of proof relies on chain rules for t-dependent subdifferentials and an appropriate boundedness condition on $\partial \lambda^t$ however, it does not require either a strong monotonicity condition or a boundedness condition on $\partial \psi$. Moreover, an initial-boundary value problem for a nonlinear parabolic equation arising from an approximation of Bean's critical-state model for type-II superconductivity is also treated as an application of our abstract theory.
On the basis properties of the functions arising in the boundary control problem of a string with a variable tension
Sergei A. Avdonin and Boris P. Belinskiy
2005, 2005(Special): 40-49 doi: 10.3934/proc.2005.2005.40 +[Abstract](743) +[PDF](218.5KB)
Abstract:
We consider the boundary control problem for a string. We say that the string is controllable if, by suitable manipulation of the exterior force, the string goes to rest. To prove our controllability results we apply the method of characteristics. Then, using the method of moments we establish a connection between the boundary control problem and the basis property of a system of functions that substitutes the system of nonharmonic exponential functions. The latter system regularly appears in the problems of controllability since the classical papers of H.O. Fattorini and D.L. Russell.
Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent
Soohyun Bae
2005, 2005(Special): 50-59 doi: 10.3934/proc.2005.2005.50 +[Abstract](706) +[PDF](208.6KB)
Abstract:
We establish that the elliptic equation $\Delta u + K(x)u^p + \mu f(x) = 0 in \mathbb{R}^n$ possesses a continuum of positive entire solutions under a set of assumptions on $K, p, \mu$ and $f$. When $K$ behaves like $1 + d|x|^( - q)$ near $\infty$ for some constants $d$ > 0 and $q$ > 0, separation and uncountable multiplicity of solutions appear for small $\mu$ > 0 provided that $n$ > 10, $p$ is large enough, and $f$ satisfies suitable decay conditions at $\infty$.
Semiconjugacy of quasiperiodic flows and finite index subgroups of multiplier groups
L. Bakker
2005, 2005(Special): 60-69 doi: 10.3934/proc.2005.2005.60 +[Abstract](546) +[PDF](210.5KB)
Abstract:
The multiplier group of a flow describes the types of generalized spacetime symmetries that the flow has. It will be shown that if an F-algebraic quasiperiodic flow is smoothly semiconjugate to flow generated by a constant vector field, then the second flow is F-algebraic quasiperiodic and its multiplier group is a finite index subgroup of the multiplier group of the first flow.
Timelike Geodesics in stationary Lorentzian manifolds with unbounded coefficients
R. Bartolo, Anna Maria Candela and J.L. Flores
2005, 2005(Special): 70-76 doi: 10.3934/proc.2005.2005.70 +[Abstract](512) +[PDF](200.5KB)
Abstract:
The aim of this note is to study the existence and multiplicity of timelike geodesics in standard stationary Lorentzian manifolds with unbounded coefficients.
Periodic trajectories in plane wave type spacetimes
R. Bartolo, Anna Maria Candela, J.L. Flores and Addolorata Salvatore
2005, 2005(Special): 77-83 doi: 10.3934/proc.2005.2005.77 +[Abstract](608) +[PDF](204.7KB)
Abstract:
In this note we study periodic trajectories in Plane Wave type spacetimes by applying some recent results about periodic orbits for a Lagrangian system in a Riemannian manifold under the action of an (eventually) unbounded potential.
Higher order boundary value problems with multiple solutions: examples and techniques
John Baxley, Mary E. Cunningham and M. Kathryn McKinnon
2005, 2005(Special): 84-90 doi: 10.3934/proc.2005.2005.84 +[Abstract](636) +[PDF](112.2KB)
Abstract:
Conditions are derived on the nonlinear function f(y) so that two boundary value problems associated with the fifth order differential equation y(5) = f(y) has one or more positive solutions. These examples are not included in any previous results.
Bifurcations of self-similar solutions of the Fokker-Plank equations
F. Berezovskaya and G. Karev
2005, 2005(Special): 91-99 doi: 10.3934/proc.2005.2005.91 +[Abstract](605) +[PDF](212.5KB)
Abstract:
A class of one-dimensional Fokker-Plank equations having a common stationary solution, which is a power function of the state of the process, was found. We prove that these equations also have generalized self-similar solutions which describe the temporary transition from one stationary state to another. The study was motivated by problems arising in mathematical modeling of genome size evolution.
Reduction and chaotic behavior of point vortices on a plane and a sphere
A.V. Borisov, A.A. Kilin and I.S. Mamaev
2005, 2005(Special): 100-109 doi: 10.3934/proc.2005.2005.100 +[Abstract](540) +[PDF](396.5KB)
Abstract:
We offer a new method of reduction for a system of point vortices on a plane and a sphere. This method is similar to the classical node elimination procedure. However, as applied to the vortex dynamics, it requires substantial modification. Reduction of four vortices on a sphere is given in more detail. We also use the Poincaré surface-of-section technique to perform the reduction a four-vortex system on a sphere.
New periodic solutions for three or four identical vortices on a plane and a sphere
A.V. Borisov, I.S. Mamaev and A.A. Kilin
2005, 2005(Special): 110-120 doi: 10.3934/proc.2005.2005.110 +[Abstract](461) +[PDF](205.8KB)
Abstract:
In this paper we describe new classes of periodic solutions for point vortices on a plane and a sphere. They correspond to similar solutions (so-called choreographies) in celestial mechanics.
Extended Riemann Bessel functions
M.T. Boudjelkha
2005, 2005(Special): 121-130 doi: 10.3934/proc.2005.2005.121 +[Abstract](650) +[PDF](149.6KB)
Abstract:
In this paper we discuss the extended Riemann Bessel functions, their integral representations and series expansions in terms of Riemann zeta functions and Bessel functions. We consider also other properties such as their differential equations, recurrence relations, a recent proof of the Hurwitz type formulae and applications.
Renormalization group calculation of asymptotically self-similar dynamics
G. A. Braga, Frederico Furtado and Vincenzo Isaia
2005, 2005(Special): 131-141 doi: 10.3934/proc.2005.2005.131 +[Abstract](663) +[PDF](209.6KB)
Abstract:
We present a systematic numerical procedure for the computation of asymptotically self-similar dynamics of physical systems whose evolution is modeled by PDEs. This approach is based on the renormalization group (RG) for PDEs, which was originally introduced by N. Goldenfeld, Y. Oono and collaborators, and was further developed by J. Bricmont, A. Kupiainen and collaborators. We explain how successive iterations of a discrete RG transformation in space and time drive the system towards a fixed point, which corresponds to a self-similar dynamics. The iteration of the RG transformation renders explicit the relative importance of the distinct physical effects being modeled in the long-time dynamics. The resulting nu- merical procedure is very efficient and provides a detailed picture of the asymptotics, including scaling exponents, profile functions, and prefactors. We illustrate the ef- fectiveness of the procedure on a set of examples of nonlinear PDEs, including cases where nonlinear effects are asymptotically irrelevant or neutral. In the latter case the asymptotic scaling laws obeyed by the dynamics frequently contain logarithmic corrections, which are detected and successfully handled by the RG procedure.
Rapidly converging phase field models via second order asymptotics
G. Caginalp and Christof Eck
2005, 2005(Special): 142-152 doi: 10.3934/proc.2005.2005.142 +[Abstract](431) +[PDF](176.9KB)
Abstract:
We consider phase field models with the objective of approximating sharp interface models. By using second order asymptotics in the interface thickness parameter, $\epsi$, we develop models in which the order $\epsi$ term is eliminated, suggesting more rapid convergence to the $\epsi$ = 0 (sharp interface) limit. In addition we use non-smooth potentials with a non-zero gradient at the roots. These changes result in an error that is 1/200 of the classical models in sample one-dimensional calculations. Alternatively, one can use 1/100 of the node points in each direction for our proposed models and still obtain the same accuracy as one would with the classical model. We expect that this will greatly facilitate two and three dimensional calculations of dendritic growth with physically realistic parameters.
Finding open-loop Nash equilibrium for variational games
Dean A. Carlson
2005, 2005(Special): 153-163 doi: 10.3934/proc.2005.2005.153 +[Abstract](553) +[PDF](195.6KB)
Abstract:
It is well known that in static games the existence of Nash equilibria is often established through the use of a fixed point theorem applied to the ”best reply mapping”. As most fixed point theorems are non-constructive these important theorems provide almost no clues for determining the equilibria. In a dynamic setting the notion of Nash equilibria must also be further qualified as open-loop or closedloop. Here we restrict our attention to the open-loop concept. Recently, based on an optimization result from Leitmann [1], a joint paper by Leitmann and the author has explored a new method for determining open-loop equilibria for a large class of N-player differential games. This ”direct method” transforms the original game into an equivalent one which, hopefully, has a solution which is easier to identify. This method implicitly involves a fixed-point map. In this paper we explore these ideas providing a “constructive method” for finding a fixed point of the best reply map. An example will illustrate the utility of our approach.
Weighted Hardy-Littlewood-Sobolev inequalities and systems of integral equations
Wenxiong Chen, Chao Jin, Congming Li and Jisun Lim
2005, 2005(Special): 164-172 doi: 10.3934/proc.2005.2005.164 +[Abstract](1213) +[PDF](216.5KB)
Abstract:
In this paper, we consider systems of integral equations related to the weighted Hardy-Littlewood-Sobolev inequality. We present the symmetry, monotonity, and regularity of the solutions. In particular, we obtain the optimal integrability of the solutions to a class of such systems. We also present a simple method for the study of regularity, which has been extensively used in various forms. The version we present here contains some new developments. It is much more general and very easy to use. We believe the method will be helpful to both experts and non-experts in the field.
Steady states of a strongly coupled prey-predator model
Xinfu Chen, Yuanwei Qi and Mingxin Wang
2005, 2005(Special): 173-180 doi: 10.3934/proc.2005.2005.173 +[Abstract](529) +[PDF](211.5KB)
Abstract:
We study an elliptic system arising from a prey-predator model, where cross diffusions are included to reflect the influences of density gradients of prey and predator toward the fluxes of underlying populations. We establish existence and non{existence of non{constant positive solutions, or the possible pattern of popula- tion distribution.
Locking-free nonconforming finite elements for planar linear elasticity
Zhangxin Chen, Qiaoyuan Jiang and Yanli Cui
2005, 2005(Special): 181-189 doi: 10.3934/proc.2005.2005.181 +[Abstract](552) +[PDF](194.0KB)
Abstract:
In this paper we introduce a nonconforming finite element method for a planar linear elasticity problem. We show that this nonconforming method is robust in that error estimates generated by it are uniform with respect to one of the Lamé elasticity constants, $\l$; i.e., it is locking-free. Applications to nonconforming $P_1$ and rotated $Q_1$ finite elements are discussed.
Dynamic parameters identification in traffic flow modeling
Rinaldo M. Colombo and Andrea Corli
2005, 2005(Special): 190-199 doi: 10.3934/proc.2005.2005.190 +[Abstract](511) +[PDF](270.9KB)
Abstract:
In this paper we review and extend some recent results in the theory of conservation laws to make them suitable for the application to problems motivated by traffic flow modeling. In particular, we consider the problem of parameter identification in continuum traffic flow models.
Singular limit of dissipative hyperbolic equations with memory
Monica Conti, Vittorino Pata and M. Squassina
2005, 2005(Special): 200-208 doi: 10.3934/proc.2005.2005.200 +[Abstract](616) +[PDF](210.6KB)
Abstract:
We consider a class of weakly damped semilinear hyperbolic equations with memory, expressed by a convolution integral. We study the passage to the singular limit when the memory kernel collapses into the Dirac mass at zero, and we establish a convergence result for a proper family of exponential attractors.
Critical point, anti-maximum principle and semipositone p-laplacian problems
E. N. Dancer and Zhitao Zhang
2005, 2005(Special): 209-215 doi: 10.3934/proc.2005.2005.209 +[Abstract](568) +[PDF](199.3KB)
Abstract:
In this paper, we use Nehari manifold to extend the anti-maximum principle of Laplacian operator to an existence theorem for p-Laplacian ($p\not=2$), then consider the existence of nonnegative solutions to semipositone quasilinear elliptic problems $-\Delta_p u=\lambda f(u), x\in \Om; u>0, x\in \Om; u=0, x\in \Po$.
Fitzhugh-Nagumo equations in a nonhomogeneous medium
Arnold Dikansky
2005, 2005(Special): 216-224 doi: 10.3934/proc.2005.2005.216 +[Abstract](558) +[PDF](506.1KB)
Abstract:
In this paper we investigate various propagation phenomena for the FitzHugh-Nagumo system (% (4) with a nonhomogeneous threshold function $a(x)$. It is studied over a range of values $b$, $d,\varepsilon $ and function $a(x).$ Numerical simulations of system show that system (\ref{fh}) exhibits different patterns of behavior and they significantly differ from those in a homogeneous medium.
An integral representation of the determinant of a matrix and its applications
Joshua Du and Jun Ji
2005, 2005(Special): 225-232 doi: 10.3934/proc.2005.2005.225 +[Abstract](696) +[PDF](177.7KB)
Abstract:
The Hadamard determinant theorem states that the ratio of the determinant of a square matrix over the complex field to the product of its main diagonal elements is less than or equal to one for a positive definite Hermitian matrix. An integral representation of this ratio for both positive definite Hermitian matrix and diagonally dominant real matrix is given in this paper. Using this new identity, an alternative proof of the famous Hadamard determinant theorem is discussed. In addition, a lower bound of determinant in terms of the product of the main diagonal elements is given. Finally, a numerical algorithm fundamentally different from current approaches in literature is also proposed for the computation of the determinant of a small and dense matrix. Numerical experiments indicate that this new approach is robust.
On some fractional differential equations in the Hilbert space
Mahmoud M. El-Borai
2005, 2005(Special): 233-240 doi: 10.3934/proc.2005.2005.233 +[Abstract](665) +[PDF](179.6KB)
Abstract:
Let $A$ be a closed linear operator defined on a dense set in the Hilbert space $H$. Fractional evolution equations of the form $\frac{d^\alpha u(t)}{dt^\alpha} = Au(t), 0 < \alpha \leq 1$, are studied in $H$, for a wide class of the operators $A$. Some properties of the solutions of the Cauchy problem for the considered equation are studied under suitable conditions . It is proved also that there exists a dense set $S$ in $H$, such that if the initial condition $u(0)$ is an element of $S$, then there exists a solution $u(t)$ of the considered Cauchy problem. Applications to general partial differential equations of the form $$ \frac{\partial^\alpha u(x,t)}{\partial t^\alpha} = \sum_{|q| \leq m} a_q(x) D^q u(x,t) $$ are given without any restrictions on the characteristic form $\sum_{|q|=m} a_\alpha(x) \xi^q$, where $D^q = D_1^{q_1} ... D_n^{q_n}, x = (x_1, ..., x_n), D_j= \frac{\partial}{\partial x_j}, \xi^q = \xi_1^{q_1}, ..., \xi_n^{q_n}, |q| = q_1 + ... + q_n$, and $q = (q_1, ..., q_n)$ is a multi index.\par}
Numerical results for floating drops
Alan Elcrat and Ray Treinen
2005, 2005(Special): 241-249 doi: 10.3934/proc.2005.2005.241 +[Abstract](564) +[PDF](249.6KB)
Abstract:
Numerical results are presented for various configurations that may be described as axisymmetric floating drops. In each case the drops are constructed by matching solutions of the differential equations for the axisymmetric capillary surfaces to solve a free boundary problem. These include both ``heavy'' and ``light'' drops in which the density of the fluid in the drop is either larger or smaller than that of the reservoir on which it rests. The reservoir may be either infinite or contained in a finite container. Of particular interest are heavy drops which have multiple necks reminiscent of classic results of Kelvin for pendent drops.
Periodic solutions in fading memory spaces
Khalil Ezzinbi, James H. Liu and Nguyen Van Minh
2005, 2005(Special): 250-257 doi: 10.3934/proc.2005.2005.250 +[Abstract](537) +[PDF](201.7KB)
Abstract:
For $A(t)$ and $f(t,x,y)$ $T$-periodic in $t$, consider the following evolution equation with infinite delay in a general Banach space $X$, $$u^\prime (t)+ A(t)u(t)=f(t,u(t),u_t),\;\; t> 0,\;\;u(s) =\phi (s),\;\;s \leq 0, $$ where the resolvent of the unbounded operator $A(t)$ is compact, and $u_t (s)=u(t+s),\; s\leq 0$. We will work with general fading memory phase spaces satisfying certain axioms, and derive periodic solutions. We will show that the related Poincar\'{e} operator is condensing, and then derive periodic solutions using the boundedness of the solutions and some fixed point theorems. This way, the study of periodic solutions for equations with infinite delay in general Banach spaces can be carried to fading memory phase spaces. In doing so, we will improve a condition of [4] and extend the results of [7,8].
Dynamics of the discrete Chaplygin sleigh
Yuri N. Fedorov and Dmitry V. Zenkov
2005, 2005(Special): 258-267 doi: 10.3934/proc.2005.2005.258 +[Abstract](489) +[PDF](246.2KB)
Abstract:
This paper studies the dynamics of the discrete Chaplygin sleigh. Properties such as discrete momentum and measure conservation are explored.
Stability and pattern in two-patch predator-prey population dynamics
Wei Feng and Jody Hinson
2005, 2005(Special): 268-279 doi: 10.3934/proc.2005.2005.268 +[Abstract](768) +[PDF](235.7KB)
Abstract:
In this paper we explore the dynamics of predator-prey interactions in two patches which are coupled by the diffusion of the predator. The purpose of this exploration to find upper and lower bounds for the populations, and to discuss the complexity and stability of the equilibriums. Some of our results relax the conditions, which are given in earlier papers, necessary for stability of the equilibrium solutions associated with this model. Numerical simulations are also provided to graphically demonstrate the population dynamics of this model utilizing some of the relaxed conditions for stability and new conditions for instability.
Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains
Ilaria Fragalà, Filippo Gazzola and Gary Lieberman
2005, 2005(Special): 280-286 doi: 10.3934/proc.2005.2005.280 +[Abstract](562) +[PDF](200.9KB)
Abstract:
For a class of anisotropic elliptic problems in bounded domains $\Omega$ we show that the convexity of $\Omega$ plays an important role in regularity and nonexistence results. Using recent results in [9] we improve some statements in [3].
Dynamical structure of one-phase model of solid combustion
Michael L. Frankel and Victor Roytburd
2005, 2005(Special): 287-296 doi: 10.3934/proc.2005.2005.287 +[Abstract](607) +[PDF](374.3KB)
Abstract:
We present results of a numerical study of complex dynamics generated by a one-phase free-boundary problem with kinetics modeling gasless combustion. We study dynamical structure generated by the problem using bifurcational diagrams obtained through the correlation dimension. In the case of periodically varying initial concentration the problem exhibits frequency locking. We also demonstrate that a finite-dimensional reduction via a method of collocations leads to a similar dynamical structure.
Spacecraft dynamics near a binary asteroid
F. Gabern, W.S. Koon and Jerrold E. Marsden
2005, 2005(Special): 297-306 doi: 10.3934/proc.2005.2005.297 +[Abstract](553) +[PDF](876.0KB)
Abstract:
We study a simple model for an asteroid pair, namely a planar system consisting of a rigid body and a sphere. This model is interesting because it is one of the simplest that captures the coupling between rotational and translational dynamics. By assuming that the binary is in a relative equilibria of the system, we construct a model for the motion of a spacecraft about this asteroid pair without affecting its motion (that is, we consider a restricted problem). This model can be studied as a perturbation of the standard Restricted Three Body Problem (RTBP). We use the stable zones near the triangular relative equilibrium points of the binary and a normal form of the Hamiltonian to compute stable periodic and quasi-periodic orbits for the spacecraft, which enable it to observe the binary while the binary orbits around the Sun.
Water-gas flow in porous media
Cedric Galusinski and Mazen Saad
2005, 2005(Special): 307-316 doi: 10.3934/proc.2005.2005.307 +[Abstract](482) +[PDF](220.6KB)
Abstract:
The goal of this paper is to establish a global existence theorem for a strongly degenerate problem modeling water-gas flows mixing compressible and incompressible fluids. The problem is strongly nonlinear and an evolution term degenerates as well as a diffusion term.
Nonlinear hemivariational inequalities with eigenvalues near zero
Leszek Gasiński and Nikolaos S. Papageorgiou
2005, 2005(Special): 317-326 doi: 10.3934/proc.2005.2005.317 +[Abstract](645) +[PDF](234.3KB)
Abstract:
In this paper we consider an eigenvalue problem for a quasilinear hemivariational inequality of the type $-\Delta_p x(z) -\lambda f(z,x(z))\in \partial j(z,x(z))$ with null boundary condition, where $f$ and $j$ satisfy ``$p-1$-growth condition''. We prove the existence of a nontrivial solution for $\lambda$ sufficiently close to zero. Our approach is variational and is based on the critical point theory for nonsmooth, locally Lipschitz functionals due to Chang [4].
Dynamics of microfluidic mixing using time pulsing
Arnaud Goullet, Ian Glasgow and Nadine Aubry
2005, 2005(Special): 327-336 doi: 10.3934/proc.2005.2005.327 +[Abstract](689) +[PDF](930.5KB)
Abstract:
Many microfluidic applications require the mixing of reagents, but efficient mixing in these laminar systems remains a challenge. In this paper, we consider further the method of pulsed flow mixing which takes advantage of time dependency rather than spatial complexity. In particular, using computational fluid dynamics (CFD) we analyze the dynamics of the flow when the two inlets are pulsed at $90^\circ$ and $180^\circ$ out of phase. Both cases achieve enhanced mixing although better results occur in the first case. This is apparent in the concentration level plots as well as in the shape of material lines which show strong repeated stretching and folding at the confluence region.
Multiple positive solutions to a three point third order boundary value problem
John R. Graef and Bo Yang
2005, 2005(Special): 337-344 doi: 10.3934/proc.2005.2005.337 +[Abstract](785) +[PDF](186.6KB)
Abstract:
The authors consider the boundary value problem $$ \begin{cases} u'''(t) = q(t)f(u), \quad 0 < t < 1, \\ u(0) = u'(p) = u''(1) = 0, \end{cases} $$ where $p \in(\frac{1}{2},1)$ is a constant. They give sufficient conditions for the existence of multiple positive solutions to this problem. In so doing, they are able to improve some recent results on this problem. Examples are included to illustrate the results.
Optimal control of a commercial loan repayment plan
Ellina Grigorieva and Evgenii Khailov
2005, 2005(Special): 345-354 doi: 10.3934/proc.2005.2005.345 +[Abstract](579) +[PDF](124.4KB)
Abstract:
We consider a controlled system of differential equations modeling a firm that takes a loan in order to expand its production activities. The objective is to determine the optimal loan repayment schedule using the variables of the business current profitability, the bank's interest rate on the loan and the cost of reinvestment of capital. The portion of the annual profit which a firm returns to the bank and the value of the total loan taken by the firm are control parameters. We consider a linear production function and investigate the attainable sets for the system analytically and numerically. Optimal control problems are stated and their solutions are found using attainable sets. Attainable sets for different values of the parameters of the system are constructed with the use of a computer program written in MAPLE. Possible economic applications are discussed.
A global semi-Lagrangian spectral model of shallow water equations with time-dependent variable resolution
Daniel Guo and John Drake
2005, 2005(Special): 355-364 doi: 10.3934/proc.2005.2005.355 +[Abstract](573) +[PDF](189.3KB)
Abstract:
A time-dependent focusing grid works together with the formulation of a semi-implicit, semi-Lagrangian spectral method for the shallow-water equations in a rotated and stretched spherical geometry. The conformal mapping of the underlying discrete grid based on the Schmidt transformation, focuses grid on a particular region or path with variable resolution. A new advective form of the vorticity-divergence equations allows for the conformal map to be incorporated while maintaining an efficient spectral transform algorithm. A shallow water model on the sphere is used to test the spectral model with variable resolution. We are able to focus on a specified location resolving local details of the flow. More importantly, we could follow the features of the flow at all time.
Riesz basis property and related results for a Rao-Nakra sandwich beam
Scott W. Hansen and Rajeev Rajaram
2005, 2005(Special): 365-375 doi: 10.3934/proc.2005.2005.365 +[Abstract](504) +[PDF](246.9KB)
Abstract:
We consider a three layer Rao-Nakra sandwich beam with distinct wave speeds. We prove that the eigenvectors form a Riesz basis for the natural energy space. In the damped case, we give precise conditions under which there is a uniform exponential decay of energy. We also consider the problem of boundary control using bending moment and lateral force control at one end. We prove that the space of exact controllability has finite co-dimension and provide sufficient conditions (related to small damping) for exact controllability to a zero energy state.
Nonexistence of positive solutions of quasilinear elliptic equations with singularity on the boundary in strip-like domains
Takahiro Hashimoto
2005, 2005(Special): 376-385 doi: 10.3934/proc.2005.2005.376 +[Abstract](704) +[PDF](215.9KB)
Abstract:
In this paper, we discuss the nonexistence of positive solutions for nonlinear elliptic equations with singularity on the boundary in infinite strip-like domains. The main results are the nonexistence in eigen-value and sub principal case and they are shown by giving an argument concerning the simplicity of the first eigenvalue for generalized eigenvalue problems combined with translation invariance of the domain. We also show another nonexistence result via a modified version of Pohozaev-type identity.
A class of integrodifferential equations and applications
Min He
2005, 2005(Special): 386-396 doi: 10.3934/proc.2005.2005.386 +[Abstract](484) +[PDF](191.9KB)
Abstract:
This work considers an abstract integrodifferential equation in Banach space: \be & &u'(t)=A(\ep)\left[u(t)+\int_0^t F(t-s)u(s)\,ds\right]+Ku(t)+f(t), \,\,t\ge0, \nonumber\\ & &u(0)=u_0, \nonumber \ee where $A(\ep)$ is a closed, linear, and non-densely defined operator, $\ep$ is a multi-parameter, and $K$ and $F(t)$ are bounded operators. The purpose of this work is to study effect of the parameter on the solution of the equation. The approach used is based on the integrated semigroup theory. Two methods are employed to determine the continuity in parameter $\ep$ of integrated semigroup, which is generated by $A(\ep)$. The main theorems on the integrated semigroup can be effectively used to obtain the similar results for the solution of the equation. The applications of these results to some equations of viscoelasticity are discussed.
Magnetic hydrodynamics equations in movingboundaries
Hiroshi Inoue
2005, 2005(Special): 397-402 doi: 10.3934/proc.2005.2005.397 +[Abstract](522) +[PDF](177.8KB)
Abstract:
We show the existence and uniqueness of the strong solutions of the zero Dirichlet problems of the coupled Navier-Stokes equations which govern the incompressible magnetic fluid. We derive the existence of a unique strong solution for suitable initial conditions which depend on the space dimensions. The proofs have been shown to apply the contraction mapping theorem by using the theory of the sub-differential operators.
A bound for ratios of eigenvalues of Schrodinger operators on the real line
Miklós Horváth and Márton Kiss
2005, 2005(Special): 403-409 doi: 10.3934/proc.2005.2005.403 +[Abstract](585) +[PDF](184.5KB)
Abstract:
We give upper estimates of ratios of eigenvalues of Schrödinger operators with nonnegative single-well potentials tending to infinity for large $|x|$, corresponding to previous estimates on a finite interval.
Traveling wave solutions in cellular neural networks with multiple time delays
Cheng-Hsiung Hsu and Suh-Yuh Yang
2005, 2005(Special): 410-419 doi: 10.3934/proc.2005.2005.410 +[Abstract](678) +[PDF](178.9KB)
Abstract:
This work investigates the existence of traveling wave solutions of the cellular neural network distributed in $\mathbb{Z}^1$ with multiple time delays. Applying the method of step with the help of the characteristic function, we can figure out an analytic solution in an explicit form with many parameters. We then focus on the mechanism for producing the so-called camel-like traveling wave solutions and study the effect of delays on the shape of solutions. Some numerical results are also provided to demonstrate the theoretical analysis.
Stability of cellular neural network with small delays
Ying Sue Huang and Chai Wah Wu
2005, 2005(Special): 420-426 doi: 10.3934/proc.2005.2005.420 +[Abstract](583) +[PDF](145.1KB)
Abstract:
We consider a system of cellular neural networks with delays. By using appropriate Lyapunov functions, we obtain sufficient conditions so that the system is globally stable when the delay is small enough.
Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations
J. Húska, Peter Poláčik and M.V. Safonov
2005, 2005(Special): 427-435 doi: 10.3934/proc.2005.2005.427 +[Abstract](706) +[PDF](200.4KB)
Abstract:
We consider the Dirichlet problem for nonautonomous second order parabolic equations with bounded measurable coefficients on bounded Lipschitz cylinders. We discuss the exponential separation between positive and sign changing solutions and its consequences on principal eigenvalues, eigenfunctions in the time-independent case, and principal Lyapunov exponent and principal Floquet bundle in the general case.
Eigenvalues and positive solutions of odes involving integral boundary conditions
Gennaro Infante
2005, 2005(Special): 436-442 doi: 10.3934/proc.2005.2005.436 +[Abstract](614) +[PDF](178.0KB)
Abstract:
Using the theory of fixed point index we obtain existence of at least one or of multiple positive solutions of some nonlocal boundary value problems subject to integral boundary conditions. We also study the existence of positive eigenvalues for these problems.
Existence of a stable set for some nonlinear parabolic equation involving critical Sobolev exponent
Michinori Ishiwata
2005, 2005(Special): 443-452 doi: 10.3934/proc.2005.2005.443 +[Abstract](545) +[PDF](245.5KB)
Abstract:
In this paper, we discuss the asymptotic behavior of some solutions for nonlinear parabolic equation in ${\Bbb R}^N$ involving critical Sobolev exponent. For the subcritical problem (with bounded domain), it is well-known that the solution which intersects the "stable set" must be a global one. But for the critical problem, it is not known whether the same conclusion holds or not. In this paper, we shall show that, in the critical case, the same conclusion actually holds true. The proof requires the concentration compactness type argument.
Nim-induced dynamical systems over Z2
Michael A. Jones and Diana M. Thomas
2005, 2005(Special): 453-462 doi: 10.3934/proc.2005.2005.453 +[Abstract](486) +[PDF](136.3KB)
Abstract:
Winning and losing positions in the well-known two-player game, Nim, are defined recursively as a two symbol sequence depending on a $k$-parameter set known as the subtraction set. In this paper, we write the recursion as a nonlinear dynamical system defined on the phase space $\mathbb Z_2^{s_k}$ with the binary sequence for Nim generated by the appropriate initial conditions. The transient dynamics and Garden of Eden points are completely determined for arbitrary-sized subtraction sets. A characterization of cycle lengths for two parameter subtraction sets is determined. Extensions of the two parameter case to an arbitrary-sized subtraction set are explored.
Mean square approximation of multi dimensional reflecting fractional Brownian motion via penalty method
S. Kanagawa, K. Inoue, A. Arimoto and Y. Saisho
2005, 2005(Special): 463-475 doi: 10.3934/proc.2005.2005.463 +[Abstract](704) +[PDF](175.8KB)
Abstract:
We investigate the mean square error of the Euler-Maruyama type approximate solution of multi dimensional reflecting fractional Brownian motion using the penalty method. Furthermore we show some examples of the reflecting fractional Brownian motion with several boundaries.
Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations
N. I. Karachalios, H. E. Nistazakis and A. N. Yannacopoulos
2005, 2005(Special): 476-486 doi: 10.3934/proc.2005.2005.476 +[Abstract](604) +[PDF](281.9KB)
Abstract:
We study the asymptotic behavior of complex discrete evolution equations of Ginzburg- Landau type. Depending on the nonlinearity and the data of the problem, we find different dynamical behavior ranging from global existence of solutions and global attractors, to blow up in finite time. We provide estimates for the blow up time, depending not only on the initial data but also on the size of the lattice. The theoretical estimates, are tested by numerical simulations.
Dynamics of heterogeneous populations and communities and evolution of distributions
Georgy P. Karev
2005, 2005(Special): 487-496 doi: 10.3934/proc.2005.2005.487 +[Abstract](453) +[PDF](215.0KB)
Abstract:
Most population models assume that individuals within a given population are identical, that is, the fundamental role of variation is ignored. Inhomogeneous models of populations and communities allow for birth and death rates to vary among individuals; recently, theorems of existence and asymptotic of solutions of such models were investigated. Here we develop another approach to modeling heterogeneous populations by reducing the model to the Cauchy problem for a special system of ODEs. As a result, the total population size and current distribution of the vector-parameter can be found in explicit analytical form or computed effectively. The developed approach is extended to the models of inhomogeneous communities.
Attractivity properties of oscillator equations with superlinear damping
János Karsai and John R. Graef
2005, 2005(Special): 497-504 doi: 10.3934/proc.2005.2005.497 +[Abstract](576) +[PDF](361.0KB)
Abstract:
The authors investigate the asymptotic behavior of solutions of the damped nonlinear oscillator equation $$x''+ a(t)|x'|^\alpha \sgn(x') + f(x)=0, $$ where $uf(u) > 0$ for $u \neq 0$, $a(t)\geq 0$, and $\alpha\geq 1$. The case $\alpha=1$ has been investigated by a number of other authors. There are also some results for the case $\alpha>1$, but they are not really based on the power $\alpha$, although it plays an essential role in the behavior of the solutions. In this paper, we give new attractivity results for the large damping case, $a(t) \geq a_0 > 0$, that improve previously known results. Our conditions involve the power $\alpha$ in such a way that our results reduce directly to known conditions in the case $\alpha=1$. Some open problems for future research are also indicated.
Characterizations of conditionally complete partially ordered sets
Paula Kemp
2005, 2005(Special): 505-509 doi: 10.3934/proc.2005.2005.505 +[Abstract](457) +[PDF](142.3KB)
Abstract:
In the mid 1950's Tarski showed that a complete lattice P has the property that every increasing increasing function from P into itself has a fixed point. Anne Davis proved the converse of this result that every lattice with the fixed point property is complete. In this paper, the author proves new equivalences for conditionally complete partially ordered sets with a type of fixed point property. Some comments about these theorems are also given in the paper.
On lane-emden type systems
Philip Korman and Junping Shi
2005, 2005(Special): 510-517 doi: 10.3934/proc.2005.2005.510 +[Abstract](493) +[PDF](120.7KB)
Abstract:
We consider a class of singular systems of Lane-Emden type \begin{equation} \nonumber \begin{cases} \Delta u + \la u^{p_1} v^{q_1}=0, & x\in D,\\ \Delta v + \la u^{p_2} v^{q_2}=0, & x\in D,\\ u=v=0, & x\in \partial D, \end{cases} \end{equation} with $p_1\le 0, \; p_2> 0, \; q_1> 0, \; q_2\le 0$, and $D$ a smooth domain in $\R^n$. In case the system is sublinear we prove existence of a positive solution. If $D$ is a ball in $\R^n$, we prove both existence and uniqueness of positive radially symmetric solution.
2-wild trajectories
Krystyna Kuperberg
2005, 2005(Special): 518-523 doi: 10.3934/proc.2005.2005.518 +[Abstract](530) +[PDF](170.1KB)
Abstract:
Every orientable boundaryless 3-manifold admits a continuous dynamical system with a discrete set of fixed points and every non-trivial semi-trajectory wild.
On the measure attractor of a cellular automaton
Petr Kůrka
2005, 2005(Special): 524-535 doi: 10.3934/proc.2005.2005.524 +[Abstract](618) +[PDF](275.4KB)
Abstract:
Given a cellular automaton $F:A^{\ZZ} \to A^{\ZZ}$, we define its small quasi-attractor $\Qq_F$ as the nonempty intersection of all shift-invariant attractors of all $F^q\sigma^p$, where $q>0$ and $p\in\ZZ$. The measure attractor $\Mm_F$ is the closure of the supports of the members of the unique attractor of $F:\MMM_{\sigma}(A^{\ZZ}) \to \MMM_{\sigma}(A^{\ZZ})$ in the space of shift-invariant Borel probability measures.
Coexistence states for a prey-predator model with cross-diffusion
Kousuke Kuto and Yoshio Yamada
2005, 2005(Special): 536-545 doi: 10.3934/proc.2005.2005.536 +[Abstract](533) +[PDF](229.9KB)
Abstract:
This paper discusses a prey-predator system with cross-diffusion. We can prove that the set of coexistence steady-states of this system contains an S or $\supset$-shaped branch with respect to a bifurcation parameter in a large cross-diffusion case. We give also some criteria on the stability of these positive steady-states. Furthermore, we find the Hopf bifurcation point on the steady-state solution branch in a certain case.
Properties of kernels and eigenvalues for three point boundary value problems
K. Q. Lan
2005, 2005(Special): 546-555 doi: 10.3934/proc.2005.2005.546 +[Abstract](605) +[PDF](198.1KB)
Abstract:
We investigate the properties of a kernel arising from a three point boundary value problem. We seek a lower bound for the kernel and evaluate the optimal values for the integrals related to the kernel. The smallest positive characteristic value for a linear second ordinary differential equation with a three point boundary condition is estimated by using our lower bound. These optimal values and the estimates for characteristic values are useful in studying the existence of nonzero positive solutions for the boundary value problem.
Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument
Irena Lasiecka and Roberto Triggiani
2005, 2005(Special): 556-565 doi: 10.3934/proc.2005.2005.556 +[Abstract](531) +[PDF](237.2KB)
Abstract:
We prove exact controllability in the energy space of semilinear wave equations with $L_2$-Neumann boundary controls. The present proof integrates a double compactness/uniqueness PDE-based argument in establishing the uniform continuous observability inequality of the linearized, dual, uncontrolled problem with the abstract operator-theoretic approach proposed in [11], [21]. The latter approach analyzes suitable families of collectively compact operators [1] and ultimately culminates with the application of a global inversion theorem (homeomorphism) [4], [17] to the original controlled semilinear problem.
Monotone local semiflows with saddle-point dynamics and applications to semilinear diffusion equations
Monica Lazzo and Paul G. Schmidt
2005, 2005(Special): 566-575 doi: 10.3934/proc.2005.2005.566 +[Abstract](623) +[PDF](231.7KB)
Abstract:
Consider a monotone local semiflow in the positive cone of a strongly ordered Banach space, for which $0$ and $\infty$ are stable attractors, while all nontrivial equilibria are unstable. We prove that under suitable monotonicity, compactness, and smoothness assumptions, the two basins of attraction, $\Bz$ and $\Bi$, are separated by a Lipschitz manifold $\M$ of co-dimension one that forms the common boundary of $\Bz$ and $\Bi$. This abstract result is applied to a class of semilinear reaction-diffusion equations with superlinear, yet subcritical reaction terms.
Partial regularity of solutions to a class of strongly coupled degenerate parabolic systems
Dung Le
2005, 2005(Special): 576-586 doi: 10.3934/proc.2005.2005.576 +[Abstract](612) +[PDF](215.4KB)
Abstract:
Using the method of heat approximation, we will establish partial regularity results for bounded weak solutions to certain strongly coupled degenerate parabolic systems.
Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem
Shin-Yi Lee, Shin-Hwa Wang and Chiou-Ping Ye
2005, 2005(Special): 587-596 doi: 10.3934/proc.2005.2005.587 +[Abstract](761) +[PDF](187.4KB)
Abstract:
We establish explicit necessary and sufficient conditions for the existence of a dead core solution of a $p$-Laplacian steady-state reaction-diffusion problem. The gap is extremely small between the explicit necessary condition and the explicit sufficient condition for the existence of a dead core solution.
An algebraic approach to building interpolating polynomial
Aihua Li
2005, 2005(Special): 597-604 doi: 10.3934/proc.2005.2005.597 +[Abstract](574) +[PDF](182.2KB)
Abstract:
In this paper, a different approach to constructing interpolating (multivariable) polynomials is given, which uses Gröbner Bases Techniques. The well known Buchberger -Möller Algorithm is applied in the computation. Furthermore, this algebraic method can be used to construct all the polynomial models of a discrete time series, by repeatedly using the same algorithm. The advantage of this method is that it makes it possible for researchers to search different types of resulting polynomials, such as those involving certain favored variables, or those with small total degrees.
A new regularity estimate for solutions of singular parabolic equations
Gary Lieberman
2005, 2005(Special): 605-610 doi: 10.3934/proc.2005.2005.605 +[Abstract](540) +[PDF](193.2KB)
Abstract:
In 1982, K. Ecker showed that solutions of the parabolic equation \[ u_t=\operatorname {div} \left( \frac {Du}{(1+|Du|^2)^{1/2}}\right) + H(x,u) \] have a very unusual regularity property: The interior regularity of $u$ is determined only by its initial regularity. In this note, we show that a similar result is true for a general class of equations. The model equation is \[ u_t=\operatorname {div} \left( |Du|^{p-2} {Du}\right) \] with $1

Variational analysis of energy-enstrophy theories on the sphere
Chjan C. Lim and Da Zhu
2005, 2005(Special): 611-620 doi: 10.3934/proc.2005.2005.611 +[Abstract](561) +[PDF](211.5KB)
Abstract:
Kraichnan's energy-enstrophy theory for 2D inviscid flows on the sphere is discussed within a variational framework. We will give necessary and sufficient conditions for the existence and uniqueness for the extremals of the energy with zero circulation under different values of the temperature parameter $\beta$. The unboundedness of the augmented energy functional in this model when $\beta$ is located in the certain intervals will be shown and related to energy catastrophe of the energy-enstrophy model.
On the largest common fixed point of a commuting family of isotone maps
Teck-Cheong Lim
2005, 2005(Special): 621-623 doi: 10.3934/proc.2005.2005.621 +[Abstract](490) +[PDF](122.8KB)
Abstract:
We prove that in a complete partially ordered set with a largest point, every commutative family of isotones has the largest common fixed point. This result for a single mapping was used recently by Ok (2004) to study fixed set theory and its applications in economics.
Geometric approach to a singular boundary value problem with turning points
Weishi Liu
2005, 2005(Special): 624-633 doi: 10.3934/proc.2005.2005.624 +[Abstract](531) +[PDF](180.7KB)
Abstract:
The singularly perturbed boundary value problem $\epsilon \ddot x=f(x,t;\epsilon)\dot x$, $x(-1;\epsilon)=A$, $x(0;\epsilon)=B$ is studied as an application of the geometric singular perturbation theory for turning points. The key ingredients are: the delay of stability loss that characterizes all possible singular orbits of the boundary value problem, and the exchange lemmas for problems with turning points as the geometric tool to show the existence of solutions shadowing singular orbits.
Exponential attractors for 2d magneto-micropolor fluid flow in bounded domain
Kei Matsuura
2005, 2005(Special): 634-641 doi: 10.3934/proc.2005.2005.634 +[Abstract](455) +[PDF](192.9KB)
Abstract:
We show the existence of an exponential attractor for the dynamical system associated with the equations of the system of the 2D magneto-micropolar fluid flow in a bounded domain. The construction of an exponential attractor relies on the abstract theory given in [2].
Accounting for nonlinearities in mathematical modelling of quantum dot molecules
Roderick Melnik, B. Lassen, L. C Lew Yan Voon, M. Willatzen and C. Galeriu
2005, 2005(Special): 642-651 doi: 10.3934/proc.2005.2005.642 +[Abstract](660) +[PDF](279.9KB)
Abstract:
Nonlinear mathematical models are becoming increasingly important for new applications of low-dimensional semiconductor structures. Examples of such structures include quasi-zero-dimensional quantum dots that have potential applications ranging from quantum computing to nano-biological devices. In this contribution, we analyze presently dominating linear models for bandstructure calculations and demonstrate why nonlinear models are required for characterizing adequately optoelectronic properties of self-assembled quantum dots.
Existence and uniqueness of solutions of a system of nonlinear PDE for phase transitions with vector order parameter
Emil Minchev
2005, 2005(Special): 652-661 doi: 10.3934/proc.2005.2005.652 +[Abstract](589) +[PDF](188.9KB)
Abstract:
The paper deals with existence and uniqueness of solutions for a system of nonlinear PDE's which describes phase transition models with vector order parameter.
Existence and location result for a fourth order boundary value problem
Feliz Minhós, T. Gyulov and A. I. Santos
2005, 2005(Special): 662-671 doi: 10.3934/proc.2005.2005.662 +[Abstract](505) +[PDF](156.7KB)
Abstract:
In the present work we prove an existence and location result for the fourth order fully nonlinear equation% \begin{equation*} u^{(iv)}=f\left( t,u,u^{\prime },u^{\prime \prime },u^{\prime \prime \prime }\right) ,\quad 0
Controllability to trajectories for semilinear thermoelastic plates
Maria Grazia Naso
2005, 2005(Special): 672-681 doi: 10.3934/proc.2005.2005.672 +[Abstract](467) +[PDF](231.2KB)
Abstract:
The controllability to trajectories for semilinear thermoelastic plates by a control source acting only on the heat equation of the system is considered. The method we use combines the analyticity of the associated semigroup of the linearized problem and Kakutani fixed point theorem.
Maximal sustainable yield in a multipatch habitat
Igor Nazarov and Bai-Lian Li
2005, 2005(Special): 682-691 doi: 10.3934/proc.2005.2005.682 +[Abstract](479) +[PDF](149.8KB)
Abstract:
We have considered a generalized $n$-patch model of harvesting population dynamics with continuous and discrete time. The main result is the condition when parameters which maximize the total ’stationary’ yield also have to stabilize the stationary point, in order to produce and maintain sustainable yield. Conditions when reserves or no-take areas are needed to increase the yield are derived.
The dipole dynamical system
P.K. Newton
2005, 2005(Special): 692-699 doi: 10.3934/proc.2005.2005.692 +[Abstract](625) +[PDF](126.0KB)
Abstract:
A dynamical system governing the collective interaction of N point-vortexdipoles is derived.Each dipole has an inherent orientation $\psi $ and generates a velocity field that decayslike $O(\mu /2 \pi r^2)$ where $ \mu $ is the dipole strength and$r$ is the distance from the dipole.The system of N-complex ordinary differentialequations plus N-real ordinary differentialequations for the dipole positions and orientationsare derived based on theassumption that each dipole moves with and tries to align itselfwith the local fluid velocity field.
Neutral one-dimensional attractor of a two-dimensional system derived from Newton's means
Tomasz Nowicki and Grezegorz Świrszcz
2005, 2005(Special): 700-709 doi: 10.3934/proc.2005.2005.700 +[Abstract](449) +[PDF](3598.0KB)
Abstract:
We investigate a special case of Newton's means as an example of a two dimensional rational dynamical system with an observed neutral behavior. We provide the reason for such a behavior and state a program for further investigations.
Equipartition times in a Fermi-Pasta-Ulam system
Simone Paleari and Tiziano Penati
2005, 2005(Special): 710-719 doi: 10.3934/proc.2005.2005.710 +[Abstract](546) +[PDF](349.0KB)
Abstract:
We investigate with numerical methods the celebrated Fermi-Pasta- Ulam model, a chain of non-linearly coupled oscillators with identical masses. We are interested in the evolution towards equipartition when energy is initially given to one or a few modes. In previous works we considered the initial energy being given on the lower part of the spectrum. Using the spectral entropy as a numerical indicator we obtained a strong indication that the relaxation time to equipartition increases exponentially with an inverse power of the specific energy. Such a scaling appears to remain valid in the thermodynamic limit. In this paper we explore the dynamics obtained with the initial excitation on the high frequency modes, and we obtain also in this case indication of exponentially long times to equipartition.
Minimum degrees of polynomial models on time series
Chuang Peng
2005, 2005(Special): 720-729 doi: 10.3934/proc.2005.2005.720 +[Abstract](637) +[PDF](198.7KB)
Abstract:
This paper studies polynomial models of time series. The focus will be on minimum degrees of polynomial modeling, in particular, the minimum degrees for arbitrary tail row. The paper proves decomposition theorems to reduce the associated matrices of time series to various matrix blocks. It introduces an augmented matrix of the associated matrix and gives a simple equivalent condition for existence of linear models. Moreover, it provides a new algorithm to get polynomial models, which improves the upper bound on the minimum degrees to $\le m-\bar l+1$ for an $m+1$ step time series with its augmented matrix of rank $\bar l$.
Unique summing of formal power series solutions to advanced and delayed differential equations
David W. Pravica and Michael J. Spurr
2005, 2005(Special): 730-737 doi: 10.3934/proc.2005.2005.730 +[Abstract](588) +[PDF](213.6KB)
Abstract:
The analytic delayed-differential equation $z^2 \psi ^{\ \! \prime } (z) \ + \ \psi (z/q) \ = \ z$ for $q>1$ has a solution which can be expressed as a formal power series. A $q$-advanced Laplace-Borel kernel provides for the construction of an analytic solution whose domain is the right half plane with vertex at the initial point $z=0$. This method is extended to provide a continuous family of solutions, of which a subfamily extends to a punctured neighborhood of $z=0$ on the logarithmic Riemann surface. Conditions are given on the asymptotics of $\psi ^{\ \! \prime } (z)$ near $z=0$ to ensure uniqueness.
Anomalous exponents and RG for nonlinear diffusion equations
Yuanwei Qi
2005, 2005(Special): 738-745 doi: 10.3934/proc.2005.2005.738 +[Abstract](556) +[PDF](191.3KB)
Abstract:
In this paper, we discuss how to combine Renormalization Group Methods (RG) and classical PDE techniques to study nonlinear diffusion equations and systems with critical nonlinearity. In particular, we demonstrate, using several examples, the successful application of RG, when sharp a priori estimates are derived, in showing universal global dynamics involving anomalous exponents for nonlinear systems in $R^n$.
Null controllability of a damped Mead-Markus sandwich beam
Rajeev Rajaram and Scott W. Hansen
2005, 2005(Special): 746-755 doi: 10.3934/proc.2005.2005.746 +[Abstract](525) +[PDF](238.5KB)
Abstract:
The Mead-Markus sandwich beam model with shear damping is shown to be null controllable modulo a one dimensional state in an arbitrarily short time. The moment method is used to obtain this result.
Subharmonic bifurcations of localized solutions of a discrete NLS equation
Vassilis Rothos
2005, 2005(Special): 756-767 doi: 10.3934/proc.2005.2005.756 +[Abstract](715) +[PDF](267.0KB)
Abstract:
Using an analytical approach, we derive an explicit formula for the subharmonic Mel'nikov potential ${\rm L}^{^{{\p}/{\q}}}$ for perturbations of twist maps. Our method based on the integrability of map and the variational approach of twist map. If ${\rm L}^{^{{\p}/{\q}}}$ is non--constant the perturbed twist map is non--integrable and all the resonant curves are destroyed for $\abs{\varepsilon}\ll 1$. We also apply our result to show the existence of such subharmonic bifurcations for a mapping representing localized oscillatory solutions of a discrete NLS equation with conservative and dissipative perturbations.
Dynamics of noninvertibility in delay equations
Evelyn Sander, E. Barreto, S.J. Schiff and P. So
2005, 2005(Special): 768-777 doi: 10.3934/proc.2005.2005.768 +[Abstract](796) +[PDF](265.8KB)
Abstract:
Models with a time delay often occur, since there is a naturally occurring delay in the transmission of information. A model with a delay can be noninvertible, which in turn leads to qualitative di erences between the dynamical properties of a delay equation and the familiar case of an ordinary di erential equation. We give speci c conditions for the existence of noninvertible solutions in delay equations, and describe the consequences of noninvertibility.
Dissipation of mean energy of discretized linear oscillators under random perturbations
Henri Schurz
2005, 2005(Special): 778-783 doi: 10.3934/proc.2005.2005.778 +[Abstract](624) +[PDF](180.9KB)
Abstract:
This paper deals with the problem of correct asymptotic dissipation of mean energy functional related to numerical integration of systems of uncoupled linear oscillators under random perturbations. It is shown that the drift-implicit trapezoidal method provides numerical approximations which possess the correct asymptotic behavior of their mean energy functional compared to that of the underlying exact solution as integration time t advances to infinity.
Existence of guided modes on periodic slabs
Stephen P. Shipman and Darko Volkov
2005, 2005(Special): 784-791 doi: 10.3934/proc.2005.2005.784 +[Abstract](583) +[PDF](189.4KB)
Abstract:
We prove the existence of bound guided modes for the Helmholtz equation on lossless penetrable periodic slabs. We handle both robust modes, for which no Bragg harmonics propagate away from slab, as well as nonrobust standing modes, which exist in the presence of propagating Bragg harmonics. The latter are made possible by symmetries of the slab structure, which prevent coupling of energy to the propagating harmonics. These modes are isolated in wavevector-frequency space, as they disappear under a perturbation of the wavevector. The main tool is a volumetric integral equation of Lippmann-Schwinger type that has a self-adjoint kernel.
An adaptive splitting algorithm for the sine-Gordon equation
Qin Sheng, David A. Voss and Q. M. Khaliq
2005, 2005(Special): 792-797 doi: 10.3934/proc.2005.2005.792 +[Abstract](565) +[PDF](6546.4KB)
Abstract:
This preliminary investigation concerns an adaptive splitting scheme for the numerical solution of two dimensional sine-Gordon equation. The dispersive wave equation allows for soliton-alike solutions, an ubiquitous phenomenon in a large variety of physical problems. The system of nonlinear differential equations obtained via the method of lines is then attached by a recurrence procedure whose solution yields second order accuracy. The numerical solution of the system is designed using the Peaceman-Rachford splitting to avoid solving a nonlinear system of equations at each step and allows more efficient implementations of the difference scheme.
Semilinear elliptic equations with generalized cubic nonlinearities
Junping Shi and R. Shivaji
2005, 2005(Special): 798-805 doi: 10.3934/proc.2005.2005.798 +[Abstract](454) +[PDF](231.1KB)
Abstract:
A semilinear elliptic equation with generalized cubic nonlinearity is studied. Global bifurcation diagrams and the existence of multiple solutions are obtained and in certain cases, exact multiplicity is proved.
Operator splitting method for friction constrained dynamical systems
Liejune Shiau and Roland Glowinski
2005, 2005(Special): 806-815 doi: 10.3934/proc.2005.2005.806 +[Abstract](566) +[PDF](199.1KB)
Abstract:
In a previous article [1] the time-discretization of those relations modeling a class of dynamical systems with friction was discussed. The main goal of this article is to address similar problems using a more sophisticated friction model giving a better description of the system behavior particularly when the velocities are close to zero. These investigations are motivated by the need for more accurate friction models in the software simulating the motion of mechanical systems, such as the remote manipulators of the Space Shuttle or of the International Space Station. In this article, we discuss the methods in the case of higher number of degrees of freedom elasto-dynamical systems, and the special case of one degree of freedom. The content can be summarized as follows: We discuss first models of the constrained motions under consideration, including a rigorous formulation involving a kind of dynamical multiplier. An iterative method allowing the computation of this multiplier will be discussed. Next, in order to treat friction, we introduce an implicit/explicit numerical scheme which is unconditionally stable, and easy to implement and generalize to more complicated systems. Indeed the above scheme can be coupled, via operator-splitting, to schemes classically used to solve differential equations from frictionless elasto-dynamics. The above schemes are validated through numerical experiments.
About global null controllability of a quasi-static thermoelastic contact system
Irina F. Sivergina and Michael P. Polis
2005, 2005(Special): 816-823 doi: 10.3934/proc.2005.2005.816 +[Abstract](712) +[PDF](206.3KB)
Abstract:
We study the null controllability properties of a system that models the temperature evolution of a one-dimensional thermoelastic rod that may come into contact with a rigid obstacle. Basically the system dynamics is described by a one-dimensional nonlocal heat equation with a nonlinear and nonlocal boundary condition of Newmann type at the free end of the rod. We study the control problem and treat the case when the control is distributed over the whole space domain.

In [8], we proved that if the initial condition is smooth and the system has a strong solution, then there is a control that brings the system to zero. The proof was based on changing the control variable and using Aubin's Compactness Lemma. In this paper, we focus on the null controllability of the weak solutions. We establish the existence of a control that steers the system to zero. Our approach consists of approximating the initial condition by the smooth functions and then proving that the obtained sequence of strong solutions converges to a weak solution of the desired type. The uniqueness of a weak solution is established only under special assumptions on the parameters of the system.
On the global attractor for the damped Benjamin-Bona-Mahony equation
Milena Stanislavova
2005, 2005(Special): 824-832 doi: 10.3934/proc.2005.2005.824 +[Abstract](505) +[PDF](217.2KB)
Abstract:
We present a new necessary and sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood-Paley projection operators. We then use this condition to prove the existence of an attractor for the damped \bbme in the phase space $H^1({\bf R})$ by showing the solutions are point dissipative and asymptotically compact. Moreover the attractor is in fact smoother and it belongs to $H^{3/2-\ve}$ for every $\ve>0$.
Colored coalescent theory
Jianjun Tian and Xiao-Song Lin
2005, 2005(Special): 833-845 doi: 10.3934/proc.2005.2005.833 +[Abstract](596) +[PDF](150.4KB)
Abstract:
We introduce a colored coalescent process which recovers random colored genealogical trees. Here a colored genealogical tree has its vertices colored black or white. Moving backward along the colored genealogical tree, the color of vertices may change only when two vertices coalesce. Explicit computations of the expectation and the cumulative distribution function of the coalescent time are carried out. For example, when $x=1/2$, for a sample of $n$ colored individuals, the expected time for the colored coalescent process to reach a black MRCA or a white MRCA, respectively, is $3-2/n$. On the other hand, the expected time for the colored coalescent process to reach a MRCA, either black or white, is $2-2/n$, which is the same as that for the standard Kingman coalescent process.
Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization
Bourama Toni
2005, 2005(Special): 846-853 doi: 10.3934/proc.2005.2005.846 +[Abstract](520) +[PDF](196.1KB)
Abstract:
We discuss polynomial 1-form small perturbation of an isochronous polynomial 1-form in the Pfaffian form $\omega_{\epsilon}=\omega_0+\epsilon \omega$ where $\omega$ is a n-degree polynomial 1-form, $\epsilon$ a small real parameter, and $\omega_0$ an isochronous 1-form with a known birational linearization $T,$ setting $\omega_0$ as the pullback 1-form $T^*\Cal I_0$ of the exact linear isochrone 1-form $\Cal I_0=dH.$ Using recursively the cohomology decompositions of $\omega$ in the related Petrov module, we construct the Bautin-like ideal of the Poincar\'e-Melnikov functions, and study the zeros of Abelian integrals over the ovals $\tilde H=T^*H=r.$ We then stabilize the sequence of the successive Melnikov functions through a multistep reduction of the system coefficients, and determine in terms of the degrees of $\tilde H$ and $\omega$ the overall upper bounds for limit cycles emerging from the polynomial deformation.
Analysis and simulations of magnetic materials
Edward Della Torre and Lawrence H. Bennett
2005, 2005(Special): 854-861 doi: 10.3934/proc.2005.2005.854 +[Abstract](575) +[PDF](207.4KB)
Abstract:
The development of mathematical models for the design of magnetic devices requires a thorough understanding of hysteresis. The classical Preisach model provided such an understanding but is unable to emulate all the properties of magnetic materials. In this paper, we summarize some of the modifications that have been proposed within the context of Preisach modelling in order to explain the observed phenomena.
Comments on radially symmetric liquid bridges with inflected profiles
Thomas I. Vogel
2005, 2005(Special): 862-867 doi: 10.3934/proc.2005.2005.862 +[Abstract](635) +[PDF](172.9KB)
Abstract:
For a liquid bridge between parallel planes which makes equal contact angles with those planes, it is already known that a pitchfork bifurcation occurs when there is an inflection in the profile curve. A geometrical argument is outlined to give an alternate and more elementary proof of this fact. In contrast to the behavior of liquid bridges between parallel planes, it is shown that a liquid bridge between spheres exists which is stable and has two inflections. Along the way, a result relating stability and \(dH/dV\) for a family of capillary surfaces is established.
Transition layers and spikes for a reaction-diffusion equation with bistable nonlinearity
Michio Urano, Kimie Nakashima and Yoshio Yamada
2005, 2005(Special): 868-877 doi: 10.3934/proc.2005.2005.868 +[Abstract](625) +[PDF](141.5KB)
Abstract:
This paper is concerned with steady-state solutions for the following reaction-diffusion equation $$ u_t= \varepsilon^2 u_{xx} + u(1-u)(u-a(x)),\quad(x,t)\in (0,1)\times(0,\infty) $$ with $u_x(0,t) = u_x(1,t) = 0$ for $t\in (0,\infty)$. Here $\varepsilon$ is a small positive parameter and $a$ is a $C^2[0,1]$ function such that $0 < a(x) < 1$ for $x\in [0,1]$ and that $\Sigma:= \{x\in(0,1); a(x) = 1/2\}$ is a nonempty finite set. It is well known that the corresponding steady-state problem admits solutions with transition layers or spikes when $\varepsilon$ is sufficiently small. We will give some information on the location of transition layers and spikes for steady-state solutions. Under certain circumstances, such solutions possess multi-layers or multi-spikes. We will also show some conditions for the appearance of multi-spikes as well as for the existence of multi-layers.
Positive radial solutions for quasilinear equations in the annulus
Haiyan Wang
2005, 2005(Special): 878-885 doi: 10.3934/proc.2005.2005.878 +[Abstract](532) +[PDF](213.3KB)
Abstract:
The paper deals with the existence of positive radial solutions for the quasilinear system $\textrm{ div} \left ( | \nabla u_i|^{p-2}\nabla u_i \right ) + f^i(u_1,...,u_n)=0,\; p>1, R_1 <|x| < R_2,\;u_i(x)=0,$ on $|x|=R_1$ and $R_2$, $i=1,...,n$, $x \in \mathbb{R}^N.$ $f^i$, $i=1,...,n,$ are continuous and nonnegative functions. Let $\vect{u}=(u_1,...,u_n),$ $\varphi(t)=|t|^{p-2}t,$ $f_0^i =\lim_{\norm{\vect{u}} \to 0} \frac{f^i(\vect{u})}{\var(\norm{\vect{u}})},$ $f_{\infty}^i =\lim_{\norm{\vect{u}} \to \infty} \frac{f^i(\vect{u})}{\var(\norm{\vect{u}})}$, $i=1,...,n,$ $\vect{f}=(f^1,...,f^n),$ $\vect{f}_0=\sum_{i=1}^n f_0^i$ and $\vect{f}_{\infty}=\sum_{i=1}^n f_{\infty}^i$. We prove that $\vect{f}_0 =0$ and $\vect{f}_{\infty}=\infty$ (superlinear) guarantee the existence of positive radial solutions for the system. We shall use fixed point theorems in a cone.
Stability and symmetry breaking of solutions of semilinear elliptic equations
Hwai-Chiuan Wang
2005, 2005(Special): 886-894 doi: 10.3934/proc.2005.2005.886 +[Abstract](606) +[PDF](126.7KB)
Abstract:
In this article, we prove that there are three unstable positive solutions of a semilinear elliptic equation in a two bumps domain or in a one hole domain in which one is axially symmetric and the other two are nonaxially symmetric.
Multiple positive solutions of some nonlinear heat flow problems
J. R. L. Webb
2005, 2005(Special): 895-903 doi: 10.3934/proc.2005.2005.895 +[Abstract](694) +[PDF](187.4KB)
Abstract:
We give improved results on the existence of multiple positive solutions for a nonlinear heat flow problem with nonlocal boundary conditions. These results utilise some recent work in \cite{jwkleig} and involve the principal eigenvalue of a related linear problem. We also investigate constants that have previously been employed in the study of positive solutions. We obtain the optimal values of these constants but show that using eigenvalues, when possible, gives better results.
An application of optimal transport paths to urban transport networks
Qinglan Xia
2005, 2005(Special): 904-910 doi: 10.3934/proc.2005.2005.904 +[Abstract](462) +[PDF](177.9KB)
Abstract:
In this article, we provide a model to study urban transport network by means of optimal transport paths recently studied by the author. Under this model, we can set up an optimal urban transport network of finite total cost which provide access to all residents from their home to their destinations. The quality of the road depends on the traffic density it carries, which make it necessary to build a large highway for heavy traffic. Moreover, we provide a reasonable pricing system for an optimal transport network, under which all residents will travel to their destinations by the network. We also studied the problem of expanding and modifying a given network.
A general class of nonlinear impulsive integral differential equations and optimal controls on Banach spaces
X. Xiang, Y. Peng and W. Wei
2005, 2005(Special): 911-919 doi: 10.3934/proc.2005.2005.911 +[Abstract](519) +[PDF](196.7KB)
Abstract:
In this paper quite general impulsive integral differential equations on Banach space are considered. Existence of $PWC-\alpha $-mild solutions is proved. Existence of optimal controls of systems governed by impulsive integral differential equations is also presented. An example is given for demonstration.
Doubly nonlinear evolution equations associated with elliptic-parabolic free boundary problems
Noriaki Yamazaki
2005, 2005(Special): 920-929 doi: 10.3934/proc.2005.2005.920 +[Abstract](489) +[PDF](235.1KB)
Abstract:
We study an abstract doubly nonlinear evolution equation associated with elliptic-parabolic free boundary problems. In this paper we show the existence and uniqueness of solution for the doubly nonlinear evolution equation. Moreover we apply our abstract results to an elliptic-parabolic free boundary problem.
Existence of monotonic traveling waves in modified RTD-based cellular neural networks
Suh-Yuh Yang and Cheng-Hsiung Hsu
2005, 2005(Special): 930-939 doi: 10.3934/proc.2005.2005.930 +[Abstract](644) +[PDF](1937.5KB)
Abstract:
We study the existence of monotonic traveling and standing wave solutions of the one-dimensional modified RTD-based cellular neural networks. Employing the techniques of monotone iteration coupled with the concept of upper and lower solutions, we can classify the monotonic traveling and standing waves with various asymptotic boundary conditions. Some numerical examples are also provided.
Wave map with potential and hypersurface flow
Jian Zhai, Jianping Fang and Lanjun Li
2005, 2005(Special): 940-946 doi: 10.3934/proc.2005.2005.940 +[Abstract](559) +[PDF](177.1KB)
Abstract:
The simplified equation of the dynamics of weak ferromagnets magnetization is related to wave maps with potential. The global existence and behavior of solutions as parameter $\epsilon\to 0$ are obtained.
Notes on the convergence and applications of surrogate optimization
Chunlei Zhang, Qin Sheng and Raúl Ordóñez
2005, 2005(Special): 947-956 doi: 10.3934/proc.2005.2005.947 +[Abstract](536) +[PDF](254.5KB)
Abstract:
Surrogate optimization is a computational procedure that uses a sequence of approximations of the objective function to predict an optimum. Even in the case when the derivative information of the objective function is not available, the surrogate functions can still be constructed only based on the function values. Moreover, the optimization method may provide required tools for computing the numerical solution of certain nonlinear differential equations in recent engineering applications, such as the optimal determination of the perturbation values in robot control dynamic systems [13, 16, 18]. In this paper, we will present a direct convergence analysis of the surrogate optimization process in one dimensional fashion via polynomial interpolations of objective function values. Numerical experiments will be given to illustrate the effectiveness of the algorithms developed. An application in multi-agent cooperative search problem will be also presented.

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