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Homogeneous coupled cell networks with s3-symmetric quotient
Manuela A. D. Aguiar, Ana Paula S. Dias, Martin Golubitsky and Maria Conceição A. Leite
2007, 2007(Special): 1-9 doi: 10.3934/proc.2007.2007.1 +[Abstract](179) +[PDF](205.5KB)
A coupled cell network represents dynamical systems (the coupled cell systems) that can be seen as a set of individual dynamical systems (the cells) with interactions between them. Every coupled cell system associated to a network, when restricted to a flow-invariant subspace defined by the equality of certain cell coordinates, corresponds to a coupled cell system associated to a smaller network, called quotient network.
  In this paper we consider homogeneous networks admitting a S3-symmetric quotient network. We assume that a codimension-one synchrony-breaking bifurcation from a synchronous equilibrium occurs for that quotient network. We aim to investigate, for different networks admitting that S3-symmetric quotient, if the degeneracy condition leading to that bifurcation gives rise to branches of steady-state solutions outside the flow-invariant subspace associated with the quotient network. We illustrate that the existence of new solutions can be justified directly or not by the symmetry of the original network. The bifurcation analysis of a six-cell asymmetric network suggests that the existence of new solutions outside the flow-invariant subspace associated with the quotient is 'forced' by the symmetry of a five-cell quotient network.
A free boundary problem for an elastic material
Toyohiko Aiki
2007, 2007(Special): 10-17 doi: 10.3934/proc.2007.2007.10 +[Abstract](203) +[PDF](166.3KB)
In this paper we investigate the dynamics of an elastic material, for example, a spring with some weight. Such dynamics is usually represented by the ordinary differential equation for the length of the spring or the partial differential equation with the linear strain on a fixed domain. The main purpose of this paper is to propose a new free boundary problem with a nonlinear strain as a mathematical model for an elastic material. Also, we establish the wellposedness for initial boundary value problem with the nonlinear strain on the cylindrical domain.
On a certain degenerate parabolic equation associated with the infinity-laplacian
Goro Akagi and Kazumasa Suzuki
2007, 2007(Special): 18-27 doi: 10.3934/proc.2007.2007.18 +[Abstract](175) +[PDF](201.1KB)
The comparison, uniqueness and existence of viscosity solutions to the Cauchy-Dirichlet problem are proved for a degenerate parabolic equation of the form $u_t$ = $\Delta_(\infty)u$, where $\Delta_(\infty)$ denotes the so-called infinity-Laplacian given by $\Delta_(\infty)u$ = $\Sigma^(N)_(i,j=1) u_x_i u_x_j u_(x_i)_x_j$ . Our proof relies on a coercive regularization of the equation, barrier function arguments and the stability of viscosity solutions.
Solvability of some volterra type integral equations in hilbert space
Onur Alp İlhan
2007, 2007(Special): 28-34 doi: 10.3934/proc.2007.2007.28 +[Abstract](184) +[PDF](155.7KB)
We consider an integral equation of Fredholm and Volterra type with spectral parameter depending on time. Conditions of solvability are established when the initial value of the parameter coincides with an eigenvalue of Fredholm operator.
A necessary and sufficient condition for the existence of periodic solutions of linear impulsive differential equations with distributed delay
Jehad O. Alzabut
2007, 2007(Special): 35-43 doi: 10.3934/proc.2007.2007.35 +[Abstract](244) +[PDF](191.0KB)
A necessary and sufficient condition is established for the existence of periodic solutions of linear impulsive differential equations with distributed delay.
Contour enhancement via a singular free boundary problem
Anna Lisa Amadori
2007, 2007(Special): 44-53 doi: 10.3934/proc.2007.2007.44 +[Abstract](217) +[PDF](198.8KB)
We study a degenerate nonlinear parabolic equation with moving boundaries which describes the technique of contour enhancement in image processing. Such problem arises from the model by Malladi and Sethian after an asymptotic expansion suggested by Barenblatt: in order to recover the phenomenon of mass concentration, a singular data is imposed at the free boundary.
Second order estimates for boundary blow-up solutions of elliptic equations
Claudia Anedda and Giovanni Porru
2007, 2007(Special): 54-63 doi: 10.3934/proc.2007.2007.54 +[Abstract](188) +[PDF](193.7KB)
We investigate blow-up solutions of the equation $\Deltau$ = $f(u)$ in a bounded smooth domain $\Omega \subset R^N$. Under appropriate growth conditions on $f(t)$ as $t$ goes to infinity we show how the mean curvature of the boundary $\partial\Omega$ appears in the second order term of the asymptotic expansion of the solution $u(x)$ as $x$ goes to $\partial\Omega$.
Computation of heteroclinic orbits between normally hyperbolic invariant 3-spheres foliated by 2-dimensional invariant Tori in Hill's problem
Lorenzo Arona and Josep J. Masdemont
2007, 2007(Special): 64-74 doi: 10.3934/proc.2007.2007.64 +[Abstract](160) +[PDF](2929.9KB)
This paper addresses a methodology to compute homoclinic and heteroclinic orbits between hyperbolic invariant tori of Hamiltonian systems. To illustrate the procedure, we focus the presentation on an aspect of libration point mission design, where zero cost transfer trajectories (heteroclinic orbits) are of special relevance.
New evolution equations for non-linear water waves in general bathymetry with application to steady travelling solutions in constant, but arbitrary, depth
G. A. Athanassoulis and K. A. Belibassakis
2007, 2007(Special): 75-84 doi: 10.3934/proc.2007.2007.75 +[Abstract](143) +[PDF](393.5KB)
A non-linear coupled-mode system of horizontal equations is derived with the aid of Luke’s [13] variational principle, which models the evolution of nonlinear water waves in intermediate depth and over a general bathymetry. The vertical structure of the wave field is exactly represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional modes, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The system fully accounts for the effects of non-linearity and dispersion. The main features of this approach are the following: (i) various standard models of water-wave propagation are recovered by appropriate simplifications of the coupled-mode system, and (ii) a small number of modes (up to 5) are enough for the precise numerical solution, provided that the two new modes (the free-surface and the sloping-bottom ones) are included in the local-mode series. In the present work, the consistent coupled-mode system is applied to the numerical investigation of families of steady travelling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate to shallow-water wave conditions.
Optimal design of an optical length of a rod with the given mass
Boris P. Belinskiy
2007, 2007(Special): 85-91 doi: 10.3934/proc.2007.2007.85 +[Abstract](213) +[PDF](158.0KB)
The optical length of an elastic rod appears to be important in problems of control of its longitudinal vibrations. We consider the problem of optimal design (optimal density distribution) of an elastic rod of a given variable modulus of elasticity with the optical length as a criterion and assuming that the total mass of the rod is given. The results provide some bounds on the optical length.
Energy estimate for the wave equation driven by a fractional Gaussian noise
Boris P. Belinskiy and Peter Caithamer
2007, 2007(Special): 92-101 doi: 10.3934/proc.2007.2007.92 +[Abstract](244) +[PDF](199.4KB)
We consider a general linear stochastic wave equation driven by fractional-in-time noise and study its energy. We provide a mild solution for the wave equation in terms its Fourier expansion. We calculate the expected energy and give asymptotic results for the expected energy for large and small times and as the Hurst parameter, H, approaches 1/2. These results are phrased in terms of the norms of powers of the differential operator times powers of the spatial covariance operator.
Periodic solutions of Birkhoff-Lewis type for the nonlinear wave equation
Luca Biasco and Laura Di Gregorio
2007, 2007(Special): 102-109 doi: 10.3934/proc.2007.2007.102 +[Abstract](166) +[PDF](214.5KB)
We prove the existence of infinitely many periodic solutions accumulating to zero for the one–dimensional nonlinear wave equation (vibrating string equation). The periods accumulate to zero and are both rational and irrational multiples of the string length.
The dirichlet problem for singluar fully nonlinear operators
Isabeau Birindelli and Francoise Demengel
2007, 2007(Special): 110-121 doi: 10.3934/proc.2007.2007.110 +[Abstract](182) +[PDF](200.1KB)
No abstract present.
Asymptotic behavior of time optimal orbital transfer for low thrust 2-body control system
Alex Bombrun and Jean-Baptiste Pomet
2007, 2007(Special): 122-129 doi: 10.3934/proc.2007.2007.122 +[Abstract](178) +[PDF](184.6KB)
In the studies of low thrust time optimal orbital transfer it was conjectured that, when the thrust modulus tends to zero, the product of this modulus by the minimum transfer time admits a finite limit. The purpose of the present note is to better frame the nature of this asymptotic behavior and to prove this conjecture.
Singular evolution on maniforlds, their smoothing properties, and soboleve inequalities
Matteo Bonforte and Gabriele Grillo
2007, 2007(Special): 130-137 doi: 10.3934/proc.2007.2007.130 +[Abstract](134) +[PDF](201.6KB)
The evolution equation $u_ = \Delta_pu$, posed on a Riemannian manifold, is studied in the singular range $p \in 2$ (1; 2). It is shown that if the manifold supports a suitable Sobolev inequality, the smoothing effect $||u(t)||\infty\leq C ||u(0)||_q^\gamma$/$t^\alpha$ holds true for suitable for $\alpha, \gamma$and that the converse holds if $p$ is sufficiently close to 2, or in the degenerate range $p$ > 2. In such ranges, the Sobolev inequality and the smoothing efect are then equivalent
Landau-Lifschitz-Gilbert equation with applied eletric current
Gaël Bonithon
2007, 2007(Special): 138-144 doi: 10.3934/proc.2007.2007.138 +[Abstract](162) +[PDF](188.1KB)
In this paper, we are concerned with a model of electric current effect in ferromagnetic materials, that is Landau-Lifschitz equation adding a transport term. We prove classical existence theorem in the general three dimensional case, and we justify a one dimensional approximation for wich we have the explicit behavior of the magnetisation.
Second order optimality conditions with applications
B. Bonnard, J.-B. Caillau and E. Trélat
2007, 2007(Special): 145-154 doi: 10.3934/proc.2007.2007.145 +[Abstract](153) +[PDF](3711.1KB)
An algorithm to compute the first conjugate point along a smooth extremal curve is presented. Under generic assumptions, the trajectory ceases to be locally optimal at such a point. An implementation of this algorithm, called cotcot, is available online and based on recent developments in geometric optimal control. It is applied to analyze the averaged optimal transfer of a satellite between elliptic orbits.
Positive Solutions of second order differential equations with integral boundary conditions
Abdelkader Boucherif
2007, 2007(Special): 155-159 doi: 10.3934/proc.2007.2007.155 +[Abstract](169) +[PDF](153.1KB)
In this paper we investigate the existence of positive solutions of second order differential equations with integral boundary conditions, and the nonlinearity is a continuous function, depending on the first derivative of the unknown function and may change sign, with respect to its second argument, infinitely many times.
Asymptotic behavior of solution of hyperbolic problems on a cylindrical domain
Bernard Brighi and S. Guesmia
2007, 2007(Special): 160-169 doi: 10.3934/proc.2007.2007.160 +[Abstract](194) +[PDF](207.0KB)
The asymptotic behavior of the hyperbolic evolution problems of order two, on a cylindrical domain $\Omega$ = $\Delta \times \omega$, with coefficients dependent on a parameter is examined. The convergence of the solution of such problems towards a solution of a problem of the same type defined in $\omega$ is proved, and the rate of convergence estimates is given. One can see this work as a singular perturbation of the hyperbolic problems in some directions.
Normal form for spatial dynamics in the Swift-Hohenberg equation
John Burke and Edgar Knobloch
2007, 2007(Special): 170-180 doi: 10.3934/proc.2007.2007.170 +[Abstract](179) +[PDF](203.1KB)
The reversible Hopf bifurcation with 1:1 resonance holds the key to the presence of spatially localized steady states in many partial differential equations on the real line. Two different techniques for computing the normal form for this bifurcation are described and applied to the Swift-Hohenberg equation with cubic/quintic and quadratic/cubic nonlinearities.
Parabolic problems with varying operators and Dirichlet and Neumann boundary conditions on varying sets
Carmen Calvo-Jurado, Juan Casado-Díaz and Manuel Luna-Laynez
2007, 2007(Special): 181-190 doi: 10.3934/proc.2007.2007.181 +[Abstract](207) +[PDF](235.8KB)
For a bounded open set $\Omega$ $\subset$ $\mathbb{R}^N$ and an arbitrary sequence $\Gamma_n$ of closed subsets of $\partial\Omega$, we study the asymptotic behavior of the solutions of linear parabolic problems posed in $\Omega$ $\times$ (0, $T$) satisfying Dirichlet boundary conditions on $\Gamma_n$ $\times$ (0,T) and Neumman boundary conditions on ($\partial\Omega$ \ $\Gamma_n$) $\times$ (0, T). The coefficients of the equations are also assumed to vary with n. We obtain a limit problem which is stable by homogenization and where it appears a Fourier-Robin boundary condition.
Dead cores and bursts for p-Laplacian elliptic equations with weights
Elisa Calzolari, Roberta Filippucci and Patrizia Pucci
2007, 2007(Special): 191-200 doi: 10.3934/proc.2007.2007.191 +[Abstract](195) +[PDF](220.8KB)
In this paper we consider p-Laplace elliptic equations with weights on domains of $\mathbb{R}^n$, which include several prototypes, and we show that there exist a dead core solution having a burst within the core. This result is obtained by using an existence theorem for ground states having compact support, proved in [4] by the authors, together with qualitative properties and an existence theorem for dead core solutions contained in a recent work of Pucci and Serrin, see [10].
Numerical solution of a time-dependent Signorini contact problem
María Teresa Cao-Rial, Peregrina Quintela and Carlos Moreno
2007, 2007(Special): 201-211 doi: 10.3934/proc.2007.2007.201 +[Abstract](178) +[PDF](245.9KB)
The purpose of this work is to study the dynamic frictionless contact problem between an elastic body and a rigid foundation. In order to model the contact we consider Signorini conditions. A numerical algorithm is proposed to approximate the solution; the algorithm involves a contact multiplier, which is a fixed point of a nonlinear equation solved by using a generalized Newton method. We use one of the Newmark methods for time discretization and a finite element method for space discretization. The convergence of the method is numerically studied, and a simple test problem is used to validate the methodology.
Relaxation approximation of the Kerr model for the impedance initial-boundary value problem
Gilles Carbou and Bernard Hanouzet
2007, 2007(Special): 212-220 doi: 10.3934/proc.2007.2007.212 +[Abstract](172) +[PDF](192.4KB)
The Kerr-Debye model is a relaxation of the nonlinear Kerr model in which the relaxation coefficient is a finite response time of the nonlinear material. We establish the convergence of the Kerr-Debye model to the Kerr model when this relaxation coefficient tends to zero.
Comparison results for a class of quasilinear evolutionary hemivariational inequalities
Siegfried Carl
2007, 2007(Special): 221-229 doi: 10.3934/proc.2007.2007.221 +[Abstract](187) +[PDF](205.8KB)
We consider a class of quasilinear evolutionary hemivariational inequalities under nonmonotone multivalued flux boundary conditions. Our main goal is to provide existence and comparison results in terms of appropriately defined sub- and supersolutions on the basis of which one can prove compactness and extremality results of the solution set within the sector of sub- and supersolutions.
Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$
Alexandre N. Carvalho and Jan W. Cholewa
2007, 2007(Special): 230-239 doi: 10.3934/proc.2007.2007.230 +[Abstract](194) +[PDF](275.6KB)
This paper is devoted to well posedness and regularity of the solutions of

$u_t_t + 2\nA^(1/2)u_t + A\u = \f(u)$

in $W^(1,p)_0 (\Omega) \times L^p(\Omega), p \in (1,\infty)$, where $\Omega \subset \mathbb {R}^N$ is a bounded smooth domain, $\n$ > 0 and -$\A$ is the Dirichlet Laplacian in $L^p(\Omega)$. We prove local well posedness result for nonlinearities $\f : \mathbb {R} \rightarrow \mathbb {R}$ satisfying $|f(s) - f(t)| \<= C|s - t|(1 + |s|^(p - 1) + |t|^(p - 1))$ with $\p < (N+p)/(N - p) (N > p)$, and show that the solutions are classical. If $f$ is dissipative and $p < (N+2)/(N - 2) (N \>= 3)$, we show that the associated semigroup has a global attractor $\cc{A}_(n,p)$ in $W^(1,p)_0(\Omega)\times L^p(\Omega)$, $p \in [2,\infty)$, which coincides with the attractor $\bb{A}_(n,2) =: \bb{A}_n$. We also obtain that $\bb{A}_n$ is compact in $C^(2+\mu)(bar(\Omega)) \times C^(1+\mu)(bar(\Omega))$ and attracts bounded subsets of $H^1_0(\Omega) \times L^2(\Omega)$ in $C^(2+\mu)(bar(\Omega)) \times C^(1+\mu)(bar(\Omega))$ for each $\mu \in (0, 1)$.

Solve the vehicle routing problem with time windows via a genetic algorithm
Yaw Chang and Lin Chen
2007, 2007(Special): 240-249 doi: 10.3934/proc.2007.2007.240 +[Abstract](196) +[PDF](190.8KB)
The objective of vehicle routing problem (VRP) is to deliver a set of customers with known demands on minimum-cost routes originating and terminating at the same depot. A vehicle routing problem with time windows (VRPTW) requires the delivery made in a specific time window given by the customers. Prins (2004) proposed a simple and effective genetic algorithm (GA) for VRP. In terms of average solution, it outperforms most published tabu search results. We implement this hybrid GA to handle VRPTW. Both the implementation and computation results will be discussed.
Modeling the motion of a cell population in the extracellular matrix
A. Chauviere, L. Preziosi and T. Hillen
2007, 2007(Special): 250-259 doi: 10.3934/proc.2007.2007.250 +[Abstract](162) +[PDF](240.3KB)
The paper aims at describing the motion of cells in fibrous tissues taking into account the interaction with the network fibers and among cells, chemotaxis, and contact guidance from network fibers. Both a kinetic model and its continuum limit are described.
Eigenvalues of homogeneous gradient mappings in Hilbert space and the Birkoff-Kellogg theorem
Raffaele Chiappinelli
2007, 2007(Special): 260-268 doi: 10.3934/proc.2007.2007.260 +[Abstract](145) +[PDF](194.2KB)
It is well known that any (nontrivial) linear compact self-adjoint operator acting in a Hilbert space possesses at least one non-zero eigenvalue. We present a generalization of this to nonlinear mappings as in the title, and discuss the relations of our results with the Birkhoff-Kellogg Theorem on one side, and with the spectral properties of self-adjoint operators on the other.
Chaotic translation semigroups
José A. Conejero and Alfredo Peris
2007, 2007(Special): 269-276 doi: 10.3934/proc.2007.2007.269 +[Abstract](187) +[PDF](193.6KB)
We characterize chaos for the translation semigroup, with a sector in the complex plane as index set, defined on a weighted function space. The results are stated in terms of the integrability of the weight function, and in terms of the existence of periodic points. We generalize previous results of [8, 15]. Some examples are also provided to complete the study.
Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations
Eduardo Cuesta
2007, 2007(Special): 277-285 doi: 10.3934/proc.2007.2007.277 +[Abstract](195) +[PDF](212.3KB)
We study he asymptotic behaviour as $t \rightarrow + \infty$ of the solutions of an abstract fractional equation $u = u_0 + \partial^( - \alpha) Au + g$, 1 < $\alpha$ < 2, where $A$ is a linear operator of sectorial type. We also show that a discretization in time of this equation based on backward Euler convolution quadrature inherits this behaviour.
Theoretical optimization of finite difference schemes
Claire David and Pierre Sagaut
2007, 2007(Special): 286-293 doi: 10.3934/proc.2007.2007.286 +[Abstract](233) +[PDF](189.2KB)
The aim of this work is to develop general optimization methods for linear finite difference schemes used to approximate linear differential equations, on the basis of a matrix equation, which enables to determine the optimal value of a parameter for a given scheme.
A new class of parabolic problems connected with Newton's polygon
Robert Denk and Leonid Volevich
2007, 2007(Special): 294-303 doi: 10.3934/proc.2007.2007.294 +[Abstract](181) +[PDF](205.9KB)
A new class of boundary value problems for parabolic operators is introduced. We discuss some linearized free boundary problems not satisfying the classical parabolicity condition. It is shown that they belong to this class and by means of the Newton polygon method the nontrivial two-sided estimates of these problems are found.
Entropy methods for reaction-diffusion systems
Laurent Desvillettes and Klemens Fellner
2007, 2007(Special): 304-312 doi: 10.3934/proc.2007.2007.304 +[Abstract](170) +[PDF](224.1KB)
In the continuation of [6], we study reversible reaction-diffusion systems via entropy methods (based on the free energy functional). We show for a particular model problem with two reacting species in which one of the diffusion constants vanishes that the solutions decay exponentially with explicit rate and constant towards the unique constant equilibrium state.
On a free-boundary problem modeling the action of a limiter on a plasma
J. I. Díaz and J. F. Padial
2007, 2007(Special): 313-322 doi: 10.3934/proc.2007.2007.313 +[Abstract](185) +[PDF](231.1KB)
In this paper we prove the existence of weak solutions for a 2D free–boundary problems arising in the magnetic confinement of a plasma in a Stellarator device which includes the action of a limiter. The model can be expressed as an inverse thin obstacle problem in which the limiter plays the role of a thin obstacle for the plasma. The inverse nature of the problem comes from the fact that the associated Grad–Shafranov equation involves some unknown nonlinear terms which must be determined by the current–carrying Stellarator condition.
Asymptotics for 1D flows with time-dependent external fields
Bernard Ducomet
2007, 2007(Special): 323-333 doi: 10.3934/proc.2007.2007.323 +[Abstract](160) +[PDF](226.0KB)
We consider a free boundary problem for the Navier–Stokes system describing one–dimensional flows of a viscous compressible radiative and reactive gas, with time-dependent external fields. For prescribed decay rates of the data, we prove precise asymptotics for the solution as $t \rightarrow \infty$.
Uniform estimate of dimension of the global attractor for a semi-discretized chemotaxis-growth system
Messoud Efendiev, Etsushi Nakaguchi and Wolfgang L. Wendland
2007, 2007(Special): 334-343 doi: 10.3934/proc.2007.2007.334 +[Abstract](164) +[PDF](231.1KB)
We study a finite-element approximation of the chemotaxis-growth system. We establish dimension estimate of global attractors for the approximate systems. Our results show that the estimates are uniform with respect to the discretization parameter and polynomial order with respect to the chemotactic coefficient in the equation.We especially emphasize that, this is just the same order (polynomial) as for the original system obtained in the preceding papers [Adv.Math.Sci.Appl. Part I and II].
An existence result for a P.D.E. with hysteresis, convection and a nonlinear boundary condition
Michela Eleuteri
2007, 2007(Special): 344-353 doi: 10.3934/proc.2007.2007.344 +[Abstract](197) +[PDF](217.7KB)
In this paper a partial differential equation containing a continuous hysteresis operator and a convective term is considered. This model equation, which appears in the context of magnetohydrodynamics, is coupled with a nonlinear boundary condition containing a memory operator. Under suitable assumptions, an existence result is achieved using an implicit time discretization scheme.
Pricing realized variance options using integrated stochastic variance options in the Heston stochastic volatility model
Lorella Fatone, Francesca Mariani, Maria Cristina Recchioni and Francesco Zirilli
2007, 2007(Special): 354-363 doi: 10.3934/proc.2007.2007.354 +[Abstract](172) +[PDF](184.5KB)
This paper presents a numerical method to price European options on realized variance. A European realized variance option is an option where payoff depends on the time of maturity, on the observed variance of the log-returns of the stock prices in a preassigned sequence of time values $t_i$, $i$ = 0, 1, . . . ,$N$. The realized variance is the variance observed in the sample of the log-returns considered, so that the value at maturity of the realized variance option depends on the discrete sample of the log-returns of the stock prices observed at the preassigned dates t$_i$, $i$= 0, 1, . . . ,$N$. The method proposed to approximate the price of these options is based on the idea of approximating the discrete sum that gives the realized variance with an integral, using as model of the dynamics of the log-return of the stock price the Heston stochastic volatility model. In this way the price of a realized variance option is approximated with the price of an integrated stochastic variance option where payoff depends on the time of maturity and on the integrated stochastic variance. The integrated stochastic variance option is priced with the method of discounted expectations. We derive an integral representation formula for the price of this last kind of options. This integral formula reduces to a one dimensional Fourier integral in the case of the most commonly traded options that have a simple payoff function. The method has been validated on some test problems. The numerical experiments show that the approach suggested in this paper gives satisfactory approximations of the prices of the realized variance options (relative error 10−$^2$, 10-$^3$). This approach also allows substantial savings of computational time when compared with the Monte Carlo method used to evaluate with approximately the same accuracy. The website contains auxiliary material that can help in the understanding of this paper and makes available to the interested users the codes that implement the numerical method proposed here to price realized variance options. The use of these codes on a computing grid has been made user friendly developing a dedicated application using the software Symphony (that is, a Service Oriented Architecture (SOAM) software of Platform Computing Toronto, Canada). The website mentioned above makes this Symphony application available to the users.
Dynamics in 30species preadtor-prey models with time delays
Wei Feng
2007, 2007(Special): 364-372 doi: 10.3934/proc.2007.2007.364 +[Abstract](167) +[PDF](742.8KB)
We study a differential equation system with diffusion and time delays which models the dynamics of predator-prey interactions within three biological species. Our main focus is on the persistence (non-extinction) of u-species which is at the bottom of the nutrient hierarchy, and the permanence effect (long-term survival of all the predators and prey) in this model. When u-species persists in the absence of its predators, we generate a condition on the interaction rates to ensure that it does not go extinction under the predation of the v- and w-species. With certain additional conditions, we can further obtain the permanence effect (long-term survival of all three species) in the ecological system. Our proven results also explicitly present the effects of all the environmental data (growth rates and interaction rates) on the ultimate bounds of the three biological species. Numerical simulations of the model are also given to demonstrate the pattern of dynamics (extinction, persistence, and permanence)in the ecological model.
Existence of positive solutions for second order differential equations arising from chemical reactor theory
Wenying Feng, Guang Zhang and Yikang Chai
2007, 2007(Special): 373-381 doi: 10.3934/proc.2007.2007.373 +[Abstract](188) +[PDF](193.6KB)
In this paper, we study second-order differential equations that represent the steady state model in an adiabatic tubular chemical reactor. Theoretical results on existence and range of positive solutions are proved by applying a fixed point theorem. At the mean time, numerical solutions are obtained by computer programming. Results from mathematical analysis are compared with the numerical solutions.
Traveling waves to a reaction-diffusion equation
Zhaosheng Feng
2007, 2007(Special): 382-390 doi: 10.3934/proc.2007.2007.382 +[Abstract](194) +[PDF](187.8KB)
In this paper, we study a nonlinear reaction–diffusion equation for its traveling waves. This equation can be regarded as a generalization of the Fisher equation and is used as a nonlinear model, in the one-dimensional situation, for studying insect and animal dispersal with growth dynamics. Applying the Lie symmetry method, we obtain two traveling wave solutions under certain parametric conditions and express them in terms of elliptic functions.
New discrete analogue of neural networks with nonlinear amplification function and its periodic dynamic analysis
Xilin Fu and Zhang Chen
2007, 2007(Special): 391-398 doi: 10.3934/proc.2007.2007.391 +[Abstract](224) +[PDF](184.3KB)
In this paper, new discrete analogue of a class of neural networks with nonlinear amplification function is obtained by analysis and approximation techniques. Using continuation theorem of coincidence degree theory, periodic solution for discrete model is studied, and sufficient condition is given to guarantee the existence of periodic solution. Moreover, global stability on periodic solution is investigated by Lyapunov method.
Nonlinear degenerate parabolic equations for a thermohydraulic model
Takesi Fukao and Masahiro Kubo
2007, 2007(Special): 399-408 doi: 10.3934/proc.2007.2007.399 +[Abstract](214) +[PDF](220.3KB)
In this paper, we consider an initial boundary value problem for a system of second order partial differential equations. This system consists of the Navier-Stokes equations and a nonlinear heat equation. More precisely, we impose a nonlinear heat flux associated with a class of maximal monotone graphs with a Neumann boundary condition. We establish the conditions required to prove the existence of a solution for the given data.
Existence results for quasilinear hemivariational inequalities at resonance
Leszek Gasiński
2007, 2007(Special): 409-418 doi: 10.3934/proc.2007.2007.409 +[Abstract](171) +[PDF](213.4KB)
In this paper we consider quasilinear hemivariational inequality at resonance. We obtain two existence theorems using a Landesman-Lazer type condition. The method of the proof is based on the nonsmooth critical point theory for locally Lipschitz functions.
Nonlinear dynamics on centre manifolds describing turbulent floods: k-$\omega$ model
D.J. Georgiev, A. J. Roberts and D. V. Strunin
2007, 2007(Special): 419-428 doi: 10.3934/proc.2007.2007.419 +[Abstract](161) +[PDF](161.8KB)
In shallow turbulent flows such as floods and tsunami vertical mixing tends to smooth out the flow characteristics in cross-sectional direction. The evolution of the average cross-flow characteristics presents considerable interest. We model such flows using the $k-\omega$ model of turbulence in the framework of the centre manifold theory. We tested the approach on an artificial diffusion problem for which an exact analytical solution is derived. Then we apply the method to model the turbulent flows and deduced the evolution equations for the average velocity, turbulent energy and its rate of dissipation.
Multiplicity results for a singular and quazilinear equation
J. Giacomoni and K. Sreeandh
2007, 2007(Special): 429-435 doi: 10.3934/proc.2007.2007.429 +[Abstract](148) +[PDF](229.7KB)
In this paper, we investigate the following quasilinear and singular problem:

-$\Delta_pu = \lambda/(u^\delta) + u^q$ in $\Omega$ $u|_(\partial\Omega) = 0 , u > 0$ in $\Omega$       (1)

where $\Omega$ is an open bounded domain with smooth boundary, 1 < $p$, $p - 1$ < $q$ and $\lambda$, $\delta$ > 0. We first prove that there exist weak solutions for $\lambda$ > 0 small in $W^(1,p)_0(\Omega) \cap C(bar(\Omega))$ if and only if $\delta < 2 + 1/(p - 1)$ . Investigating the radial symmetric case $(\Omega = B_R(0))$, we prove by a shooting method the global multiplicity of solutions to $(P)$ in $C(bar(\Omega))$ with $0 < \delta$, 1 < $p$ and $p - 1$ < $q$.

Turing patterns on growing spheres: the exponential case
Julijana Gjorgjieva and Jon Jacobsen
2007, 2007(Special): 436-445 doi: 10.3934/proc.2007.2007.436 +[Abstract](175) +[PDF](790.6KB)
We consider Turing patterns for reaction-diffusion systems on the surface of a growing sphere. In particular, we are interested in the effect of dynamic growth on the pattern formation. We consider exponential isotropic growth of the sphere and perform a linear stability analysis and compare the results with numerical simulations.
The thermistor problem with degenerate thermal conductivity and metallic conduction
María Teresa González Montesinos and Francisco Ortegón Gallego
2007, 2007(Special): 446-455 doi: 10.3934/proc.2007.2007.446 +[Abstract](240) +[PDF](228.9KB)
The aim of this work is to establish the existence of a capacity solution to the thermistor problem supposing that the thermal and the electrical conductivities are not bounded below by a positive constant value. Furthermore, the thermal conductivity vanishes at points where the temperature is null. These assumptions on data include the case of practical interest of the Wiedemann–Franz law with metallic conduction and lead us to very complex mathematical situations.
Optimal control of a nonlinear model of economic growth
Ellina Grigorieva and Evgenii Khailov
2007, 2007(Special): 456-466 doi: 10.3934/proc.2007.2007.456 +[Abstract](239) +[PDF](195.5KB)
A nonlinear control model of a firm describing the change of production and accumulated $R&D$ investment is investigated. An optimal control problem with $R&D$ investment rate as a control parameter is solved. Optimal dynamics of economic growth of a firm versus the current cost of innovation is studied. It is analytically determined that dependent on the model parameters, the optimal control must be of one of the following types : a) piecewise constant with at most two switchings, b) piecewise constant with two switching and containing a singular arc. The intervals on which switching from regular to singular arcs occur are found numerically. Finally, optimal investment strategies and production activities are compared with econometric data of an actual firm.
On differential variational inequalities and projected dynamical systems - equivalence and a stability result
J. Gwinner
2007, 2007(Special): 467-476 doi: 10.3934/proc.2007.2007.467 +[Abstract](190) +[PDF](203.8KB)
   The purpose of this note is two-fold. Firstly we deal with projected dynamical systems that recently have been introduced and investigated in finite dimensions to treat various time dependent (dis)equilibrium and network problems in operations research. Here at the more general level of a Hilbert space, we show that a projected dynamical system is equivalent in finding the “slow” solution (the solution of minimal norm) of a differential variational inequality, a class of evolution inclusions studied much earlier. This equivalence follows easily from a precise geometric description of the directional derivative of the metric projection in Hilbert space. By our approach, we can easily characterize a stationary point of a projected dynamical system as a solution of a related variational inequality.
   Secondly we are concerned with stability of the solution set to differential variational inequalities. Here we present a novel upper set convergence result with respect to perturbations in the data. In particular, we admit perturbations of the associated set-valued maps and the constraint set, where we impose weak convergence assumptions on the perturbed set-valued maps and employ Mosco convergence as set convergence.
Comparisons of eigenvalues of second order elliptic operators
François Hamel, Emmanuel Russ and Nikolai Nadirashvili
2007, 2007(Special): 477-486 doi: 10.3934/proc.2007.2007.477 +[Abstract](168) +[PDF](237.9KB)
To any second order elliptic operator $L =$ −div$(A\nabla)$ + $v * \nabla + V$ in a bounded $C^2$ domain $\Omega$ with Dirichlet boundary condition, we associate a second order elliptic operator $L$* in divergence form in the Euclidean ball $\Omega$* centered at 0 and having the same Lebesgue measure as $\Omega$. In $\Omega$, the symmetric matrix field $A$ is in $W^(1,\infty)(\Omega)$, the vector field $v$ is in $L^\infty(\Omega \mathbb{R}^n)$ and $V$ is a continuous function in $bar(\Omega)$. In $\Omega$*, the coefficients of $L$* are radial, they preserve some quantities associated to the coefficients of $L$, and we can construct the operator $L$* in such a way that its principal eigenvalue is not too much larger than that of $L$. In particular, we generalize the Rayleigh-Faber-Krahn inequality for the principal eigenvalue of the Dirichlet Laplacian. The proofs use a new rearrangement technique, different from the Schwarz symmetrization and interesting by itself.
Nonexistence of global solutions of nonlinear Schrodinger equations in non star-shaped domains
Takahiro Hashimoto
2007, 2007(Special): 487-494 doi: 10.3934/proc.2007.2007.487 +[Abstract](204) +[PDF](178.7KB)
In this paper, we discuss the nonexistence of global solutions of mixed problems of the nonlinear Schödinger equations with power nonlinearity. When the domain is whole space, there are many results concerning the nonexistence of global solutions ( or existence of blow-up solutions ) for the equation. For the case of a general domain, there are few studies of blowing-up conditions. The main purpose of this paper is to discuss the nonexistence of global solutions in a deformed tube-shaped domain which is not star-shaped.
Lie group study of finite difference schemes
Emma Hoarau, Claire David, Pierre Sagaut and Thiên-Hiêp Lê
2007, 2007(Special): 495-505 doi: 10.3934/proc.2007.2007.495 +[Abstract](217) +[PDF](384.7KB)
Differential equations arising in fluid mechanics are usually derived from the intrinsic properties of mechanical systems, in the form of conservation laws, and bear symmetries, which are not generally preserved by a finite difference approximation, and lead to inaccurate numerical results. This paper deals with the analysis of symmetry group of finite difference equations, which is based on the differential approximation. We develop a new scheme, the related differential approximation of which is invariant under the symmetries of the original differential equations. A comparison of numerical performance of this scheme, with standard ones and a higher order one has been realized for the Burgers equation.
Multi-compartment models
Abderrahman Iggidr, Josepha Mbang, Gauthier Sallet and Jean-Jules Tewa
2007, 2007(Special): 506-519 doi: 10.3934/proc.2007.2007.506 +[Abstract](240) +[PDF](216.5KB)
We consider models with a general structure which, for example, encompasses the so-called DI, SP or DISP models with mass action incidence. We give a very simple formule for the basic reproduction ratio $R_0$. If $R_0 \<= 1$ we prove that the disease free equilibrium is globally asymptotically stable on the nonnegative orthant. If $R_0$ > 1, we prove the existence of a unique endemic equilibrium in the positive orthant and give an explicit formula. We prove the global asymptotic stability of the endemic equilibrium, when $R_0$ > 1 for SP model.
Scalar conservation law with discontinuous flux in a bounded domain
Julien Jimenez
2007, 2007(Special): 520-530 doi: 10.3934/proc.2007.2007.520 +[Abstract](222) +[PDF](233.4KB)
We consider the Dirichlet problem for a first-order hyperbolic equation with a convection term discontinuous with respect to the space variable. We introduce a definition of a weak entropy solution to the corresponding problem and then we prove existence and uniqueness of the entropy solution for a class of flux functions. The existence property is obtained by regularization of the flux function while for the uniqueness result we use the method of doubling variables and a Rankine-Hugoniot condition along the line of discontinuity.
On the dynamics of a degenerate damped semilinear wave equation in \mathbb{R}^N : the non-compact case
Nikos I. Karachalios and Athanasios N Lyberopoulos
2007, 2007(Special): 531-540 doi: 10.3934/proc.2007.2007.531 +[Abstract](182) +[PDF](228.5KB)
We show the existence of a global attractor for a degenerate, linearly damped, semilinear wave equation in $mathbb{R}^N$ under a new condition concerning a variable non-negative diffusivity. In particular, we show the asymptotic compactness of the induced semiflow by combining the energy equation method with appropriate tail estimates.
On stochastic fractional Volterra equations in Hilbert space
Anna Karczewska and Carlos Lizama
2007, 2007(Special): 541-550 doi: 10.3934/proc.2007.2007.541 +[Abstract](157) +[PDF](227.6KB)
In this paper, stochastic Volterra equations, particularly fractional, in Hilbert space are studied. Sufficient conditions for existence of strong solutions are provided.
Existence of periodic solutions for enzyme-catalysed reactions with periodic substrate input
Guy Katriel
2007, 2007(Special): 551-557 doi: 10.3934/proc.2007.2007.551 +[Abstract](164) +[PDF](175.6KB)
Considering a basic enzyme-catalysed reaction, in which the rate of input of the substrate varies periodically in time, we give a necessary and sufficient condition for the existence of a periodic solution of the reaction equations. The proof employs the Leray-Schauder degree, applied to an appropriately constructed homotopy.
Asymptotic equivalence of dynamic systems on time scales
B. Kaymakcalan, R. Mert and A. Zafer
2007, 2007(Special): 558-567 doi: 10.3934/proc.2007.2007.558 +[Abstract](196) +[PDF](230.7KB)
Time scale approach allows one to treat the continuous, discrete, as well as more general systems simultaneously. In this paper we establish a Levinson type theorem and a Yakubovich type result on asymptotic equivalence of linear dynamic equations and linear and quasilinear dynamic equations, respectively.
Fixed points and complete lattices
Paula Kemp
2007, 2007(Special): 568-572 doi: 10.3934/proc.2007.2007.568 +[Abstract](231) +[PDF](140.3KB)
Tarski proved in 1955 that every complete lattice has the fixed point property. Later, Davis proved the converse that every lattice with the fixed point property is complete. For a chain complete ordered set, there is the well known Abian-Brown fixed point result. As a consequence of the Abian-Brown result, every chain complete ordered set with a smallest element has the fixed point property. In this paper, a new characterization of a complete lattice is given. Also, fixed point theorems are given for decreasing functions where the partially ordered set need not be dense as is the usual case for fixed point results for decreasing functions.
Boundary gradient estimates for subsonic solutions of compressible transonic potential flows
Eun Heui Kim
2007, 2007(Special): 573-579 doi: 10.3934/proc.2007.2007.573 +[Abstract](189) +[PDF](168.1KB)
We establish boundary gradient estimates for a solution for selfsimilar transonic potential flow on the subsonic region. The gradient estimates enable us to verify the ellipticity of the solution in the region. We consider both convex and general domains. We provide the sufficient conditions to show that the flow becomes transonic.
Nonlinear semigroup methods in problems with hysteresis
Jana Kopfová
2007, 2007(Special): 580-589 doi: 10.3934/proc.2007.2007.580 +[Abstract](191) +[PDF](190.9KB)
Results from a nonlinear semigroup theory are applied to get ex- istence and uniqueness for PDEs with hysteresis. The hysteresis nonlinearity considered is of the generalized play operator type, but can be easily extended to a generalized Prandtl-Ishlinskii operator of play type, both possibly discontinuous.
Exact boundary controllability of Maxwell's equations with weak conductivity in the heterogeneous medium inside a general domain
S. S. Krigman
2007, 2007(Special): 590-601 doi: 10.3934/proc.2007.2007.590 +[Abstract](161) +[PDF](249.3KB)
This paper considers the question of control of Maxwell’s Equations (ME) in a non-homogeneous medium with positive conductivity by means of boundary surface currents applied over the entire boundary. The domain is assumed to be a bounded simply connected "star-shaped" region in R3 with smooth boundary. Using the Hilbert Uniqueness Method (HUM) of Lions, the exact boundary controllability over a sufficiently long time period is established in this case, provided that both the size of conductivity term and the spatial gradient of conductivity to electric permittivity ratio are small enough to satisfy a certain technical inequality. The restriction on conductivity size remains even when the medium is homogeneous.
Asymptotic behavior of solutions to semilinear wave equations with dissipative structure
Hideo Kubo
2007, 2007(Special): 602-613 doi: 10.3934/proc.2007.2007.602 +[Abstract](171) +[PDF](204.0KB)
We study the asymptotic behavior of solutions to semilinear wave equations. The aim of this note is to find a condition on the nonlinearity which guarantees that the corresponding equation has dissipative structure, and to show that asymptotic behavior of the global solution is possibly different from the free solution.
Periodic stability of elliptic-parabolic variational inequalities with time-dependent boundary double obstacles
Masahiro Kubo and Noriaki Yamazaki
2007, 2007(Special): 614-623 doi: 10.3934/proc.2007.2007.614 +[Abstract](172) +[PDF](220.9KB)
We consider periodic problems of elliptic-parabolic variational inequalities with time-dependent boundary double obstacles. In this paper we assume that the given boundary obstacles change periodically in time. Then, we prove the existence, uniqueness and asymptotic stability of a periodic solution to our problem.
Optimal constants for two point boundary value problems
K. Q. Lan and G. C. Yang
2007, 2007(Special): 624-633 doi: 10.3934/proc.2007.2007.624 +[Abstract](167) +[PDF](207.0KB)
The upper and lower bounds of the smallest positive characteristic value $\mu_1$ of a linear differential equation of the form

$u''(t) + \mug(t)u(t)$ = 0 a.e. on [0, 1],

subject to the general separated boundary conditions (BCs) are estimated. It is shown that $m$ < $\mu_1$ < $M(a, b)$, where $m$ and $M(a, b)$ are computable definite integrals related to the kernels arising from the above boundary value problems. The mimimum values for $M(a, b)$ are discussed when $g \stackrel{-}{=}$ 1 and $g(s) = 1/s^\alpha (\alpha > 0)$ for some of these BCs. All of these values obtained here are useful in studying the existence of nonzero positive solutions for the nonlinear differential equations of the form

$u''(t) + g(t)f(t, u(t)) = 0$ a.e. on [0, 1],

subject to the above BCs.

Nodal properties of radial solutions for a class of polyharmonic equations
Monica Lazzo and Paul G. Schmidt
2007, 2007(Special): 634-643 doi: 10.3934/proc.2007.2007.634 +[Abstract](156) +[PDF](283.4KB)
This paper is concerned with the equation $\Delta^(m)u = f(|x|, u)$, where $\Delta$ is the Laplace operator in $\mathbb{R}^N, N \in \mathbb{N}, m \in \mathbb{N}, and f \in C^(0,1 - )(\mathbb{R}_+ \times \mathbb{R}, \mathbb{R})$. Specifically, we analyze the nodal properties of radial solutions on a ball, under Dirichlet or Navier boundary conditions. We obtain precise information about the number of sign changes and the nature of the zeros of the solutions and their iterated Laplacians.
Coupling of scalar conservation laws in stratified porous media
Laurent Lévi and Julien Jimenez
2007, 2007(Special): 644-654 doi: 10.3934/proc.2007.2007.644 +[Abstract](194) +[PDF](242.3KB)
We carry out the mathematical analysis of a quasilinear parabolic-hyperbolic problem in a multidimensional bounded domain $\Omega = \Omega_h \cup \Omega_p$, where $\Omega = \Omega \ \Omega_h$. We start by providing the definition of a weak solution $u$ through an entropy inequality on the whole $\Omega$ by using the classical Kuzhkov pairs. The uniqueness proof begins by focusing on the behavior of a weak solution in $\Omega_h$ and then in $\Omega_p$. The existence property uses a discontinuous vanishing viscosity method in accordance with the layer.
Hamiltonian dynamics of atom-diatomic molecule complexes and collisions
F. J. Lin
2007, 2007(Special): 655-666 doi: 10.3934/proc.2007.2007.655 +[Abstract](109) +[PDF](212.3KB)
For a polyatomic molecule at zero total angular momentum, this paper shows that an internal motion with nonzero internal angular momentum within a (generalized) Eckart frame will produce a net rotation of the (generalized) Eckart frame in the center-of-mass frame. For a polyatomic molecule at nonzero total angular momentum, an internal motion within a generalized Eckart frame with nonzero orbital angular momentum will produce a net rotation of the generalized Eckart frame in the center-of-mass frame. Specifically, at zero total angular momentum, an internal rotation of a diatomic molecule within an atom-diatomic molecule system has nonzero internal rotational angular momentum and produces a counter-rotary net rotation of the orientation of the system (and of its generalized Eckart frame) in the center-of mass frame. Beyond a net overall rotation of an atom-diatomic molecule complex in the center-of-mass frame, a net rotation of the scattering angle of an atom colliding with a rotating diatomic molecule is obtained. A rotation in the recoil angle of an atom departing from a dissociating triatomic molecule has been observed.
Positive steady state of a food chain system with diffusion
Jing Liu, Xiaodong Liu, Sining Zheng and Yanping Lin
2007, 2007(Special): 667-676 doi: 10.3934/proc.2007.2007.667 +[Abstract](245) +[PDF](196.7KB)
This paper deals positive steady state for predator-prey microorganisms in a flow reactor with diffusion and flow. The coexistence conditions for the predator-prey populations are established. The main method used here is the fixed point index theory.
Uniqueness of radially symmetric large solutions
Julián López-Gómez
2007, 2007(Special): 677-686 doi: 10.3934/proc.2007.2007.677 +[Abstract](170) +[PDF](226.7KB)
In this paper we discuss the uniqueness of the large solutions and metasolutions in a general class of radially symmetric singular boundary value problems.
Traveling wave solutions of a generalized curvature flow equation in the plane
Bendong Lou
2007, 2007(Special): 687-693 doi: 10.3934/proc.2007.2007.687 +[Abstract](211) +[PDF](198.7KB)
We study a generalized curvature flow equation in the plane: $V =F(k,$ n, $x)$, where for a simple plane curve $\Gamma$ and for any $P \in \Gamma, k$ denotes the curvature of $\Gamma$ at $P$, n denotes the unit normal vector at $P$ and $V$ denotes the velocity in direction n, $F$ is a smooth function which is 1-periodic in $x$. For any given $\alpha \in ( - \pi/2, \pi/2)$, we prove the existence and uniqueness of a planar-like traveling wave solution of $V = F(k,$n,$x)$, that is, a curve: $y = v$*$(x) + c$*$t$ traveling in $y$-direction in speed $c$*, the graph of $v$*$(x)$ is in a bounded neighborhood of the line $x$tan$\alpha$. Also, we show that the graph of $v$*$(x)$ is periodic in the direction (cos$\alpha$, sin$\alpha$).
Positive solutions for a nonlocal fourth order equation of Kirchhoff type
To Fu Ma
2007, 2007(Special): 694-703 doi: 10.3934/proc.2007.2007.694 +[Abstract](163) +[PDF](208.0KB)
Existence and multiplicity of positive solutions for the fourth order equation

$u'''' - M(\eq^1_0 |u'|^2 dx) u'' = q(x)f(x, u, u'),$

which models simply supported extensible beams, are considered using fixed point theorems in cones of ordered Banach spaces.

Blow up and decay bounds in guasi linear parabolic problems
Monica Marras and Stella Vernier Piro
2007, 2007(Special): 704-712 doi: 10.3934/proc.2007.2007.704 +[Abstract](155) +[PDF](179.2KB)
Aim of this paper is to investigate a class of quasilinear parabolic problems whose solutions may blow up at some finite time. We establish conditions on data sufficient to preclude blow up and to insure that the solution and its spatial gradient decay exponentially for all $t > 0$.
Exponential attractors for a quasilinear parabolic equation
Kei Matsuura and Mitsuharu Otani
2007, 2007(Special): 713-720 doi: 10.3934/proc.2007.2007.713 +[Abstract](159) +[PDF](184.5KB)
Global attractors and exponential attractors are constructed for some quasilinear parabolic equations. The construction for the exponential attractor is carried out within the framework of the method of $l$-trajectories.
Global convergence of a memory gradient method with closed-form step size formula
Nora Merabet
2007, 2007(Special): 721-730 doi: 10.3934/proc.2007.2007.721 +[Abstract](163) +[PDF](191.7KB)
The memory gradient method is used to solve large scale unconstrained optimization problems. We investigate a closed-form step size formula given by a finite number of iterates ofWeiszfeld’s algorithm to compute the step size for a memory gradient method. This formula can be classified as a no-line search procedure since no stopping criteria is involved to ensure convergence, unlike the classical line search procedures. We show the global convergence of the memory gradient method, under weaker conditions.
Navier-Stokes problems modeled by evolution hemivariational inequalities
Stanislaw Migórski and Anna Ochal
2007, 2007(Special): 731-740 doi: 10.3934/proc.2007.2007.731 +[Abstract](221) +[PDF](217.0KB)
In this paper we study an inequality problem for the evolution Navier-Stokes type operators related to the model of motion of a viscous incompressible fluid in a bounded domain. The equations are nonlinear Navier-Stokes ones for the velocity and pressure with non-standard boundary conditions. We assume the nonslip boundary condition together with a Clarke subdifferential relation between the pressure and the normal components of the velocity. The existence of weak solutions to the model is proved by applying the regularized Galerkin method.
Limit of kinetic term for a Stefan problem
Piotr B. Mucha
2007, 2007(Special): 741-750 doi: 10.3934/proc.2007.2007.741 +[Abstract](205) +[PDF](231.8KB)
We investigate solutions to the one-phase quasi-stationary Stefan problem with the surface tension and kinetic term. Main results show existence of unique regular solutions with a suitable bound which enables to obtain the limit as the kinetic term is vanishing. Our problem is considered in anisotropic Besov spaces locally in time.
Almost uniqueness result for reversed variational inequalities
Dimitri Mugnai
2007, 2007(Special): 751-757 doi: 10.3934/proc.2007.2007.751 +[Abstract](155) +[PDF](171.2KB)
In this note we show that reversed variational inequalities cannot be studied in a general abstract framework as it happens for classical variational inequalities with Stampacchia’s Lemma. Indeed, we provide two different situations for reversed variational inequalities which are of the same type from an abstract point of view, but which behave quite differently.
On $L^2$-Boundedness of a $C_0$-Semigroup generated by the perturbed oseen-type operator arising from flow around a rotating body
Jiří Neustupa
2007, 2007(Special): 758-767 doi: 10.3934/proc.2007.2007.758 +[Abstract](215) +[PDF](233.0KB)
We deal with a $C_0$–semigroup generated by the linearized problem for perturbations of a flow of an incompressible viscous fluid around a rotating body. Although the uniform growth bound of the semigroup is non–negative, we derive a sufficient condition for the uniform boundedness of the semigroup.
Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method
Christos V. Nikolopoulos and Georgios E. Zouraris
2007, 2007(Special): 768-778 doi: 10.3934/proc.2007.2007.768 +[Abstract](260) +[PDF](186.8KB)
We consider the following non-local elliptic boundary value problem:

− $w''(x) = \lambda (f(w(x)))/((\eq^1_(-1) f(w(z)) dz)^2) \all x \in$ (−1, 1),
$w'(1) + \alpha w(1) = 0$, $w'$(−1) − $\alpha w$(−1)$ = $0,

where $\alpha$ and $\lambda$ are positive constants and $f$ is a function satisfying $f(s)$ > 0, $f'(s) < 0, f''(s) > 0$ for $s > 0, \eq^\infty_0 f(s)ds < \infty.$ The solution of the equation represents the steady state of a thermistor device. The problem has a unique solution for a critical value $\lambda$* of the parameter $\lambda$, at least two solutions for $\lambda < \lambda$* and has no solution for $\lambda > \lambda$*. We apply a finite element and a finite volume method in order to find a numerical approximation of the solution of the problem from the space of continuous piecewise quadratic functions, for the case that $\lambda < \lambda$* and for the stable branch of the bifurcation diagram. A comparison of these two methods is made regarding their order of convergence for $f(s) = e^( - s)$ and $f(s) = (1 + s)^( - 2)$. Also, for the same equation but with Dirichlet boundary conditions, a situation where the solution is unique for $\lambda < \lambda$*, a similar comparison of the finite element and the finite volume method is presented.

Construction of solutions to a global Eikonal equation
Chadi Nour
2007, 2007(Special): 779-783 doi: 10.3934/proc.2007.2007.779 +[Abstract](162) +[PDF](149.3KB)
We give a new and simple proof to the main result of [8] in which we derived a geometric necessary and sufficient condition for the existence of solutions to a global eikonal equation.
Degenerate Hopf instability in oscillatory reaction-diffusion equations
Toshi Ogawa
2007, 2007(Special): 784-793 doi: 10.3934/proc.2007.2007.784 +[Abstract](184) +[PDF](462.3KB)
Degenerate critical points on a 2-compnent reaction diffusion system with a global inhibition (GI) are studied. They can be an organizing center for a variety of spatially non-uniform oscillations. The Hopf-bifurcation point for the 0-mode oscillation (spatially uniform oscillation) is controlled by the strength of GI so that the eairlier destabilization of the higher mode oscillations can happen.
Modelling of interfaces in unsaturated porous media
Mario Ohlberger and Ben Schweizer
2007, 2007(Special): 794-803 doi: 10.3934/proc.2007.2007.794 +[Abstract](175) +[PDF](365.5KB)
In this contribution we discuss interface conditions for unsaturated flow in porous media. Our aim is to provide a concise collection of the arguments that lead to the standard models for interfaces that either separate two porous media or a porous medium and void space. We furthermore present a regularization procedure for these interface conditions. In a singular limit, a nonlinear boundary condition of third kind can provide approximate solutions to the outflow condition of Signorini type.
Large time behaviour of solutions of nonlinear ode describing hysteresis
Takanobu Okazaki
2007, 2007(Special): 804-813 doi: 10.3934/proc.2007.2007.804 +[Abstract](157) +[PDF](375.2KB)
We consider a nonlinear ordinary differential system which describes hysteresis input-output relations. The main part of this system is governed by subdifferential operator and it is used to present various hysteresis effects.

  In real phenomena, many hysteresis branches are observed. We are interested in verifying our system to express such branches. Our main objective of this talk is to investigate the precise behaviour of orbits of solutions of our system and show some numerical simulations.
Singular boundary value problems, via Sperner's lemma
Panos K. Palamides and Alex P. Palamides
2007, 2007(Special): 814-823 doi: 10.3934/proc.2007.2007.814 +[Abstract](177) +[PDF](210.3KB)
In this paper existence results of positive solutions, by means of a new approaches, for the $3^(the)$ order three-point BVP

$x'''(t) = \alpha (t) f(t, x(t), x' (t) , x'' (t)), x (0) = x' (n) = x'' (1) = 0,$

Regular weak solutions to 3-D Cahn-Hilliard system in elastic solids
Irena Pawłow and Wojciech M. Zajączkowski
2007, 2007(Special): 824-833 doi: 10.3934/proc.2007.2007.824 +[Abstract](165) +[PDF](227.8KB)
In this paper we prove the existence and regularity of weak solutions to a three-dimensional (3–D) Cahn-Hilliard system coupled with nonstationary elasticity. Such nonlinear parabolic-hyperbolic system arises as a model of phase separation in deformable alloys.
Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains
Lukáš Poul
2007, 2007(Special): 834-843 doi: 10.3934/proc.2007.2007.834 +[Abstract](220) +[PDF](220.7KB)
We prove existence of a weak solution to the Navier-Stokes-Fourier system on a bounded Lipschitz domain in $\mathbb{R}^3$. The key tool is the existence theory for weak solutions developed by Feireisl for the case of bounded smooth domains. We prove our result by inserting an additional limit passage where smooth domains approximate the Lipschitz one. Results on sensitivity of solutions with respect to the convergence of spatial domains are shortly discussed at the end of the paper.
Global attractor for a Klein-Gordon-Schrodinger type system
Marilena N. Poulou and Nikolaos M. Stavrakakis
2007, 2007(Special): 844-854 doi: 10.3934/proc.2007.2007.844 +[Abstract](197) +[PDF](208.5KB)
In this paper we prove the existence and uniqueness of solutions for the following evolution system of Klein-Gordon-Schrodinger type

$i\psi_t + k\psi_(xx) + i\alpha\psi$ = $\phi\psi + f(x)$,

$\phi_(tt)$ - $\phi_(xx) + \phi + \lambda\phi_t$ = -$Re\psi_x + g(x)$,

$\psi(x,0)=\psi_0(x), \phi(x,0)$ = $\phi_0, \phi_t(x,0)=\phi_1(x)$

$\phi(x,t)=\phi(x,t)=0$, $x\in\partial\Omega, t>0$

where $x \in \Omega, t > 0, k > 0, \alpha > 0, \lambda > 0, f(x)$ and $g(x)$ are the driving terms and $\Omega$ (bounded) $\subset \mathbb{R}$. Also we prove the continuous dependence of solutions of the system on the initial data as well as the existence of a global attractor.

Three state relays
Oleg Rasskazov and Gary Friedman
2007, 2007(Special): 855-863 doi: 10.3934/proc.2007.2007.855 +[Abstract](124) +[PDF](199.5KB)
We consider a hysteresis operator that arises as a three state generalization of a bi-stable relay. Basic properties and a geometric interpretation of the three-state relay are considered. Analogously to Preisach operator, which can be introduced as an aggregation of all possible non-ideal relays, we consider a "Super-Preisach" operator, that is an aggregation of all possible three-state relays.
Reduction principle in the theory of stability of difference equations
Andrejs Reinfelds and Klara Janglajew
2007, 2007(Special): 864-874 doi: 10.3934/proc.2007.2007.864 +[Abstract](165) +[PDF](172.7KB)
The reduction principle is generalized to the case of the nonautonomous difference equations in Banach space whose right-handed side is allowed to be noninvertible and whose linear part satisfies weaker condition than exponential dichotomy.
On the influence of the kernel of the bi-harmonic operator on fourth order equations with exponential growth
Frédéric Robert
2007, 2007(Special): 875-882 doi: 10.3934/proc.2007.2007.875 +[Abstract](159) +[PDF](209.5KB)
Continuing the analysis of [1, 9, 10], we discuss in this note the influence of the Kernel of the bi-harmonic operator $\Delta^2$ on the behavior of families of solutions to $\Delta^2u = e^(4u)$ on a four-dimensional domain of the Euclidean space. We also make a remark on the Paneitz-type equation in the context of compact Riemannian manifolds.
Interval oscillation of a second order nonlinear differential equation with a damping term
Yuri V. Rogovchenko and Fatoş Tuncay
2007, 2007(Special): 883-891 doi: 10.3934/proc.2007.2007.883 +[Abstract](192) +[PDF](194.1KB)
Using generalized Riccati transformations, we derive new interval oscillation criteria for a class of second order nonlinear differential equations with damping. Our theorems prove to be efficient in many cases where known results fail to apply.
Modelling of thermodynamics of martensitic transformation in shape-memory alloys
Tomáš Roubíček
2007, 2007(Special): 892-902 doi: 10.3934/proc.2007.2007.892 +[Abstract](153) +[PDF](253.8KB)
Existence of a solution to the thermo-visco-elasto-"plastic-type" system involving also higher capillarity/viscosity terms and describing thermodynamics of activated martensitic transformation at large strains is proved by a careful successive passage to a limit in a suitably regularized Galerkin approximation.
Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity
Hironobu Sasaki
2007, 2007(Special): 903-911 doi: 10.3934/proc.2007.2007.903 +[Abstract](152) +[PDF](213.6KB)
We consider the scattering problem for the nonlinear Klein-Gordon equation. The nonlinear term of the equation behaves like a power term. We show some scattering state, which improves the known results in some sense. Our proof is based on the Strichartz type estimates.
Mountain pass solutions to semilinear problems with critical nonlinearity
Ian Schindler and Kyril Tintarev
2007, 2007(Special): 912-919 doi: 10.3934/proc.2007.2007.912 +[Abstract](201) +[PDF](198.1KB)
The mountain pass statement for semilinear elliptic equations −$\Deltau = f(u)$ in $\mathbb{R}^N, N > 2$, with a critical exponent nonlinearity, namely $C_1 <= f(s)s/|s|^2^* <= C_2$, satisfies the (PS)$_c$ condition provided that the critical sequences are bounded and that the nonlinearity either has log-periodic oscillations or dominates its asymptotic values (relative to $|s|^2^*$ ) at zero and at infinity.
On the unusual Fucik spectrum
Natalija Sergejeva
2007, 2007(Special): 920-926 doi: 10.3934/proc.2007.2007.920 +[Abstract](204) +[PDF](219.6KB)
We construct the spectra for some second order boundary value problem of Fucik type. This spectrum differs essentially from the known Fucik spectra.
Solvability for phase field systems of Penrose-Fife type associated with $p$-laplacian diffusions
Ken Shirakawa
2007, 2007(Special): 927-937 doi: 10.3934/proc.2007.2007.927 +[Abstract](199) +[PDF](254.6KB)
In this paper, we deal with systems of nonlinear evolution equations, which are mathematical models of phase transitions, classified as "Penrose-Fife type". In the presented models, $p$-Laplacians, with 1 $<= p$ < 2, are adopted to describe the diffusions in exchanges. As the main conclusions, some theorems, concerned with the existence and the uniqueness of solutions of our systems, will be proved, under appropriate assumptions.
Waiting time of propagation and the backward motion of interfaces in thin-film flow theory
Andrey Shishkov
2007, 2007(Special): 938-945 doi: 10.3934/proc.2007.2007.938 +[Abstract](157) +[PDF](210.2KB)
Method of difference-functional inequalities in the study of initial propagation of the supports of generalized solutions of higher order quasilinear parabolic equations is described. As application, we find some flatness conditions on the initial function, which guarantee appearance of a waiting time in the propagation and backward motion of interfaces phenomena in thin-film flow type equations. Some lower estimate of backward motion speed is obtained too.
Regular bursting emerging from coupled chaotic neurons
Jianzhong Su, Humberto Perez-Gonzalez and Ming He
2007, 2007(Special): 946-955 doi: 10.3934/proc.2007.2007.946 +[Abstract](147) +[PDF](474.0KB)
In this note, we study the change of collective behavior of two synaptically coupled bursting systems as the strength of coupling increases. The two cells present chaotic bursting behavior when not coupled. But as the strength increases past a certain value, the behavior of two cells becomes synchronized regular bursting motions. It shows that regular oscillations can emerge from connecting intrinsically chaotic oscillators with synapses. The method of analysis is similar to that of Fast Threshold Modulation theory.
Numerical solutions for a coupled parabolic equations arising induction heating processes
Dong Sun, V. S. Manoranjan and Hong-Ming Yin
2007, 2007(Special): 956-964 doi: 10.3934/proc.2007.2007.956 +[Abstract](191) +[PDF](1209.1KB)
In this paper we study the numerical solution for a coupled parabolic equations. The system is derived from an induction heating process. An implicit finite-difference scheme for a coupled parabolic system is proposed and analyzed. Some numerical experiments are performed. We found that the numerical solutions do match the theoretical results obtained from the previous study. Moreover, some numerical results show new phenomenon which has not been proved up to now.
Partial flat core properties associated to the $p$-laplace operator
Shingo Takeuchi
2007, 2007(Special): 965-973 doi: 10.3934/proc.2007.2007.965 +[Abstract](172) +[PDF](217.1KB)
This paper deals with singular perturbation problems for quasilinear elliptic equations with the $p$-Laplace operator, e.g., −$\epsilon_pu = u^(p − 1)|a(x) − u|^(q − 1)(a(x) − u)$, where $\Epsilon$ is a positive parameter, $p$ > 1, $q$ > 0 and $a(x)$ is a positive continuous function. It is proved that any positive solution converges to $a(x)$ uniformly in any compact subset as $\epsilon \rightarrow 0$. In particular, when $q$ < $p$−1 and $\epsilon$ is small enough, the solutions coincide with $a(x)$ on one or more than one subdomain where $a(x)$ is constant, and hence there appear flat cores partially in the whole domain. These results are proved by comparison principles.
Positive solutions of elliptic equations with a critical oscillatory nonlinearity
Kyril Tintarev
2007, 2007(Special): 974-981 doi: 10.3934/proc.2007.2007.974 +[Abstract](186) +[PDF](206.9KB)
We prove existence of a counterpart of the Talenti solution in the critical semilinear problem −$\Delatu = f(u)$ in $\mathbb{R}^N$, $N$ > 3, where the nonlinearity $f$ oscillates about the critical "stem" $f(s) = s^((N+2)/(N − 2))$ : specifically, $f(2^((N − 2)/2j)s) = 2^(( N + 2)/ 2 j)f(s)$ for all $j \in \mathbb{Z}$, $s \in \mathb{R}.
Conservative numerical scheme in complex arithmetic for coupled nonlinear Schrödinger equations
M. D. Todorov and C. I. Christov
2007, 2007(Special): 982-992 doi: 10.3934/proc.2007.2007.982 +[Abstract](205) +[PDF](651.2KB)
For the Coupled Nonlinear Schrodinger Equations (CNLSE) we construct a conservative fully implicit numerical scheme using complex arithmetic which allows to reduce the computational time fourfold. We obtain various results numerically and investigate the role of the nonlinear and linear coupling. For nontrivial but moderate nonlinear coupling parameter, we find that the polarization of the system changes, but no other effects are present. For large values of the nonlinear coupling parameter, additional (faster) solitons are created during the collision of the initial ones. The linear coupling is shown to manage the self-focusing/dispersion and to make the additional solitons appear for smaller nonlinear coupling. These seem to be new effects, not reported in the literature.
Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: the clamped case
Roberto Triggiani
2007, 2007(Special): 993-1004 doi: 10.3934/proc.2007.2007.993 +[Abstract](217) +[PDF](209.8KB)
We provide sharp regularity results for hyperbolic-dominated thermoelastic plate-like systems under the action of an interior point control exercised in the mechanical equation, in the case of clamped/Dirichlet B.C.
The changes of air gap in inductive engines as vibration indicator aided by mathematical model and artificial neural network
Boguslaw Twarog, Robert Pekala, Jacek Bartman and Zbigniew Gomolka
2007, 2007(Special): 1005-1012 doi: 10.3934/proc.2007.2007.1005 +[Abstract](172) +[PDF](690.6KB)
The method of analyzing vibration of electric engines or electro- magnetic generators proposed in the work is based on the analyzing of course current of load. In considerations were used the method based on specialized mathematics model and advanced calculation technique. It allow to create of patterns for artificial neural networks. These patterns represented different states of machine for the diagnostic and they are enable to define precisely the changes caused by failure. Received experiments showed that the designed architecture of the net enables to achieve good properties of generalization correct answer for entrance date which weren't a part of training process.
Crystal dissolution and precipitation in porous media: L$^1$-contraction and uniqueness
T. L. van Noorden, I. S. Pop and M. Röger
2007, 2007(Special): 1013-1020 doi: 10.3934/proc.2007.2007.1013 +[Abstract](215) +[PDF](180.0KB)
In this note we continue the analysis of the pore-scale model for crystal dissolution and precipitation in porous media proposed in [C. J. van Duijn and I. S. Pop, Crystal dissolution and precipitation in porous media: pore scale analysis, J. Reine Angew. Math. 577 (2004), 171–211]. There the existence of weak solutions was shown. We prove an L1-contraction property of the pore-scale model. As a direct consequence we obtain the uniqueness of (weak) solutions.
Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcations
Alexandre Vidal
2007, 2007(Special): 1021-1030 doi: 10.3934/proc.2007.2007.1021 +[Abstract](197) +[PDF](901.9KB)
The biological models - particularly the ecological ones - must be understood through the bifurcations they undergo as the parameters vary. However, the transition between two dynamical behaviours of a same system for diverse values of parameters may be sometimes quite involved. For instance, the analysis of the non generic motions near the transition states is the first step to understand fully the bifurcations occurring in complex dynamics.

  In this article, we address the question to describe and explain a double bursting behaviour occuring for a tritrophic slow–fast system. We focus therefore on the appearance of a double homoclinic bifurcation of the fast subsystem as the predator death rate parameter evolves.

  The first part of this article introduces the slow–fast system which extends Lotka–Volterra dynamics by adding a superpredator. The second part displays the analysis of singular points and bifurcations undergone by fast dynamics. The third part is devoted to the flow analysis near the homoclinic points. Finally, the fourth part is concerned with the main results about the existence of periodic orbits of different periods as the two homoclinic orbits are close enough to each other.
Blow up for some nonlinear parabolic problems with convection under dynamical boundary conditions
Joachim von Below and Gaëlle Pincet Mailly
2007, 2007(Special): 1031-1041 doi: 10.3934/proc.2007.2007.1031 +[Abstract](168) +[PDF](241.7KB)
Blow up phenomena for solutions of nonlinear parabolic problems including a convection term $\partial_tu = div(a(x)\Deltau) + f(x, t, u,\Deltau)$ in a bounded domain under dissipative dynamical time lateral boundary conditions $\sigma\partial_tu + \partial_vu = 0$ are investigated. It turns out that under natural assumptions in the proper superlinear case no global weak solutions can exist. Moreover, for certain classes of nonlinearities the blow up times can be estimated from above as in the reaction diffusion case [6]. Finally, as a model case including a sign change of the convection term, the occurrence of blow up is investigated for the one–dimensional equation $\partial_tu = \partial^2_xu − u\partial_xu + u^p.
On the existence of fixed-sign solutions for a system of generalized right focal problems with deviating arguments
Patricia J.Y. Wong
2007, 2007(Special): 1042-1051 doi: 10.3934/proc.2007.2007.1042 +[Abstract](193) +[PDF](197.9KB)
We consider the following system of third-order three-point generalized right focal boundary value problems

$u^(''')_ i (t) = f_i(t, u_1(\phi_1(t)), u_2(\phi_2(t)), · · · , u_n(\phi_n(t))), t \in [a, b]$ $u_i(a) = u^'_i(z_i) = 0$, \gamma_i u_i(b) + \delta_iu^('')_i (b) = 0$

where $i$ = 1, 2, · · · , $n$, $1/2 (a + b) < z_i < b, \gamma_i > 0$, and $\phi_i$ are deviating arguments. By using some fixed point theorems, we establish the existence of one or more fixed-sign solutions $u = (u_1, u_2, · · · , u_n)$ for the system, i.e., for each 1 $<=$ $i$ $<=$ $n$, $\theta_iui(t) >= 0$ for $t \in [a, b]$, where $\theta_i \in$ {1,−1} is fixed. An example is also presented to illustrate the results obtained.

Accounting for the effect of internal viscosity in dumbbell models for polymeric fluids and relaxation of DNA
Xiao-Dong Yang and Roderick V. N. Melnik
2007, 2007(Special): 1052-1060 doi: 10.3934/proc.2007.2007.1052 +[Abstract](198) +[PDF](211.4KB)
The coarse-graining approach is one of the most important modeling methods in research of long-chain polymers such as DNA molecules. The dumbbell model is a simple but efficient way to describe the behavior of polymers in solutions. In this paper, the dumbbell model with internal viscosity (IV) for concentrated polymeric liquids is analyzed for the steady-state and time-dependent elongational flow and steady-state shear flow. In the elongational flow case, by analyzing the governing ordinary differential equations the contribution of the IV to the stress tensor is discussed for fluids subjected to a sudden elongational jerk. In the shear flow case, the governing stochastic differential equation of the finitely extensible nonlinear elastic dumbbell model is solved numerically. For this case, the extensions of DNA molecules for different shear rates are analyzed, and the comparison with the experimental data is carried out to estimate the contribution of the effect of internal viscosity.
Types of solutions and multiplicity results for second order nonlinear boundary value problems
Inara Yermachenko and Felix Sadyrbaev
2007, 2007(Special): 1061-1069 doi: 10.3934/proc.2007.2007.1061 +[Abstract](182) +[PDF](221.9KB)
We study the nonlinear BVP

$x'' = f(t, x, x')$,     (i) $x(0)cos\alpha - x'(0)sin\alpha = 0$, $x(1)cos\Beta - x'(1)sin\Beta = 0$,    (ii)

provided that $f$ : [0,1] $\times R^2 \rightarrow R$ is continuous together with the partial derivatives $f_(x'), 0 <= \alpha < \pi, 0 < \Beta <= \pi.$ If a quasi-linear ($F$ is bounded) equation

$(L_2x)(t) := d/(dt) (e^(2mt)x') + e^(2mt)k^2x = F (t,x,x')$    (iii)

can be constructed so that any solution of the problem (iii), (ii) solves also the BVP (i), (ii), then we say that the problem (i), (ii) allows for ($L_2x$)-quasilinearization. We show that if the problem (i), (ii) allows for quasilinearization with respect to essentially different linear parts then the problem (i), (ii) has multiple solutions.

Seventeen limit cycles bifurcations of a fifth system
Yunming Zhou, Desheng Shang and Tonghua Zhang
2007, 2007(Special): 1070-1081 doi: 10.3934/proc.2007.2007.1070 +[Abstract](194) +[PDF](227.6KB)
In this paper, we study a kind of cubic system perturbed by degree five. By using multi-parameter perturbation theory and qualitative analysis, we obtained seventeen limit cycles with two different distributions(see Fig 5).

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