melnikov method for discontinuous planar systems abstract this paper treats the occurrence of homoclinic solutions in planar systems with discontinuous right hand side more precisely we deal with a t periodic perturbed system such that the unperturbed system is an autonomous possessing homoclinic orbit by means of the so called non smooth melnikov function there is shown the existence of a homoclinic solution for a perturbed system the non smooth melnikov function is derived and the method of how to find it in concrete problems is also introduced

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

construction of exact solutions to eigenvalue problems by the asymptotic iteration method we apply the asymptotic iteration method aim ciftci hall and saad 2003 j phys a math gen 36 11807 to solve new classes of second order homogeneous linear differential equation in particular solutions are found for a general class of eigenvalue problems which includes schrodinger problems with coulomb harmonic oscillator or poschl teller potentials as well as the special eigenproblems studied recently by bender et al 2001 j phys a math gen 34 9835 and generalized in the present paper to arbitrary dimension

bifurcation analysis on a class of z2 equivariant cubic switching systems showing eighteen limit cycles abstract in this paper the method developed for computing the lyapunov constants of planar switching systems associated with an elementary singular point is applied to study bifurcation of limit cycles in a cubic switching system a complete classification on the center conditions and 16 limit cycles of this system are obtained around the two foci 1 0 and 1 0 further with the method an example of cubic switching systems is constructed to show the existence of 18 small amplitude limit cycles bifurcating from centers this is a new lower bound on the maximal number of small amplitude limit cycles obtained in such cubic switching systems finally a method is present to show the realization of the 18 limit cycles

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

jumps of the fundamental solution matrix in discontinuous systems and applications abstract this paper deals with differential equations with discontinuous right hand side the concept of a solution for a discontinuous system is defined on the basis of differential inclusions using filippov s method we study in particular the behaviour of solutions crossing a discontinuity surface transversally a formula characterizing jumps of the fundamental solution matrix is derived as an application of it the concept of poincare mapping is defined for such systems

melnikov transforms bernoulli bundles and almost periodic perturbations in this paper we study nonlinear time varying perturbations of an autonomous vector field in the plane r2 we assume that the unperturbed equation i e the given vector field has a homoclinic orbit and we present a gen eralization of the melnikov method which allows us to show that the perturbed equation has a transversal homoclinic trajectory the key to our generalization is the concept of the melnikov transform which is a linear transformation on the space of perturbation functions the appropriate dynamical setting for studying these perturbation is the concept of a skew product flow the concept of transversality we require is best understood in this context under conditions whereby the perturbed equation admits a transversal homoclinic trajectory we also study the dynamics of the perturbed vector field in the vicinity of this trajectory in the skew product flow we show the dynamics near this trajectory can have the exotic behavior of the bernoulli shift the exact description of this dynamical phenomenon is in terms of a flow on a fiber bundle which we call the bernoulli bundle we allow all perturbations which are bounded and uniformly continuous in time thus our theory includes the classical periodic perturbations studied by melnikov quasi periodic and almost periodic perturbations as well as toroidal perturbations which are close to quasi periodic perturbations

new global bifurcation diagrams for piecewise smooth systems transversality of homoclinic points does not imply chaos abstract in this paper we consider some piecewise smooth 2 dimensional systems having a possibly non smooth homoclinic t we assume that the critical point 0 lies on the discontinuity surface 0 we consider 4 scenarios which differ for the presence or not of sliding close to 0 and for the possible presence of a transversal crossing between t and 0 we assume that the systems are subject to a small non autonomous perturbation and we obtain 4 new bifurcation diagrams in particular we show that in one of these scenarios the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinics

an adaptive petrov galerkin formulation for the compressible euler and navier stokes equations in this paper we combine a stable petrov galerkin formulation for the compressible euler and navier stokes equations with an h adaptive remeshing refinement including directional stretching and stretching ratio in the mesh regeneration procedure the result is an accurate and efficient scheme appropriated to solve problems presenting shocks and boundary layers

stabilized finite element method for the radial dirac equation a challenging difficulty in solving the radial dirac eigenvalue problem numerically is the presence of spurious unphysical eigenvalues among the genuine ones that are neither related to mathematical interpretations nor to physical explanations many attempts have been made and several numerical methods have been applied to solve the problem using the finite element method fem the finite difference method or other numerical schemes unfortunately most of these attempts failed to overcome the difficulty as a fem approach this work can be regarded as a first promising scheme to solve the spuriosity problem completely our approach is based on an appropriate choice of trial and test function spaces we develop a streamline upwind petrov galerkin method to the equation and derive an explicit stability parameter

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

melnikov method for homoclinic bifurcation in nonlinear impact oscillators based on an inverted pendulum impacting on rigid walls under external periodic excitation a class of nonlinear impact oscillators is discussed for its homoclinic bifurcation the melnikov method established for smooth dynamical systems is extended to be applicable to the nonsmooth one for nonlinear impact systems closed form solutions between impacts are generally unavailable the absence of closed form solutions makes difficulties in estimation of the gap between the stable manifold and unstable manifold in this paper we give a method to compute the melnikov functions up to the n t h order so as to obtain conditions of parameters for the persistence of homoclinic cycles which are formed via the identification given by the impact rule

melnikov s method for a general nonlinear vibro impact oscillator abstract melnikov s method is presented for a general nonlinear vibro impact oscillator and melnikov function for homoclinic orbits is obtained analytically a duffing vibro impact oscillator is given to illustrate the application of the procedures the obtained results are verified by the phase portrait poincare surface of section and bifurcation diagrams

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part ii analysis for p1 and q1 finite elements abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers a review and a comparison of the most methods which have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers are given in part i v john p knobloch on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review comput methods appl mech engrg 196 2007 2197 2215 in the present paper the most promising of these sold methods are investigated in more detail for p 1 and q 1 finite elements in particular the dependence of the results on the mesh the data of the problems and parameters of the methods are studied analytically and numerically furthermore the numerical solution of the nonlinear discrete problems is discussed and the capability of adaptively refined grids for reducing spurious oscillations is examined our conclusion is that also for simple problems any of the sold methods generally provides solutions with non negligible spurious oscillations

the asymptotic iteration method for the angular spheroidal eigenvalues the asymptotic iteration method is applied to calculate the angular spheroidal eigenvalues ml c it is shown that this method asymptotically gives accurate results over the full range of parameter values l m and c

melnikov theory for a class of planar hybrid systems abstract in this paper we extend the well known melnikov theory for smooth systems to a class of two zonal planar hybrid piecewise smooth systems in this class each zone of differentiability is separated by a straight line and in these zones the dynamics are governed by a smooth system when an orbit reaches the separation line then a reset map applies before entering the orbit in the other zone the family of non generic systems of this class which has a continuum of periodic orbits is considered then we study the periodic orbits of the continuum that persist under a perturbation to achieve this objective first we build an appropriate poincare map after that we introduce a function of one variable whose zeros provide us the periodic orbits that remain after the perturbation this function will be called the melnikov function because it is very close to the so called melnikov function for smooth systems finally we apply the obtained results to some piecewise linear systems continuous as well as discontinuous ones

on an iteration method for eigenvalue problems we discuss a recently proposed asymptotic iteration method for eigenvalue problems we analyse its rate of convergence the use of adjustable parameters to improve it and the relationship with an alternative method based on the same ideas

finite element methods for linear hyperbolic problems abstract we give a survey of some recent work by the authors on finite element methods for convectiondiffusion problems and first order linear hyperbolic problems

a stable petrov galerkin method for convection dominated problems in this paper a new petrov galerkin formulation for solving convection dominated problems is presented the method developed achieves the quasi optimal convergence rates when the solution is regular and provides the necessary stability to avoid spurious oscillations when strong gradients are present such important properties allow the use of p refinement to improve the solution in regions with discontinuities because of the stability engendered by the new petrov galerkin method in this matter a proper evaluation of the intrinsic time scale function appearing in the design of this method is crucial to guarantee the required accuracy

concurrent homoclinic bifurcation and hopf bifurcation for a class of planar filippov systems abstract this paper investigates both homoclinic bifurcation and hopf bifurcation which occur concurrently in a class of planar perturbed discontinuous systems of filippov type firstly based on a geometrical interpretation and a new analysis of the so called successive function sufficient conditions are proposed for the existence and stability of homoclinic orbit of unperturbed systems then with the discussion about poincare map bifurcation analyses of homoclinic orbit and parabolic parabolic pp type pseudo focus are presented it is shown that two limit cycles can appear from the two different kinds of bifurcation in planar filippov systems

on finite element methods for 3d time dependent convection diffusion reaction equations with small diffusion the paper studies finite element methods for the simulation of time dependent convection diffusion reaction equations with small diffusion the supg method a sold method and two types of fem fct methods the methods are assessed in particular with respect to the size of the spurious oscillations in the computed solutions at a 3d example with nonhomogeneous dirichlet boundary conditions and homogeneous neumann boundary conditions

bifurcation and chaos near sliding homoclinics we study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface we assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations following a functional analytic approach we construct a melnikov like function m in such a way that if m has a simple zero at some point then the system has solutions that behave chaotically applications of this result to quasi periodic systems are also given

impact oscillators with homoclinic orbit tangent to the wall abstract homoclinic bifurcation for a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation is analyzed under the hypothesis that the unperturbed system has a homoclinc orbit tangent to the wall consequently the impact surface cannot be chosen as the poincare section to measure the distance between the perturbed stable and unstable manifolds furthermore compared to the case that the unperturbed homoclinic orbit intersects the wall transversally more cases are involved as the parameters vary thus the analysis of the homoclinic orbit tangent to the wall is much more difficult in this paper by using a method of melnikov type we derive sufficient conditions under which the perturbed stable and unstable manifolds intersect transversally as an application an impact oscillator of duffing type is studied in detail

comment on a solution of the bosonic and algebraic hamiltonians by using an aim we analyze a recently proposed variant of the asymptotic iteration method for bosonic hamiltonians koc et al 2010 j phys a math theor 43 395208 we show that the main equations proposed by the authors are inconsistent and only apply to trivial problems we conjecture how the authors obtained their results for nontrivial models

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

a solution of the bosonic and algebraic hamiltonians by using an aim we apply the notion of an asymptotic iteration method aim to determine the eigenvalues of the bosonic hamiltonians that include a wide class of quantum optical models we consider the solutions of the hamiltonians which are even the polynomials of the fourth order with respect to the boson operators we also demonstrate the applicability of the method for obtaining the eigenvalues of the simple lie algebraic structures the eigenvalues of the multi boson hamiltonians have been obtained by transforming them into the form of the single boson hamiltonian in the framework of the aim

homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations nonautomonous ordinary differential equations depending on two parameters 1 and 2 are considered in rn it is assumed that when both parameters are zero the differential equation is autonomous with a hyperbolic equilibrium and a homoclinic solution no restriction is placed on the dimension of the phase space rn or on the dimension of intersection of the stable and unstable manifolds by means of the method of lyapunov schmidt a bifurcation function h is constructed between two finite dimensional spaces where the zeros of h correspond to homoclinic solutions at nonzero parameter values the independent variables of h consist of scalars 1 2 and a vector where is a phase angle and corresponds to directions other than along the original homoclinic solution tangent to both the stable and unstable manifolds when is fixed the equation h 0 yields in general several bifurcation curves through the origin in the 1 2 plane along which there exists a homoclinic solution when is varied these become a number of wedge shaped regions the theory is applied to two examples one in r6 where the invariant manifolds meet in dimension three and a second in r4 where these manifolds agree

iterative local solvers for distributed krylov schwarz method applied to convection diffusion problems abstract nowadays supercomputers can be used to solve large scale problems that come from simulation of industrial or research problems however those machines are usually inacessible to most industries and university laboratories around the world in this work we present an iterative solver a krylov schwarz method ksm to be used in a collection of workstations under pvm the subdomain problems are solved by using many methods in order to show how the choice of the local solvers affects the overall performance of the distributed ksm

finite element method for solving the dirac eigenvalue problem with linear basis functions abstract in this work we will treat the spurious eigenvalues obstacle that appears in the computation of the radial dirac eigenvalue problem using numerical methods the treatment of the spurious solution is based on applying petrov galerkin finite element method the significance of this work is the employment of just continuous basis functions thus the need of a continuous function which has a continuous first derivative as a basis as in 2 3 is no longer required the petrov galerkin finite element method for the dirac eigenvalue problem strongly depends on a stability parameter that controls the size of the diffusion terms added to the finite element formulation for the problem the mesh dependent parameter is derived based on the given problem with the particular basis functions

a new stabilized finite element formulation for scalar convection diffusion problems the streamline and approximate upwind petrov galerkin method abstract a new stabilized and accurate finite element formulation for convection dominated problems is herein developed the basis of the new formulation is the choice of a new upwind function the upwind function chosen for the new method provokes its degeneration into the supg or cau methods depending on the approximate solution s regularity the accuracy and stability of the new formulation for the linear and scalar advection diffusion equation is demonstrated in several numerical examples

variational sturmian approximation a nonperturbative method of solving time independent schrodinger equation a variationally improved sturmian approximation for solving time independent schrodinger equation is developed this approximation is used to obtain the energy levels of a quartic anharmonic oscillator a quartic potential and a gaussian potential the results are compared with those of the perturbation theory the wkb approximation and the accurate numerical values

quasi exact solution of the rabi hamiltonian a method is suggested to obtain the quasi exact solution of the rabi hamiltonian it is conceptually simple and can be easily extended to other systems the analytical expressions are obtained for eigenstates and eigenvalues in terms of orthogonal polynomials it is also demonstrated that the rabi system in a particular case coincides with the quasi exactly solvable poschl teller potential

improvements of the mizukami hughes method for convection diffusion equations abstract we consider the mizukami hughes method for the numerical solution of scalar two dimensional steady convection diffusion equations using conforming triangular piecewise linear finite elements we propose several modifications of this method to eliminate its shortcomings the improved method still satisfies the discrete maximum principle and gives very accurate discrete solutions in convection dominated regime which is illustrated by several numerical experiments in addition we show how the mizukami hughes method can be applied to convection diffusion reaction equations and to three dimensional problems

nonsmooth homoclinic orbits melnikov functions and chaos in discontinuous systems abstract we first study the problem of chaotic behaviour in time perturbed discontinuous systems whose unperturbed part has a piecewise c 1 homoclinic solution transversally crossing the discontinuity manifolds we show that if a certain melnikov function has a simple zero at some point then the system has solutions that behave chaotically the melnikov function is explicitly constructed for two dimensional systems and extends the usual melnikov function for the smooth case in the second part we extend these results to sliding homoclinic bifurcations we also mention some possibilities for further research

hp cloud approximation of the dirac eigenvalue problem the way of stability we apply hp cloud method to the radial dirac eigenvalue problem the difficulty of occurrence of spurious eigenvalues among the genuine ones in the computation is resolved the method of treatment is based on assuming hp cloud petrov galerkin scheme to construct the weak formulation of the problem which adds a consistent diffusivity to the variational formulation the size of the artificially added diffusion term is controlled by a stability parameter tau the derivation of tau assumes the limit behavior of the eigenvalues at infinity the parameter tau is applicable for generic basis functions this is combined with the choice of appropriate intrinsic enrichments in the construction of the cloud shape functions c 2014 elsevier inc all rights reserved

bifurcation of periodic orbits in a class of planar filippov systems abstract in this paper we discuss the perturbations of a general planar filippov system with exactly one switching line when the system has a limit cycle we give a condition for its persistence when the system has an annulus of periodic orbits we give a condition under which limit cycles are bifurcated from the annulus we also further discuss the stability and bifurcations of a nonhyperbolic limit cycle when the system has an annulus of periodic orbits we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching

third harmonic generation as a third order quasi exactly solvable system the two degrees of freedom in the usual third harmonic generation effective hamiltonian can be separated both classically and quantum mechanically into a harmonic oscillator and a nonlinear oscillator in turn the quantum hamiltonian of the nonlinear oscillator can be written as a cubic polynomial in the generators of a quasi exactly solvable lie algebra and the physically relevant eigenstates are precisely those that can be determined exactly although in general not explicitly since the standard jeffreys wentzel kramers brillouin methods are not easily applicable to general third order differential equations we use a bohr sommerfeld quantization of the classical orbits to obtain approximate explicit formulas for the corresponding quantum eigenvalues

reply to the comment on a solution of the bosonic and algebraic hamiltonians by using an aim we reply to the comment made by fernandez 2011 j phys a math theor 44 228001 on our recent paper koc et al 2010 j phys a math theor 43 395208 we have shown that the main equations are consistent and the results can be applied for solving a significant number of problems we admit that one or two steps missing in the original paper may lead to a different interpretation

an algebraic variational multiscale multigrid method based on plain aggregation for convection diffusion problems a new computational framework referred to as algebraic variational multiscale multigrid method is proposed representing the initial step of our work on merging the variational multiscale method with algebraic multigrid projection methods this new approach allows for a separation of resolved scales into various scale groups in a purely algebraic way that is with no need to utilize further discretizations beyond the basic one by this means it conveniently facilitates the application of modeling terms either to all scale groups or only to selected scale groups for instance only to the fine scales of the problem variational multigrid techniques form the basis of our new approach after presenting the cornerstones of the framework for an abstract problem formulation the close relationship of the variational multiscale method to projection methods in general and to variational multigrid techniques are discussed the algebraic variational multiscale multigrid method is then particularly analyzed for convection diffusion problems the present initial study focuses on exploiting the methodical aspects of the new framework by developing a fine scale discontinuity capturing approach to diminish oscillations at sharp layers our technique including a fine scale discontinuity capturing term is applied to numerical examples of strongly convection dominated convection diffusion problems the results demonstrate that it enables the diminuation of local oscillations at parabolic boundary and interior layers by about 60 80 percent without any notable smearing effect

exponential dichotomies heteroclinic orbits and melnikov functions the interest in the behaviour of the solutions to perturbed autonomous systems with bounded perturbative term like

asymptotic iteration method for eigenvalue problems an asymptotic iteration method for solving second order homogeneous linear differential equations of the form y 0 x y s0 x y is introduced where 0 x 0 and s0 x are c functions applications to schrodinger type problems including some with highly singular potentials are presented

boundary element methods for transient convective diffusion part i general formulation and 1d implementation a general formulation of higher order boundary element methods bem is presented for time dependent convective diffusion problems in one and multi dimensions free space time dependent convective diffusion fundamental solutions originally proposed by carslaw and jaeger are used to obtain the boundary integral formulation linear quadratic and quartic time interpolation functions are introduced in this paper for approximate representation of time dependent boundary temperatures and normal fluxes closed form time integration of the kernels is mandatory to attain both accuracy and efficiency of the numerical approach a complete set of time integrals for the one dimensional formulation is presented here for the first time in the literature