analytical approximation of cuspidal loops using a nonlinear time transformation method abstract in this work we consider cuspidal loops i e homoclinic orbits to cuspidal singular points we develop an iterative procedure founded on the nonlinear time transformation method to estimate such codimension three global bifurcations up to any wanted order not only in the space of parameters but also in the phase plane as far as we know this is the first time in the literature that this theoretical result is achieved for these global connections the existence and uniqueness of the perturbed solution obtained are proved to illustrate the effectiveness of the method we study cuspidal loops in two normal forms of degenerate takens bogdanov bifurcations excellent agreement is found between our analytical predictions and the corresponding numerical continuations

numerical computation of heteroclinic orbits we give a numerical method for the computation of heteroclinic orbits connecting two saddle points in 2 these can be computed to very high period due to an integral phase condition and an adaptive discretization we can also compute entire branches one dimensional continua of such orbits the method can be extended to compute an invariant manifold that connects two fixed points in n as an example we compute branches of traveling wave front solutions to the huxley equation using weighted sobolev spaces and the general theory of approximation of nonlinear problems we show that the errors in the approximate wave speed and in the approximate wave front decay exponentially with the period

the dynamics of traveling waves for a nonlinear belousov zhabotinskii system abstract in this paper we consider the existence of traveling wave fronts in a belousov zhabotinskii system with delay by traveling wave transformation and time scale transformation we change the belousov zhabotinskii system with delay into a singularly perturbed differential system by applying geometric singular perturbation theory we construct a locally invariant manifold for the associated traveling wave equation and obtain the traveling wave fronts for the equation by using the fredholm orthogonality finally we discuss the asymptotic behaviors of traveling wave solutions by applying the asymptotic theory

bifurcation of plane voidage waves in fluidized beds abstract a two dimensional fluidized bed model is investigated from a bifurcation point of view it is shown that bifurcation to periodic travelling waves occurs when the basic state of uniform fluidization becomes unstable the nature of these waves may be one or two dimensional here the main focus is on the bifurcation of oblique travelling waves i e plane voidage waves propagating into various directions in space these waves can be described by a two dimensional dynamical system which can be analyzed extensively using phase plane methods several stationary and periodic solutions as well as homoclinic and heteroclinic orbits are shown to exist in the model studied

a melnikov method for homoclinic orbits with many pulses we present an extension of the melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near integrable systems the melnikov function in this situation is the sum of the usual melnikov functions evaluated with some appropriate phase delays we show that a nonfolding condition which involves the logarithmic derivative of the melnikov function must be satisfied in addition to the usual transversality conditions in order for homoclinic orbits with more than one pulse to exist

influence of counter rotating von karman flow on cylindrical rayleigh benard convection the axisymmetric flow in an aspect ratio one cylinder whose upper and lower bounding disks are maintained at different temperatures and rotate at equal and opposite velocities is investigated in this combined rayleigh benard von karman problem the imposed temperature gradient is measured by the rayleigh number ra and the angular velocity by the reynolds number re although fluid motion is present as soon as re not equal 0 a symmetry breaking transition analogous to the onset of convection takes place at a finite rayleigh number higher than that for re 0 for re 95 it is a hopf bifurcation to a limit cycle the steady states and limit cycle are connected via a pair of saddle node infinite period bifurcations except very near the takens bogdanov codimension two point where the scenario includes global bifurcations detailed phase portraits and bifurcation diagrams are presented as well as the evolution of the leading part of the spectrum over the parameter ranges 0 re 120 and 0 ra 30 000

perturbations of symmetric elliptic hamiltonians of degree four in this paper four parameter unfoldings x of symmetric elliptic hamiltonians of degree four are studied we prove that in a compact region of the period annulus of x0 the displacement function of x is sign equivalent to its principal part which is given by a family induced by a chebychev system and we describe the bifurcation diagram of x in a full neighborhood of the origin in the parameter space where at most two limit cycles can exist for the corresponding systems

the lindstedt poincare technique as an algorithm for computing periodic orbits the lindstedt poincare technique in perturbation theory is used to calculate periodic orbits of perturbed differential equations it uses a nearby periodic orbit of the unperturbed differential equation as the first approximation we derive a numerical algorithm based upon this technique for computing periodic orbits of dynamical systems the algorithm unlike the lindstedt poincare technique does not require the dynamical system to be a small perturbation of a solvable differential equation this makes it more broadly applicable the algorithm is quadratically convergent it works with equal facility as examples show irrespective of whether the periodic orbit is attracting or repelling or a saddle one of the examples presents what is possibly the most accurate computation of hill s orbit of lunation since its justly celebrated discovery in 1878

exact results for the kuramoto model with a bimodal frequency distribution we analyze a large system of globally coupled phase oscillators whose natural frequencies are bimodally distributed the dynamics of this system has been the subject of long standing interest in 1984 kuramoto proposed several conjectures about its behavior ten years later crawford obtained the first analytical results by means of a local center manifold calculation nevertheless many questions have remained open especially about the possibility of global bifurcations here we derive the system s stability diagram for the special case where the bimodal distribution consists of two equally weighted lorentzians using an ansatz recently discovered by ott and antonsen we show that in this case the infinite dimensional problem reduces exactly to a flow in four dimensions depending on the parameters and initial conditions the long term dynamics evolves to one of three states incoherence where all the oscillators are desynchronized partial synchrony where a macroscopic group of phase locked oscillators coexists with a sea of desynchronized ones and a standing wave state where two counter rotating groups of phase locked oscillators emerge analytical results are presented for the bifurcation boundaries between these states similar results are also obtained for the case in which the bimodal distribution is given by the sum of two gaussians

fast and accurate evaluation of nonlocal coulomb and dipole dipole interactions via the nonuniform fft we present a fast and accurate algorithm for the evaluation of nonlocal long range coulomb and dipole dipole interactions in free space the governing potential is simply the convolution of an interaction kernel u mathbf x and a density function rho mathbf x psi mathbf x 2 for some complex valued wave function psi mathbf x permitting the formal use of fourier methods these are hampered by the fact that the fourier transform of the interaction kernel widehat u mathbf k has a singularity and or widehat rho mathbf k ne0 at the origin mathbf k bf 0 in fourier phase space thus accuracy is lost when using a uniform cartesian grid in mathbf k which would otherwise permit the use of the fft for evaluating the convolution here we make use of a high order discretization of the fourier integral accelerated by the nonuniform fast fourier transform nufft by adopting spherical and polar phase space discretizations in three and two dimensions respectively

application of the trefftz method in plane elasticity problems direct and indirect approximations using sets of non singular complete trefftz functions i e the complete systems of solutions have been successfully applied to harmonic problems 1 in this paper the procedure is applied to a more complex situation plane elasticity problems the examples show that the present method can avoid the difficulties relating to singular integration and has good accuracy compared with traditional boundary elements

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

persistent properties of bifurcations abstract this paper reviews results about bifurcations which occur in families of differential equations persistent properties are defined to be those which remain when the family of equations is perturbed we provide a list of such properties which is relevant for numerical studies of dynamical systems

knotted periodic orbits in suspensions of smale s horseshoe extended families and bifurcation sequences abstract we consider knotted periodic orbits in the natural suspension of smale s horseshoe map as constructed by holmes and williams 1985 and holmes 1986 we identify quartets and octets of periodic orbits which are isotopic knots and indicate how they are related via the sequences of bifurcations in which they are created in two parameter families such as the henon map

an efficient calculation of the far field radiated by non uniformly sampled planar fields complying nyquist theorem an efficient and accurate method to calculate the far field radiated by non uniformly sampled planar apertures using the non uniform fast fourier transform nufft is shown the sampled fields in the planar aperture have to comply the nyquist theorem the method takes into account the amplitude of the unit cell radiation pattern which allows to compute more accurately the copolar and crosspolar components of the far field due to the generality of the nufft it also allows to overcome some limitations of the fft when computing the radiated fields of reflectarrays with periodicity larger than half a wavelength because the nufft is precision dependent a discussion of how its accuracy can affect the computed radiated fields is carried out numerical examples are provided to show the accuracy and performance of the nufft with regard to the fft and direct evaluation of the fields finally a study of computing times comparing the fft nufft and direct evaluation is presented

fast algorithms for helmholtz green s functions the formal representation of the quasi periodic helmholtz green s function obtained by the method of images is only conditionally convergent and thus requires an appropriate summation convention for its evaluation instead of using this formal sum we derive a candidate green s function as a sum of two rapidly convergent series one to be applied in the spatial domain and the other in the fourier domain as in ewald s method we prove that this representation of green s function satisfies the helmholtz equation with the quasi periodic condition and furthermore leads to a fast algorithm for its application as an operator we approximate the spatial series by a short sum of separable functions given by gaussians in each variable for the series in the fourier domain we exploit the exponential decay of its terms to truncate it we use fast and accurate algorithms for convolving functions with this approximation of the quasi periodic green s function the resulting method yields a fast solver for the helmholtz equation with the quasi periodic boundary condition the algorithm is adaptive in the spatial domain and its performance does not significantly deteriorate when green s function is applied to discontinuous functions or potentials with singularities we also construct helmholtz green s functions with dirichlet neumann or mixed boundary conditions on simple domains and use a modification of the fast algorithm for the quasi periodic green s function to apply them the complexity in dimension d 2 of these algorithms is graphic 1 d log c log 1 d where is the desired accuracy is proportional to the number of wavelengths contained in the computational domain and c is a constant we illustrate our approach with examples 1 embed inline graphic 1 gif

bifurcation diagram and stability for a one parameter family of planar vector fields we consider the 1 parameter family of planar quintic systems x y3 x3 y x my5 introduced by a bacciotti in 1985 it is known that it has at most one limit cycle and that it can exist only when the parameter m is in 0 36 0 6 in this paper using the bendixon dulac theorem we give a new unified proof of all the previous results we shrink this to 0 547 0 6 and we prove the hyperbolicity of the limit cycle we also consider the question of the existence of polycycles the main interest and difficulty for studying this family is that it is not a semi complete family of rotated vector fields when the system has a limit cycle we also determine explicit lower bounds of the basin of attraction of the origin finally we answer an open question about the change of stability of the origin for an extension of the above systems

three dimensional green s functions in anisotropic piezoelectric solids abstract explicit expressions for three dimensional extended green s displacements in general anisotropic piezoelectric solids are derived a very efficient procedure for the numerical evaluation of the derivatives of the extended green s displacements is also proposed numerical comparisons are carried out for a transversely isotropic piezoelectric solid for which exact closed form solutions are available it is found that the extended green s displacements and their derivatives obtained with the present explicit formulation are in perfect agreement with the exact closed form solutions these green s functions can be used in the boundary integral equations for piezoelectric solids of general anisotropy and for subsequent numerical solutions of these equations by means of the boundary element method

fast convolution with the free space helmholtz green s function we construct an approximation of the free space green s function for the helmholtz equation that splits the application of this operator between the spatial and the fourier domains as in ewald s method for evaluating lattice sums in the spatial domain we convolve with a sum of decaying gaussians with positive coefficients and in the fourier domain we multiply by a band limited kernel as a part of our approach we develop new quadratures appropriate for the singularity of green s function in the fourier domain the approximation and quadratures yield a fast algorithm for computing volumetric convolutions with green s function in dimensions two and three the algorithmic complexity scales as where is selected accuracy is the number of wavelengths in the problem d is the dimension and c is a constant the algorithm maintains its efficiency when applied to functions with singularities in contrast to the fast multipole method as our approximation makes a transition to that of the free space green s function for the poisson equation we illustrate our approach with examples

efficient thermal field computation in phase field models we solve the phase field equations in two dimensions to simulate crystal growth in the low undercooling regime the novelty is the use of a fast solver for the free space heat equation to compute the thermal field this solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient free space heat equation with a smooth source term the computational cost and memory requirements of the new solver are reasonable and no artificial boundary conditions are needed this allows one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field which can result in significant computational savings in the low undercooling regime

period doubling cascades of attractors a prerequisite for horseshoes this paper shows that if a horseshoe is created in a natural manner as a parameter is varied then the process of creation involves the appearance of attracting periodic orbits of all periods furthermore each of these orbits will period double repeatedly with those periods going to infinity

field inhomogeneity correction based on gridding reconstruction for magnetic resonance imaging spatial variations of the main field give rise to artifacts in magnetic resonance images if disregarded in reconstruction with non cartesian k space sampling they often lead to unacceptable blurring data from such acquisitions are usually reconstructed with gridding methods and optionally restored with various correction methods both types of methods essentially face the same basic problem of adequately approximating an exponential function to enable an efficient processing with fast fourier transforms nevertheless they have commonly addressed it differently so far in the present work a unified approach is pursued the principle behind gridding methods is first generalized to nonequispaced sampling in both domains and then applied to field inhomogeneity correction three new algorithms which are compatible with a direct conjugate phase and an iterative algebraic reconstruction are derived in this way from a straightforward embedding of the data into a higher dimensional space their evaluation in simulations and phantom experiments with spiral k space sampling shows that one of them promises to provide a favorable compromise between fidelity and complexity compared with existing algorithms moreover it allows a simple choice of key parameters involved in approximating an exponential function and a balance between the accuracy of reconstruction and correction

variational formulations for tfem of piezoelectricity abstract the paper presents a family of variational formulations of trefftz finite elements wherein the assumed internal displacement and electric potential fields a priori fulfil the governing differential equations of the problem over the element sub domain while the inter element continuity and the boundary conditions are enforced using a modified variational principle together with an independent frame field defined on each element boundary it is based on four free energy densities each with two kinds of independent variables as basic independent variables i e d e e e d and e based on the assumed intra element and frame fields an element stiffness matrix equation is obtained which is implemented into computer programs for numerical analysis some numerical examples are considered to show the application of the proposed formulation

application of symmetric indirect trefftz method to free vibration problems in 2d a symmetric indirect trefftz method is developed to solve the free vibration problem of a 2d membrane it is proved that in this approach the spurious eigensolution exists and an auxiliary matrix is constructed to help extraction of the spurious solution using the generalized singular value decomposition in addition to the spurious eigensolution this regular formulation suffers from its ill posed nature i e the numerical instability in order to deal with the numerical instability the tikhonov s regularization method in conjunction with the generalized singular value decomposition is suggested the proposed approach has some merits when compared with other regular boundary element formulations reported so far namely the capacity of representing eigenmodes and the ability to deal with a multiply connected domain of genus 1 several numerical examples are demonstrated to show the validity of the current approach copyright 2003 john wiley sons ltd

knotted periodic orbits in suspensions of smale s horseshoe period multiplying and cabled knots abstract following earlier work on knotted periodic orbits in a suspension of smale s 1 horseshoe diffeomorphism holmes and williams 2 we define a notion of iterated horseshoe knots we show that approximately half the horseshoe knots are cabled and that among these cablings are infinitely many non isotopic iterated torus knots we obtain uniqueness results for certain families of resonant torus knots related to cascades of period doubling and period multiplying bifurcations which occur as a horseshoe is created by passing through a family of diffeomorphisms such as the henon maps we give formulae from which linking numbers can be computed and provide closed form expressions for the type numbers of certain resonant iterated torus knots

fast iterative image reconstruction for mri in the presence of field inhomogeneities in magnetic resonance imaging magnetic field inhomogeneities cause distortions in images that are reconstructed by conventional fast fourier transform fft methods several noniterative image reconstruction methods are used currently to compensate for field inhomogeneities but these methods assume that the field map that characterizes the off resonance frequencies is spatially smooth recently iterative methods have been proposed that can circumvent this assumption and provide improved compensation for off resonance effects however straightforward implementations of such iterative methods suffer from inconveniently long computation times this paper describes a tool for accelerating iterative reconstruction of field corrected mr images a novel time segmented approximation to the mr signal equation we use a min max formulation to derive the temporal interpolator speedups of around 60 were achieved by combining this temporal interpolator with a nonuniform fast fourier transform with normalized root mean squared approximation errors of 0 07 the proposed method provides fast accurate field corrected image reconstruction even when the field map is not smooth

nonuniform fast fourier transforms using min max interpolation the fast fourier transform fft is used widely in signal processing for efficient computation of the ft of finite length signals over a set of uniformly spaced frequency locations however in many applications one requires nonuniform sampling in the frequency domain i e a nonuniform ft several papers have described fast approximations for the nonuniform ft based on interpolating an oversampled fft this paper presents an interpolation method for the nonuniform ft that is optimal in the min max sense of minimizing the worst case approximation error over all signals of unit norm the proposed method easily generalizes to multidimensional signals numerical results show that the min max approach provides substantially lower approximation errors than conventional interpolation methods the min max criterion is also useful for optimizing the parameters of interpolation kernels such as the kaiser bessel function

the slow passage through a hopf bifurcation delay memory effects and resonance this paper explores analytically and numerically in the context of the fitzhugh nagumo model of nerve membrane excitability an interesting phenomenon that has been described as a delay or memory effect it can occur when a parameter passes slowly through a hopf bifurcation point and the system s response changes from a slowly varying steady state to slowly varying oscillations on quantitative observation it is found that the transition is realized when the parameter is considerably beyond the value predicted from a straightforward bifurcation analysis which neglects the dynamic aspect of the parameter variation this delay and its dependence on the speed of the parameter variation are described the model involves several parameters and particular singular limits are investigated one in particular is the slow passage through a low frequency hopf bifurcation where the system s response changes from a slowly varying steady state to slowly varying relaxation oscillations we find in this case the onset o

sparse fourier transform via butterfly algorithm this paper introduces a fast algorithm for computing sparse fourier transforms with spatial and fourier data supported on curves or surfaces this problem appears naturally in several important applications of wave scattering digital signal processing and reflection seismology the main idea of the algorithm is that the interaction between a frequency region and a spatial region is approximately low rank if the product of their widths are bounded by the maximum frequency based on this property we can approximate the interaction between these two boxes accurately and compactly using a small number of equivalent sources the overall structure of the algorithm follows the butterfly algorithm the computation is further accelerated by exploiting the tensor product property of the fourier kernel in two and three dimensions the proposed algorithm is accurate and has the optimal complexity we present numerical results in both two and three dimensions

bifurcation of limit cycles from quadratic isochrones abstract for a one parameter family of plane quadratic vector fields x e depending analytically on a small real parameter e we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous or linearizable center techniques are developed for treating the bifurcations of all orders and these are applied to prove the following results for the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three for one class of nonlinear isochrones at most one continuous family of limit cycles can emerge whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge moreover for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center

trefftz method for kirchhoff plate bending problems kirchhoff plate bending problems were studied by both the trefftz direct and indirect approximations in which non singular complete trefftz functions are used as the weighting and or trial functions the trefftz direct method involved only the quantities of engineering interest numerical results are given to show the efficiency and the excellent accuracy of the present method

numerical computation and continuation of invariant manifolds connecting fixed points a numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in mathbb r n is given extending the results of an earlier paper comput appl math 26 1989 pp 159 170 by the authors basically a boundary value problem on the real line is truncated to a finite interval the method applies in particular to the computation of heteroclinic orbits the emphasis is on the systematic computation of such orbits by continuation using the fact that the linearized operator of our problem is fredholm in appropriate banach spaces the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval several applications are considered including the computation of traveling wave solutions to reaction diffusion problems computations were done using the software package auto

a study of the limit cycles associated with a generalized codimension 3 lienard oscillator in this paper melnikov and petrov methods are used to study the limit cycles associated with the degenerate bifurcations of a representative generalized codimension 3 lienard oscillator it is shown that the coexistence of multiple limit cycles is possible and that jumping phenomena exist between limit cycles moreover it is found that the stable limit cycles oscillations and the unstable limit cycles oscillations can occur alternatively

the hybrid trefftz finite element model and its application to plate bending this paper presents a new hybrid element approach and applies it to plate bending in contrast to more conventional models the formulation is based on displacement fields which fulfil a priori the non homogeneous lagrange equation trefftz method the interelement continuity is enforced by using a stationary principle together with an independent interelement displacement the final unknowns are the nodal displacements and the elements may be implemented without any difficulty in finite element libraries of standard finite element programs the formulation only calls for integration along the element boundaries which enables arbitrary polygonal or even curve sided elements to be generated where relevant known local solutions in the vicinity of a singularity or stress concentration may be used as an optional expansion basis to obtain for example particular singular corner elements elements presenting circular holes etc thus a high degree of accuracy may be achieved without a troublesome mesh refinement another important advantage of the formulation is the possibility of generating by a single element subroutine a large number of various elements triangles quadrilaterals etc presenting an increasing degree of accuracy the paper summarizes the results of numerical studies and shows the excellent accuracy and efficiency of the new elements the conclusions present some ideas concerning the adaptive version of the new elements extension to nonlinear problems and some other developments

integral equation methods for unsteady stokes flow in two dimensions we present an integral equation formulation for the unsteady stokes equations in two dimensions this problem is of interest in its own right as a model for slow viscous flow but perhaps more importantly as an ingredient in the solution of the full incompressible navier stokes equations using the unsteady green s function the velocity evolves analytically as a divergence free vector field this avoids the need for either the solution of coupled field equations as in fully implicit pde based marching schemes or the projection of the velocity field onto a divergence free field at each time step as in operator splitting methods in addition to discussing the analytic properties of the operators that arise in the integral formulation we describe a family of high order accurate numerical schemes and illustrate their performance with several examples

dynamical systems analysis of the maasch saltzman model for glacial cycles abstract this article is concerned with the internal dynamics of a conceptual model proposed by maasch and saltzman 1990 to explain central features of the glacial cycles observed in the climate record of the pleistocene epoch it is shown that in most parameter regimes the long term system dynamics occur on certain intrinsic two dimensional invariant manifolds in the three dimensional state space these invariant manifolds are slow manifolds when the characteristic time scales for the total global ice mass and the volume of the north atlantic deep water are well separated and they are center manifolds when these characteristic time scales are comparable in both cases the reduced dynamics on these manifolds are governed by bogdanov takens singularities and the bifurcation curves associated to these singularities organize the parameter regions in which the model exhibits glacial cycles in addition knowledge of the reduced systems and their bifurcations is useful for understanding the effects of slowly varying parameters which cause passage through hopf bifurcations and of orbital milankovitch forcing both are central to the mechanism proposed by maasch and saltzman for the mid pleistocene transition in their model

on the numerical solution of the heat equation i we describe a fast solver for the inhomogeneous heat equation in free space following the time evolution of the solution in the fourier domain it relies on a recently developed spectral approximation of the free space heat kernel coupled with the non uniform fast fourier transform unlike finite difference and finite element techniques there is no need for artificial boundary conditions on a finite computational domain the method is explicit unconditionally stable and requires an amount of work of the order o nm log n where n is the number of discretization points in physical space and m is the number of time steps we refer to the approach as the fast recursive marching frm method

rotationally invariant quadratures for the sphere we construct near optimal quadratures for the sphere that are invariant under the icosahedral rotation group these quadratures integrate all n 1 2 linearly independent functions in a rotationally invariant subspace of maximal order and degree n the nodes of these quadratures are nearly uniformly distributed and the number of nodes is only marginally more than the optimal n 1 2 3 nodes using these quadratures we discretize the reproducing kernel on a rotationally invariant subspace to construct an analogue of lagrange interpolation on the sphere this representation uses function values at the quadrature nodes in addition the representation yields an expansion that uses a single function centred and mostly concentrated at nodes of the quadrature thus providing a much better localization than spherical harmonic expansions we show that this representation may be localized even further we also describe two algorithms of complexity for using these grids and representations finally we note that our approach is also applicable to other discrete rotation groups

contact topology and hydrodynamics iii knotted orbits we employ the relationship between contact structures and beltrami fields derived in part i of this series to construct a steady nonsingular solution to the euler equations on a riemannian s3 whose flowlines trace out closed curves of all possible knot and link types using careful contacttopological controls we can make such vector fields real analytic and transverse to the tight contact structure on s3 sufficient review of concepts is included to make this paper independent of the previous works in this series

complete bifurcation diagram and global phase portraits of lienard differential equations of degree four abstract li and llibre in j differential equations 252 2012 3142 3162 proved that a lienard system of degree four d x d t y a x b x 2 c x 3 x 4 d y d t x has at most one limit cycle moreover the limit cycle is stable and hyperbolic if it exists based on their works the aim of this paper is to give the complete bifurcation diagram and global phase portraits in the poincare disc of this system further first we analyze the equilibria at both finity and infinity then a necessary and sufficient condition for existence of separatrix loop is founded by the rotation property moreover a necessary and sufficient condition of the existence of limit cycles is also obtained finally we show that the complete bifurcation diagram includes one hopf bifurcation surface and one bifurcation surface of separatrix loop

a boundary integral formulation and 2d fundamental solutions for piezoelectric media abstract first based on the basic equations of piezoelectricity the boundary integral formulation is derived by utilizing the reciprocal work theorem second for the plane problem of piezoelectric media one general solution in terms of harmonic functions is derived and the fundamental solutions are therefore obtained numerical calculations by boundary element method bem are performed to give the stress concentration coefficient of an infinite piezoelectric plane with a circular hole as well as the stress and electric intensity factors of a central crack locating on an infinite plate results are found to agree well with the exact solutions

the general solution of three dimensional problems in piezoelectric media abstract the governing equations of three dimensional problems in transversely isotropic piezo electric material are composed of four second order partial differential equations in which the displacement components and electric potential functions are the essential unknowns in this paper the so called general solutions of these equations are obtained with the aid of a set of new potential functions which are used to express the components of displacements and the electric potential functions as an application of the general solution the problem of a half space piezoelectrics acted by a concentrated lateral shear force is solved

higher order bifurcations of limit cycles abstract this paper shows that asymmetrically perturbed symmetric hamiltonian systems of the formx y y x x3 1y 2x2 3xy 4x2y with analytic j e o e have at most two limit cycles that bifurcate for smalle 0 from any period annulus of the unperturbed system this fact agrees with previous results of petrov dangelmayr and guckenheimer and chicone and iliev but shows that the result of three limit cycles for the asymmetrically perturbed exterior duffing oscillator recently obtained by jebrane and oladek is incorrect the proofs follow by deriving an explicit formula for thekth order melnikov function mk h and using a picard fuchs analysis to show that in each case mk h has at most two zeros moreover the method developed in this paper for determining the higher order melnikov functions also applies to more general perturbations of these systems

bifurcational precedences in the braids of periodic orbits of spiral 3 shoes in driven oscillators we conjecture the existence of a 3 striped spiral horseshoe under one iterate of the poincare map that arises in the analysis of a class of driven oscillators these systems represent escape from a smooth potential well under periodic forcing we assume topological conjugacy between the flow of the differential equation and an idealized suspension of a spiral 3 shoe and deduce bifurcational precedences by consideration of intertwining in the braid of periodic orbits we argue that many significant subharmonic bifurcations observed in such systems can then be understood in terms of the creation of this 3 shoe

approximate factorizations of fourier matrices with nonequispaced knots abstract we propose a new algorithm for the multiplication of vectors with fourier matrices of the form a tf e 2 i x k v j n n 2 1 j k n 2 where both x k and v j are arbitrary knots in n 2 n 2 the algorithm is based on an approximate factorization of the transform matrix a tf into sparse matrices which entries are chosen to minimize the frobenius norm of certain error matrices numerical experiments demonstrate that the approximation error introduced by our new algorithm is about 10 2 times smaller than the approximation error of previously reported algorithms with the gaussian or b splines as window functions and as good as a previous algorithm with the kaiser bessel function as window function

trefftz method fitting boundary conditions the paper presents various ways of fitting the boundary conditions in the t complete functions method the authors point out the distinct advantages of the orthogonal collocation in comparison to the equidistant collocation and the integral fit the convergence of the collatz error measures and the conditioning of the solution matrices are investigated in detail

infinite wavelength analysis for two phase flow a three parameter computer assisted study of global bifurcations abstract a computer assisted analysis of the long wave solutions of a two phase flow model in packed beds is presented based on the numerical study of global bifurcations a time dependent pulsing flow arises spontaneously for relatively high flow rates of gas and liquid through a packed bed a recently developed nonlinear model for this phenomenon invokes a travelling wave assumption and associates the observed wavelike pulses with the existence of long period periodic solutions in the travelling wave frame in parameter space such periodic solutions are located in the neighborhood of certain infinite period global bifurcations such as homoclinic and double heteroclinic loops in order to completely map out the travelling wave flow behavior for a given set of fluids and packing it is necessary to perform a three parameter analysis of these global bifurcations using techniques for the continuation of homoclinic and heteroclinic connections we have examined the relation between homoclinic heteroclinic both codimension 1 and double heteroclinic codimension 2 connections in the model in parameter space the families of these bifurcations interact and undergo further qualitative changes these changes include a crossing of homoclinic connection branches and the birth of a pair of double heteroclinic connections we discuss briefly how double heteroclinic loops apparently arise from a local codimension three singularity where a double zero eigenvalue occurs simultaneously with the vanishing of a higher second order term we also discuss the relation between the nature of the predicted solutions and experimental observations this computational study of global bifurcations provides a concise description of the operating region in which pulsing flow can be observed in packed beds

knotted periodic orbits in suspensions of annulus maps we consider a class of suspensions of diffeomorphisms of the annulus as flows in the orientable 3 manifold t 2 x i using results of birman williams topology 22 47 82 1983 contemp math 20 1 60 1983 we construct a knotholder or template that carries the set of periodic orbits of the flow we define rotation numbers and show that any orbit of period q and rotation number p q can be arranged as a positive braid on p strands this yields existence and uniqueness results for families of resonant torus knots p braids that are p q torus knots of period q p which correspond to order preserving birkhoff periodic orbits of the diffeomorphism we show that all other q periodic p braids have higher genus and we establish bounds on the genera of such knots we obtain existence and uniqueness results for a number of other non resonant torus knots including non order preserving q s q torus knots of rotation number 1

analysis of degenerate hopf bifurcations for a nonlinear model of calcium metabolism results of the in viva studies related to calcium metabolism have been generally analysed within the framework of linear mathematical models e g classical compartmental analysis 11 this approach assumes the maintenance of the extracellular calcium concentration at a nearly constant value under physiological conditions plasma calcium homeostasis and the adequacy of a sum of exponential terms the analytical solution of a first order linear differential system to fit kinetic data such as the time course of tracer concentration nonlinear models which exhibit a self oscillatory behaviour have been proposed recently a model of the hormonal control of calcium homeostasis in chicken includes the consideration of a growth function and takes into account complex interrelationships among the calciumregulating hormones and between these hormones and specific metabolic functions 2 another model constructed in our laboratory is a self oscillating model for calcium metabolism in the rat it includes some nonlinear processes involved in bone calcium metabolism and adequately reproduces the kinetic behaviour of radiocalcium in plasma including circadian variations 3 in contrast to linear systems for which the simple dynamic behaviour patterns are predictable and their analytical solutions are commonly applied in calcium metabolism field it is wellestablished that nonlinear systems have potentially numerous and complex spatio temporal expressions excitability periodic oscillations chaos etc therefore the relevance of the proposed nonlinear models and their significance with regard to the control of calcium metabolism require a concomitant knowledge of the set of the dynamic patterns that in theory such nonlinear structures are able to produce the variety of the dynamic behaviours exhibited by the self oscillating model we have established in rat or rather its nonlinear sub structure 4 51 motivated this theoretical work our model of calcium metabolism has been constructed in the general context of compartmental formalism fig 1 briefly this model was based on the explicit consideration the nonlinear properties of the phase transition and nucleation mechanisms associated with bone calcification and their formulation through a single second order autocatalytic process 5 in addition to its ability to generate periodic temporal variations fitting to experimental data fig 2 the model correctly approximates the growth curve of calcium mass as a function of age and proposes a dynamic organization for bone calcium metabolism which distinguishes

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

from template analysis to generating partitions i periodic orbits knots and symbolic encodings abstract we present a detailed algorithm to construct symbolic encodings for chaotic attractors of three dimensional flows it is based on a topological analysis of unstable periodic orbits embedded in the attractor and follows the approach proposed by lefranc et al phys rev lett 73 1994 1364 for each orbit the symbolic names that are consistent with its knot theoretic invariants and with the topological structure of the attractor are first obtained using template analysis this information and the locations of the periodic orbits in the section plane are then used to construct a generating partition by means of triangulations we provide numerical evidence of the validity of this method by applying it successfully to sets of more than 1500 periodic orbits extracted from numerical simulations and obtain partitions whose border is localized with a precision of 0 01 a distinctive advantage of this approach is that the solution is progressively refined using higher period orbits which makes it robust to noise and suitable for analyzing experimental time series furthermore the resulting encodings are by construction consistent in the corresponding limits with those rigorously known for both one dimensional and hyperbolic maps

multiple bifurcation problems for chemical reactors abstract this paper presents a mathematical strategy for exploring the dynamical behavior of chemical reactors our intent is to stimulate further development of the mathematics with which we approach these problems as well as to illustrate how it can be usefully applied to models of chemical reactors in the process of accomplishing this task we extend and correct published work on the models we consider at the same time we point to the mathematical problems which arise many of the results presented rely upon computer studies that hopefully will be surplanted by rigorous arguments in the future

an efficient meshless technique for the solution of transversely isotropic two dimensional piezoelectricity in this paper a new meshless technique is presented for the solution of two dimensional piezoelectricity problem the technique derivation is based on the solution of the corresponding analog equation by transforming the original set of differential equations into three poisson equations with unknown right hand side terms boundary discretisation of the resulting integral equations is eliminated by the use of the method of fundamental solutions the right hand side terms are represented in the solution as particular solutions expressed in terms of radial basis functions the problem solution is then rewritten in its new form which involves complementary solution and particular solution the governing partial differential operator for piezoelectricity is applied on the obtained solution form and forced to be satisfied at a set of domain points whereas the prescribed boundary conditions are satisfied at another set of boundary points the proposed technique is implemented into computer code where several numerical examples with different boundary conditions are tested the results demonstrated excellent agreement with those obtained from analytical and fem solutions

trefftz indirect methods for plane piezoelectricity in this paper a unified theory for the general solutions of three dimensional elasticity is employed to transform the governing equations of plane piezoelectricity into three generalized laplace equations the latter equations are uncoupled and can be pursued separately by introducing three generalized complex variables the complete solution sets for plane piezoelectricity are constructed by adopting the constituents in the solution sets as the trial functions the trefftz collocation and the trefftz galerkin methods are formulated both methods fall into the category of trefftz indirect methods as the coefficients of their trial functions are non physical numerical examples are presented to illustrate the efficacy of the formulations copyright 2005 john wiley sons ltd

knotted periodic orbits in suspensions of smale s horseshoe torus knots and bifurcation sequences we construct a suspension of smale s horseshoe diffeomorphism of the two dimensional disc as a flow in an orientable three manifold such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the duffing equation from this suspension we construct a knot holder or template a branched two manifold with a semiflow in such a way that the periodic orbits are isotopic to those in the full three dimensional flow we discuss some of the families of knotted periodic orbits carried by this template in particular we obtain theorems of existence uniqueness and non existence for families of torus knots we relate these families to resonant hamiltonian bifurcations which occur as horseshoes are created in a one parameter family of area preserving maps and we also relate them to bifurcations of families of one dimensional quadratic like maps which can be studied by kneading theory thus using knot theory kneading theory and hamiltonian bifurcation theory we are able to connect a countable subsequence of one dimensional bifurcations with a subsequence of area preserving bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary one implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area preserving family there are no universal routes to chaos

three coupled oscillators mode locking global bifurcations and toroidal chaos abstract many systems exhibit behaviour which can be described by three coupled oscillators provided the coupling is not too strong such systems can be modelled by maps of the two torus to itself in this paper we describe and explain those aspects of the bifurcation diagram for two parameter families of torus maps that involve change of mode locking type we introduce the concepts of partial and full mode locking for the coupled oscillators these notions correspond to the presence of one or two rational relations between the frequencies respectively numerical investigation of a particular family of torus maps reveals an intricate web of global bifurcations in order to explain the results we first show that we can approximate maps in the neighbourhood of a rational translation to arbitrary order by the time 1 map of a flow on the torus then we analyse those condimension 1 and 2 bifurcations for flows on the torus which change the set of frequency ratios we find a large variety of bifurcation diagrams in particular many involving homotopically non trivial saddle connections next we ask how the picture changes when the time 1 maps of a family of flows are perturbed to a general family of diffeomorphisms we find toroidal chaos in the neighbourhood of all these bifurcations meaning that there exist orbits which perform a pseudo random sequence of rotations in different directions around the torus the corresponding behaviour for the coupled oscillators is that the frequency ratios perform a random walk finally we show how all these ingredients can be put together to give global scenarios for bifurcation for families of torus maps which we believe to be of general applicability to physical systems with three weakly coupled modes of oscillation

completeness and nonuniqueness ofgeneral solutions of transversely isotropic elasticity abstract in this paper we give general solutions of transversely isotropic elasticity their completenessand nonuniqueness are proved we point out that famous lekhnitskii hu nowacki solutions and elliott lodge solutions are complete if the elastic region is z convex

short note the type 3 nonuniform fft and its applications the nonequispaced or nonuniform fast fourier transform nufft arises in a variety of application areas including imaging processing and the numerical solution of partial differential equations in its most general form it takes as input an irregular sampling of a function and seeks to compute its fourier transform at a nonuniform sampling of frequency locations this is sometimes referred to as the nufft of type 3 like the fast fourier transform the amount of work required is of the order o nlogn where n denotes the number of sampling points in both the physical and spectral domains in this short note we present the essential ideas underlying the algorithm in simple terms we also illustrate its utility with application to problems in magnetic resonance imagin and heat flow

the fractional fourier transform and quadratic field magnetic resonance imaging the fractional fourier transform frft is revisited in the framework of strongly continuous periodic semigroups to restate known results and to explore new properties of the frft we then show how the frft can be used to reconstruct magnetic resonance mr images acquired under the presence of quadratic field inhomogeneity particularly we prove that the order of the frft is a measure of the distortion in the reconstructed signal moreover we give a dynamic interpretation to the order as time evolution of a function we also introduce the notion of r a space as an extension of the fourier or k space in mr and we use it to study the distortions introduced in two common mr acquisition strategies we formulate the reconstruction problem in the context of the frft and show how the semigroup theory allows us to find new reconstruction formulas for discrete sampled signals finally the results are supplemented with numerical examples that show how it performs in a standard 1d mr signal reconstruction

fast transform from an adaptive multi wavelet representation to a partial fourier representation we present a fast algorithm to compute the partial transformation of a function represented in an adaptive pseudo spectral multi wavelet representation to a partial fourier representation such fast transformations are useful in many contexts in physics and engineering where changes of representation from a piece wise polynomial basis to a fourier basis the algorithm is demonstrated for a gaussian in one and in three dimensions for 2d we apply this approach to a gaussian in a periodic domain the accuracy and the performance of this method is compared with direct summation

generalized finite element analysis with t complete boundary solution functions the use of a complete and nonsingular set of trefftz functions in the solution of quasi harmonic equations is demonstrated and shown to be often superior to the more conventional singularity distribution in boundary type approximation procedures for coupling separate domains of such solution and indeed of deriving equivalent finite elements are demonstrated

trefftz solutions for piezoelectricity by lekhnitskii s formalism and boundary collocation method in this paper a solution procedure for plane piezoelectricity is developed by trefftz boundary collocation method starting with the general plane piezoelectricity solution derived by lekhnitskii s formalism the basic sets of trefftz functions which satisfy the homogeneous governing equations are derived moreover special sets of trefftz functions are derived for impermeable elliptical voids impermeable sharp cracks and permeable sharp cracks with arbitrary orientations with respect to the material poling direction the functions in the special sets satisfy not only the homogeneous governing equations but also the boundary conditions at the peripheries of the pertinent defects by adopting trefftz functions as the trial functions multi domain trefftz boundary collocation method is formulated numerical examples are presented to illustrate the efficacy of the formulation copyright 2005 john wiley sons ltd

on the global structure of period doubling flows abstract for a wide class of period doubling flows on r3 we analyze the global structure of the invariant manifolds and the topology of the bifurcating periodic orbits we emphasize aspects of the dynamics which are not visible in an analysis of the associated poincare return map the global manifold structure implies constraints for the subsequent bifurcational behavior of the flow the period doubled orbits are classified using the theory of iterated torus knots this classification reveals an infinite number of topologically distinct period doubling flows and raises the important question as to which sequences of torus knots appear in the asymptotic limit of a period doubling cascade if the asymptotic sequence of torus knots produced is not unique then the self similarity of the poincare map does not imply a self similar structure for the flow in particular at the accumulation point for the cascade one would not expect to observe a universal power spectrum for the flow this possibility is of some experimental interest

computation of normal forms for high dimensional non linear systems and application to non planar non linear oscillations of a cantilever beam a new and efficient computation of the normal forms is developed in this paper for high dimensional non linear systems and the computational method is applied to non planar non linear oscillations of a cantilever beam the method developed here has the advantage of directly calculating the coefficients of the normal forms and the associated near identity non linear transformations for three different cases that is 1 the case of two pairs of pure imaginary eigenvalues 2 the case of one non semisimple double zero and a pair of pure imaginary eigenvalues and 3 the case of two non semisimple double zero eigenvalues the final partial differential equations of various resonant cases appear in a canonical form whose solutions can be conveniently obtained using polynomial equations with the aid of the maple software a symbolic program for computing the normal forms of high dimensional non linear systems is given comparing the method developed here with other methods of computing the normal forms it is understood that we may respectively obtain the normal forms the coefficients of the normal forms and the associated near identity non linear transformations for three resonant cases by using a same main maple symbolic program moreover the method is easy to apply to engineering problems the normal forms of the averaged equations and their coefficients for non planar non linear oscillations of the cantilever beam under combined parametric and forcing excitations are calculated for two resonant cases

fast fourier transforms for nonequispaced data a group of algorithms is presented generalizing the fast fourier transform to the case of noninteger frequencies and nonequispaced nodes on the interval pi pi the schemes of this paper are based on a combination of certain analytical considerations with the classical fast fourier transform and generalize both the forward and backward ffts each of the algorithms requires o n cdot log n n cdot log 1 varepsilon arithmetic operations where varepsilon is the precision of computations and n is the number of nodes the efficiency of the approach is illustrated by several numerical examples

general solutions for coupled equations for piezoelectric media abstract three general solutions are obtained for the coupled dynamic equations for a transversely isotropic piezoelectric medium these solutions are expressed in terms of the two functions and f where satisfies a second degree partial differential equation and f a sixth degree partial differential equation respectively if the terms concerning the derivatives of time are removed the results become three general solutions for the corresponding equilibrium equations in which the function f can be represented by functions fi i 1 2 3 each of which satisfies a second degree partial differential equation by utilizing a generalized almansi theorem and the solution wang and zheng int j solids structures32 105 115 1995 obtained is proved to be consistent with one case of one of the three general solutions when the constants e15 e31 e33 0 the piezo electric coupling is absent then two of the solutions reduce to the elasticity general solutions for a transversely isotropie medium one of which is the result hu acta scientia sin 2 2 145 151 1953 obtained the other one has not been published last the solution in the limiting explicit form for the problem for a half space with concentrated loads at the boundary is obtained by utilizing the general solutions

computing periodic orbits and their bifurcations with automatic differentiation this paper formulates several algorithms for the direct computation of periodic orbits as solutions of boundary value problems the algorithms emphasize the use of coarse meshes and high orders of accuracy convergence theorems are given in the limit of increasing order with a fixed mesh the algorithms are implemented with the use of matlab and adol c a software package for automatic differentiation automatic differentiation enables accurate computation of high order derivatives of functions without the truncation errors inherent in finite difference calculations we embed the algorithms in a continuation framework and extend them to compute saddle node bifurcations of periodic orbits directly we present data from numerical studies of four test problems making some comparisons with other methods for computing periodic orbits these results demonstrate that high order methods based upon automatic differentiation are capable of high precision with small meshes

the boundary contour method for piezoelectric media with linear boundary elements this paper presents a development of the boundary contour method bcm for piezoelectric media first the divergence free property of the integrand of the piezoelectric boundary element is proved secondly the boundary contour method formulation is derived and potential functions are obtained by introducing linear shape functions and green s functions computer methods in applied mechanics and engineering 1998 158 65 for piezoelectric media the bcm is applied to the problem of piezoelectric media finally numerical solutions for illustrative examples are compared with exact ones and those of the conventional boundary element method bem the numerical results of the bcm coincide very well with the exact solution and the feasibility and efficiency of the method are verified copyright 2003 john wiley sons ltd

interaction between fold and hopf curves leads to new bifurcation phenomena this paper presents a numerical investigation of the complex phenomena which can occur at an interaction between fold and hopf curves in a two parameter problem qualitative information about steady state and periodic solutions can be obtained by computing the bifurcation set consisting of fold curves and curves of hopf points this paper studies the evolution of the bifurcation set with respect to a third parameter for two mathematical models the brusselator trimolecular reaction scheme and a tubular reactor model we find that a limit point of a branch of b points coincides with a cusp point of a fold curve at such a limit point branches of hopf curves can disappear or can be detached from the fold curve bifurcation of two fold curves at a cusp point and bifurcation of hopf curves has been detected too

new developments concerning piezoelectric materials with defects abstract the problem of a piezoelectric body with an elliptic cavity is revisited within the frame work of two dimensional electro elasticity in contrast to our previous formulation the present analysis is based on the use of exact electric boundary conditions at the rim of the hole thus avoiding the common assumption of electric impermeability expressions for the elastic and electric variables induced inside and outside the cavity are derived in closed form in terms of complex potentials comparisons between present exact and previous approximate models are effected to establish the limitations associated with the assumption of the impermeable hole in particular when it becomes a slit crack

solution of helmholtz equation by trefftz method the application of the trefftz method for calculating wave forces on offshore structures is presented indirect and direct formulations using complete and non singular systems of trefftz functions for the helmholtz equation are posed in this paper an effective technique using different interpolation functions for the velocity potential and wave force are suggested to improve the computational accuracy of the wave force the numerical examples show that the present method is highly efficient and accurate