defining functions for manifolds of matrices abstract we consider defining systems for manifolds of possibly nonsquare matrices the functions are algebraic expressions in the entries of the matrix the systems are local in the sense of being defined on the sets of a predefined cover of the manifold with open and dense sets we use bordered matrix techniques to characterize the manifold of matrices with a given rank in the case of square matrices we give conditions to express that zero is an eigenvalue with a given algebraic multiplicity and geometric multiplicity 1 in both cases the obtained systems have full linear rank hence they define the underlying manifolds in a regular way and can be used analytically to obtain local parametrizations of the manifold or numerically to set up newton systems with local quadratic convergence

structural instability analyses based on generalised path following this paper describes how quasi static conservative instability problems can be completely described using generalised path following procedures for augmented equilibrium formulations in particul

characterizing contextual equivalence in calculi with passivation we study the problem of characterizing contextual equivalence in higher order languages with passivation to overcome the difficulties arising in the proof of congruence of candidate bisimilarities we introduce a new form of labeled transition semantics together with its associated notion of bisimulation which we call complementary semantics complementary semantics allows to apply the well known howe s method for proving the congruence of bisimilarities in a higher order setting even in the presence of an early form of bisimulation we use complementary semantics to provide a coinductive characterization of contextual equivalence in the ho pp calculus an extension of the higher order p calculus with passivation obtaining the first result of this kind we then study the problem of defining a more effective variant of bisimilarity that still characterizes contextual equivalence along the lines of sangiorgi s notion of normal bisimilarity we provide partial results on this difficult problem we show that a large class of test processes cannot be used to derive a normal bisimilarity in ho pp but we show that a form of normal bisimilarity can be defined for ho pp without restriction

multiparametric bifurcation analysis of a basic two stage population model in this paper we investigate long term dynamics of the most basic model for stage structured populations in which the per capita transition from the juvenile into the adult class is density dependent the model is represented by an autonomous system of two nonlinear differential equations with four parameters for a single population we find that the interaction of intra adult competition and intra juvenile competition gives rise to multiple attractors one of which can be oscillatory a detailed numerical study reveals a rich bifurcation structure for this two dimensional system originating from a degenerate bogdanov takens bt bifurcation point when one parameter is kept constant depending on the value of this fixed parameter the corresponding triple critical equilibrium has either an elliptic sector or it is a topological focus which is demonstrated by the numerical normal form analysis it is shown that the canonical unfolding of the codimension three bt point reveals the underlying dynamics of

time finite element discretization of hamilton s law of varying action hamilton s law of varying action is used as a variational source for the derivation of finite element discretization procedure in the time domain three different versions of the proposed algorithms are presented and verified for accuracy and stability the first one is the high precision finite time element analogous to the standard finite elements with cutoff frequency the second version is the step by step one time element from which the unconditionally stable with slightly altered accuracy third algorithm is derived the new operator connected with the proposed algorithms bears attractive properties of much greater accuracy than other existing stable methods and easy computer implementation thus the work herein shows that the reservations expressed against the use of finite elements in time domain seem unjustified

convergence cones near bifurcation the performance of predictor solver continuation methods near bifurcation points is considered a general theory is developed for the convergence of two commonly used solvers namely newton s method and the chord method near singular points of a nonlinear equation this theory provides a uniform treatment of continuation for simple bifurcation multiple bifurcation and multiple limit point bifurcation problems for these types of bifurcation it is shown that there are conical domains of attraction for the above iterative methods centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations

numerical analysis of complex instability behaviour using incremental iterative strategies the paper describes how quasi static conservative instability problems can be analysed in a multi parametric space using generalised path following procedures for augmented equilibrium problems

breather solutions for inhomogeneous fpu models using birkhoff normal forms abstract we present results on spatially localized oscillations in some inhomogeneous nonlinear lattices of fermi pasta ulam fpu type derived from phenomenological nonlinear elastic network models proposed to study localized protein vibrations the main feature of the fpu lattices we consider is that the number of interacting neighbors varies from site to site and we see numerically that this spatial inhomogeneity leads to spatially localized normal modes in the linearized problem this property is seen in 1 d models and in a 3 d model with a geometry obtained from protein data the spectral analysis of these examples suggests some non resonance assumptions that we use to show the existence of invariant subspaces of spatially localized solutions in quartic birkhoff normal forms of the fpu systems the invariant subspaces have an additional symmetry and this fact allows us to compute periodic orbits of the quartic normal form in a relatively simple way

a geometric framework for the numerical study of singular points while bifurcation theory has developed rapidly in recent years there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems this paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter dependent equation f z lambda 0 the presentation draws from differential geometry and singularity theory and provides a basis for various numerical methods used to detect and compute certain types of bifurcation points

computation of singularities in large nonlinear systems a general approach is proposed for the numerical computation of one parameter singularities in the sense of golubitsky and schaeffer singularities and groups in bifurcation theory vol i springer verlag berlin 1985 the singularities of codimension three or less are organized in a new way our scheme is close to the numerics and in some cases gives explicit expressions for the defining equations obtained inductively where no such expressions were previously available the numerical computation by the lyapunov schmidt reduction summarizes work by many authors during the last 20 years treating all cases in a unified way a numerical example is provided where singularities with codimension up to three are detected and computed in a parameterized boundary value problem

on the computation of multi dimensional solution manifolds of parametrized equations a new algorithm is presented for computing vertices of a simplicial triangulation of thep dimensional solution manifold of a parametrized equationf x 0 wheref is a nonlinear mapping fromr n tor m p n m 1 an essential part of the method is a constructive algorithm for computing moving frames on the manifold that is of orthonormal bases of the tangent spaces that vary smoothly with their points of contact the triangulation algorithm uses these bases together with a chord form of the gauss newton process as corrector to compute the desired vertices the jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches several numerical examples show that the method is very efficient in computing triangulations even around singularities such as limit points and bifurcation points this opens up new possibilities for determining the form and special features of such solution manifolds

numerical continuation of bifurcations of limit cycles in matlab rm cl matcont and matcont are matlab continuation packages for the interactive numerical study of a range of parameterized nonlinear dynamical systems in particular odes matcont is an interactive graphical package and rm cl matcont is a commandline version both packages allow us to compute curves of equilibria limit points hopf points limit cycles flip fold and torus bifurcation points of limit cycles we discuss computational details of the continuation of limit cycles and flip fold and torus bifurcations of limit cycles in matcont and rm cl matcont using orthogonal collocation instead of the more commonly used fully extended systems we use minimally extended systems we further describe the use of the matlab sparse matrix routines and the initialization and adaptation of the bordering vectors that are essential in minimally extended systems finally we compare the use of the minimally and the fully extended systems in the matlab environment

iterative computation of the smallest singular value and the corresponding singular vectors of a matrix abstract influenced by some techniques used for computing singular points of nonlinear equations a generalized inverse iteration method is proposed for approximating the smallest singular value n and the associated left and right singular vectors u v of a matrix a r n n in the practically relevant case n 0 the method is mathematically equivalent to inverse iteration with aa t however unlike classic inverse iteration the new method works with matrices b k r n 1 n 1 obtained by bordering a in such a way that the b k have uniformly bounded condition numbers this allows using iterative krylov type solvers for large problems if n 1 n the singular vector approximations convergence linearly with factor n n 1 k obtained as a byproduct has a relative error k n n which goes to zero r linearly with factor 2 some numerical examples confirm the theoretical results and show that the algorithm works reliable also for almost singular matrices and when using krylov solvers

improved minimal augmentation procedure for the direct computation of critical points this paper presents a new numerical procedure for the direct computation of critical points for elastic beam structures undergoing large displacements and rotations compared to the approach described by wriggers et al comput methods appl mech engrg 70 1988 329 int j numer methods engrg 30 1990 155 two main modifications are introduced first following eriksson comput methods appl mech engrg 114 1994 77 comput methods appl mech engrg 156 1998 45 comput methods appl mech engrg 179 1999 265 int j struct stability dynam 1 1 2001 the condition of criticality is expressed by a scalar equation instead of a vectorial one next the present procedure does not use exclusively the extended system obtained from the equilibrium equations and the criticality condition but also introduces intermediate iterations based purely on equilibrium equations under load or displacement control eight numerical examples presenting bifurcation and limit points are used in order to compare the performances of this new method and the one presented in comput methods appl mech engrg 70 1988 329 int j numer methods engrg 30 1990 155

bisimulation can t be traced in the concurrent language ccs two programs are considered the same if they are bisimilar several years and many researchers have demonstrated that the theory of bisimulation is mathematically appealing and useful in practice however bisimulation makes too many distinctions between programs we consider the problem of adding operations to ccs to make bisimulation fully abstract we define the class of gsos operations generalizing the style and technical advantages of ccs operations we characterize gsos congruence in as a bisimulation like relation called ready simulation bisimulation is strictly finer than ready simulation and hence not a congruence for any gsos language

space time spectral element methods for one dimensional nonlinear advection diffusion problems abstract the following space time galerkin spectral element methods are developed and applied to solve the burgers equation with small viscosity a coupled methods consisting of an explicit method for hyperbolic dominated equations and an implicit method for parabolic dominated equations b two splitting methods which solve the hyperbolic substep explicitly and the parabolic one implicitly one uses spectral elements in the explicit part and the other uses the adams bashforth multistep method a subcycling technique in which several convective steps are taken for each implicit viscous step was also investigated for the two splitting methods a stability analysis of the four methods is performed and subsequent results are debated a convergence study and a comparison of computer execution time for the four methods is made and the results are discussed comparative study leads to the conclusion that the space time spectral element splitting method with subcycling is superior to the other methods presented in terms of robustness and computer execution time the number of subcycles should be kept low 2 3 in order to avoid significant loss of accuracy the coupled explicit method is also applied to the solution of the one dimensional coupled continuity momentum and energy equations for non isothermal flow of an ideal gas with temperature dependent properties in a cylindrical duct of variable radius

accelerated projection and iterated projection methods with applications to nonlinear integral equations a new approach is developed to numerically approximate fixed points for a class of nonlinear operators by projection and iterated projection methods it is shown that isolated solutions can be determined to any desired accuracy for a fixed choice of the projection neither the structure nor the rank of the projection is altered to achieve increased accuracy instead the fixed point is decomposed by the given projection into a finite and an infinite dimensional component successive approximations to the infinite dimensional component are employed to construct a sequence of accelerated finite dimensional systems their solutions define a sequence of higher order approximations to the finite dimensional component and consequently to the fixed point error expressions are obtained and several examples of solving nonlinear integral equations are given

uncertainty quantification of limit cycle oscillations different computational methodologies have been developed to quantify the uncertain response of a relatively simple aeroelastic system in limit cycle oscillation subject to parametric variability the aeroelastic system is that of a rigid airfoil supported by pitch and plunge structural coupling with nonlinearities in the component in pitch the nonlinearities are adjusted to permit the formation of a either a subcritical or supercritical branch of limit cycle oscillations uncertainties are specified in the cubic coefficient of the torsional spring and in the initial pitch angle of the airfoil stochastic projections of the time domain and cyclic equations governing system response are carried out leading to both intrusive and non intrusive computational formulations non intrusive formulations are examined using stochastic projections derived from wiener expansions involving haar wavelet and b spline bases while wiener hermite expansions of the cyclic equations are employed intrusively and non intrusively application of the b spline stochastic projection is extended to the treatment of aerodynamic nonlinearities as modeled through the discrete euler equations the methodologies are compared in terms of computational cost convergence properties ease of implementation and potential for application to complex aeroelastic systems

numerical periodic normalization for codim 1 bifurcations of limit cycles explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous odes are derived they include second order coefficients for the fold limit point bifurcation as well as third order coefficients for the flip period doubling and neimark sacker torus bifurcations the formulas are independent of the dimension of the phase space and involve solutions of certain boundary value problems on the interval 0 t where t is the period of the critical cycle as well as multilinear functions from the taylor expansion of the right hand sides near the cycle the formulas allow us to distinguish between sub and supercritical bifurcations in agreement with earlier asymptotic expansions of the bifurcating solutions our formulation makes it possible to use robust numerical boundary value algorithms based on orthogonal collocation rather than shooting techniques which greatly expands its applicability the actual implementation is described in detail we include three numerical examples in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients

a newton based method for the calculation of the distance to instability in this paper a new fast algorithm for the computation of the distance of a stable matrix to the unstable matrices is provided the method uses newton s method to find a two dimensional jordan block corresponding to a pure imaginary eigenvalue in a certain two parameter hamiltonian eigenvalue problem introduced by byers r byers a bisection method for measuring the distance of a stable matrix to the unstable matrices siam j sci statist comput 9 1988 875 881 this local method is augmented by a test step previously used by other authors to produce a global method numerical results are presented for several examples and comparison is made with the methods of boyd and balakrishnan s boyd v balakrishnan a regularity result for the singular values of a transfer matrix and a quadratically convergent algorithm for computing its l norm systems control lett 15 1990 1 7 and he and watson c he g a watson an algorithm for computing the distance to instability siam j matrix anal appl 20 1999 101 116

a conservative look at operational semantics with variable binding we set up a formal framework to describe transition system specifications in the style of plotkin this framework has the power to express many sortedness general binding mechanisms and substitutions among other notions such as negative hypotheses and unary predicates on terms the framework is used to present a conservativity format in operational semantics which states sufficient criteria to ensure that the extension of a transition system specification with new transition rules does not affect the semantics of the original terms

existence and boundedness of a minimiser for a constrained minimisation problem on rn with limiting exponent we use the concentration compactness principle to study the existence of a minimiser of the minimisation problem where u u 1 u n we also prove the boundedness of the minimiser of l 1 by using the reverse holder inequality

on the computation of manifolds of foldpoints for parameter dependent problems for parameter dependent nonlinear equations in r n a new augmentation method for computing submanifolds of foldpoints is presented the underlying theory is based on a characterization developed earlier of foldpoints on solution manifolds of parameterized equations the method allows for a locally convergent iterative computation of foldpoints of a certain class for a continuation along paths of such foldpoints and for a computation of simplicial approximations of higher dimensional submanifolds of foldpoints of the same type

continuation of bifurcations in periodic delay differential equations using characteristic matrices in this paper we describe a method for continuing periodic solution bifurcations in periodic delay differential equations first the notion of characteristic matrices of periodic orbits is introduced and equivalence with the monodromy operator is demonstrated an alternative formulation of the characteristic matrix is given which can be computed efficiently defining systems of bifurcations are constructed in a standard way including the characteristic matrix and its derivatives for following bifurcation curves in two parameters the pseudo arclength method is used combined with newton iteration two test examples an interrupted machining model and a traffic model with driver reaction time conclude the paper the algorithm has been implemented in the software tool pdde cont

numerical liapunov schmidt spectral methods for k determined problems we describe several generalized liapunov schmidt methods in the bordered systems formulation for operator equations and collect information concerning classification of bifurcation functions we present a short outline of spectral methods in this bifurcation context and show the convergence of the discrete defining equations and nondegeneracy conditions for spectral finite element and difference methods to the counterparts of the original operator for k determined problems this result is then applied to a model for turing patterns in a two component mixture

propagation and stability of waves of electrical activity in the cerebral cortex nonlinear equations are introduced to model the behavior of the waves of cortical electrical activity that are responsible for signals observed in electroencephalography these equations incorporate nonlinearities axonal and dendritic lags excitatory and inhibitory neuronal populations and the two dimensional nature of the cortex while rendering nonlinear features far more tractable than previous formulations both analytically and numerically the model equations are first used to calculate steady state levels of cortical activity for various levels of stimulation dispersion equations for linear waves are then derived analytically and an analytic expression is found for the linear stability boundary beyond which a seizure will occur the effects of boundary conditions in determining global eigenmodes are also studied in various geometries and the corresponding eigenfrequencies are found numerical results confirm the analytic ones which are also found to reproduce existing results in the relevant limits thereby elucidating the limits of validity of previous approximations

matrices with rank deficiency two in eigenvalue problems and dynamical systems let k alpha beta alpha beta real be a family of real square matrices several computational problems are equivalent to the calculation of a pair alpha 0 beta 0 of parameter values for which k alpha 0 beta 0 has rank deficiency 2 among these problems are the computation of a conjugate pair of complex eigenvalues of a given real matrix the computation of a hopf bifurcation point on a branch of stationary solutions to a parametrized differential equation and the computation of a takens bogdanov point in the two dimensional solution manifold of a set of nonlinear equations developing ideas of griewank and reddien the authors define scalar functions g 1 alpha beta g 2 alpha beta that vanish in alpha 0 beta 0 a nondegeneracy condition ndc which expresses the fact that alpha 0 beta 0 is in a natural sense an isolated point in alpha beta space is introduced it is proved that under certain conditions on the family k alpha be

spectral element method in time for rapidly actuated systems in this paper the spectral element se method is applied in time to find the entire time periodic or transient solution of time dependent differential equations the time periodic solution is computed by enforcing periodicity of the element set of particular interest are periodic forcing functions possessing high frequency content to maintain the spectral accuracy for such forcing functions an h refinement scheme is employed near the semi discontinuity without increasing the number of degrees of freedom time discretization by spectral elements is applied initially to a standard form of a set of linear first order differential equations subject to harmonic excitation and an excitation admitting rapid variation other case studies include the application of the se approach to parabolic and hyperbolic partial differential equations the first order form of these equations is obtained through semi discretization using conventional finite element spectral element and finite difference schemes element clustering h refinement is applied to maintain the high accuracy and efficiency in the region of the forcing function admitting rapid variation the convergence in time of the method is demonstrated in some cases machine precision is obtained with 25 degrees of freedom per cycle finally the method is applied to a weakly nonlinear problem with time periodic solution to demonstrate its future applicability to the analysis of limit cycle oscillations in aeroelastic systems

numerical computation of bifurcations in large equilibrium systems in matlab the continuation of invariant subspaces cis algorithm produces a smoothly varying basis for an invariant subspace r s of a parameter dependent matrix a s we have incorporated the cis algorithm into cl matcont a matlab package for the study of dynamical systems and their bifurcations using subspace reduction we extend the functionality of cl matcont to large scale computations of bifurcations of equilibria in this paper we describe the algorithms and functionality of the resulting matlab bifurcation package cl matcontl the novel features include new cis based continuous well scaled test functions for codimension 1 and 2 bifurcations detailed description of locators for large problems and examples of bifurcation analysis in large sparse problems

a space time coupled p version least squares finite element formulation for unsteady two dimensional navier stokes equations this paper presents a p version least squares finite element formulation for two dimensional unsteady fluid flow described by navier stokes equations where the effects of space and time are coupled the dimensionless form of the navier stokes equations are first cast into a set of first order differential equations by introducing auxiliary variables this permits the use of c0 element approximation the element properties are derived by utilizing the p version approximation functions in both space and time and then minimizing the error functional given by the space time integral of the sum of squares of the errors resulting from the set of first order differential equations this results in a true space time coupled least squares minimization procedure the application of least squares minimization to the set of coupled first order partial differential equations results in finding a solution vector which makes gradient of error functional with respect to a null vector this is accomplished by using newton s method with a line search a time marching procedure is developed in which the solution for the current time step provides the initial conditions for the next time step equilibrium iterations are carried out for each time step until the error functional and each component of the gradient of the error functional with respect to nodal degrees of freedom are below a certain prespecified tolerance the space time coupled p version approximation functions provide the ability to control truncation error which in turn permits very large time steps what literally requires hundreds of time steps in uncoupled conventional time marching procedures can be accomplished in a single time step using the present space time coupled approach the generality success and superiority of the present formulation procedure is demonstrated by presenting specific numerical examples for transient couette flow and transient lid driven cavity the results are compared with the analytical solutions and those reported in the literature the formulation presented here is ideally suited for space time adaptive procedures the element error functional values provide a mechanism for adaptive h p or hp refinements

efficient numerical path following beyond critical points the paper presents a numerical pathfollowing technique based on an implicit change of parametrization using qr decomposition an automatic steplength control for tangent continuation is derived on a theoretical basis in the suggested approach turning points do not play any exceptional role for the actual computation of simple bifurcation points a special algorithm based on the augmented system of moore is developed this treatment includes imperfect bifurcations and permits an easy way of branch switching an efficient linear equation solver is worked out in connection with the qr decomposition finally numerical comparisons are included that clearly document the efficiency of the newly developed algorithms

analysis of bifurcations of limit cycles with lyapunov exponents and numerical normal forms in this paper we focus on the combination of normal form and lyapunov exponent computations in the numerical study of the three codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous odes the normal form formulas are independent of the dimension of the phase space and involve solutions of certain linear boundary value problems the formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations where 3 tori and 4 tori can be present we apply our techniques to the study of a known laser model a novel model from population biology and a model of mechanical vibrations these models exhibit limit point neimark sacker period doubling neimark sacker and double neimark sacker bifurcations lyapunov exponents are computed to numerically confirm the results of the normal form analysis in particular with respect to the existence of stable invariant tori of various dimensions conversely the normal forms are essential to understand the significance of the lyapunov exponents

equilibrium subsets for multi parametric structural analysis the paper discusses multi parametric formulations for structural equilibrium problems by seeking subsets of equilibrium states which also fulfil extra conditions more detailed information on structural behaviour can be obtained these conditions can be related to analyses of e g critical behaviour imperfections or optimisation the paper shows a general problem setting and some interesting augmentations a solution method for the special case of one dimensional equilibrium subsets is described the algorithm is a generalisation of a path following algorithm where new methods are developed for the evaluation of the path tangent and for the isolation of special solution points examples are used to show the properties and the possibilities of the ideas

on solving nonlinear equations with simple singularities or nearly singular solutions the paper begins with a review of the convergence theory for newton s method near simple and other regular singularities followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy then several ways to accelerate the slow convergence of newton s method are described in some detail these are richardson extrapolation overrelaxation bordering by a singularity condition and appending the usual linear model by a quadratic term the last two approaches require limited second derivative information which can be obtained analytically by differencing or through secant updating numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy

an algebraic geometric method to calculate branches near a bifurcation point abstract systems of nonlinear algebraic equations with a parameter arises in many branches of mathematics and engineering computation of the solution curve is usually hindered by the existence of a bifurcation point where the rank of the corresponding matrix is not full and usually there is another branch curve passing it we introduce a method from algebraic geometry to find the branch curve a computational example is provided

a derivation of a macroscopic field theory of the brain from the quasi microscopic neural dynamics abstract the purpose of this paper is twofold first we present a semi quantitative nonlinear field theory of the brain under realistic anatomical connectivity conditions describing the interaction between functional units within the brain this macroscopic field theory is derived from the quasi microscopic conversion properties of neural populations occurring at synapses and somas the quasi microscopic models by wilson cowan 1972 1973 and nunez 1974 can be derived from these functional units are treated as inhomogeneities within a nonlinear one dimensional neural tissue second for the case of the kelso experiment the field equation is treated analytically and numerically and can be reduced to a set of ordinary differential equations which corresponds to a model by jirsa et al 1994 1995 this phenomenological model reproduces the spatio temporal phenomena experimentally observed here the most prominent property of the neural tissue is the parametric excitation the macroscopic field parameters can be expressed by quasi microscopic neural parameters

characterization and computation of singular points with maximum rank deficiency characterizations are given of points at which a mapping has a rank drop possibly greater than one multiple bifurcation points can then be computed without needing a full unfolding of the singularity nondegeneracy conditions are studied in detail

bialgebras for structural operational semantics bialgebras and distributive laws are an abstract categorical framework to study various flavors of structural operational semantics this paper aims to introduce the reader to the basics of bialgebras for operational semantics and to sketch the state of the art in this research area

the approximate solution of defining equations for generalized turning points a set of defining equations for turning points is shown to produce a test function that can be computed iteratively to double accuracy an inexact newton method is then derived that gives a method for the efficient calculation of turning points applications are given to discretized boundary value problems using conjugate gradients

floquet theory as a computational tool we describe how classical floquet theory may be utilized in a continuation framework to construct an efficient fourier spectral algorithm for approximating periodic orbits at each continuation step only a single square matrix whose size equals the dimension of the phase space needs to be factorized the rest of the required numerical linear algebra just consists of back substitutions with this matrix the eigenvalues of this key matrix are the floquet exponents whose crossing of the imaginary axis indicates bifurcation and change in stability hence we also describe how the new periodic orbits created at a period doubling bifurcation point may be efficiently computed using our approach

computation of hopf bifurcation with bordered matrices hopf bifurcation for dynamical systems dot x f x lambda is characterized by rank deficiency two of a 2 omega 2 i where a f x x lambda in mathbb r n times n is the jacobian at the hopf point and pm i omega are the hopf eigenvalues replacing w 2 by nu and allowing nu leq 0 we introduce real and imaginary hopf matrices a with hopf numbers nu defining a codimension 2 manifold mathcal m h nu subset mathbb r n times n times mathbb r which can be described by two scalar equations alpha a nu 0 beta a nu 0 given by the solution of certain linear bordered systems these two scalar equations form a suitable augmentation of f x lambda 0 for the numerical computation of branches of hopf points in two parameter systems in contrast to chu govaerts and spence siam j numer anal 31 1994 pp 524 539 we suggest an a dependent bordering having certain advantages we construct a scalar test function for hopf points in the

comparing operational models of name passing process calculi we study three operational models of name passing process calculi coalgebras on pre sheaves indexed labelled transition systems and history dependent automata the coalgebraic model is considered both for presheaves over the category of finite sets and injections and for its subcategory of atomic sheaves known as the schanuel topos each coalgebra induces an indexed labelled transition system such transition systems are characterised relating the coalgebraic approach to an existing model of name passing further we consider internal labelled transition systems within the sheaf topos and axiomatise a class that is in precise correspondence with the coalgebraic and the indexed labelled transition system models by establishing and exploiting the equivalence of the schanuel topos with a category of named sets these internal labelled transition systems are also related to the theory of history dependent automata

efficient finite differences for parameter dependent singularities in two point boundary value problems we will consider the implementation of finite differences to the numerical computation of simple turning points cubic turning points simple bifurcation points and bifurcation points with corank 2 in b v ps for ordinary differential equations the singularity will be characterized by an equation or a set of equations which will be augmented with the original system to produce a regular problem block elimination will be employed for saving on the amount of work while richardson extrapolation will be used to produce accurate results numerical examples will be presented to show the efficiency of the above described algorithm

characterization and computation of generalized turning points a minimal set of augmenting equations is given so that general parameter dependent problems together with these equations will be nonsingular at points of simple rank deficiency applications to de

the approximation of generalized turn ng points by projections methods with superconvergence to the critical parameter a procedure is given that generates characterizations of singular manifolds for mildly nonlinear mappings between banach spaces this characterization is used to develop a method for determining generalized turning points by using projection methods as a discretization applications are given to parameter dependent two point boundary value problems in particular collocation at gauss points is shown to achieve superconvergence in approximating the parameter at simple turning points

limit cycles bifurcations and accuracy of the milling process time finite element analysis tfea is used to determine the accuracy stability and limit cycle behavior of the milling process predictions are compared to traditional euler simulation and experiments the tfea method forms an approximate solution by dividing the time in the cut into a finite number of elements the approximate solution is then matched with the exact solution for free vibration to obtain a discrete linear map stability is then determined from the characteristic multipliers of the map map fixed points correspond to stable periodic solutions which are used to evaluate surface location error bifurcations and limit cycle behavior are predicted from a non linear tfea formulation experimental cutting tests are used to confirm theoretical predictions

the approximation of generalized turning points newline by krylov subspace methods certain types of singular solutions of nonlinear parameter dependent operator equations were characterized by griewank and reddien 5 6 as regular solutions of suitable augmented systems for their numerical approximation an approach based on the use of krylov subspaces is here presented the application to boundary value problems is illustrated by numerical examples

derper an algorithm for the continuation of periodic solutions in ordinary differential equations abstract an algorithm for the continuation of periodic solutions of ode was derived and tested the algorithm is based on the shooting method and a standard continuation algorithm variational variables are used to compute the jacobi matrix the starting point on the periodic orbit is adaptively changed in the algorithm by fixing one component of the solution stability of periodic solutions along the continuous branch of solutions is determined by computing characteristic multipliers without difficulty the algorithm crosses limit points and at bifurcation points double period bifurcation points and points of tori bifurcation it proceeds along the original branch of solutions the usefulness of the algorithm is demonstrated in two examples one of them describes two stirred tank reactors with mutual mass exchange the other one is the lorenz model of turbulence

superlinearly convergent algorithms for solving singular equations and smooth reformulations of complementarity problems we propose a new algorithm for solving smooth nonlinear equations in the case where their solutions can be singular compared to other techniques for computing singular solutions a distinctive feature of our approach is that we do not employ second derivatives of the equation mapping in the algorithm and we do not assume their existence in the convergence analysis important examples of once but not twice differentiable equations whose solutions are inherently singular are smooth equation based reformulations of the nonlinear complementarity problems reformulations of complementarity problems serve both as illustration of and motivation for our approach and one of them we consider in detail we show that the proposed method possesses local superlinear quadratic convergence under reasonable assumptions we further demonstrate that these assumptions are in general not weaker and not stronger than regularity conditions employed in the context of other superlinearly convergent newton type algorithms for solving complementarity problems which are typically based on nonsmooth reformulations therefore our approach appears to be an interesting complement to the existing ones

computation of periodic solution bifurcations in odes using bordered systems we consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous odes namely the fold the period doubling or flip bifurcation and the torus or neimark sacker bifurcation in the fold and flip cases we append one scalar equation to the standard periodic bvp that defines the periodic solution in the torus case four scalar equations are appended evaluation of these scalar equations and their derivatives requires the solution of linear bvps whose sparsity structure after discretization is identical to that of the linearization of the periodic bvp therefore the calculations can be done using existing numerical linear algebra techniques such as those implemented in the software auto and colsys

continuous bordering of matrices and continuous matrix decompositions let cal m n be the set of all real n times n matrices of rank ge n 1 we prove that for n ge 2 there are no continuous vector fields l r cal m n rightarrow bbb r n such that the bordered matrix is regular for all a in cal m n this result has some relevance for the numerical analysis of steady state bifurcation as a by product we show that there is no nonvanishing continuous vector field u cal m n n 1 rightarrow bbb r n with au a 0 for all a in cal m n n 1 where cal m n n 1 subset cal m n is the set of all matrices of rank deficiency one this implies that there is no singular value decomposition of a depending continuously on a in any matrix set which contains cal m n n 1 as another application we prove that in general there is no global analytic singular value decomposition for analytic matrix valued functions of more than one real variable

a congruence rule format for name passing process calculi abstract we introduce a gsos like rule format for name passing process calculi specifications in this format correspond to theories in nominal logic the intended models of such specifications arise by initiality from a general categorical model theory for operational semantics given in this rule format a natural behavioural equivalence a form of open bisimilarity is a congruence

compositional sos and beyond a coalgebraic view of open systems in this paper we address the issue of providing a structured coalgebra presentation of transition systems with algebraic structure on states determined by an equational specification more precisely we aim at representing such systems as coalgebras for an endofunctor on the category of algebras the systems we consider are specified by using arbitrary sos rules which in general do not guarantee that bisimilarity is a congruence we first show that the structured coalgebra representation works only for systems where transitions out of complex states can be derived from transitions out of corresponding component states this decomposition property of transitions indeed ensures that bisimilarity is a congruence for a system not satisfying this requirement next we propose a closure construction which adds context transitions i e transitions that spontaneously embed a state into a bigger context or vice versa the notion of bisimulation for the enriched system coincides with the notion of dynamic bisimilarity for the original one i e with the coarsest bisimulation which is a congruence this is sufficient to ensure that the structured coalgebra representation works for the systems obtained as result of the closure construction

matrix manifolds and the jordan structure of the bialternate matrix product the bialternate product of matrices was introduced at the end of the 19th century and recently revived as a computational tool in problems where real matrices with conjugate pairs of pure imaginary eigenvalues are important i e in stability theory and hopf bifurcation problems we give a complete description of the jordan structure of the bialternate product 2a in of an n n matrix a thus extending several partial results in the literature we use these results to obtain regular local defining systems for some manifolds of matrices which occur naturally in applications in particular for manifolds with resonant conjugate pairs of pure imaginary eigenvalues such defining systems can be used analytically to obtain local parameterizations of the manifolds or numerically to set up newton systems with local quadratic convergence we give references to explicit numerical applications and implementations in software we expect that the analysis provided in this paper can be used to further improve such implementations

fourier methods for quasi periodic oscillations quasi periodic oscillations and invariant tori play an important role in the study of forced or coupled oscillators this paper presents two new numerical methods for the investigation of quasi periodic oscillations both algorithms can be regarded as generalizations of the averaging and the harmonic spectral balance methods the algorithms are easy to implement and require only minimal a priori knowledge of the system most importantly the methods do not depend on an a priori co ordinate transformation the methods are applied to a number of illustrative examples from non linear electrical engineering and the results show that the methods are efficient and reliable in addition these examples show that the presented algorithms can also continue through regions of sub harmonic phase locked resonance even though they are designed only for the quasi periodic case copyright 2006 john wiley sons ltd

a neural mass model with direct and indirect excitatory feedback loops identification of bifurcations and temporal dynamics neural mass modeling is a part of computational neuroscience that was developed to study the general behavior of a neuronal population this type of mesoscopic model is able to generate output signals that are comparable to experimental data such as electroencephalograms classically neural mass models consider two interconnected populations excitatory pyramidal cells and inhibitory interneurons however many authors have included an excitatory feedback on the pyramidal cell population two distinct approaches have been developed a direct feedback on the main pyramidal cell population and an indirect feedback via a secondary pyramidal cell population in this letter we propose a new neural mass model that couples these two approaches we perform a detailed bifurcation analysis and present a glossary of dynamical behaviors and associated time series our study reveals that the model is able to generate particular realistic time series that were never pointed out in either simulated or experimental data finally we aim to evaluate the effect of balance between both excitatory feedbacks on the dynamical behavior of the model for this purpose we compute the codimension 2 bifurcation diagrams of the system to establish a map of the repartition of dynamical behaviors in a direct versus indirect feedback parameter space a perspective of this work is from a given temporal series to estimate the parameter value range especially in terms of direct versus indirect excitatory feedback

computation of cusp singularities for operator equations and their discretizations we discuss the direct calculation of cusp singularities as solutions of a minimally augmented defining system which is nonsingular under the canonical cusp conditions the underlying operator equation may have any finite fredholm index and bounds on the discretization error are derived for the case of projection methods