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

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

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 critical boundaries on equilibrium manifolds the study of various equilibrium phenomena leads to nonlinear equations 1 f y u 0 where y in r n is a vector of behavior or state variables u in r p a vector of p geqq 2 parameters or controls and f d subset r n times r p to r n a sufficiently differentiable map the solution set of 1 in r n times r p is often called the equilibrium surface of the problem and in many applications it is of interest to determine the critical boundary that is the set of solutions of 1 where the derivative d y f x is singular for example in structural problems these points may represent buckling points after characterizing the various properties of the problem we present three different numerical methods which allow for a computational trace of paths in the critical boundary these methods represent extensions of the earlier developed locally parametrized continuation method for tracing regular paths on the equilibrium surface they permit for instance a direct computational deter

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

nondeterministic space is closed under complementation in this paper we show that nondeterministic space s n is closed under complementation for s n greater than or equal to log n it immediately follows that the context sensitive languages are closed under complementation thus settling a question raised by kuroda inform and control 7 1964 pp 207 233

boundary sets of regular and context free languages we investigate the descriptional and computational complexity of boundary sets of regular and context free languages for a letter a the right left respectively a boundary set of a language l consists of the words in l whose a predecessor or a successor w r t the prefix suffix respectively relation is not in l for regular languages described by deterministic finite automata dfas we give tight bounds on the number of states for accepting boundary sets moreover the question whether the boundary sets of a regular language is finite is shown to be nl complete for dfas while it turns out to be pspace complete for nondeterministic devices boundary sets for context free languages are not necessarily context free anymore here we find a subtle difference of right and left a boundary sets while right a boundary sets of deterministic context free languages stay deterministic context free we give an example of a deterministic context free language whose a boundary set is already non context free in fact the finiteness problem for a boundary sets of context free languages becomes undecidable

numerical computation of nonsimple turning points and cusps a method is developed for the numerical computation of a double turning point corresponding to a cusp catastrophe of a nonlinear operator equation depending on two parameters an augmented system containing the original equation is introduced for which the cusp point is an isolated solution an efficient implementation of newton s method in the finite dimensional case is presented results are given for some chemical engineering problems and this direct method is compared with some other techniques to locate cusp points

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

analysis on takens bogdanov points for delay differential equations abstract the paper analyzes the takens bogdanov points for the nonlinear system of differential equations with one constant delay and two parameters by representing the delay differential equations as abstract ordinary differential equations in their phase spaces the quadratic takens bogdanov point is defined and an enlarged system for it is produced based on the descriptions for the eigenspace associated with the double zero eigenvalue we reduce the enlarged system to a finite dimensional algebraic equation the quadratic takens bogdanov point together with the corresponding values of parameters is proved to be a regular solution of the reduced enlarged system and then can be computed by the standard newton iteration directly

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

simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of nxn complex matrices a b under a simultaneous similarity roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set then we construct a finite number of rational functions in the entries of a and b whose values are constant on all pairs similar to a b the values of the functions phi sub i a b i equals 1 s determine a finite number of similarity classes

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

evaluating higher derivative tensors by forward propagation of univariate taylor series this article considers the problem of evaluating all pure and mixed partial derivatives of some vector function defined by an evaluation procedure the natural approach to evaluating derivative tensors might appear to be their recursive calculation in the usual forward mode of computational differentiation however with the approach presented in this article much simpler data access patterns and similar or lower computational counts can be achieved through propagating a family of univariate taylor series of a suitable degree it is applicable for arbitrary orders of derivatives also it is possible to calculate derivatives only in some directions instead of the full derivative tensor explicit formulas for all tensor entries as well as estimates for the corresponding computational complexities are given

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

automatic correction of ma and sonka s thinning algorithm using p simple points ma and sonka proposed a fully parallel 3d thinning algorithm which does not always preserve topology we propose an algorithm based on p simple points which automatically corrects ma and sonka s algorithm as far as we know our algorithm is the only fully parallel curve thinning algorithm which preserves topology

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

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

a singular value inequality for block matrices abstract let a be an n n submatrix of a nonsingular n m n m matrix m and let s be the inverse of the schur complement of a in m let p min m n we obtain upper bounds on the p smallest singular values of a in terms of the corresponding ones in s and vice versa

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

the codimension of singular matrix pairs abstract the singular pairs of n n matrices those satisfying det a b 0 form a closed set of codimension n 1 inside the space of all matrix pairs the same holds for singular symmetric pairs for hermitian pairs the singular ones form a closed set of codimension n 1 or n 2 according as n is odd or even the irreducible components of these closed sets are determined by various basic singular summands

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

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

defining functions for multiple hopf bifurcations let a u alpha f u u alpha u in r n alpha in r k be a family of real n times n matrices arising as the jacobian matrices of equilibrium solutions to the dynamical system stackrel u f u alpha an equilibrium point is called a hopf point if a has a conjugate pair of pure imaginary eigenvalues pm i omega omega 0 it is called a double hopf point if there are two such pairs pm i omega 1 pm i omega 2 and a 1 1 resonant double hopf point if in addition omega 1 omega 2 defining functions are obtained for the numerical detection computation and continuation of hopf double hopf and 1 1 resonant double hopf points they are based on a combination of matrix biproduct and bordered matrix methods example computations are done in a fairly realistic and complicated neural model problem with n 13 and k 29 however to make the methods applicable to large scale problems e g discretized boundary value problems we reduce the state space to a subspace that essentially contains the generalized eigenspaces of the eigenvalues with largest real part

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

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

digital topology and geometry in medical imaging a survey digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids such methods have been widely used in many medical imaging applications including image segmentation visualization manipulation interpolation registration surface tracking object representation correction quantitative morphometry etc digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability fidelity and efficiency this paper presents a comprehensive yet compact survey on results principles and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications specifically this paper reviews methods related to distance analysis and path propagation connectivity surface tracking image segmentation boundary and centerline detection topology preservation and local topological properties skeletonization and object representation correction and quantitative morphometry a common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity path propagation and neighborhood analysis

three dimensional topology preserving reduction on the 4 subfields this paper discusses thinning on 3d binary images with the 4 subfield approach although a thinning algorithm concerns binary images the algorithm itself can be represented as a set of three color reduction templates a thinning algorithm is topology preserving if the set of all three color templates is topology preserving sufficient and necessary conditions of time complexity o n were proposed for verifying the topological soundness of a 3d 4 subfield thinning algorithm of n three color templates theories and techniques for computerizing such conditions were discussed two 4 subfield thinning algorithms on 3d images one for generating medial curves and the other one for generating medial surfaces are proposed and proved to preserve topology by our sufficient and necessary conditions

computing skeletons in three dimensions skeletonization will probably become as valuable a tool for shape analysis in 3d as it is in 2d we present a topology preserving 3d skeletonization method which computes both surface and curve sk

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

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

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

a method for the computation of nonsimple turning points corresponding to cusps abstract a direct method is described for the approximation of nonsimple turning points corresponding to cusp points of nonlinear operator equations depending on two parameters the procedure is based on the application of a special projection method to the computation of simple turning points of a suitable augmented system numerical examples illustrate the features of the proposed algorithm

on a numerical liapunov schmidt spectral method and its application to biological pattern formation spectral expansions are used to provide a basis which preserves continuous symmetries we show that spectral methods satisfy the conditions for convergence of numerical liapunov schmidt methods an explicit algorithm for the calculation of stationary bifurcation scenarios near primary instabilities in general continuous symmetric equations is given the above convergence is extended to gamma equivalent discrete and original bifurcation scenarios the method is applied to a biologically motivated reaction diffusion system with spherical symmetry forming patterns a specific singularity of a generic steady state bifurcation is investigated in detail

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

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

solution of bordered singular systems in numerical continuation and bifurcation abstract in numerical continuation and bifurcation problems linear systems with coefficient matrices in the block form m a b c t d arise naturally here a r n n b c r n m d r m m and n may be large but m is small a usually has a special structure banded block banded sparse and b c d are dense so that it is advisable to use a specialized solver for a and to solve with m by some block method unfortunately a is often also a nearly singular matrix in fact made nonsingular only by roundoff and truncation errors on the other hand m is usually nonsingular but can be ill conditioned and in certain situations will degenerate to singularity as well we describe numerical tests for this problem using the mixed block elimination method of govaerts and pryce 1993 for solving bordered linear systems with possibly nearly singular blocks a to this end we compute by newton s method a triple point bifurcation point in a parameterized reaction diffusion equation the brusselator the numerical tests show that the linear systems are solved in a stable way in spite of the use of a black box solver sgbtrs from lapack for a nearly singular matrix

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

folds in solutions of two parameter systems and their calculation part i this paper is concerned with paths of turning points in solutions of nonlinear systems having two parameters it is well known that these paths are solutions of a particular extended system of nonlinear equations in this paper both regular points and simple turning points in the extended system are related to the local geometry of the solution surface of the original nonlinear system a description is given of numerical methods both for solving the extended system and for calculating certain quantities which determine the local geometry of the solution surface applications to perturbed bifurcation to the formation of isolas and to the calculation of the multiplicity of solutions are also discussed numerical examples are given

the codimension of degenerate pencils abstract let dn dn r denote the codimension of the set of pairs of n n hermitian really symmetric matrices a b for which det i a xb p x is a reducible polynomial we prove that dn r n 1 dn n 1 n odd dn n n even we conjecture that the equality holds in all three inequalities we prove this conjecture for n 2 3

the dimension of matrices matrix pencils with given jordan kronecker canonical forms abstract the set of n by n matrices with a given jordan canonical form defines a subset of matrices in complex n 2 dimensional space we analyze one classical approach and one new approach to count the dimension of this set the new approach is based upon and meant to give insight into the staircase algorithm for the computation of the jordan canonical form as well as the occasional failures of this algorithm we extend both techniques to count the dimension of the more complicated set defined by the kronecker canonical form of an arbitrary rectangular matrix pencil a b

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

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

two symmetrical thinning algorithms for 3d binary images based on p simple points in this paper we propose two 3d symmetrical thinning algorithms based on the parallel deletion of p simple points the first one permits surface skeletons to be obtained the second one permits curve skeletons to be extracted and as far as we know this is the only symmetrical curve thinning algorithm which preserves topology these algorithms have been conceived in order to obtain precise results on simple specific objects parallelepipeds consequently we can predict the number of deletion iterations the number of deleted points and the skeleton of these objects obtained by the two algorithms which is hardly ever possible to assess with other algorithms even on these simple objects

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

numerical methods for the generalized hopf bifurcation the presence of a generalized hopf gh point hopf point where the first lyapunov coefficient vanishes deeply influences the qualitative behavior of a dynamical system further information on this behavior can be obtained from the second lyapunov coefficient which depends on derivatives of order up to five we describe two computational procedures implemented in the software package content to compute and numerically continue gh points this is applied to a biochemical model and two examples from neurobiology we make some comparisons between the two methods for computing gh points and discuss briefly the merits of three methods for computing curves of hopf points

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

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

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

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

generalized deflated block elimination a stable algorithm is presented to solve a nonsingular bordered system of the form left begin array 20 c a b c t d end array right left begin array 20 c x y end array right left begin array 20 c f g end array right where b and c are n by m matrices and the n by n matrix a could be nearly singular with at most mu small singular values the algorithm needs only a solver for a and the solution to an m mu by m mu dense linear system it is thus well suited for problems for which a has easily exploitable structures and m mu ll n such as in continuation methods bifurcation problems and constrained optimization

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

minimal nfa problems are hard finite automata fa s are of fundamental importance in theory and in applications the following basic minimization problem is studied given a dfa deterministic fa find a minimum equivalent no

computation of takens bogdanov type bifurcations with arbitrary codimension let dot u f u alpha u 0 u 0 describe a dynamical system where u in r n is the state variable alpha in r k is a parameter variable and f u alpha takes values in r n here n can be large but k is usually small say leqslant 10 a solution u alpha to the equilibrium equations f u alpha 0 is a takens bogdanov type bifurcation of codimension m if the jacobian f u u alpha contains an m dimensional jordan block corresponding to a zero eigenvalue and has no other eigenvalues equal to zero this includes the cases of turning point bifurcation m 1 takens bogdanov bifurcation m 2 and triple point bifurcation m 3 a unified direct method is proposed based on ideas of a griewank and g w reddien in the case of takens bogdanov points for both the detection and computation of parameter values a at which thkens bogdanov type bifurcations of any given codimension occur as an application we compute tkens bogdanov type bifurcations with codi