local well posedness for the maxwell schrodinger equation time local well posedness for the maxwell schrodinger equation in the coulomb gauge is studied in sobolev spaces by the contraction mapping principle the lorentz gauge and the temporal gauge cases are also treated by the gauge transform

computing hopf bifurcations ii three examples from neurophysiology in guckenheimer myers and sturmfels siam j numer anal 34 1997 we present algorithms for detecting hopf bifurcations in two parameter families of vector fields based on classical algebraic constructions in addition to their utility as augmented systems for use with standard newton type continuation methods they are shown to be particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension the present study examines the performance of these methods on test problems selected from models of current research interest in neurophysiology implementation issues are examined and the numerical properties of the proposed methods are compared with several alternative algorithms for hopf pathfollowing that appear in the literature

rank order smoothers with two dimensional data model abstract development of the theory of rank order smoothers is motivated by hardware implementation as well as statistical considerations a recently proposed class of rank order smoothers based on minimization of polyhedral convex functions involving one dimensional data models employs minimization algorithms derived from dynamic programming computations proceed by storing and manipulating arrays of breakpoints of piecewise linear convex functions such an approach has certain advantages over alternatives foremost among these is the attainment of a fixed point or root signal in a single application of the minimization algorithm other advantages are the intelligibility of the connection between the rough data and the root signal smooth derived from it the absence of any significant problem of how to deal with end values compatibility with real time applications and storage and computation time requirements of the order of the number of pieces of rough data when experimental comparison was made to median smoothers better smoothing was observed on the negative side such smoothers have been largely restricted to types that behave fundamentally like median smoothers i e have the same sorts of fixed points or root signals it is shown that by letting the data model be two dimensional and formulating the dynamic programming method so that variables are eliminated in pairs then the root signal sets of such smoothers are greatly expanded while the various advantages are retained

a linpack style condition estimator for the equation ax xb t c estimation of the condition number of ax xb t c requires an approximation of the smallest l 2 singular value of the linear transformation x rightarrow ax xb t the approximation is obtained from a step of inverse iteration applied to a heuristically chosen starting matrix in the style of linpack the cost of the estimator is kept down by using the same matrix factorizatlons to make the estimate as are used to solve ax xb t c computational experiments verify its reliability

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 harmonic balance method for bifurcation analysis of large scale nonlinear mechanical systems abstract the harmonic balance hb method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems the objective of this paper is to exploit the method for bifurcation analysis i e for the detection and tracking of bifurcations of nonlinear systems to this end an algorithm that combines the computation of the floquet exponents with bordering techniques is developed a new procedure for the tracking of neimark sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed the hb method is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinear vibration isolation device

the method of resultants for computing real solutions of polynomial systems a new method for determining the real solutions to a set of polynomial equations is presented it is based on the theory of multiresultants the inherently unstable calculation of the determinant is replaced by a stable minimization procedure which is able to take advantage of the sparseness of the resultant matrix two numerical examples illustrate the method the paper contains preliminary work which demonstrates the feasibility of the given approach

discretizing dynamical systems with generalized hopf bifurcations we consider the discretizations of parameter dependent continuous time dynamical systems we show that the general one step methods shift a generalized hopf bifurcation and turn it into a generalized neimark sacker point we analyze the effect of discretization methods on the emanating hopf curve in particular we obtain estimates for the eigenvalues of the discretized system along this curve a detailed analysis of the discretized first lyapunov coefficient is also given the results are illustrated by a numerical example dynamical consequences are discussed

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

an algorithm for the computation of hopf bifurcation points in comparison with other methods we present an algorithm for the computation of a hopf bifurcation point based on a direct method i e an augmented time independent system is solved the algorithm is designed to deal efficiently with problems representing a finite difference discretization of a partial differential system in this case the bandstructure of the jacobian matrix is exploited we also compare this technique with some other methods in regard to its efficiency and reliability

the bifurcations of separatrix contours and chaos abstract bifurcations of systems close to systems having contours composed of separatrices of a pair of saddle points are studied and cases of the appearance of nontrivial hyperbolic sets are pointed out

local numerical analysis of hopf bifurcation this paper contributes to a local numerical analysis of hopf bifurcation in the sense of a bordered systems approach however the method proposed enables us to decide about stability exchange betw

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

an algorithm for melnikov functions and application to a chaotic rotor in this work we study a dynamical system with a complicated nonlinearity which describes oscillation of a turbine rotor and give an algorithm to compute melnikov functions for analysis of its chaotic behavior we first derive the rotor model whose nonlinear term brings difficulties to investigating the distribution and qualitative properties of its equilibria this nonlinear model provides a typical example of a system for which the homoclinic and heteroclinic orbits cannot be analytically determined in order to apply melnikov s method to make clear the underlying conditions for chaotic motion we present a generic algorithm that provides a systematic procedure to compute melnikov functions numerically substantial analysis is done so that the numerical approximation precision at each phase of the computation can be guaranteed using the algorithm developed in this paper it is straightforward to obtain a sufficient condition for chaotic motion under damping and periodic external excitation whenever the rotor parameters are given

a method for the numerical computation of hopf bifurcation this paper deals with the computation of hopf bifurcations in parameter dependent ordinary differential equations an algorithm based on an indirect method is presented while the conventional method approaches the hopf bifurcation point from the stationary solution the proposed method detects the hopf bifurcation points from the dynamic solution a computation using the proposed algorithm is implemented for finding the hopf bifurcation point of several cases and the comparison with the results obtained by other methods is given the proposed method is applicable for both autonomous and non autonomous systems

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

the cauchy problem for coupled yang mills and scalar fields in the temporal gauge we study the cauchy problem for minimally coupled classical yang mills and scalar fields inn 1 dimensional space time in the temporal gauge we prove the existence and uniqueness of solutions for small time intervals and for anyn we then develop a general theory of solutions in local spaces and extend the previous local in time results to this more general setting in space time dimensions two and three we prove the existence of global in time solutions by the method of a priori estimates both in global and local spaces in space time dimension four our estimates yield only partial results on the global existence problem

t points a codimension two heteroclinic bifurcation the local bifurcation structure of a heteroclinic bifurcation which has been observed in the lorenz equations is analyzed the existence of a particular heteroclinic loop at one point in a two dimensional parameter space a t point implies the existence of a line of heteroclinic loops and a logarithmic spiral of homoclinic orbits as well as countably many other topologically more complicatedt points in a small neighborhood in parameter space

boolean versus continuous dynamics on simple two gene modules we investigate the dynamical behavior of simple modules composed of two genes with two or three regulating connections continuous dynamics for mrna and protein concentrations is compared to a boolean model for gene activity using a generalized method we study within a single framework different continuous models and different types of regulatory functions and establish conditions under which the system can display stable oscillations these conditions depend only on general features such as the ratio of the relevant time scales the degree of cooperativity of the regulating interactions and the logical structure of the interactions our results combine and generalize the findings of several disconnected previous studies

global existence and asymptotic behavior of solutions for the maxwell schrodinger equations in three space dimensions in this paper we study the global existence and asymptotic behavior of solutions for the maxwell schrodinger equations under the coulomb gauge condition in three space dimensions with the final states given att this leads to the construction of the modified wave operator for certain scattered data it is also shown that for the initial data in the range of the modified wave operator the initial value problem of the maxwell schrodinger equations has the global solutions in time

regular sets and rank order processors rank order processors rops can be specified in terms of regular sets and consequently studied as finite state automata or sequential machines a necessary and sufficient condition for a regular set to represent an rop is given examples are given to illustrate the advantages of such a specification in terms of a primary focus on root signals and input output relations one result is a practical equivalent of the recursive median smoother that is easier to compute another is a demonstration that some rops although not implementable as finite autoregressive algorithms involving only the input output signal values are implementable as finite state sequential machines e g those with a modified stack filter structure the demonstration is constructive and coincidentally shows how the idea of a nondeterministic automaton relates to the subject matter

smoothing effects for schrodinger equations with electro magnetic potentials and applications to the maxwell schrodinger equations abstract we consider schrodinger equations in r 1 2 with electro magnetic potentials the potentials belong to h 1 and typically they are time independent or determined as solutions to inhomogeneous wave equations we prove kato type smoothing estimates for solutions we also apply this result to the maxwell schrodinger equations in the lorentz gauge and prove unique solvability of this system in the energy space

on the sensitivity of direct form and nested structures of fir filters the sensitivity analysis of nested structures of fir filters is extended to show that direct form structures have sensitivities identical to those of the nested structures the key to having identical coefficient sensitivity is to use the same scale representation for the coefficients

numerical computation of periodic orbits that bifurcate from stationary solutions of ordinary differential equations the computation of periodic orbits of autonomous ordinary differential equations is considered a new method especially suited for computing limit cycles that bifurcate from stationary solutions is established this novel approach is very easy to implement the method requiring the programming of the right hand side only another topic of this paper is the demonstration of how a direct method should be applied in the computation of hopf bifurcation points these methods solve boundary value problems by means of standard software only the methods are tested on four examples arising in different application areas including chemistry and biology

analytical search for bifurcation surfaces in parameter space the method of resultants can be used to compute hopf bifurcations in ode systems we discuss this method from the applicant s point of view furthermore we show by theory and examples that the method may be extended to cover other bifurcation situations as well among them are the real hopf situation which plays an important role in the transition to shil nikov chaos as well as some higher codimension bifurcations such as takens bogdanov gavrilov guckenheimer and double hopf bifurcations the method yields an analytical test function that can be solved analytically or by computer algebra systems in contrast to common analytical techniques based on eigenvalue computation which can only be applied to systems of size n 4 the method is applicable for systems of intermediate size n 10 we illustrate the power of the method by discussing examples from different disciplines of science a lorenz like oscillator two coupled oscillators and a five species food chain

the hopf bifurcation theorem for quasilinear differential algebraic equations we prove a generalization of the hopf bifurcation theorem for quasilinear differential equations daes i e equations of the form a g where the matrix a has constant but not full rank and hence the system cannot be made into an explicit ode the paper includes an appendix by j ernsthausen addressing the numerical calculation of the hopf points in the dae setting

continuation techniques and interactive software for bifurcation analysis of odes and iterated maps physics abstract we present a numerical technique for the analysis of local bifurcations which is based on the continuation of structurally unstable invariant sets in a suitable phase parameter space the invariant sets involved in our study are equilibrium points and limit cycles of autonomous odes periodic solutions of time periodic nonautonomous odes fixed points and periodic orbits of iterated maps the more general concept of a continuation strategy is also discussed it allows the analysis of various singularities of generic systems and of their mutual relationships the approach is extended to codimension three singularities we introduce several bifurcation functions and show how to use them to construct well posed continuation problems the described continuation technique is supported by an interactive graphical program called locbif we discuss briefly the concepts of the locbif interface and give some examples of typical applications

equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function some practical models contain so complicated mathematical expressions that it is hard to determine the number and distribution of all equilibria not mentioning the qualitative properties and bifurcations of those equilibria the three node recurrent neural network system with two free weight parameters originally introduced by ruiz owens and townley in 1997 is such a system for which the equation of equilibria involves transcendental function and its iterates not computing coordinates of its equilibria in this paper we display an effective technique to determine the number and distribution of its equilibria without full information about equilibria our method enables to further study qualitative properties of those equilibria and discuss their saddle node pitchfork and hopf bifurcations by approximating center manifolds

fir filter structures having low sensitivity and roundoff noise a class of structures for fir filters is presented which exhibits reduced coefficient sensitivity and superior roundoff noise properties as compared to the direct form realization it is shown that using fixed point arithmetic these structures achieve the same accuracy and about the same roundoff noise as those obtained in the floating point implementationn the optimum structure to achieve the minimum roundoff noise can be found in most cases near optimum results are easily obtained by simple permutations and combinations of the impulse response coefficients while a serial form realization of these structures requires a certain amount of software complexity a parallel form on the other hand does not require additional complexity

long range scattering for the maxwell schrodinger system in the lorenz gauge without any restriction on the size of data abstract this paper concerns the scattering theory for the maxwell schrodinger ms system in the lorenz gauge or more precisely the existence of the modified wave operators for this system in r 3 1 space time we construct solutions to the ms system which behave as free maxwell and schrodinger waves with prescribed asymptotic states when t without any restriction on the size thereof since this system belongs to the borderline between the short range case and the long range case we need modification of phase for the schrodinger function

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

bifurcation analysis of a predator prey system with nonmonotonic functional response we consider a predator prey system with nonmonotonic functional response p x frac mx ax 2 bx 1 by allowing b to be negative b 2 sqrt a p x is concave up for small values of x 0 a

parameter space structure of continuous time recurrent neural networks a fundamental challenge for any general theory of neural circuits is how to characterize the structure of the space of all possible circuits over a given model neuron as a first step in this direction this letter begins a systematic study of the global parameter space structure of continuous time recurrent neural networks ctrnns a class of neural models that is simple but dynamically universal first we explicitly compute the local bifurcation manifolds of ctrnns we then visualize the structure of these manifolds in net input space for small circuits these visualizations reveal a set of extremal saddle node bifurcation manifolds that divide ctrnn parameter space into regions of dynamics with different effective dimensionality next we completely characterize the combinatorics and geometry of an asymptotically exact approximation to these regions for circuits of arbitrary size finally we show how these regions can be used to calculate estimates of the probability of encountering different kinds of dynamics in ctrnn parameter space

chaos in a low order model of magnetoconvection abstract in the limit of tall thin rolls weakly nonlinear convection in a vertical magnetic field is described by an asymptotically exact third order set of ordinary differential equations these equations are shown here to have three codimension two bifurcation points a takens bogdanov bifurcation at which a gluing bifurcation is created a point at which the gluing bifurcation is replaced by a pair of homoclinic explosions between which there are lorenz like chaotic trajectories and a new type of bifurcation point at which the first of a cascade of period doubling bifurcation lines originates the last two bifurcation points are analysed in terms of a one dimensional map the equations also have a t point at which there is a heteroclinic connection between a saddle and a pair of saddle foci emerging from this point is a line of sil nikov bifurcations involving homoclinic connections to a saddle focus

classification and unfoldings of degenerate hopf bifurcations abstract this paper initiates the classification up to symmetry covariant contact equivalence of perturbations of local hopf bifurcation problems which do not satisfy the classical non degeneracy conditions the only remaining hypothesis is that i should be simple eigenvalues of the linearized right hand side at criticality then the lyapunov schmidt method allows a reduction to a scalar equation g x 0 where g x g x a definition is given of the codimension of g and a complete classification is obtained for all problems with codimension 3 together with the corresponding universal unfoldings the perturbed bifurcation diagrams are given for the cases with codimension 2 and for one case with codimension 3 for this last case one of the unfolding parameters is a modal parameter such that the topological codimension equals in fact 2 formulas are given for the calculation of the taylor coefficients needed for the application of the results and finally the results are applied to two simple problems a model of glycolytic oscillations and the fitzhugh nerve equations

a precise calculation of bifurcation points for periodic solution in nonlinear dynamical systems this paper presented a precise yet efficient algorithm to determine bifurcation points of periodic solutions in nonlinear dynamical systems period doubling and symmetry breaking points were obtained to an accuracy of 10 11 the bifurcation points can be directly determined without a tedious trail and error process calculations and analysis on bifurcations of periodic solutions has played a pivotal role in nonlinear dynamics researches this paper presents an efficient and precise approach based on the incremental harmonic balance ihb method to exactly determine the critical points at which periodic solutions exhibit bifurcations the presented method is based on a fact that some zero harmonic coefficients vary as non zeros during the bifurcation of a periodic solution equating one of these coefficients to an infinitesimal quantity the bifurcation value can be determined by the ihb method with the control parameter varying numerical examples show that the critical values for the control parameter where a symmetry breaking or period doubling happens can be calculated very accurately importantly each bifurcation value can be directly obtained without a tedious trail and error process

analytical calculation of fragmentation transitions in adaptive networks in adaptive networks fragmentation transitions have been observed in which the network breaks into disconnected components we present an analytical approach for calculating the transition point in general adaptive network models using the example of an adaptive voter model we demonstrate that the proposed approach yields good agreement with numerical results

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

a class of rank order smoothers smoothers or nonlinear filters that make estimates purely on the basis of rank and temporal order determinations among members of the input data are inherently robust and compatible with vlsi implementations the classic example is the moving median smoother but it is possible to define others which have significantly different attributes both in relation to the statistical estimation problem and in relation to hardware implications attributes offered in the latter category which are especially desirable in systolic vlsi include essentially recursive operation and elimination of the interactions between rank and temporal ordering requirements which are a feature of median smoothers experimental results indicate significantly improved response to noisy steps for these smoothers in comparison to median smoothers of comparable rectangular pulse detection capabilities

stack filters the median and other rank order operators possess two properties called the threshold decomposition and the stacking properties the first is a limited superposition property which leads to a new architecture for these filters the second is an ordering property which allows an efficient vlsi implementation of the threshold decomposition architecture motivated by the success of rank order filters in a wide variety of applications and by the ease with which they can now be implemented we consider in this paper a new class of filters called stack filters they share the threshold decomposition and stacking properties of rank order filters but are otherwise unconstrained they are shown to form a very large class of easily implemented nonlinear filters which includes the rank order operators as well as all compositions of morphological operators the convergence properties of these filters are investigated using techniques similar to those used to determine root signal behavior of median filters the results obtained include necessary conditions for a stack filter to preserve monotone regions or edges in signals the output distribution for these filters is also found all the stack filters of window width 3 are determined along with their convergence properties among these filters are found two which we have named asymmetric median filters they share all the properties of median filters except that they remove impulses of one sign only that is one removes only positive going edges the other removes only negative going edges while the median filter removes impulses of both signs this investigation of the properties of stack filters thus produces several new useful and easily implemented filters

adaptive networks coevolution of disease and topology d epartement de physique de g enie physique et d optique universit e laval qu ebec qu ebec canada g1v 0a6 dated september 13 2010 adaptive networks have been recently introduced in the context of disease propagation on complexnetworks they account for the mutual interaction between the network topology and the states ofthe nodes until now existing models have been analyzed using low complexity analytic formalisms revealing nevertheless some novel dynamical features however current methods have failed toreproduce with accuracy the simultaneous time evolution of the disease and the underlying networktopology in the framework of the adaptive sis model of gross et al phys rev lett 96 208701 2006 we introduce an improved compartmental formalism able to handle this coevolutionarytask successfully with this approach we analyze the interplay and outcomes of both dynamicalelements process and structure on adaptive networks featuring di erent degree distributions at theinitial stage

computing hopf bifurcations i this paper addresses the problems of detecting hopf bifurcations in systems of ordinary differential equations and following curves of hopf points in two parameter families of vector fields the established approach to this problem relies upon augmenting the equilibrium condition so that a hopf bifurcation occurs at an isolated regular point of the extended system we propose two new methods of this type based on classical algebraic results regarding the roots of polynomial equations and properties of kronecker products for matrices in addition to their utility as augmented systems for use with standard newton type continuation methods they are also particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension

state description for the root signal set of median filters median filtering is a simple digital technique for smoothing signals one main characteristic of the filter is that it maps the input signal space into a root signal space where signals invariant to median filters are called roots of the signal in this paper we develop the theory for the root signal set of median filters a tree structure for the root signal set is obtained for binary signals the number of roots r n for a signal of length n and window size filter 2s 1 is exactly represented by the difference equation r n r n 1 r n s a general solution is obtained in a z domain approach finally a method for faster one dimensional median filter operation is introduced

using machine learning to predict catastrophes in dynamical systems nonlinear dynamical systems which include models of the earth s climate financial markets and complex ecosystems often undergo abrupt transitions that lead to radically different behavior the ability to predict such qualitative and potentially disruptive changes is an important problem with far reaching implications even with robust mathematical models predicting such critical transitions prior to their occurrence is extremely difficult in this work we propose a machine learning method to study the parameter space of a complex system where the dynamics is coarsely characterized using topological invariants we show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner we are able to predict with high accuracy the locations of critical transitions in parameter space

bifurcation analysis of an ecological food chain model with switching predator the present paper deals with the mathematical analysis of a prey predator model where the prey population is assumed to live in two distinct habitats and the predator consumes prey from both these habitats with a switching tendency the dynamical behavior of the system is investigated from the point of view of stability and bifurcation aspects it has been found that the mortality rate of the predator plays an important role in controlling the system dynamics finally a detailed study of the hopf bifurcating solutions is carried out and the criteria for subcritical and supercritical hopf bifurcation derived numerical simulations are performed in support of analytical results

sweeping algebraic curves for singular solutions many problems give rise to polynomial systems these systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters using predictor corrector methods we track the solution paths a point along a solution path is critical when the jacobian matrix is rank deficient the simplest case of quadratic turning points is well understood but these methods no longer work for general types of singularities in order not to miss any singular solutions along a path we propose to monitor the determinant of the jacobian matrix we examine the operation range of deflation and relate the effectiveness of deflation to the winding number computational experiments on systems coming from different application fields are presented

lyapunov inverse iteration for identifying hopf bifurcations in models of incompressible flow the identification of instability in large scale dynamical systems caused by hopf bifurcation is difficult because of the problem of identifying the rightmost pair of complex eigenvalues of large sparse generalized eigenvalue problems a new method developed in k meerbergen and a spence siam j matrix anal appl 31 2010 pp 1982 1999 avoids this computation instead performing an inverse iteration for a certain set of real eigenvalues that requires the solution of a large scale lyapunov equation at each iteration in this study we refine the lyapunov inverse iteration method to make it more robust and efficient and we examine its performance on challenging test problems arising from fluid dynamics various implementation issues are discussed including the use of inexact inner iterations and the impact of the choice of iterative solution for the lyapunov equations and the effect of eigenvalue distribution on performance numerical experiments demonstrate the robustness of the algorithm

coevolutionary networks with homophily and heterophily we have investigated a simple coevolutionary network model incorporating three processes changes of opinions homophily and heterophily in this model each node holds one of g opinions and changes its opinion as in the voter model homophily is the tendency for connections to form between individuals of the same opinions and heterophily is the opposite effect if there is no heterophily this model corresponds to the holme and newman model phys rev e 74 056108 2006 we show that the behavior of this model without heterophily can be understood in terms of a mean field approximation we also find that this model with heterophily exhibits topologically complicated behaviors such as the small world property

robustness of behaviorally induced oscillations in epidemic models under a low rate of imported cases this paper is concerned with the robustness of the sustained oscillations predicted by an epidemic ode model defined on contact networks the model incorporates the spread of awareness among individuals and moreover a small inflow of imported cases these cases prevent stochastic extinctions when we simulate the epidemics and hence they allow to check whether the average dynamics for the fraction of infected individuals are accurately predicted by the ode model stochastic simulations confirm the existence of sustained oscillations for different types of random networks with a sharp transition from a nonoscillatory asymptotic regime to a periodic one as the alerting rate of susceptible individuals increases from very small values this abrupt transition to periodic epidemics of high amplitude is quite accurately predicted by the hopf bifurcation curve computed from the ode model using the alerting rate and the infection transmission rate for aware individuals as tuning parameters

homoclinic phenomena in laser models abstract a description of the principal bifurcations which lead to the appearance of the lorenz attractor is given for the 3d normal form for codimension 3 bifurcations of equilibria and periodic orbits in systems with symmetry we pay special attention to two bifurcation points corresponding to the formation of a homoclinic butterfly of a saddle with unit saddle index and to a homoclinic butterfly with zero separatrix value

self organization of complex networks as a dynamical system to understand the dynamics of real world networks we investigate a mathematical model of the interplay between the dynamics of random walkers on a weighted network and the link weights driven by a resource carried by the walkers our numerical studies reveal that under suitable conditions the co evolving dynamics lead to the emergence of stationary power law distributions of the resource and link weights while the resource quantity at each node ceaselessly changes with time we analyze the network organization as a deterministic dynamical system and find that the system exhibits multistability with numerous fixed points limit cycles and chaotic states the chaotic behavior of the system leads to the continual changes in the microscopic network dynamics in the absence of any external random noises we conclude that the intrinsic interplay between the states of the nodes and network reformation constitutes a major factor in the vicissitudes of real world networks

consensus formation on adaptive networks the structure of a network can significantly influence the properties of the dynamical processes that take place on them while many studies have been paid to this influence much less attention has been devoted to the interplay and feedback mechanisms between dynamical processes and network topology on adaptive networks adaptive rewiring of links can happen in real life systems such as acquaintance networks where people are more likely to maintain a social connection if their views and values are similar in our study we consider different variants of a model for consensus formation our investigations reveal that the adaptation of the network topology fosters cluster formation by enhancing communication between agents of similar opinion although it also promotes the division of these clusters the temporal behavior is also strongly affected by adaptivity while on static networks it is influenced by percolation properties on adaptive networks both the early and late time evolutions of the system are determined by the rewiring process the investigation of a variant of the model reveals that the scenarios of transitions between consensus and polarized states are more robust on adaptive networks

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

on the existence and computation of lu factorizations with small pivots let a be an n by n matrix which may be singular with a one dimensional null space and consider the lu factorization of a when a is exactly singular we show conditions under which a pivoting strategy will produce a zero n th pivot when a is not singular we show conditions under which a pivoting strategy will produce an nth pivot that is o g or o k a where is the smallest singular value of a and k a is the condition number of a these conditions are expressed in terms of the elements of a in general but reduce to conditions on the elements of the singular vectors corresponding to when a is nearly or exactly singular they can be used to build a 2 pass factorization algorithm which is guaranteed to produce a small n th pivot for nearly singular matrices as an example we exhibit an lu factorization of the n by n upper triangular matrix

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

the cauchy problem for the coupled maxwell schrodinger equations the cauchy problem for the coupled maxwell schrodinger equations in rd in the lorentz gauge is considered the viscosity method is used to establish local existence in one and two space dimensions global solutions are obtained

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

median filtering by threshold decomposition median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to further passes of the median filter in this paper it is shown that median filtering an arbitrary level signal to its root is equivalent to decomposing the signal into binary signals filtering each binary signal to a root with a binary median filter and then reversing the decomposition this equivalence allows problems in the analysis and the implementation of median filters for arbitrary level signals to be reduced to the equivalent problems for binary signals since the effects of median filters on binary signals are well understood this technique is a powerful new tool

on models with non rough poincare homoclinic curves abstract the possibility of an a priori complete description of finite parameter models including systems with structurally unstable poincare homoclinic curves is studied the main result reported here is that systems having a countable set of moduli of equivalence and systems having infinitely many degenerate periodic and homoclinic orbits are dense in the newhouse regions of non stability we discuss the question of correctly setting a problem for the analysis of models of such type

the bialternate matrix product revisited the well known bialternate product of two square matrices is re examined together with another matrix product defined by means of the permanent function and having similar properties old and new results concerning both products are presented in a unified manner a simple and elegant relation with the kronecker product of matrices is also given

bifurcation analysis of modified leslie gower predator prey model with michaelis menten type prey harvesting abstract in the present paper we discuss bifurcation analysis of a modified leslie gower prey predator model in the presence of nonlinear harvesting in prey we give a detailed mathematical analysis of the model to describe some significant results that may arise from the interaction of biological resources the model displays a complex dynamics in the prey predator plane the permanence stability and bifurcation saddle node bifurcation transcritical hopf andronov and bogdanov takens of this model are discussed we have analyzed the effect of prey harvesting and growth rate of predator on the proposed model by considering them as bifurcation parameters as they are important from the ecological point of view the local existence and stability of the limit cycle emerging through hopf bifurcation is given the emergence of homoclinic loops has been shown through simulation when the limit cycle arising though hopf bifurcation collides with a saddle point this work reflects that the feasible upper bound of the rate of harvesting for the coexistence of the species can be guaranteed numerical simulations using matlab are carried out to demonstrate the results obtained

unstable network fragmentation in co evolution of potts spins and system topology abstract we investigate co evolution of discrete q state potts model and the underlying network topology where spin changes and link re wiring follow the same canonical ensemble dynamics in a constant temperature it means that there are no absorbing frozen states present in our model depending on the temperature t and probability of link dynamics p the system can exist in one of three states ordered disordered and ordered clusters fragmented network with the last being unstable and slowly relaxing into ordered state the transition from ordered clusters to globally ordered system is characterized by non exponential slow growth of the order parameter we investigate this process analytically and explain the transition characteristics as the result of the dominance of activity of surface nodes in each ordered cluster as opposed to bulk nodes that are inactive

a direct method for the computation of hopf bifurcation points we present a method for the numerical computation of a hopf bifurcation point of an evolution equation an augmented time independent system of equations has to be solved for which the hopf bifurcation point is an isolated solution in the case of a system of ordinary differential equations we propose an efficient implementation of newton s method for the solution of the resulting system of algebraic equations if the method is applied directly to a partial differential system with one space variable a two point boundary value problem is obtained we also compare this technique with a related method and results are given for two systems of partial differential equations describing chemical reacting systems

numerical computation of hopf bifurcation points for parabolic diffusion reaction differential equations a numerical method for the direct computation of hopf bifurcation points for a system of parabolic diffusion reaction partial differential equations is described a finite difference discretization in space is used to approximate these equations by a system of first order ordinary differential equations the hopf bifurcation of the latter system is computed by using kubicek s method the results are improved by richardson extrapolation for approximation of the space operator a standard three point difference formula as well as the stormer numerov technique are used the method proposed is illustrated on an example describing heat and mass transfer in a porous catalyst particle

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

an arnoldi based iterative scheme for nonsymmetric matrix pencils arising in finite element stability problems abstract a method for computing the desired eigenvalues and corresponding eigenvectors of a large scale nonsymmetric complex generalized eigenvalue problem is described this scheme is primarily intended for the normal mode analysis and the stability characterization of the stationary states of parameterized time dependent partial differential equations in particular when a finite element method is used for the numerical discretization the algorithm which is based on the previous work of saad may be succintly described as a multiple shift and invert restarted arnoldi procedure which uses reorthogonalization and automatic shift selection to provide stability and convergence while minimizing the overall computational effort the application and efficiency of the method is illustrated using two representative test problems

fragmentation transitions in multistate voter models adaptive models of opinion formation among humans can display a fragmentation transition where a social network breaks into disconnected components here we investigate this transition in a class of models with arbitrary number of opinions in contrast to previous work we do not assume that opinions are equidistant or arranged on a one dimensional conceptual axis our investigation reveals detailed analytical results on fragmentations in a three opinion model which are confirmed by agent based simulations furthermore we show that in certain models the number of opinions can be reduced without affecting the fragmentation points

new algorithms for the evaluation of complex bifurcation points in ordinary differential equations a comparative numerical study four iterative algorithms two of them new for the evaluation of complex hopf bifurcation points in ordinary differential equations are compared a comparison of effectiveness of the proposed algorithms is made for two examples taken from chemical reaction engineering the applicability of the algorithms for the solution of various types of problems is also discussed

generalized models as a universal approach to the analysis of nonlinear dynamical systems we present a universal approach to the investigation of the dynamics in generalized models in these models the processes that are taken into account are not restricted to specific functional forms therefore a single generalized models can describe a class of systems which share a similar structure despite this generality the proposed approach allows us to study the dynamical properties of generalized models efficiently in the framework of local bifurcation theory the approach is based on a normalization procedure that is used to identify natural parameters of the system the jacobian in a steady state is then derived as a function of these parameters the analytical computation of local bifurcations using computer algebra reveals conditions for the local asymptotic stability of steady states and provides certain insights on the global dynamics of the system the proposed approach yields a close connection between modelling and nonlinear dynamics we illustrate the investigation of generalized models by considering examples from three different disciplines of science a socioeconomic model of dynastic cycles in china a model for a coupled laser system and a general ecological food web

approximation of hopf bifurcation we make several assumptions on a nonlinear evolution problem ensuring the existence of a hopf bifurcation under a fairly general approximation condition we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions

global existence and uniqueness of solutions to the maxwell schrodinger equations the time local and global well posedness for the maxwell schrodinger equations is considered in sobolev spaces in three spatial dimensions the strichartz estimates of koch and tzvetkov type are used for obtaining the solutions in the sobolev spaces of low regularities one of the main results is that the solutions exist time globally for large data

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

early fragmentation in the adaptive voter model on directed networks we consider voter dynamics on a directed adaptive network with fixed out degree distribution a transition between an active phase and a fragmented phase is observed this transition is similar to the undirected case if the networks are sufficiently dense and have a narrow out degree distribution however if a significant number of nodes with low out degree is present then fragmentation can occur even far below the estimated critical point due to the formation of self stabilizing structures that nucleate fragmentation this process may be relevant for fragmentation in current political opinion formation processes

modeling symbiosis by interactions through species carrying capacities abstract we introduce a mathematical model of symbiosis between different species by taking into account the influence of each species on the carrying capacities of the others the modeled entities can pertain to biological and ecological societies or to social economic and financial societies our model includes three basic types symbiosis with direct mutual interactions symbiosis with asymmetric interactions and symbiosis without direct interactions in all cases we provide a complete classification of all admissible dynamical regimes the proposed model of symbiosis turned out to be very rich as it exhibits four qualitatively different regimes convergence to stationary states unbounded exponential growth finite time singularity and finite time death or extinction of species

detection and computation of high codimension bifurcations in diffuse predator prey systems abstract in this paper we consider reaction diffusion model with modified leslie gower and holling type ii functional response and by varying simultaneously three model parameters our model exhibits different types of complex dynamics this allowed us to delimit several bifurcation surfaces with higher codimension corresponding to turing turing transcritical turing bogdanov taken turing hopf andronov turing saddle node moreover by varying at least three bifurcation parameters we show that small variations in the ratio of the diffusion coefficients can significantly alter bifurcation structure finally the paper ends with the emergence of spatio temporal patterns via numerical simulations these simultaneous parameter variations leaded to spatio temporal local and global bifurcations which are always catastrophic our results give new insights about how simultaneous changes in environmental and life history parameters drive different distribution dynamics of predator prey populations this only underscores the importance of including global bifurcations in the analysis of food chain models such results can be used to improve ecological decision making on species conservation

multiopinion coevolving voter model with infinitely many phase transitions we consider an idealized model in which individuals changing opinions and their social network coevolve with disagreements between neighbors in the network resolved either through one imitating the opinion of the other or by reassignment of the discordant edge specifically an interaction between x and one of its neighbors y leads to x imitating y with probability 1 and otherwise i e with probability x cutting its tie to y in order to instead connect to a randomly chosen individual building on previous work about the two opinion case we study the multiple opinion situation finding that the model has infinitely many phase transitions in the large graph limit with infinitely many initial opinions moreover the formulas describing the end states of these processes are remarkably simple when expressed as a function of 1