the optimal sampling pattern for linear control systems in digital control systems the state is sampled at given sampling instants and the input is kept constant between two consecutive instants with the optimal sampling problem we mean the selection of sampling instants and control inputs such that a given function of the state and input is minimized in this paper we formulate the optimal sampling problem and we derive a necessary condition for the optimality of a set of sampling instants in the linear quadratic regulator problem since the numerical solution of the optimal sampling problem is very time consuming we also propose a new quantization based sampling strategy that is computationally tractable and capable of achieving near optimal cost finally and probably most interesting of all we prove that the quantization based sampling is optimal in first order systems for a large number of samples experiments demonstrate that quantization based sampling has near optimal performance even when the system has a higher order however it is still an open question whether quantization based sampling is asymptotically optimal in any case

strange nonchaotic attractors with wada basins a b s t r a c t we demonstrate strange nonchaotic attractors with wada basins snaswb and verify the abundance of snaswb in a quasiperiodically forced holmes map we identify the routes to the creation of the snaswb in a two parameter space the snaswb are characterized by the maximal lyapunov exponent by the estimation of the phase sensitivity exponent and by the singular continuous spectra we observe that the snas basins are totally wada basins in a large range of parameters the topological structures of the snas wada basins are distinguished by the basin cell method we investigate the underlying mechanism for the abundance of snaswb which is responsible for different types of basin cell in the absence of forcing this suggests that the snas cannot be predicted reliably for specific initial conditions these snaswb can thus be expected to occur more commonly in dynamical systems

onset of cooperative entrainment in limit cycle oscillators with uniform all to all interactions bifurcation of the order function abstract the concept of order function was previously proposed as the key to a general theory of macroscopic mutual or cooperative entrainment in large populations of limit cycle oscillators with weak interactions such that every element is linked to all the other as well as with weakly dispersed intrinsic frequencies that is limit cycle oscillators that can be modeled by globally coupled phase oscillators with distributed frequencies following previous work a bifurcation theory of the order function is developed on the basis of its self consistent functional equation to elucidate in particular generic scaling behavior of such systems at the onset of cooperative entrainment among other results when the onset is not abrupt the critical exponent of fundamental order parameters turns out to generically differ from the conventional value 1 2 taken by the well studied sinusoidal coupling model as well as by typical mean fields models of thermodynamic phase transitions to which coupled oscillator models investigated here are analogous the theory also reveals what happens in nongeneric cases moreover a criterion is found of whether the bifurcation is normal or inverted all these analytical results and predictions are verified not only by numerically solving the equation of the order function but also by numerical simulations although this paper is mainly concerned with the critical behaviors noncritical regimes are also explored to demonstrate overall power of the order function theory by reproducing simulation results such as average frequency spectra the theory however keeps some room to be further generalized a finding which suggests this is put forth

self organized partially synchronous dynamics in populations of nonlinearly coupled oscillators we analyze a minimal model of a population of identical oscillators with a nonlinear coupling a generalization of the popular kuramoto model in addition to well known for the kuramoto model regimes of full synchrony full asynchrony and integrable neutral quasiperiodic states ensembles of nonlinearly coupled oscillators demonstrate two novel nontrivial types of partially synchronized dynamics self organized bunch states and self organized quasiperiodic dynamics the analysis based on the watanabe strogatz ansatz allows us to describe the self organized bunch states in any finite ensemble as a set of equilibria and the self organized quasiperiodicity as a two frequency quasiperiodic regime an analytic solution in the thermodynamic limit of infinitely many oscillators is also discussed

optimized regression for efficient function evaluation in many applications of regression one is concerned with the efficiency of the estimated function in addition to the accuracy of the regression for efficiency it is common to represent the estimated function as a rectangular lattice of values a lookup table lut that can be linearly interpolated for any needed value typically a lut is constructed from data with a two step process that first fits a function to the data then evaluates that fitted function at the nodes of the lattice we present an approach termed lattice regression that directly optimizes the values of the lattice nodes to minimize the post interpolation training error additionally we propose a second order difference regularizer to promote smoothness we demonstrate the effectiveness of this approach on two image processing tasks that require both accurate regression and efficient function evaluations inverse device characterization for color management and omnidirectional super resolution for visual homing

model reduction via parametrized locally invariant manifolds some examples a method for model reduction in non linear ode systems is demonstrated through computational examples the method does not require an implicit separation of time scales in the fine dynamics to be effective from the computational standpoint the method has the potential of serving as a subgrid modeling tool from the physical standpoint it provides a model for interpreting and describing history dependence in coarse grained response of an autonomous system

noise induced escape from bifurcating attractors symplectic approach in the weak noise limit the effect of noise is studied in one dimensional maps undergoing transcritical tangent and pitchfork bifurcations the attractors of the noiseless map become metastable states in the presence of noise in the weak noise limit a symplectic two dimensional map is associated with the original one dimensional map the consequences of their noninvertibility on the phase space structures are discussed heteroclinic orbits are identified which play a key role in the determination of the escape rates from the metastable states near bifurcations the critical slowing down justifies the use of a continuous time approximation replacing maps by flows which allows the analytic calculation of the escape rates this method provides the universal scaling behavior of the escape rates at the bifurcations

algorithms for optimal discontinuous piecewise linear and constant l 2 fits to continuous functions with adjustable nodes in one and two dimensions in this paper a direct variational approach with nonstandard variations is used to generate algorithms to determine optimal discontinuous piecewise linear and piecewise constant l2 fits to a continuous function of one or two variables with adjustable nodes in the one variable case the algorithm is fast and robust the mesh cannot tangle and the resulting fits are continuous a e in the two variable case on an adjustable triangular grid the algorithm is less robust but gives good results for particular functions possessing a single steep feature the extension to higher dimensions is straightforward

a subdivision algorithm for the computation of unstable manifolds and global attractors each invariant set of a given dynamical system is part of the global attractor therefore the global attractor contains all the potentially interesting dynamics and in particular it contains every global unstable manifold for this reason it is of interest to have an algorithm which allows to approximate the global attractor numerically in this article we develop such an algorithm using a subdivision technique we prove convergence of this method in a very general setting and moreover we describe the qualitative convergence behavior in the presence of a hyperbolic structure the algorithm can successfully be applied to dynamical systems of moderate dimension and we illustrate this fact by several numerical examples

an efficient method for computing invariant manifolds of planar maps abstract an efficient method for computing invariant manifolds of mappings in the plane is described and compared to a simple method in common use the emphasis is on numerically producing smoothly resolved segments of invariant manifolds of arbitrary length while requiring a minimum number of calls to the map itself a brief estimate of errors is given and applications are discussed qualitative and quantitative comparisons between the two methods are made using an example problem the present method is applied to computing unstable manifolds for the well know periodically forced duffing system and the henon map for which the manifold is a strange attractor

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

entropy and bifurcations in a chaotic laser we compute bounds on the topological entropy associated with a chaotic attractor of a semiconductor laser with optical injection we consider the poincare return map to a fixed plane and are able to compute the stable and unstable manifolds of periodic points globally even though it is impossible to find a plane on which the poincare map is globally smoothly defined in this way we obtain the information that forms the input of the entropy calculations and characterize the boundary crisis in which the chaotic attractor is destroyed this boundary crisis involves a periodic point with negative eigenvalues and the entropy associated with the chaotic attractor persists in a chaotic saddle after the bifurcation

stochastic euler bernoulli beam driven by additive white noise global random attractors and global dynamics abstract this paper concerns with the long time dynamical behavior of a stochastic euler bernoulli beam driven by additive white noise by verifying the existence of absorbing set and obtaining the stabilization estimation of the dynamical system induced by the beam the existence of global random attractors that attracts all bounded sets in phase space is proved furthermore the finite hausdorff dimension for the global random attractors is attained in light of the relationship between global random attractor and random invariant probability measure the global dynamics of the beam are analyzed according to numerical simulation on global random basic attractors and global random point attractors

complex balancing motions of an inverted pendulum subject to delayed feedback control we show that an inverted pendulum that is balanced on a cart by linear delayed feedback control may exhibit small chaotic motion about the upside down position in periodic windows associated with this chaotic regime we find periodic orbits of arbitrarily high period that correspond to complex balancing motion of the pendulum with bounded velocity of the cart this result shows that complex balancing is possible in a controlled mechanical system with a geometric nonlinearity even when the control law is only linear this is in contrast to other proposed models that require a nonlinearity of the controller such as round off due to digitization we find the complex motion by studying homoclinic bifurcations of a reduced three dimensional vector field near a triple zero eigenvalue singularity more generally the dynamics we describe must be expected in any system with a triple zero singularity and reflection symmetry

computation of the dominant lyapunov exponent via spatial integration using matrix norms in a previous paper aston p j dellnitz m 1999 comput meth appl mech engng 170 223 237 we introduced a new method for computing the dominant lyapunov exponent of a chaotic map by using spatial integration involving a matrix norm we conjectured that this sequence of integrals decayed proportional to 1 n we now prove this conjecture and derive a bound on the next term in the asymptotic

accumulating regions of winding periodic orbits in optically driven lasers we investigate the route to locking in class b lasers subject to optically injected light for injection strengths and detunings near a codimension two saddle node hopf point this is the parameter region where the adler approximation is not valid and where yeung and strogatz recently reported a self similar cascade of periodic orbits in the case of a solid state laser we explain this cascade as an accumulation of large regions bounded by saddle node bifurcations of periodic orbits but also containing further bifurcations such as period doubling torus bifurcations and small pockets of chaos in the vicinity of the simultaneous saddle node and hopf bifurcations successive periodic orbits wind more and more near the point in phase space where the saddle node bifurcation is about to occur this leads to a self similar period adding cascade by varying the linewidth enhancement parameter from zero the case of a solid state or co2 laser to values larger than 1 the case of semiconductor lasers we show how the accumulating regions of winding periodic orbits change rapidly this explains why the period adding cascade has not been found in injected semiconductor lasers moreover we are able to identify certain regions with complex dynamics in injected semiconductor lasers as remainders of the accumulating regions 2002 elsevier science b v all rights reserved

desynchronization transitions in nonlinearly coupled phase oscillators abstract we consider the nonlinear extension of the kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter a bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed we demonstrate that for small ensembles it is typically mediated by stable cluster states that disappear with creation of heteroclinic cycles while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs

growing 1d and quasi 2d unstable manifolds of maps we present a new 1d algorithm for computing the global one dimensional unstable manifold of a saddle point of a map our method can be generalized to compute two dimensional unstable manifolds of maps with three dimensional state spaces this is shown here with a quasi 2d q2d algorithm for the special case of a quasiperiodically forced map which allows for a substantial simplification of the general case described in krauskopf and osinga 1998 the key idea is to grow the manifold in steps which consist of finding a new point on the manifold at a prescribed distance from the last point the speed of growth is determined only by the curvature of the manifold and not by the dynamics the performance of the 1d algorithm is demonstrated with a constructed test example and it is then used to compute one dimensional manifolds of a map modeling mixing in a stirring tank with the q2d algorithm we compute two dimensional unstable manifolds in the quasiperiodically forced henon map

one dimensional unstable eigenfunction and manifold computations in delay differential equations in this paper we present a new numerical technique for computing the unstable eigenfunctions of a saddle periodic orbit in a delay differential equation this is used to obtain the necessary starting data for an established algorithm for computing one dimensional 1d unstable manifolds of an associated saddle fixed point of a suitable poincare map to illustrate our method we investigate an intermittent transition to chaos in a delay system describing a semiconductor laser subject to phase conjugate feedback

genuinely multidimensional upwinding for the 2d shallow water equations a multidimensional upwinding technique is applied to the simulation of 2d shallow water flows it is adapted from fluctuation splitting methods recently proposed for the solution of the euler system of equations on unstructured triangular grids the basis of the numerical method is stated and the particular adaptation to the shallow water system is described numerical results of interest to hydraulic engineers are presented despite the complexities advantages related to the use of a discretisation based on triangles would seem to make the schemes competitive with those currently in use

constructing constrained invariant sets in multiscale continuum systems we present a method that we name the constrained invariant manifold method a visualization tool to construct stable and unstable invariant sets of a map or flow where the invariant sets are constrained to lie on a slow invariant manifold the construction of stable and unstable sets constrained to an unstable slow manifold is exemplified in a singularly perturbed model arising from a structural mechanical system consisting of a pendulum coupled to a viscoelastic rod additionally we extend the step and stagger method d sweet h nusse and j yorke phys rev lett 86 2261 2001 to calculate a delta pseudoorbit on a chaotic saddle constrained to the slow manifold in order to be able to compute the lyapunov exponents of the saddle

grading functions and mesh redistribution we consider the problem of determining an appropriate grading of a mesh for piecewise polynomial interpolation and for approximate solution of two point boundary value problems by finite difference or finite element methods an analysis of optimality of the mesh and optimal grading functions in various norms and seminorms for the interpolation problem leads to the formulation of an adaptive mesh redistribution algorithm for boundary value problems in one dimension error estimates are given and numerical results presented to demonstrate the performance of the scheme and compare alternative redistribution criteria

moving finite element methods for evolutionary problems i theory abstract in this paper the first of two on the subject we present a unified approach to moving and fixed finite element methods for evolutionary problems in terms of projections the central theoretical results are concerned with moving finite elements for one dimensional scalar problems particularly hyperbolic equations with shocks but the viewpoint extends to general systems in any number of dimensions

a hybrid method for computing lyapunov exponents in this paper we propose a numerical method for computing all lyapunov coefficients of a discrete time dynamical system by spatial integration the method extends an approach of aston and dellnitz comput methods appl mech eng 170 223 237 1999 who use a box approximation of an underlying ergodic measure and compute the first lyapunov exponent from a spatial average of the norms of the jacobian for the iterated map in the hybrid method proposed here we combine this approach with classical qr oriented methods by integrating suitable r factors with respect to the invariant measure in this way we obtain approximate values for all lyapunov exponents assuming somewhat stronger conditions than those of oseledec multiplicative theorem these values satisfy an error expansion that allows to accelerate convergence through extrapolation

computation of stable and unstable manifolds of hyperbolic trajectories in two dimensional aperiodically time dependent vector fields in this paper we develop two accurate and fast algorithms for the computation of the stable and unstable manifolds of hyperbolic trajectories of two dimensional aperiodically time dependent vector fields first we develop a benchmark method in which all the trajectories composing the manifold are integrated from the neighborhood of the hyperbolic trajectory this choice although very accurate is not fast and has limited usage a faster and more powerful algorithm requires the insertion of new points in the manifold as it evolves in time its numerical implementation requires a criterion for determining when to insert those points in the manifold and an interpolation method for determining where to insert them we compare four different point insertion criteria and four different interpolation methods we discuss the computational requirements of all of these methods we find two of the four point insertion criteria to be accurate and robust one is a variant of a criterion originally proposed by hobson the other is a slight variant of a method due to dritschel and ambaum arising from their studies of contour dynamics the preferred interpolation method is also due to dritschel these methods are then applied to the computation of the stable and unstable manifolds of the hyperbolic trajectories of several aperiodically time dependent variants of the duffing equation

moving finite element least squares and finite volume approximations of steady and time dependent pdes in multidimensions abstract we review recent advances in galerkin and least squares methods for approximating the solutions of first and second order pdes with moving nodes in multidimensions these methods use unstructured meshes and minimise the norm of the residual of the pde over both solutions and nodal positions in a unified manner both finite element and finite volume schemes are considered as are transient and steady problems for first order scalar time dependent pdes in any number of dimensions residual minimisation always results in the methods moving the nodes with the often inconvenient approximate characteristic speeds for second order equations however the moving finite element mfe method moves the nodes usefully towards high curvature regions in the steady limit for pdes derived from a variational principle the mfe method generates a locally optimal mesh and solution this also applies to least squares minimisation the corresponding moving finite volume mfv method based on the l2 norm does not have this property however although there does exist a finite volume method which gives an optimal mesh both for variational principles and least squares

uniform estimates of attracting sets of extended lurie systems using lmis this research presents a systematic procedure to obtain estimates via extended lyapunov functions of attracting sets of a class of nonlinear systems as well as an estimate of their stability regions the considered class of nonlinear systems called in this note the extended lurie system consists of nonlinear systems like those of the lurie problem where one of the nonlinear functions can violate the sector conditions of the lurie problem around the origin in case of nonautonomous systems the concept of absolute stability is extended and uniform estimates of the attracting set are obtained two classical nonlinear systems the forced duffing equation and the van der pol system are analyzed with the proposed procedure

a procedure for finding numerical trajectories on chaotic saddles abstract examples are common in dynamical systems in which there are regions containing chaotic sets that are not attractors if almost every trajectory eventually leaves some region but the region contains a chaotic set then typical trajectories will behave chaotically for a while and then will leave the region and so we will observe chaotic transients the main objective that will be addressed is the dynamic restraint problem given a region that contains a chaotic set but does not contain an attractor find a chaotic trajectory numerically that remains in the regionfor an arbituarily long period of time systems with horseshoes have such regions as do systems with fractal basin boundaries as does the henon map for suitably chosen parameters we present a numerical technique for finding trajectories which will stay in such chaotic sets for arbitrarily long periods of time and it leads to a saddle straddle trajectory the method is called the pim triple procedure since it is based on so called pim triples a pim proper interior maximum triple is three points a c b in a straight line segment such that the interior point c i e c is between a and b has the maximum escape time that is its escape time from the region is greater than the escape time of both a and b proper means the segment from a to b is smaller than a previously obtained segment we show rigorously that the pim triple procedure works in ideal situations we find it works well even in less than ideal cases this procedure can also be used for the computation of lyapunov exponents furthermore the accessible pim triple procedure a refined pim triple procedure for finding accessible trajectories on the chaotic saddle will also be discussed

taylor approximation of invariant fiber bundles for nonautonomous difference equations abstract invariant fiber bundles generalize invariant manifolds to nonautonomous difference equations in this paper we develop a method to calculate their taylor approximation which is of crucial importance e g for an application of the reduction principle in a nonautonomous setting

discretization of the frobenius perron operator using a sparse haar tensor basis the sparse ulam method the global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated frobenius perron operator for systems with low dimensional long term dynamics efficient techniques exist for a numerical approximation of the most important eigenfunctions cf m dellnitz and o junge siam j numer anal 36 1999 pp 491 515 they are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor ulam s method in this paper we develop a numerical technique which makes ulam s approach applicable to systems with higher dimensional long term dynamics it is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids c zenger sparse grids in parallel algorithms for partial differential equations kiel 1990 vieweg braunschweig 1991 pp 241 251 h j bungartz and m griebel acta numer 13 2004 pp 147 269 here we use a sparse haar tensor basis as the underlying approximation space we develop the technique establish statements about its complexity and convergence and present two numerical examples

rigorous computation of topological entropy with respect to a finite partition we present a method to compute rigorous upper bounds for the topological entropy h t a of a continuous map t with respect to a fixed coarse partition of the phase space a long trajectories are not used rather a single application of t to the phase space produces a topological markov chain which contains all orbits of t plus some additional spurious orbits by considering the markov chain as a directed graph and labelling the arcs according to the fixed partition one constructs a sofic shift with topological entropy greater than or equal to h t a to exactly compute the entropy of the sofic shift we produce a subshift of finite type with equal entropy via a standard technique the exact entropy calculation for subshifts is then straightforward we prove that the upper bounds converge monotonically to h t a as the topological markov chains become increasingly accurate the entire procedure is completely automatic 2001 elsevier science b v all rights reserved msc 37b40 94a17 37m25

design and optimization of color lookup tables on a simplex topology an important computational problem in color imaging is the design of color transforms that map color between devices or from a device dependent space e g rgb cmyk to a device independent space e g cielab and vice versa real time processing constraints entail that such nonlinear color transforms be implemented using multidimensional lookup tables luts furthermore relatively sparse luts with efficient interpolation are employed in practice because of storage and memory constraints this paper presents a principled design methodology rooted in constrained convex optimization to design color luts on a simplex topology the use of n simplexes i e simplexes in n dimensions as opposed to traditional lattices recently has been of great interest in color lut design for simplex topologies that allow both more analytically tractable formulations and greater efficiency in the lut in this framework of n simplex interpolation our central contribution is to develop an elegant iterative algorithm that jointly optimizes the placement of nodes of the color lut and the output values at those nodes to minimize interpolation error in an expected sense this is in contrast to existing work which exclusively designs either node locations or the output values we also develop new analytical results for the problem of node location optimization which reduces to constrained optimization of a large but sparse interpolation matrix in our framework we evaluate our n simplex color luts against the state of the art lattice e g international color consortium profiles and simplex based techniques for approximating two representative multidimensional color transforms that characterize a cmyk xerographic printer and an rgb scanner respectively the results show that color luts designed on simplexes offer very significant benefits over traditional lattice based alternatives in improving color transform accuracy even with a much smaller number of nodes

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

fitting optimal piecewise linear functions using genetic algorithms constructing a model for data in r sup 2 is a common problem in many scientific fields including pattern recognition computer vision and applied mathematics often little is known about the process which generated the data or its statistical properties for example in fitting a piecewise linear model the number of pieces as well as the knot locations may be unknown hence the method used to build the statistical model should have few assumptions yet still provide a model that is optimal in some sense such methods can be designed through the use of genetic algorithms we examine the use of genetic algorithms to fit piecewise linear functions to data in r sup 2 the number of pieces the location of the knots and the underlying distribution of the data are assumed to be unknown we discuss existing methods which attempt to solve this problem and introduce a new method which employs genetic algorithms to optimize the number and location of the pieces experimental results are presented which demonstrate the performance of our method and compare it to the performance of several existing methods we conclude that our method represents a valuable tool for fitting both robust and nonrobust piecewise linear functions

sequential linear interpolation of multidimensional functions we introduce a new approach that we call sequential linear interpolation sli for approximating multidimensional nonlinear functions the sli is a partially separable grid structure that allows us to allocate more grid points to the regions where the function to be interpolated is more nonlinear this approach reduces the mean squared error mse between the original and approximated function while retaining much of the computational advantage of the conventional uniform grid interpolation to obtain the optimal grid point placement for the sli structure we appeal to an asymptotic analysis similar to the asymptotic vector quantization vq theory in the asymptotic analysis we assume that the number of interpolation grid points is large and the function to be interpolated is smooth closed form expressions for the mse of the interpolation are obtained from the asymptotic analysis these expressions are used to guide us in designing the optimal sli structure for cases where the assumptions underlying the asymptotic theory are not satisfied we develop a postprocessing technique to improve the mse performance of the sli structure the sli technique is applied to the problem of color printer characterization where a highly nonlinear multidimensional function must be efficiently approximated our experimental results show that the appropriately designed sli structure can greatly improve the mse performance over the conventional uniform grid

hopf normal form with s_n symmetry and reduction to systems of nonlinearly coupled phase oscillators abstract coupled oscillator models where n oscillators are identical and symmetrically coupled to all others with full permutation symmetry s n are found in a variety of applications much but not all work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators in this paper we show this is restrictive and we characterize generic multi way interactions between oscillators that are typically present except at the very lowest order near a hopf bifurcation where the oscillations emerge we examine a network of identical weakly coupled dynamical systems that are close to a supercritical hopf bifurcation by considering two parameters the strength of coupling and an unfolding parameter for the hopf bifurcation for small enough 0 there is an attractor that is the product of n stable limit cycles this persists as a normally hyperbolic invariant torus for sufficiently small 0 using equivariant normal form theory we derive a generic normal form for a system of coupled phase oscillators with s n symmetry for fixed n and taking the limit 0 1 we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that to lowest order is the well known kuramoto sakaguchi system of coupled oscillators the next order of approximation generically includes terms with up to four interacting phases regardless of n using a normalization that maintains nontrivial interactions in the limit n we show that the additional terms can lead to new phenomena in terms of coexistence of two cluster states with the same phase difference but different cluster size

strange attractors that are not chaotic abstract it is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic here we use the word strange to refer to the geometry or shape of the attracting set while the word chaotic refers to the dynamics of orbits on the attractor in particular the exponential divergence of nearby trajectories we first give examples for which it can be demonstrated that there is a strange nonchaotic attractor these examples apply to a class of maps which model nonlinear oscillators continuous time which are externally driven at two incommensurate frequencies it is then shown that such attractore are persistent under perturbations which preserve the original system type i e there are two incommensurate external driving frequencies this suggests that for systems of the typw which we have considered nonchaotic strange attractors may be expected to occur for a finite interval of parameter values on the other hand when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit thus we conjecture that in general continuous time systems flows which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space

moving finite element methods for evolutionary problems ii applications abstract in this the second of two papers on the subject we present applications of the moving finite element method to a number of test problems key features are linear elements a direct approach to parallelism and node overtaking avoiding penalty functions rapid inversion of the mass matrix by preconditioned conjugate gradients and explicit euler time stepping the resulting codes are fast and efficient and are able to follow fronts and similar features with great accuracy the paper includes a substantial section on changes of dependent variable and front tracking techniques for non linear diffusion problems test problems include non linear hyperbolic conservation laws and non linear parabolic equations in one and two dimensions

optimal discretizations in adaptive finite element electromagnetics one of the primary objectives of adaptive finite element analysis research is to determine how to effectively discretize a problem in order to obtain a sufficiently accurate solution efficiently therefore the characterization of optimal finite element solution properties could have significant implications on the development of improved adaptive solver technologies ultimately the analysis of optimally discretized systems in order to learn about ideal solution characteristics can lead to the design of better feedback refinement criteria for guiding practical adaptive solvers towards optimal solutions efficiently and reliably a theoretical framework for the qualitative and numerical study of optimal finite element solutions to differential equations of macroscopic electromagnetics is presented in this study for one two and three dimensional systems the formulation is based on variational aspects of optimal discretizations for helmholtz systems that are closely related to the underlying stationarity principle used in computing finite element solutions to continuum problems in addition the theory is adequately general and appropriate for the study of a range of electromagnetics problems including static and time harmonic phenomena moreover finite element discretizations with arbitrary distributions of element sizes and degrees of approximating functions are assumed so that the implications of the theory for practical h p hp and r type finite element adaption in multidimensional analyses may be examined copyright 2001 john wiley sons ltd

numerical computation of nonlinear normal modes in mechanical engineering this paper reviews the recent advances in computational methods for nonlinear normal modes nnms different algorithms for the computation of undamped and damped nnms are presented and their respective advantages and limitations are discussed the methods are illustrated using various applications ranging from low dimensional weakly nonlinear systems to strongly nonlinear industrial structures

controlling the unsteady analogue of saddle stagnation points it is well known that saddle stagnation points are crucial flow organizers in steady autonomous flows due to their accompanying stable and unstable manifolds these have been extensively investigated experimentally numerically and theoretically in situations related to macro and micromixers in order to either restrict or enhance mixing saddle points are also important players in the dynamics of mechanical oscillators in which such points and their associated invariant manifolds form boundaries of basins of attraction corresponding to qualitatively different types of behavior the entity analogous to a saddle point in an unsteady nonautonomous flow is a time varying hyperbolic trajectory with accompanying stable and unstable manifolds which move in time within the context of nearly steady flows the unsteady velocity perturbation required to ensure that such a hyperbolic saddle trajectory follows a specified trajectory in space is derived and shown to be equivalent to that which can be obtained

a new moving mesh algorithm for the finite element solution of variational problems the paper is devoted to the description and application of a new iterative mesh optimization algorithm for the finite element solution of variational problems set in infinite dimensional spaces the optimality criterion is that the mesh should be such that the variational energy functional evaluated at the finite element approximation be minimized such a criterion has a relatively long history in the finite element literature the chief merit of the procedure presented in this paper is that each node of the mesh and the corresponding nodal value of the discrete approximation are updated by solving sequentially local minimization problems with very few degrees of freedom it is shown that this procedure reduces the energy functional monotonically without the need to solve the global discrete problem at intermediate stages applications to partial differential equations are considered numerical results in two dimensions are obtained by incorporating the algorithm into bank s well known pltmg elliptic solver

numerical methods for two parameter local bifurcation analysis of maps we discuss new and improved algorithms for the bifurcation analysis of fixed points and periodic orbits cycles of maps and their implementation in matcont a matlab toolbox for continuation and bifurcation analysis of dynamical systems this includes the numerical continuation of fixed points of iterates of the map with one control parameter detecting and locating their bifurcation points i e limit point period doubling and neimark sacker and their continuation in two control parameters as well as detection and location of all codimension 2 bifurcation points on the corresponding curves for all bifurcations of codim 1 and 2 the critical normal form coefficients are computed both numerically with finite directional differences and using symbolic derivatives of the original map using a parameter dependent center manifold reduction explicit asymptotics are derived for bifurcation curves of double and quadruple period cycles rooted at codim 2 points of cycles with arbitrary period these asymptotics are implemented into the software and allow one to switch at codim 2 points to the continuation of the double and quadruple period bifurcations we provide two examples illustrating the developed techniques a generalized henon map and a juvenile adult competition model from mathematical biology

analysis of an algorithm for generating locally optimal meshes for l 2 approximation by discontinuous piecewise polynomials this paper discusses the problem of constructing a locally optimal mesh for the best l 2 approximation of a given function by discontinuous piecewise polynomials in the one dimensional case it is shown that under certain assumptions on the approximated function baines algorithm m j baines math comp 62 1994 pp 645 669 for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh the rate of convergence is investigated a two dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self adjusting numerical results in one and two dimensions are presented

normal form transforms separate slow and fast modes in stochastic dynamical systems modelling stochastic systems has many important applications normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics we explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes the thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems thus we not only must reduce the dimensionality of the dynamics but also endeavour to separate emph all slow processes from emph all fast time processes both deterministic and stochastic quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system the results will help us accurately model interpret and simulate multiscale stochastic systems

a local convergence proof for the minvar algorithm for computing continuous piecewise linear approximations the class of continuous piecewise linear pl functions represents a useful family of approximants because invertibility can be readily imposed and if a pl function is invertible then it can be inverted in closed form many applications arising for example in control systems and robotics involve the simultaneous construction of a forward and inverse system model from data most approximation techniques require that separate forward and inverse models be trained whereas an invertible continuous pl affords simultaneously the forward and inverse system model in a single representation the minvar algorithm computes a continuous pl approximation to data local convergence of minvar is proven for the case when the data generating function is itself a pl function and available directly rather than through data

approximating stable and unstable manifolds in experiments we introduce a procedure to reveal invariant stable and unstable manifolds given only experimental data we assume a model is not available and show how coordinate delay embedding coupled with invariant phase space regions can be used to construct stable and unstable manifolds of an embedded saddle we show that the method is able to capture the fine structure of the manifold is independent of dimension and is efficient relative to previous techniques

boundary crisis in quasiperiodically forced systems abstract we study boundary crisis in quasiperiodically forced dissipative systems using the henon map as a characteristic example the crisis is due to a homoclinic tangency of the stable and unstable manifolds of an accessible invariant circle of saddle type on the basin boundary of the attractor numerical evidence shows that the type of boundary crisis can change as the saddle circle loses its accessibility due to the quasiperiodic forcing this implies the existence of codimension two double crisis vertices where a curve of boundary crisis and a curve of interior crisis meet we argue that bifurcation points of higher codimension must exist in the full parameter space

estimating long term behavior of flows without trajectory integration the infinitesimal generator approach the long term distributions of trajectories of a flow are described by invariant densities i e fixed points of an associated transfer operator in addition global slowly mixing structures such as almost invariant sets which partition phase space into regions that are almost dynamically disconnected can also be identified by certain eigenfunctions of this operator indeed these structures are often hard to obtain by brute force trajectory based analyses in a wide variety of applications transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system the computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories in this paper we propose to directly work with the infinitesimal generator instead of the operator completely avoiding trajectory integration we propose two different discretization schemes a cell based discretization and a spectral coll

constants of motion for superconducting josephson arrays abstract we show that series arrays of n identical overdamped josephson junctions have extremely degenerate dynamics in particular we prove that such arrays have n 3 constants of motion for all n 3 the analysis is based on a coordinate transformation that reduces the governing equations to an n 3 parameter family of low dimensional systems in the weak coupling limit the reduced equations can be analyzed completely either all solutions approach the synchronous state or they converge to a continuous family of incoherent oscillations depending on a certain parameter value at the transitional value of this parameter the system becomes completely integrable then the phase space is foliated by invariant two dimensional tori for any n 3 the infinite n limit of the system is an integro partial differential equation with rigorously low dimensional dynamics it supports solitons in the integrable case and chaotic waves in the non integrable case

computing unstable manifolds of periodic orbits in delay differential equations we present the first algorithm for computing unstable manifolds of saddle type periodic orbits with one unstable floquet multiplier in systems of autonomous delay differential equations ddes with one fixed delay specifically we grow the one dimensional unstable manifold wu q of an associated saddle fixed point q of a poincare map defined by a suitable poincare section starting close to q along the linear approximation to wu q given by the associated eigenfunction our algorithm grows the manifold as a sequence of points where the distance between points is governed by the curvature of the one dimensional intersection curve wu q of wu q with our algorithm makes it possible to study global bifurcations in ddes we illustrate this with the break up of an invariant torus and a subsequent crisis bifurcation to chaos in a dde model of a semiconductor laser with phase conjugate feedback