an online method for interpolating linear parametric reduced order models a two step online method is proposed for interpolating projection based linear parametric reduced order models roms in order to construct a new rom for a new set of parameter values the first step of this method transforms each precomputed rom into a consistent set of generalized coordinates the second step interpolates the associated linear operators on their appropriate matrix manifold real time performance is achieved by precomputing inner products between the reduced order bases underlying the precomputed roms the proposed method is illustrated by applications in mechanical and aeronautical engineering in particular its robustness is demonstrated by its ability to handle the case where the sampled parameter set values exhibit a mode veering phenomenon

the 4m mixed memory markov model algorithm for finding genes in prokaryotic genomes in this paper we present a new algorithm called 4m mixed memory markov model for finding genes from the genomes of prokaryotes this is achieved by modeling the known coding regions of the genome as a set of sample paths of a multistep markov chain call it and the known non coding regions as a set of sample paths of another multistep markov chain call it the new feature of the 4m algorithm is that different states are allowed to have different memory lengths in contrast to a fixed multistep markov model used in genemark in its various versions at the same time compared with an algorithm like glimmer3 that uses an interpolation of markov models of different memory lengths the statistical significance of the conclusions drawn from the 4m algorithm is quite easy to quantify thus when a whole genome annotation is carried out and several new genes are predicted it is extremely easy to rank these predictions in terms of the confidence one has in the predictions the basis of the 4m algorithm is a simple rank condition satisfied by the matrix of frequencies associated with a markov chain the 4m algorithm is validated by applying it to 75 organisms belonging to practically all known families of bacteria and archae the performance of the 4m algorithm is compared with those of glimmer3 genemark2 5d and genemarkhmm2 6g it is found that in a vast majority of cases the 4m algorithm finds many more genes than it misses compared with any of the other three algorithms next the 4m algorithm is used to carry out whole genome annotation of 13 organisms by using 50 of the known genes as the training input for the coding model and 20 of the known non genes as the training input for the non coding model after this all of the open reading frames are classified it is found that the 4m algorithm is highly specific in that it picks out virtually all of the known genes while predicting that only a small number of the open reading frames whose status is unknown are genes

levels of knotting of spatial handlebodies if h is a spatial handlebody i e a handlebody embedded in the 3 sphere a spine of h is a graph s3 such that h is a regular neighbourhood of usually h is said to be unknotted if it admits a planar spine this suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties building on this remark we define several levels of knotting of spatial handlebodies and we provide a complete description of the relationships between these levels focusing our attention on the case of genus 2 we also relate the knotting level of a spatial handlebody h to classical topological properties of its complement m s3 h such as its cut number more precisely we show that if h is not highly knotted then m admits special cut systems for m and we discuss the extent to which the converse implication holds along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies these obstructions are based on recent quandle coloring invariants for spatial handlebodies on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links on the analysis of appropriate coverings of handlebody complements and on the study of the classical alexander elementary ideals of their fundamental groups

bifurcation phenomena in coupled chemical oscillators normal form analysis and numerical simulations abstract a class of diffusively coupled chemical oscillators is mapped into a problem of two interacting hopf bifurcations the normal form analysis predicts a cascade of steady state limit cycle 2 torus 3 torus bifurcations as well as the coexistence of two stable limit cycles numerical simulations of the original system confirm these predictions and in particular show that this system provides an example of bifurcation leading to a stable quasiperiodic regime with three incommensurate frequencies

a direct approach to conformational dynamics based on hybrid monte carlo recently a novel concept for the computation ofessentialfeatures of the dynamics of hamiltonian systems such as molecular dynamics has been proposed the realization of this concept had been based on subdivision techniques applied to the frobenius perron operator for the dynamical system the present paper suggests an alternative but related concept that merges the conceptual advantages of the dynamical systems approach with the appropriate statistical physics framework this approach allows us to define the phrase conformation in terms of the dynamical behavior of the molecular system and to characterize the dynamical stability of conformations in a first step the frequency of conformational changes is characterized in statistical terms leading to the definition of some markov operatortthat describes the corresponding transition probabilities within the canonical ensemble in a second step a discretization oftvia specific hybrid monte carlo techniques is shown to lead to astochasticmatrixp with these theoretical preparaions an identification algorithm for conformations to be presented in a later paper is applicable it is demonstrated that the discretization oftcan be restricted to few essential degrees of freedom so that the combinatorial explosion of discretization boxes is prevented and biomolecular systems can be attacked numerical results for the n pentane molecule and the triribonucleotide adenylyl 3 5 cytidylyl 3 5 cytidin are given and interpreted

stability and non linear responses of a rotor bearing system with pedestal looseness abstract vibration characteristics of a rotor bearing system with pedestal looseness are investigated a non linear mathematical model containing stiffness and damping forces with tri linear forms is considered the shooting method is used to obtain the periodic solutions of the system stability of these periodic solutions is analyzed by using the floquet theory period doubling bifurcation and naimark sacker bifurcation are found finally the governing equations are integrated using the fourth order runge kutta method different forms of periodic quasi periodic and chaotic vibrations are observed by taking the rotating speed and imbalance as the control parameter three kinds of routes to or out of chaos that is period to chaos quasi periodic route and intermittence are found

time domain model order reduction using general orthogonal polynomials for k power bilinear systems abstractin this paper we propose a model order reduction mor method based on general orthogonal polynomials for k power bilinear systems in the time domain constructing proper projection matrices by solving a series of linear equations a reduced k power bilinear system is produced which preserves the original coupled structure it can match several expansion coefficients of the original output then the error bound of our algorithm is also investigated moreover the stability of the reduced system is discussed as well finally two numerical examples are provided to illustrate the effectiveness of our algorithm

a metric between probability distributions on finite sets of different cardinalities and applications to order reduction in this paper we define a metric distance between probability distributions on two distinct finite sets of possibly different cardinalities the metric is defined in terms of a joint distribution on the product of the two sets which has the two given distributions as its marginals and has minimum entropy computing the metric exactly turns out to be np hard therefore an efficient greedy algorithm is presented for finding an upper bound on the distance we then study the problem of optimal order reduction in the metric defined here it is shown that every optimal reduced order approximation must be an aggregation of the original distribution and that optimal reduced order approximation is equivalent to finding an aggregation with maximum entropy this problem also turns out to be np hard so again a greedy algorithm is constructed for finding a suboptimal reduced order approximation taken together all the results presented here permit the approximation of an independent and identically distributed i i d process over a set of large cardinality by another i i d process over a set of smaller cardinality in future work attempts will be made to extend this work to markov processes over finite sets

frequency limited h controller order reduction for linear parameter varying systems abstractin this paper a controller order reduction method for linear parameter varying systems is presented the proposed method is based on the frequency weighted balanced truncation technique which has the advantage to reduce the order in a specific frequency range the approach is discussed and is proved to preserve the closed loop stability with a guaranteed upper error bound effectiveness and performance of the obtained reduced order controller are investigated by applying it to an automotive semi active suspension control the obtained simulation results show that objectives such as the road handling and the passenger comfort realised with the reduced order controller are kept in the same performance level as with the full order controller moreover a comparison with an other order reduction method is shown and confirms the advantage of the developed method

a locally parametrized reduced order model for the linear frequency domain approach to time accurate computational fluid dynamics for transonic flows governed by the time accurate navier stokes equations small approximately periodic perturbations can be calculated accurately by transition to the frequency domain and truncating the fourier expansion after the first harmonic this is referred to as the linear frequency domain lfd method in this paper a parametric trajectory of reduced order models roms for the lfd solver is presented to this end several local projection based roms which are essentially specified by suitable low order subspaces are computed by the method of proper orthogonal decomposition pod in an offline stage the claimed trajectory is obtained locally by interpolating the given local subspaces considered as sample points in the grassmann manifold it is shown that the manifold interpolation technique is subject to certain restrictions moreover it turns out that the application of computing accurate roms for the lfd solver requires a special choice of underlying inner product necessitating a non eu

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

normalized cuts and image segmentation we propose a novel approach for solving the perceptual grouping problem in vision rather than focusing on local features and their consistencies in the image data our approach aims at extracting the global impression of an image we treat image segmentation as a graph partitioning problem and propose a novel global criterion the normalized cut for segmenting the graph the normalized cut criterion measures both the total dissimilarity between the different groups as well as the total similarity within the groups we show that an efficient computational technique based on a generalized eigenvalue problem can be used to optimize this criterion we applied this approach to segmenting static images as well as motion sequences and found the results to be very encouraging

two periodical and quasi periodical wave solutions of the kuramoto sivashinsky equation and their stability and bifurcations abstract a nonlinear evolution equation is considered which is often encountered in modelling the behaviour of perturbations in various active dissipative media e g in problems of fluid film flow hydrodynamics wave solutions regular in space and both periodical and quasi periodical in time generating from steady state and steady state travelling waves have been found numerically stability of some solutions has been investigated and bifurcation analysis has been carried out the analysis has demonstrated that there is a sequence of bifurcations of solutions stable with respect to disturbances of the same spatial period and it has been shown that the bifurcations are related to the loss of some symmetries of the initial solution it is also shown that the regions of quiet time behaviour and those of intensive variations are separated in the temporal period of solutions with the bifurcation parameter increasing

bifurcations of tori and phase locking in a dissipative system of differential equations abstract in a system of ordinary differential equations obtained through a seven mode truncation of the plane incompressible navier stokes equations a two dimensional torus undergoes first two period doubling bifurcations and then a transition to a strange attractor this strange attractor of liapunov dimension larger than three in a wide parameter interval is characterized by a power spectrum which retains the two fundamental frequencies of the original torus superimposed on a broad jagged background as the liapunov dimension goes down towards two an interesting phenomenon of phase locking occurs which gives rise to an alternation of chaotic and periodic behavior

lumpable hidden markov models model reduction and reduced complexity filtering this paper is concerned with filtering of hidden markov processes hmp which possess or approximately possess the property of lumpability this property is a generalization of the property of lumpability of a markov chain which has been previously addressed by others in essence the property of lumpability means that there is a partition of the atomic states of the markov chain into aggregated sets which act in a similar manner as far as the state dynamics and observation statistics are concerned we prove necessary and sufficient conditions on the hmp for exact lumpability to hold for a particular class of hidden markov models hmm namely finite output alphabet models conditions for lumpability of all hmp representable by a specified hmm are given the corresponding optimal filter algorithms for the aggregated states are then derived the paper also describes an approach to efficient suboptimal filtering for hmp which are approximately lumpable by this we mean that the hmm generating the process may be approximated by a lumpable hmm this approach involves directly finding a lumped hmm which approximates the original hmm well in a matrix norm sense an alternative approach for model reduction based on approximating a given hmm by an exactly lumpable hmm is also derived this method is based on the alternating convex projections algorithm some simulation examples are presented which illustrate the performance of the suboptimal filtering algorithms

realizing homology boundary links with arbitrary patterns hornology boundary links have become an increasingly important class of links largely due to their significance in the ongoing concordance classification of links tim cochran and jerome levine defined an algebraic object called a pattern associated to a homology boundary link which can be used to study the deviance of a homology boundary link from being a boundary link since a pattern is a set of nx elements which normally generates the free group of rank nxs any invariants which detect non trivial patterns can be applied to the purely algebraic question of when such a set is a set of conjugates of a generating set for the free group we will give a constructive geometric proof that all patterns are realized by some homology boundary link lrl in srl 2 we shall also prove an analogous existence theorem for calibrations of e links a more general and less understood class of links tha homology boundary links 1 1 ntroduction a lin1z of m components is a smooth oriented submanifold l x1 km of sn 2 that is an ordered disjoint union of m manifolds each of which is piecewiselinearly homeomorphic to sn if m 1 l is usually called a 1fnot l is a boundary lin1z if there exist m disjoint seifert surfaces i e oriented submanifolds v1 vm of sn 2 such that avi ki for i 1 m knots and links are of interest since they repeatedly arise in the classification of manifolds an especially significant equivalence relation on links is c oncordance two links lo and l1 are concordant if there exists a smooth oriented submanifold of m components c c1 cm of sn 2 x i such that i c is piecewise linearly homeomorphic to lo x i and ii ac n sn 2 x i lli for i c 0 1 the classification of knot concordance groups was obtained in the rnid 1960 s by m kervaire and j levine 12 14 among the things they prove is that the knot concordance group is trivial when n is even and is an infinitely generated group when n is odd the techniques used in the concordance classification of knots have been found to extend in a compatible manner to the class of bou ndary links 2 13 19 however the extension of these ideas to links in general has been a much more difficult and less successful task in the process another class of links called homology boundary links has arisen a hornology boundary link of m components is a link which admits m disjoint generalized seifelzt surfaces y1 ym such that received by the editors may 16 1995 and in revised form october 30 1995 1991 mathematics s lbject classification primary 57q45 57m07 57m15 a 1998 american mathematic 1l so iety 87 this content downloaded from 207 46 13 28 on tue 30 aug 2016 05 27 13 utc all use subject to http about jstor org terms

structure preserving model reduction of nonlinear building thermal models abstract this paper proposes an aggregation based model reduction method for nonlinear models of multi zone building thermal dynamics the full order model which is already a lumped parameter approximation quickly grows in state space dimension as the number of zones increases an advantage of the proposed method apart from being applicable to the nonlinear thermal models is that the reduced model obtained has the same structure and physical intuition as the original model the key to the methodology is an analogy between a continuous time markov chain and the linear part of the thermal dynamics a recently developed aggregation based method of markov chains is employed to aggregate the large state space of the full order model into a smaller one simulations are provided to illustrate tradeoffs between modeling error and computation time

the average cost of markov chains subject to total variation distance uncertainty abstract this paper addresses the problem of controlling a markov chain so as to minimize the long run expected average cost per unit time when the invariant distribution is unknown but we know it belongs to a given uncertain set the mathematical model used to describe this set is the total variation distance uncertainty we show that the equilibrium control policy which yields higher probability to the states with low cost and lower probability to the states with the high cost is an optimal control policy that minimizes the average cost recognition of such a policy may be of value in practical situations with constraints consistent to those studied here when the invariant distribution is uncertain and deriving online an optimal control policy is required

adaptive rational krylov subspaces for large scale dynamical systems the rational krylov space is recognized as a powerful tool within model order reduction techniques for linear dynamical systems however its success has been hindered by the lack of a parameter free procedure which would effectively generate the sequence of shifts used to build the space in this paper we propose an adaptive computation of these shifts the whole procedure only requires us to inject some initial rough estimate of the spectral region of the matrix while further information is automatically generated during the process the approach is a full generalization to the nonsymmetric case of the idea first proposed in druskin et al 2010 18 and it is used for two important problems in control the approximation of the transfer function and the numerical solution of large lyapunov equations the procedure can be naturally extended to other related problems such as the solution of the sylvester equation and parametric or higher order systems several numerical experiments are proposed to assess the quality of the rational projection space over its most natural competitors

aggregation of graph models and markov chains by deterministic annealing we consider the problem of simplifying large weighted directed graphs by aggregating nodes and edges this problem is recast as a clustering resource allocation problem and a solution method that incorporates features of the deterministic annealing da algorithm is proposed the novelty in our method is a quantitive measure of dissimilarity that allows us to compare directed graphs of possibly different sizes i e the original and the aggregated graphs the approach we propose is insensitive to initial conditions and less likely to converge to poor local minima than lloyd type algorithms we apply our graph aggregation clustering method to markov chains where low order markov chains that approximate high order chains are obtained through appropriate aggregation of state transition matrices we further develop a decentralized computational scheme to improve tractability of the algorithm

parameter dependent model order reduction l in this paper we consider the optimal model reduction problem where the plant 2 model depends on parameters that are measurable such cases occur in many on line as well as off line applications and the question that arises is how to update the reduced order model without complete re solution of the problem a method of approximation of the updated reduced order model which is based on its series expansion is given a similar approach is used to develop a new algorithm for the numerical solution of the nominal problem the algorithm compares well in terms of computation effort and convergence properties with homotopic methods which are the common way of solving the equations

identification of almost invariant aggregates in reversible nearly uncoupled markov chains the topic of the present paper has been motivated by a recent computational approach to identify metastable chemical conformations and patterns of conformational changes within molecular systems after proper discretization such conformations show up as almost invariant aggregates in reversible nearly uncoupled markov chains numcs most of the former work on this subject treated the direct problem given the aggregates analyze the loose coupling in connection with the computation of the stationary distribution aggregation disaggregation techniques in contrast to that the present paper focuses on the inverse problem given the system as a whole identify the almost invariant aggregates together with the small probabilities of transitions between them a robust algorithm is worked out on the basis of some detailed perturbation analysis and illustrated at a simple molecular system

interpolatory projection methods for parameterized model reduction we provide a unifying projection based framework for structure preserving interpolatory model reduction of parameterized linear dynamical systems i e systems having a structured dependence on parameters that we wish to retain in the reduced order model the parameter dependence may be linear or nonlinear and is retained in the reduced order model moreover we are able to give conditions under which the gradient and hessian of the system response with respect to the system parameters is matched in the reduced order model we provide a systematic approach built on established interpolatory mathcal h 2 optimal model reduction methods that will produce parameterized reduced order models having high fidelity throughout a parameter range of interest for single input single output systems with parameters in the input output maps we provide reduced order models that are optimal with respect to an mathcal h 2 otimes mathcal l 2 joint error measure the capabilities of these approaches are illustrated by several numerical examples from technical applications

computation continuation and bifurcation of torus solutions for dissipative maps and ordinary differential equations abstract a periodic solution can be considered as a fixed point of a poincare map we shall demonstrate how we can generalize this fixed point equivalence to quasi periodic solutions with two periods thus the quasi periodic solution or bi periodic solution will be a fixed point of a more general poincare map the stability of the bi periodic solution is then equivalent to the stability of the fixed point we demonstrate how we can follow the fixed point in dependence of any control parameter in the system furthermore we demonstrate both how the bi periodic solution may bifurcate to more complicated solutions and how it may arise as a bifurcation from a periodic solution our work is algorithmic in spirit and our statements are based on numerical evidence

nonlinear normal modes for the toda chain abstract the toda chain is a nonlinear mass spring chain which can in principle be integrated analytically by the action angle theory of classical mechanics we show that the transformation from physical variables to action variables can be implemented very simply on a computer we study the correspondence between action variables and the motions of the chain and find the role of the action variables to be very similar to the role of normal mode amplitudes in the harmonic chain this similarity leads to a partly rigorous partly heuristic normal mode analysis for the toda chain new results are obtained when this normal mode analysis is applied to certain perturbations of the toda chain such as the fermi pasta ulam chain or the toda chain with a mass impurity

seifert matrices and boundary link cobordisms to an m component boundary link of odd dimension a matrix is associated by taking the seifert pairing on a seifert surface of the link an algebraic description of the set of boundary link cobordism classes of boundary links is obtained by using this matrix invariant

quasi periodic solutions calculated with the simple shooting technique abstract a shooting type numerical method for calculating quasi periodic solutions in a multiexcited system by using derivatives and with an improved interpolation technique is presented three examples of ordinary differential equations and iterative maps show the efficiency of the new algorithm this algorithm can also be effectively applied to a periodic system

reduced complexity filtering for partially observed nearly completely decomposable markov chains this paper provides a systematic method of obtaining reduced complexity approximations to aggregate filters for a class of partially observed nearly completely decomposable markov chains it is also shown why an aggregate filter adapted from courtois 1977 aggregation scheme has the same order of approximation as achieved by the algorithm proposed in this paper this algorithm can also be used systematically to obtain reduced complexity approximations to the full order fitter as opposed to algorithms adapted from other aggregation schemes however the computational savings in computing the full order filters are substantial only when the large scale markov chain has a large number of weakly interacting blocks or superstates with small individual dimensions some simulations are carried out to compare the performance of our algorithm with algorithms adapted from various other aggregation schemes on the basis of an average approximation error criterion in aggregate slow filtering these studies indicate that the algorithms adapted from other aggregation schemes may become ad hoc under certain circumstances the algorithm proposed in this paper however always yields reduced complexity filters with a guaranteed order of approximation by appropriately exploiting the special structures of the system matrices

oja s algorithm for graph clustering markov spectral decomposition and risk sensitive control given a positive definite matrix m and an integer n m 1 oja s subspace algorithm will provide convergent estimates of the first n m eigenvalues of m along with the corresponding eigenvectors it is a common approach to principal component analysis this paper introduces a normalized stochastic approximation implementation of oja s subspace algorithm as well as new applications to the spectral decomposition of a reversible markov chain recall that this means that the stationary distribution satisfies the detailed balance equations meyn tweedie 2009 equivalently the statistics of the process in steady state do not change when time is reversed stability and convergence of oja s algorithm are established under conditions far milder than that assumed in previous work applications to graph clustering markov spectral decomposition and multiplicative ergodic theory are surveyed along with numerical results

equivalence of laguerre based model order reduction and moment matching the equivalence between the classical moment matching and the laguerre based reduction approaches is shown in addition this equivalence is generalized to include a family of coefficients known as generalized markov parameters

continuation of quasi periodic solutions with two frequency harmonic balance method the continuation of quasi periodic solutions for autonomous or forced nonlinear systems is presented in this paper the association of the asymptotic numerical method a robust continuation method and a two frequency harmonic balance method is performed thanks to a quadratic formalism there is no need for a priori knowledge of the solution the two pulsations can be unknown and can vary along the solution branch and the double fourier series are computed without needing a harmonic selection a norm criterion on fourier coefficients can confirm a posteriori the accuracy of the solution branch on a forced system frequency locking regions are approximated without blocking the continuation process the continuation of these periodic solutions can be done independently on an autonomous system an example of solution is shown where the number of fourier coefficients is increased to improve the accuracy of the solution

quasiperiodically forced dynamical systems with strange nonchaotic attractors abstract we discuss the existence and properties of strange nonchaotic attractors of quasiperiodically forced nonlinear dynamical systems we do this by examining a particular model differential equation cos e cos 2 t where is a two frequency quasiperiodic function of t when 0 the analysis of the equation is facilitated since then it can be related to the schrodinger equation with quasiperiodic potential we show that the equation does indeed exhibit strange nonchaotic attractors and we consider the important question of whether these attractors are typical in the sense that they exist on a set of positive lebesgue measure in parameter space the equation also exhibits two and three frequency quasiperiodic behavior we also show that the strange nonchaotic attractors have distinctive frequency spectrum this property might make them experimentally observable

a parameterised model order reduction method for parametric systems based on laguerre polynomials abstractthis paper presents a laguerre polynomials based parametrised model order reduction method for the parametric system in time domain the method allows that the parametric dependence in system matrices is nonaffine the method is presented via reducing an approximate polynomial parametric system based on taylor expansion and laguerre polynomials resulting in a parametric reduced system that can accurately approximate the time response of the original parametric system over a wide range of parameter the reduced parametric system obtained by proposed method can be implemented by two algorithms algorithm 1 is a direct way that is suitable for single input multi output parametric systems algorithm 2 is presented based on a connection to the krylov subspace which is efficient and suitable for multi input multi output parametric systems the effectiveness of the proposed method is illustrated with two benchmarks in practical applications

optimal kullback leibler aggregation via information bottleneck in this paper we present a method for reducing a regular discrete time markov chain dtmc to another dtmc with a given typically much smaller number of states the cost of reduction is defined as the kullback leibler divergence rate between a projection of the original process through a partition function and a dtmc on the correspondingly partitioned state space finding the reduced model with minimal cost is computationally expensive as it requires an exhaustive search among all state space partitions and an exact evaluation of the reduction cost for each candidate partition our approach deals with the latter problem by minimizing an upper bound on the reduction cost instead of minimizing the exact cost the proposed upper bound is easy to compute and it is tight if the original chain is lumpable with respect to the partition then we express the problem in the form of information bottleneck optimization and propose using the agglomerative information bottleneck algorithm for searching a suboptimal partition greedily rather than exhaustively the theory is illustrated with examples and one application scenario in the context of modeling bio molecular interactions

a real time procedure for affinely dependent parametric model order reduction using interpolation on grassmann manifolds summary model order reduction helps to reduce the computational time in dealing with large dynamical systems for example during simulation control optimization in many cases the considered model depends on parameters model order reduction techniques are therefore preferred to symbolically preserve this dependence or to be adaptive to the change of the model caused by the variation in the values of the parameters in this paper we first present the application of the interpolation technique on grassmann manifolds to this problem we then improve the method for the models whose system matrices depend affinely on parameters by considerably reducing the computational complexity on the basis of analyzing the structure of sums of singular value decompositions and decomposing the whole procedure into offline and online stages a numerical example is shown to illustrate the method as well as to prove its effectiveness copyright 2012 john wiley sons ltd

the kullback leibler divergence rate between markov sources in this work we provide a computable expression for the kullback leibler divergence rate lim sub n spl rarr spl infin 1 nd p sup n spl par q sup n between two time invariant finite alphabet markov sources of arbitrary order and arbitrary initial distributions described by the probability distributions p sup n and q sup n respectively we illustrate it numerically and examine its rate of convergence the main tools used to obtain the kullback leibler divergence rate and its rate of convergence are the theory of nonnegative matrices and perron frobenius theory similarly we provide a formula for the shannon entropy rate lim sub n spl rarr spl infin 1 nh p sup n of markov sources and examine its rate of convergence

computing arnol d tongue scenarios a famous phenomenon in circle maps and synchronisation problems leads to a two parameter bifurcation diagram commonly referred to as the arnol d tongue scenario one considers a perturbation of a rigid rotation of a circle or a system of coupled oscillators in both cases we have two natural parameters the coupling strength and a detuning parameter that controls the rotation number frequency ratio the typical parameter plane of such systems has arnol d tongues with their tips on the decoupling line opening up into the region where coupling is enabled and in between these arnol d tongues quasi periodic arcs in this paper we present unified algorithms for computing both arnol d tongues and quasi periodic arcs for both maps and odes the algorithms generalise and improve on the standard methods for computing these objects we illustrate our methods by numerically investigating the arnol d tongue scenario for representative examples including the well known arnol d circle map family a periodically forced oscillator caricature and a system of coupled van der pol oscillators

the nonlinear dynamics of the damped and driven toda chain 1 energy bifurcation diagrams abstract the dynamics of a periodically driven damped toda chain with periodic boundary conditions is investigated with the help of energy cross sections it can be shown that besides periodic motions of different periodicity quasiperiodic and aperiodic chaotic motions exist the method of energy bifurcation diagrams is introduced which allows a survey of the resonance properties of the chain

the lanczos arnoldi algorithm and controllability we present methods for computing the controllable space of a linear dynamic system the methods are based on orthogonalizing krylov sequences and can use much less storage than other methods with little or no loss of numerical stability

stability and lyapunov stability of dynamical systems a differential approach and a numerical method abstract a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system as well as for the lyapunov exponents and the corresponding eigenvectors is derived the rate of convergence of the lyapunov eigenvectors is shown to be exponential the eigenvectors of the stability matrix can be grouped into sets each spanning a subspace which converges at an exponential rate it is demonstrated that generically the real parts of the eigenvalues of the stability matrix equal the corresponding lyapunov exponents this statement has been tested numerically the values of the lyapunov exponents i are shown to be related to the corresponding finite time values of the lyapunov exponents e g those deduced from a finite time numerical simulation i t by i t i b i i t t the b i s are constants and i t are noise terms of zero mean this observation leads to a method of extrapolation which has been used to predict lyapunov exponents from a finite amount of data it is shown that the use of the standard numerical methods to compute lyapunov exponents introduces an error a i t in the value of i t where the a i s are constants thus the standard method has a rate of convergence which is the same as that of the exact i t s finally we have shown how one can compute the eigenvectors associated with each of the eigenvalues of the stability matrix as well as the lyapunov eigenvectors

time domain model order reduction of general orthogonal polynomials for linear input output systems for a class of large linear input output systems we present a new model order reduction algorithm based on general orthogonal polynomials in the time domain the main idea of the algorithm is first to expand the unknown state variables in the space spanned by orthogonal polynomials then the coefficient terms of polynomial expansion are calculated by a recurrence formula the basic procedure is to use the coefficient terms to generate a projection matrix many classic methods with orthogonal polynomials are special cases of the general approach the proposed approach has a good computational efficiency and preserves the stability and passivity under certain condition numerical experiments are reported to verify the theoretical analysis

computation of quasi periodic solutions of forced dissipative systems ii abstract the problem of finding a t periodic solution of a forced dissipative system of ordinary differential equations is most conveniently reformulated as a fixed point problem of a poincare map mapping the phase space at t 0 into the phase space at t t and the stability of the periodic response is equivalent to the stability of the fixed point it is shown how the problem of determining quasi periodic solutions of forced systems with two forcing frequencies may be reformulated as a fixed point problem of a new type of poincare map that also opens the possibility of treating quasi periodic solutions with more than two frequencies as well as treating bifurcating quasi periodic solutions of autonomous equations three examples are considered one in which the exact solution is known and two others where the quasi periodic solutions have been determined by other means

the nonlinear dynamics of the damped and driven toda chain ii fourier and lyapunov analysis of tori abstract the dynamics of a periodically damped and driven toda chain with periodic boundary conditions is investigated with the help of spectral energy bifurcation diagrams and lyapunov diagrams it can be shown that quasiperiodic motions on tori tk of dimension k 3 exist the technical problems concerning the fourier and lyapunov analysis of the motions on tori tk k 2 are addressed

robust perron cluster analysis in conformation dynamics the key to molecular conformation dynamics is the direct identification of metastable conformations which are almost invariant sets of molecular dynamical systems once some reversible markov operator has been discretized a generalized symmetric stochastic matrix arises this matrix can be treated by perron cluster analysis a rather recent method involving a perron cluster eigenproblem the paper presents an improved perron cluster analysis algorithm which is more robust than earlier suggestions numerical examples are included

ℋ model reduction for linear parameter varying systems with distributed delay this paper is concerned with the model reduction for linear parameter varying lpv systems with both discrete and distributed delays for a given stable system our attention is focused on the construction of reduced order models which approximate the original system well in an norm sense first a sufficient condition is proposed for the asymptotic stability with an performance of the error system by using the parameter dependent lyapunov functional method then the decoupling technique is applied such that there does not exist any product term between the lyapunov matrices and the system matrices in the parametrised linear matrix inequality plmi constraints thus a new sufficient condition is obtained based on the new condition two different approaches are developed to solve the model reduction problem one is the convex linearisation approach and the other is the projection approach finally a numerical example is provided to demonstrate the effectiveness of the proposed design method

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

link concordance invariants and homotopy theory we show that each of a sequence of concordance invariants for codimension two links of spheres ins n 2 defined by kent orr is identically zero forn 1 for classical links n 1 the same proof shows that these invariants vanish if and only if milnor s bar mu vanish a result obtained independently and earlier by orr himself we offer sufficient conditions for the vanishing of orr s invariant not covered by the above we discuss how this relates to positive results

optimal kullback leibler aggregation via spectral theory of markov chains this paper is concerned with model reduction for complex markov chain models the kullback leibler divergence rate is employed as a metric to measure the difference between the markov model and its approximation for a certain relaxation of the bi partition model reduction problem the solution is shown to be characterized by an associated eigenvalue problem the form of the eigenvalue problem is closely related to the markov spectral theory for model reduction this result is the basis of a heuristic proposed for the m ary partition problem resulting in a practical recursive algorithm the results are illustrated with examples

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

metastability in stochastic dynamics of disordered mean field models we study a class of markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding gibbs measures we derive the analog of the wentzell freidlin theory in this case showing that any transition can be decomposed with probability exponentially close to one into a deterministic sequence of admissible transitions for these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor the distributions of the rescaled transition times are shown to converge to the exponential distribution we exemplify our results in the context of the random field curie weiss model