blind spectral unmixing based on sparse nonnegative matrix factorization nonnegative matrix factorization nmf is a widely used method for blind spectral unmixing su which aims at obtaining the endmembers and corresponding fractional abundances knowing only the collected mixing spectral data it is noted that the abundance may be sparse i e the endmembers may be with sparse distributions and sparse nmf tends to lead to a unique result so it is intuitive and meaningful to constrain nmf with sparseness for solving su however due to the abundance sum to one constraint in su the traditional sparseness measured by l0 l1 norm is not an effective constraint any more a novel measure termed as s measure of sparseness using higher order norms of the signal vector is proposed in this paper it features the physical significance by using the s measure constraint smc a gradient based sparse nmf algorithm termed as nmf smc is proposed for solving the su problem where the learning rate is adaptively selected and the endmembers and abundances are simultaneously estimated in the proposed nmf smc there is no pure index assumption and no need to know the exact sparseness degree of the abundance in prior yet it does not require the preprocessing of dimension reduction in which some useful information may be lost experiments based on synthetic mixtures and real world images collected by aviris and hydice sensors are performed to evaluate the validity of the proposed method

minimal dynamical systems from pdes using sobolev eigenfunctions abstract a modified karhunen loeve transform is derived using a weighted sobolev norm this approach optimizes the approximation of the higher derivative terms in numerical simulations of partial differential equation and reduces the sizes of the associated system of ordinary differential equations the kuramoto sivashinsky equation is used as an example to demonstrate how a galerkin procedure based on eigenfunctions determined by the sobolev optimality criterion improves the performance of the standard karhunen loeve procedure we present an example of a simulation which required a 19 mode expansion using the k l approach and show how this is reduced to a 7 mode simulation using the sobolev eigenfunctions

collaborative sparse regression using spatially correlated supports application to hyperspectral unmixing this paper presents a new bayesian collaborative sparse regression method for linear unmixing of hyperspectral images our contribution is twofold first we propose a new bayesian model for structured sparse regression in which the supports of the sparse abundance vectors are a priori spatially correlated across pixels i e materials are spatially organized rather than randomly distributed at a pixel level this prior information is encoded in the model through a truncated multivariate ising markov random field which also takes into consideration the facts that pixels cannot be empty i e there is at least one material present in each pixel and that different materials may exhibit different degrees of spatial regularity second we propose an advanced markov chain monte carlo algorithm to estimate the posterior probabilities that materials are present or absent in each pixel and conditionally to the maximum marginal a posteriori configuration of the support compute the minimum mean squared error estimates of the abundance vectors a remarkable property of this algorithm is that it self adjusts the values of the parameters of the markov random field thus relieving practitioners from setting regularization parameters by cross validation the performance of the proposed methodology is finally demonstrated through a series of experiments with synthetic and real data and comparisons with other algorithms from the literature

supervised nonlinear spectral unmixing using a postnonlinear mixing model for hyperspectral imagery this paper presents a nonlinear mixing model for hyperspectral image unmixing the proposed model assumes that the pixel reflectances are nonlinear functions of pure spectral components contaminated by an additive white gaussian noise these nonlinear functions are approximated using polynomial functions leading to a polynomial postnonlinear mixing model a bayesian algorithm and optimization methods are proposed to estimate the parameters involved in the model the performance of the unmixing strategies is evaluated by simulations conducted on synthetic and real data

unsupervised post nonlinear unmixing of hyperspectral images using a hamiltonian monte carlo algorithm this paper presents a nonlinear mixing model for hyperspectral image unmixing the proposed model assumes that the pixel reflectances are post nonlinear functions of unknown pure spectral components contaminated by an additive white gaussian noise these nonlinear functions are approximated using second order polynomials leading to a polynomial post nonlinear mixing model a bayesian algorithm is proposed to estimate the parameters involved in the model yielding an unsupervised nonlinear unmixing algorithm due to the large number of parameters to be estimated an efficient hamiltonian monte carlo algorithm is investigated the classical leapfrog steps of this algorithm are modified to handle the parameter constraints the performance of the unmixing strategy including convergence and parameter tuning is first evaluated on synthetic data simulations conducted with real data finally show the accuracy of the proposed unmixing strategy for the analysis of hyperspectral images

asymptotic solution of stiff pdes with the csp method the reaction diffusion equation the computational singular perturbation csp method is employed for the solution of stiff pdes and for the acquisition of the most important physical understanding the usefulness of the method is demonstrated by analyzing a transient reaction diffusion problem it is shown that in the regions where the solution exhibits smooth spatial slopes a simple nonstiff system of equations can be used instead of the full governing equations from the simplified system which is numerically provided by csp and whose structure varies with space and time important physical information comes to light the relation of this method to the class of asymptotic expansion methods is explored it is shown that the csp results are identical to the ones obtained by the asymptotic methods the identifications of the nondimensional parameters and the tedious manipulations needed by the asymptotic methods are performed by programmable numerical or analytic computations specified by csp preliminary numerical results are presented validating the theoretical aspects of the proposed algorithm and providing a measure of its usefulness and its accuracy

physical understanding of complex multiscale biochemical models via algorithmic simplification glycolysis in saccharomyces cerevisiae abstract large scale models of cellular reaction networks are usually highly complex and characterized by a wide spectrum of time scales making a direct interpretation and understanding of the relevant mechanisms almost impossible we address this issue by demonstrating the benefits provided by model reduction techniques we employ the computational singular perturbation csp algorithm to analyze the glycolytic pathway of intact yeast cells in the oscillatory regime as a primary object of research for many decades glycolytic oscillations represent a paradigmatic candidate for studying biochemical function and mechanisms using a previously published full scale model of glycolysis we show that due to fast dissipative time scales the solution is asymptotically attracted on a low dimensional manifold without any further input from the investigator csp clarifies several long standing questions in the analysis of glycolytic oscillations such as the origin of the oscillations in the upper part of glycolysis the importance of energy and redox status as well as the fact that neither the oscillations nor cell cell synchronization can be understood in terms of glycolysis as a simple linear chain of sequentially coupled reactions

chaotic dynamics of coherent structures abstract a variety of chaotic flows evolving in relatively high dimensional spaces are considered it is shown through the use of an optimal choice of basis functions which are consequence of the karhunen loeve procedure that an accurate description can be given in a relatively low dimensional space particular examples of this procedure which are presented are the ginzburg landau equation turbulent convection in an unbounded domain and turbulent convection in a bounded domain

invariant manifolds in singular perturbation problems for ordinary differential equations based on fenichel s geometric idea invariant manifold theory is applied to singular perturbation problems this approach clarifies the nature of outer and inner solutions a specific condition is given to ensure the existence of heteroclinic connections between normally hyperbolic invariant manifolds a method to approximate the connections is also presented

dirichlet and periodic type boundary value problems for painleve ii it is established that under certain conditions the dirichlet problem on a bounded interval for the painleve ii equation is uniquely solvable and solutions are constructed in an iterative manner moreover conditions for the existence of periodic solutions are set down

blind source separation in nonlinear mixtures separability and a basic algorithm in this paper a novel approach for performing blind source separation bss in nonlinear mixtures is proposed and their separability is studied it is shown that this problem can be solved under a few assumptions which are satisfied in most practical applications the main idea can be considered as transforming a time invariant nonlinear bss problem to local linear ones varying along the time using the derivatives of both sources and observations taking into account the proposed idea numerous algorithms can be developed performing the separation in this regard an algorithm supported by simulation results is also proposed in this paper it can be seen that the algorithm well separates the mixed sources however as the conventional linear bss methods the nonlinear bss suffers from ambiguities which are discussed in this paper

relaxation oscillations in r3 abstract the existence of periodic relaxation oscillations in singularly perturbed systems with two slow and one fast variable is analyzed geometrically it is shown that near a singular periodic orbit a return map can be defined which has a one dimensional slow manifold with a stable invariant foliation under a natural hyperbolicity assumption on the singular periodic orbit this allows to prove the existence of a periodic relaxation orbit for small values of the perturbation parameter additionally the existence of an invariant torus is proved for the periodically forced van der pol oscillator the analysis is based on methods from geometric singular perturbation theory the blow up method is used to analyze the dynamics near the fold curves

archetypal analysis of spatio temporal dynamics abstract a comparison is made between the principal component or karhunen loeve decomposition of two sets of spatio temporal data one numerical the other experimental and a new procedure called archetypal analysis cutler and breiman 1994 archetypes characterize the convex hull of the data set and the data set can be reconstructed in terms of these values archetypes may be more appropriate than kl when the data do not have elliptical distributions and are often well suited to studying regimes in which the system spends a long time near a steady state punctuated with quick excursions to outliers in the data set which may represent intermittency other advantages and disadvantages of each method are discussed

estimating the granularity coefficient of a potts markov random field within a markov chain monte carlo algorithm this paper addresses the problem of estimating the potts parameter jointly with the unknown parameters of a bayesian model within a markov chain monte carlo mcmc algorithm standard mcmc methods cannot be applied to this problem because performing inference on requires computing the intractable normalizing constant of the potts model in the proposed mcmc method the estimation of is conducted using a likelihood free metropolis hastings algorithm experimental results obtained for synthetic data show that estimating jointly with the other unknown parameters leads to estimation results that are as good as those obtained with the actual value of on the other hand choosing an incorrect value of can degrade estimation performance significantly to illustrate the interest of this method the proposed algorithm is successfully applied to real bidimensional sar and tridimensional ultrasound images

boundary value problems on the half line for a generalised painleve ii equation abstract two types of boundary value problem on the infinite half line i 0 are investigated for a class of generalised painleve ii equations existence results are obtained via the method of upper and lower solutions together with a diagonal argument

natural tangent dynamics with recurrent biorthonormalizations a geometric computational approach to dynamical systems exhibiting slow manifolds and periodic chaotic limit sets abstract this article develops a geometric method referred to as natural tangent dynamics with recurrent biorthonormalizations ntdrb that unifies into a single computational technique the analysis of dynamical systems possessing either a slow manifold associated with an equilibrium point or exhibiting nontrivial limit sets limit cycles and chaotic attractors for the first class of systems the main focus relies on the identification of the slow fast timescales while for the latter dynamics within nontrivial attractors the central issue is the temporal dichotomy between unstable stable components as described by the structure of the lyapunov spectrum the method is based on vector and covector dynamics in the tangent and cotangent bundles induced by the phase space flow and on the periodic application of a biorthonormalization procedure which contrasts the natural tendencies of vectors and covectors to align towards the most unstable slow subspaces the ntdrb method provides the most convenient and natural vector and covector bases for i the characterization of the fast timescales controlling the dynamic relaxation towards a slow manifold ii the identification of the stable and unstable components of the dynamics on a limit set iii the construction of a low dimensional reduced model reproducing the asymptotic behavior of the original systems

canard cycles in the presence of slow dynamics with singularities we study the cyclicity of limit periodic sets that occur in families of vector flelds of slow fast type the limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point at this turning point a stability change takes place on one side of the turning point the dynamics point strongly towards the curve of singularities on the other side the dynamics point away from the curve of singularities the presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point i e orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve all existing results deal with a non zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities in this paper we study what happens when the slow dynamics exhibit singularities in particular our study includes the cyclicity of the slow fast 2 saddle cycle formed by a regular saddle connection the fast part and a part of the curve of singularities the slow part we see that the relevant information is no longer merely contained in the slow divergence integral this paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow fast vector flelds on a 2 manifold m we are interested in families of vector flelds x possibly depending on other parameters as well where the unperturbed fast vector fleld x0 has a curve of singular points called a critical curve we call a point p on normally attracting resp normally repelling when dx0 p has a strictly negative resp strictly positive eigenvalue corresponding to an eigendirection not tangent to when has both normally attracting points and normally repelling points it may occur that x0 has orbits connecting two such points let f be such a fast orbit so that the limit and fi limit lie on we study the limit periodic set lf formed by f and the piece of going from the limit of f to the fi limit of f of course the nonzero eigenvalue of dx0 p must bifurcate along this piece of assume that this happens in a unique point p called a turning point one also says normal hyperbolicity of x0 is lost in p in such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits such orbits are called canard cycles

asymptotic analysis of two reduction methods for systems of chemical reactions abstract this paper concerns two methods for reducing large systems of chemical kinetics equations namely the method of intrinsic low dimensional manifolds ildms due to maas and pope combust flame 88 1992 239 and an iterative method due to fraser j chem phys 88 1988 4732 and further developed by roussel and fraser j chem phys 93 1990 1072 both methods exploit the separation of fast and slow reaction time scales to find low dimensional manifolds in the space of species concentrations where the long term dynamics are played out the asymptotic expansions of these manifolds e 0 where e measures the ratio of the reaction time scales are compared with the asymptotic expansion of m e the slow manifold given by geometric singular perturbation theory it is shown that the expansions of the ildm and m e agree up to and including terms of o e the former has an error at o e 2 that is proportional to the local curvature of m 0 the error vanishes if and only if the curvature is zero everywhere the iterative method generates term by term the asymptotic expansion of m e starting from m 0 the ith application of the algorithm yields the correct expansion coefficient at o e i while leaving the lower order coefficients invariant thus after l applications the expansion is accurate up to and including the terms of o e l the analytical results are illustrated on a planar system from enzyme kinetics michaelis menten henri and a model planar system due to davis and skodje

nonparametric detection of nonlinearly mixed pixels and endmember estimation in hyperspectral images mixing phenomena in hyperspectral images depend on a variety of factors such as the resolution of observation devices the properties of materials and how these materials interact with incident light in the scene different parametric and nonparametric models have been considered to address hyperspectral unmixing problems the simplest one is the linear mixing model nevertheless it has been recognized that the mixing phenomena can also be nonlinear the corresponding nonlinear analysis techniques are necessarily more challenging and complex than those employed for linear unmixing within this context it makes sense to detect the nonlinearly mixed pixels in an image prior to its analysis and then employ the simplest possible unmixing technique to analyze each pixel in this paper we propose a technique for detecting nonlinearly mixed pixels the detection approach is based on the comparison of the reconstruction errors using both a gaussian process regression model and a linear regression model the two errors are combined into a detection statistics for which a probability density function can be reasonably approximated we also propose an iterative endmember extraction algorithm to be employed in combination with the detection algorithm the proposed detect then unmix strategy which consists of extracting endmembers detecting nonlinearly mixed pixels and unmixing is tested with synthetic and real images

bayesian curve fitting using mcmc with applications to signal segmentation we propose some bayesian methods to address the problem of fitting a signal modeled by a sequence of piecewise constant linear in the parameters regression models for example autoregressive or volterra models a joint prior distribution is set up over the number of the changepoints knots their positions and over the orders of the linear regression models within each segment if these are unknown hierarchical priors are developed and as the resulting posterior probability distributions and bayesian estimators do not admit closed form analytical expressions reversible jump markov chain monte carlo mcmc methods are derived to estimate these quantities results are obtained for standard denoising and segmentation of speech data problems that have already been examined in the literature these results demonstrate the performance of our methods

nonlinear spectral unmixing of hyperspectral images using gaussian processes this paper presents an unsupervised algorithm for nonlinear unmixing of hyperspectral images the proposed model assumes that the pixel reflectances result from a nonlinear function of the abundance vectors associated with the pure spectral components we assume that the spectral signatures of the pure components and the nonlinear function are unknown the first step of the proposed method estimates the abundance vectors for all the image pixels using a bayesian approach an a gaussian process latent variable model for the nonlinear function relating the abundance vectors to the observations the endmembers are subsequently estimated using gaussian process regression the performance of the unmixing strategy is first evaluated on synthetic data the proposed method provides accurate abundance and endmember estimations when compared to other linear and nonlinear unmixing strategies an interesting property is its robustness to the absence of pure pixels in the image the analysis of a real hyperspectral image shows results that are in good agreement with state of the art unmixing strategies and with a recent classification method

algorithmic asymptotic analysis of the nf κb signaling system the results of a detailed step by step asymptotic analysis of the nf b signaling system are reported in the case where the system exhibits limit cycle behavior the analysis is based on the dimensional form of the model and exploits the various fast slow time scale gaps that develop as the solution evolves along the limit cycle it is shown that under the action of fast time scales of dissipative character the limit cycle is confined on a low dimensional surface in the phase space the cycle can be divided in three parts each one related to a different characteristic time scale the first part refers to the slow rate of nf b release in the cytoplasm the second part refers to the even slower rate by which the free nf b enters into the nucleus the last part refers to the fast rate by which the nuclear nf b is first produced and then depleted it is demonstrated that these and many other findings regarding the fast variables the equilibrated fast processes the rate limiting and or driving steps etc can be acquired by simple algorithmic tools

semi supervised linear spectral unmixing using a hierarchical bayesian model for hyperspectral imagery this paper proposes a hierarchical bayesian model that can be used for semi supervised hyperspectral image unmixing the model assumes that the pixel reflectances result from linear combinations of pure component spectra contaminated by an additive gaussian noise the abundance parameters appearing in this model satisfy positivity and additivity constraints these constraints are naturally expressed in a bayesian context by using appropriate abundance prior distributions the posterior distributions of the unknown model parameters are then derived a gibbs sampler allows one to draw samples distributed according to the posteriors of interest and to estimate the unknown abundances an extension of the algorithm is finally studied for mixtures with unknown numbers of spectral components belonging to a know library the performance of the different unmixing strategies is evaluated via simulations conducted on synthetic and real data

invariant grids for reaction kinetics abstract in this paper we construct low dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold mim mim is based on a formulation of the condition of invariance as an equation and its solution by newton iterations a grid based version of mim is developed the method of invariant grids we describe the newton method and the relaxation method for the invariant grids construction the problem of the grid correction is fully decomposed into the problems of the grid s nodes correction the edges between the nodes appear only in the calculation of the tangent spaces this fact determines high computational efficiency of the method of invariant grids the method is illustrated by two examples the simplest catalytic reaction michaelis menten mechanism and the hydrogen oxidation the algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed generalizations to open systems are suggested the set of methods covered makes it possible to effectively reduce description in chemical kinetics

differential equations of electrodiffusion the equations governing one dimensional steady state electrodiffusion are considered when there are arbitrarily many mobile ionic species present in any number of valence classes possibly also with a uniform distribution of fixed charges exact constant field solutions and new formulas of goldman hodgkin katz type are found all of these formulas are exact unlike the usual approximate ones corresponding boundary conditions on the ionic concentrations are identified the question of uniqueness of constant field solutions with such boundary conditions is considered and is reposed in terms of an autonomous ordinary differential equation of order n 1 for the electric field where n is the number of valence classes when there are no fixed charges the equation can be integrated once to give the nonautonomous equation of order n considered previously in the literature including in the case n 2 the form of painleve s second equation considered first in the context of electrodiffusion by one of us when n 1 the new equation is a form of lie nard s equation uniqueness of the constant field solution is established in this case

simplified csp analysis of a stiff stochastic ode system abstract we develop a simplified computational singular perturbation csp analysis of a stochastic dynamical system we focus on the case of parametric uncertainty and rely on polynomial chaos pc representations to quantify its impact we restrict our attention to a system that exhibits distinct timescales and that tends to a deterministic steady state irrespective of the random inputs a detailed analysis of eigenvalues and eigenvectors of the stochastic system jacobian is conducted which provides a relationship between the pc representation of the stochastic jacobian and the jacobian of the galerkin form of the stochastic system the analysis is then used to guide the application of a simplified csp formalism that is based on relating the slow and fast manifolds of the uncertain system to those of a nominal deterministic system two approaches are specifically developed with the resulting simplified csp framework the first uses the stochastic eigenvectors of the uncertain system as csp vectors whereas the second uses the eigenvectors of the nominal system as csp vectors numerical experiments are conducted to demonstrate the results of the stochastic eigenvalue and eigenvector analysis and illustrate the effectiveness of the simplified csp algorithms in addressing the stiffness of the system dynamics

a comparative study of computation of lyapunov spectra with different algorithms abstract in this paper we make a detailed numerical comparison between three algorithms for the computation of the full lyapunov spectrum as well as the associated eigenvectors of general dynamical systems they are a the standard method b a differential formulation of the standard method and c a new algorithm which does not require rescaling and reorthogonalization we also bring out the relations among these methods moreover we give a simplified formulation of the new algorithm when the dimensionality of the system is 4 we find that there is reasonable agreement among the lyapunov spectra obtained using the three algorithms in most cases however the standard method seems to be the most efficient followed by the new method and the differential version of the standard method in that order as far as the cpu time for the computation of the lyapunov spectra is concerned the new method is hardly suitable for finding the eigenvectors whereas the other procedures give nearly identical numerical results

a mathematical framework for critical transitions bifurcations fast slow systems and stochastic dynamics abstract bifurcations can cause dynamical systems with slowly varying parameters to transition to far away attractors the terms critical transition or tipping point have been used to describe this situation critical transitions have been observed in an astonishingly diverse set of applications from ecosystems and climate change to medicine and finance the main goal of this paper is to give an overview which standard mathematical theories can be applied to critical transitions we shall focus on early warning signs that have been suggested to predict critical transitions and point out what mathematical theory can provide in this context starting from classical bifurcation theory and incorporating multiple time scale dynamics one can give a detailed analysis of local bifurcations that induce critical transitions we suggest that the mathematical theory of fast slow systems provides a natural definition of critical transitions since noise often plays a crucial role near critical transitions the next step is to consider stochastic fast slow systems the interplay between sample path techniques partial differential equations and random dynamical systems is highlighted each viewpoint provides potential early warning signs for critical transitions since increasing variance has been suggested as an early warning sign we examine it in the context of normal forms analytically numerically and geometrically we also consider autocorrelation numerically hence we demonstrate the applicability of early warning signs for generic models we end with suggestions for future directions of the theory

the quasi steady state assumption a case study in perturbation the quasi steady state assumption qssa of biochemistry is studied as an approximation that is important in itself and also that exemplifies an approach to ode systems with an initial fast transient simple estimates of the two relevant time scales of the underlying problem are made these estimates lead to a dimensionless formulation containing a different small parameter from that which is customary earlier results on the qssa are surveyed in the context of the new nondimensionalization it is shown that more comprehensive and simpler error estimates can now be made some general methodological hints are drawn from this example

on boundary value problems in three ion electrodiffusion the existence of solutions to a class of two point boundary value problems in three ion electrodiffusion is investigated via an integro differential formulation boundedness by upper and lower solutions corresponding to associated boundary value problems is considered and illustrated by painleve ii solutions of a constrained version of the original boundary value problems

approximation of slow and fast dynamics in multiscale dynamical systems by the linearized relaxation redistribution method in this paper we introduce a fictitious dynamics for describing the only fast relaxation of a stiff ordinary differential equation ode system towards a stable low dimensional invariant manifold in the phase space slow invariant manifold sim as a result the demanding problem of constructing sim of any dimensions is recast into the remarkably simpler task of solving a properly devised ode system by stiff numerical schemes available in the literature in the same spirit a set of equations is elaborated for local construction of the fast subspace and possible initialization procedures for the above equations are discussed the implementation to a detailed mechanism for combustion of hydrogen and air has been carried out while a model with the exact chapman enskog solution of the invariance equation is utilized as a benchmark

operator splitting for chemical reaction systems with fast chemistry abstract splitting methods are frequently used for large scale chemical reaction systems the main reason is the reduced computational cost for solving the subproblems in comparison to the time integration of the full problem however a splitting error is introduced the most popular splitting schemes are the first order lie trotter splitting and the second order strang splitting however in case of stiff differential equations the strang splitting suffers from order reduction and both schemes have order one for stiff differential equations therefore the step size restrictions due to the low order can result in a prohibitive small step size hence splitting schemes with order larger than one are necessary for stiff differential equations the richardson extrapolation of the lie trotter splitting is a second order scheme in this paper we examine differential equations with a fast chemical source term and a slow transport term thus all stiffness of the system is related to the chemical source term we show that the richardson extrapolation does not suffer from order reduction in this case furthermore we perform a stability analysis of the extrapolated splitting scheme for a linear test problem thereby the operators of the test problem do not commute

nonlinearity detection in hyperspectral images using a polynomial post nonlinear mixing model this paper studies a nonlinear mixing model for hyperspectral image unmixing and nonlinearity detection the proposed model assumes that the pixel reflectances are nonlinear functions of pure spectral components contaminated by an additive white gaussian noise these nonlinear functions are approximated by polynomials leading to a polynomial post nonlinear mixing model we have shown in a previous paper that the parameters involved in the resulting model can be estimated using least squares methods a generalized likelihood ratio test based on the estimator of the nonlinearity parameter is proposed to decide whether a pixel of the image results from the commonly used linear mixing model or from a more general nonlinear mixing model to compute the test statistic associated with the nonlinearity detection we propose to approximate the variance of the estimated nonlinearity parameter by its constrained cramer rao bound the performance of the detection strategy is evaluated via simulations conducted on synthetic and real data more precisely synthetic data have been generated according to the standard linear mixing model and three nonlinear models from the literature the real data investigated in this study are extracted from the cuprite image which shows that some minerals seem to be nonlinearly mixed in this image finally it is interesting to note that the estimated abundance maps obtained with the post nonlinear mixing model are in good agreement with results obtained in previous studies

local analysis near a folded saddle node singularity abstract folded saddle nodes occur generically in one parameter families of singularly perturbed systems with two slow variables we show that these folded singularities are the organizing centers for two main delay phenomena in singular perturbation problems canards and delayed hopf bifurcations we combine techniques from geometric singular perturbation theory the blow up technique and from delayed hopf bifurcation theory complex time path analysis to analyze the flow near such folded saddle nodes in particular we show the existence of canards as intersections of stable and unstable slow manifolds to derive these canard results we extend the singularly perturbed vector field into the complex domain and study it along elliptic paths this enables us to extend the invariant slow manifolds beyond points where normal hyperbolicity is lost furthermore we define a way in way out function describing the maximal delay expected for generic solutions passing through a folded saddle node singularity branch points associated with the change from a complex to a real eigenvalue structure in the variational equation along the critical slow manifold make our analysis significantly different from the classical delayed hopf bifurcation analysis where these eigenvalues are complex only

residual component analysis of hyperspectral images application to joint nonlinear unmixing and nonlinearity detection this paper presents a nonlinear mixing model for joint hyperspectral image unmixing and nonlinearity detection the proposed model assumes that the pixel reflectances are linear combinations of known pure spectral components corrupted by an additional nonlinear term affecting the end members and contaminated by an additive gaussian noise a markov random field is considered for nonlinearity detection based on the spatial structure of the nonlinear terms the observed image is segmented into regions where nonlinear terms if present share similar statistical properties a bayesian algorithm is proposed to estimate the parameters involved in the model yielding a joint nonlinear unmixing and nonlinearity detection algorithm the performance of the proposed strategy is first evaluated on synthetic data simulations conducted with real data show the accuracy of the proposed unmixing and nonlinearity detection strategy for the analysis of hyperspectral images

the g scheme a framework for multi scale adaptive model reduction the numerical solution of mathematical models for reaction systems in general and reacting flows in particular is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics however the dynamics can develop very slow and very fast time scales separated by a range of active time scales an opportunity to reduce the complexity of the problem arises when the fast active and slow active time scales gaps becomes large we propose a numerical technique consisting of an algorithmic framework named the g scheme to achieve multi scale adaptive model reduction along with the integration of the differential equations des the method is directly applicable to initial value odes and by using the method of lines pdes we assume that the dynamics is decomposed into active slow fast and when applicable invariant subspaces the g scheme introduces locally a curvilinear frame of reference defined by a set of orthonormal basis vectors with corresponding coordinates attached to this decomposition the evolution of the curvilinear coordinates associated with the active subspace is described by non stiff des whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem adjusting the active des dynamically during the time integration is the most significant feature of the g scheme since the numerical integration is accomplished by solving a number of des typically much smaller than the dimension of the original problem with corresponding saving in computational work to demonstrate the effectiveness of the g scheme we present results from illustrative as well as from relevant problems

computational singular perturbation analysis of stochastic chemical systems with stiffness abstract computational singular perturbation csp is a useful method for analysis reduction and time integration of stiff ordinary differential equation systems it has found dominant utility in particular in chemical reaction systems with a large range of time scales at continuum and deterministic level on the other hand csp is not directly applicable to chemical reaction systems at micro or meso scale where stochasticity plays an non negligible role and thus has to be taken into account in this work we develop a novel stochastic computational singular perturbation scsp analysis and time integration framework and associated algorithm that can be used to not only construct accurately and efficiently the numerical solutions to stiff stochastic chemical reaction systems but also analyze the dynamics of the reduced stochastic reaction systems the algorithm is illustrated by an application to a benchmark stochastic differential equation model and numerical experiments are carried out to demonstrate the effectiveness of the construction

nearest matrix with prescribed eigenvalues and its applications consider an n n matrix a and a set consisting of k n prescribed complex numbers lippert 2010 in a challenging article studied geometrically the spectral norm distance from a to the set of matrices whose spectra included specified set and constructed a perturbation matrix with minimum spectral norm such that a had in its spectrum this paper presents an easy practical computational method for constructing the optimal perturbation by improving and extending the methodology necessary definitions and lemmas of previous related works also some conceivable applications of this issue are provided

bayesian estimation of linear mixtures using the normal compositional model application to hyperspectral imagery this paper studies a new bayesian unmixing algorithm for hyperspectral images each pixel of the image is modeled as a linear combination of so called endmembers these endmembers are supposed to be random in order to model uncertainties regarding their knowledge more precisely we model endmembers as gaussian vectors whose means have been determined using an endmember extraction algorithm such as the famous n finder n findr or vertex component analysis vca algorithms this paper proposes to estimate the mixture coefficients referred to as abundances using a bayesian algorithm suitable priors are assigned to the abundances in order to satisfy positivity and additivity constraints whereas conjugate priors are chosen for the remaining parameters a hybrid gibbs sampler is then constructed to generate abundance and variance samples distributed according to the joint posterior of the abundances and noise variances the performance of the proposed methodology is evaluated by comparison with other unmixing algorithms on synthetic and real images

on a painleve ii model in steady electrolysis application of a backlund transformation a model equation of painleve ii type was introduced by bass in 1964 in connection with a boundary value problem which describes the electric field distribution in a region x 0 occupied by an electrolyte this is possibly the earliest explicit physical application of a painleve equation to be found in the literature here we return to this problem informed by the subsequent discovery of a backlund transformation for painleve ii this enables us to construct exact representations for the electric field and ion distributions for boundary value problems wherein the ratio of fluxes of the positive and negative ions at the boundary adopts one of an infinite sequence of values

nonlinear unmixing of hyperspectral data based on a linear mixture nonlinear fluctuation model spectral unmixing is an important issue to analyze remotely sensed hyperspectral data although the linear mixture model has obvious practical advantages there are many situations in which it may not be appropriate and could be advantageously replaced by a nonlinear one in this paper we formulate a new kernel based paradigm that relies on the assumption that the mixing mechanism can be described by a linear mixture of endmember spectra with additive nonlinear fluctuations defined in a reproducing kernel hilbert space this family of models has clear interpretation and allows to take complex interactions of endmembers into account extensive experiment results with both synthetic and real images illustrate the generality and effectiveness of this scheme compared with state of the art methods

telescopic projective methods for parabolic differential equations projective methods were introduced in an earlier paper c w gear i g kevrekidis projective methods for stiff differential equations problems with gaps in their eigenvalue spectrum nec research institute report 2001 029 available from http www neci nj nec com homepages cwg projective pdf abbreviated version to appear in sisc as having potential for the efficient integration of problems with a large gap between two clusters in their eigenvalue spectrum one cluster containing eigenvalues corresponding to components that have already been damped in the numerical solution and one corresponding to components that are still active in this paper we introduce iterated projective methods that allow for explicit integration of stiff problems that have a large spread of eigenvalues with no gaps in their spectrum as arise in the semi discretization of pdes with parabolic components

stretching based diagnostics and reduction of chemical kinetic models with diffusion a new method for diagnostics and reduction of dynamical systems and chemical kinetic models is proposed the method makes use of the local structure of the normal stretching rates by projecting the dynamics onto the local directions of maximal stretching the approach is computationally very simple as it implies the spectral analysis of a symmetric matrix notwithstanding its simplicity stretching based analysis derives from a geometric basis grounded on the pointwise applications of concepts of normal hyperbolicity theory as a byproduct a simple reduction method is derived equivalent to a local embedding algorithm which is based on the local projection of the dynamics onto the most unstable and or slow modes compared to the time scale dictated by the local tangential dynamics this method provides excellent results in the analysis and reduction of dynamical systems displaying relaxation towards an equilibrium point limit cycles and chaotic attractors several numerical examples deriving from typical models of reaction diffusion kinetics exhibiting complex dynamics are thoroughly addressed the application to typical combustion models is also analyzed

unstable dimension variability a source of nonhyperbolicity in chaotic systems the hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set a familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable manifolds at points on the chaotic set here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible respectively noninvertible maps of dimension 3 respectively 2 and higher in particular we investigate a situation first considered by abraham and smale in 1970 for different purposes in which the dimension of the unstable and stable tangent spaces are not constant over the chaotic set we call this unstable dimension variability a simple two dimensional map that displays behavior typical of this phenomenon is presented and analyzed

robust stability and instability of biochemical networks with parametric uncertainty parameter perturbations in dynamical models of biochemical networks affect the qualitative dynamical behaviour observed in the model since this qualitative behaviour is in many cases the key model output used to explain biological function the robustness analysis of the model s behaviour with respect to parametric uncertainty is a crucial step in systems biology research in this paper we develop a new method for robustness analysis of the dynamical behaviour as a first step we provide a characterization of non robust perturbations as a system of polynomial equalities and inequalities in the second step we apply the positivstellensatz and handelman representation of polynomials to check for the non existence of solutions to this system which can be relaxed to solving a linear program thereby a solution to the linear program yields a robustness certificate for the considered dynamical behaviour with these robustness certificates we propose an algorithm to compute a lower robustness bound corresponding to a level of parametric uncertainty up to which no local bifurcations can occur the applicability of the proposed method to biochemical network models is illustrated by analysing the robustness of oscillations in a model of the nf kb signalling pathway the results may be used to define a level of confidence in the observed model behaviour under parametric uncertainty making them valuable for evaluating dynamical models of biological networks

modal reduction of pde models by means of snapshot archetypes abstract a new method for constructing low dimensional reduced models of dissipative partial differential equations is proposed the original pde u t f u is projected onto a linear subspace spanned by the so called snapshot archetypes that are selected spatial profiles of u x t the selection rule of the snapshot archetypes characterizes the method two different selection methods are proposed we provide an energetic criterion for the minimum number of archetypes needed for an accurate approximation of the asymptotic dynamics this approach is tested for several pde systems such as the kuramoto sivashinsky equation the arneodo elezgaray reaction diffusion model and the self ignition dynamics of a coal stockpile the latter two systems exhibit a rich bifurcative structure and are suitable for checking the robustness of the snapshot archetype reduced models with respect to parameter variations

the hidden dynamics of the rossler attractor abstract the fast slow dynamics of the rossler model in the chaotic regime are compared to the dynamics of the reduced slow model governing the flow along the slow invariant manifold on which the trajectory is restrained to evolve during the slow part of the attractor it is shown that the dynamics of the reduced model incorporate the slow time scales of the full model however instead of the fast dissipative time scale of the full model that restrains the trajectory on the slow invariant manifold the reduced model generates a new time scale that i relates to the curvature of the manifold and ii becomes faster as higher order corrections are incorporated in the approximation of the manifold and the reduced model it is shown that this new time scale does not characterize the motion of the solution on the manifold but it can be employed as a signal for the strengthening or weakening of the manifold depending on whether it relates to components of the reduced model that tend to lead its solution towards equilibrium or away from it finally it is demonstrated that the new time scale introduced by the reduced model has a significant role in determining the maximum accuracy that can be delivered by the singular perturbation methodology

on a neumann boundary value problem for the painleve ii equation in two ion electro diffusion abstract a two point neumann boundary value problem for a two ion electro diffusion model reducible to the painleve ii equation is investigated the problem is unconventional in that the model equation involves yet to be determined boundary values of the solution in prior work by thompson the existence of a solution was established subject to an inequality on the physical parameters here a two dimensional shooting method is used to show that this restriction may be removed a practical algorithm for the solution of the boundary value problem is presented in an appendix

nonlinear hyperspectral unmixing with robust nonnegative matrix factorization we introduce a robust mixing model to describe hyperspectral data resulting from the mixture of several pure spectral signatures the new model extends the commonly used linear mixing model by introducing an additional term accounting for possible nonlinear effects that are treated as sparsely distributed additive outliers with the standard nonnegativity and sum to one constraints inherent to spectral unmixing our model leads to a new form of robust nonnegative matrix factorization with a group sparse outlier term the factorization is posed as an optimization problem which is addressed with a block coordinate descent algorithm involving majorization minimization updates simulation results obtained on synthetic and real data show that the proposed strategy competes with the state of the art linear and nonlinear unmixing methods

on the leading term of the outer asymptotic expansion of van der pol s equation it is proved rigorously that the leading term in the outer asymptotic expansion of the time dependent solution describing relaxation oscillations of the van der pol equation is correct this is accomplished by constructing rigorous estimates of the difference between the exact solution and the outer asymptotic solution as constructed by j d cole these estimates are both rigorous and numerically computable

flow curvature manifolds for shaping chaotic attractors i rossler like systems poincare recognized that phase portraits are mainly structured around fixed points nevertheless the knowledge of fixed points and their properties is not sufficient to determine the whole structure of chaotic attractors in order to understand how chaotic attractors are shaped by singular sets of the differential equations governing the dynamics flow curvature manifolds are computed we show that the time dependent components of such manifolds structure rossler like chaotic attractors and may explain some limitation in the development of chaotic regimes

hyperspectral unmixing in presence of endmember variability nonlinearity or mismodeling effects this paper presents three hyperspectral mixture models jointly with bayesian algorithms for supervised hyperspectral unmixing based on the residual component analysis model the proposed general formulation assumes the linear model to be corrupted by an additive term whose expression can be adapted to account for nonlinearities nls endmember variability ev or mismodeling effects mes the nl effect is introduced by considering a polynomial expression that is related to bilinear models the proposed new formulation of ev accounts for shape and scale endmember changes while enforcing a smooth spectral spatial variation the me formulation considers the effect of outliers and copes with some types of ev and nl the known constraints on the parameter of each observation model are modeled via suitable priors the posterior distribution associated with each bayesian model is optimized using a coordinate descent algorithm which allows the computation of the maximum a posteriori estimator of the unknown model parameters the proposed mixture and bayesian models and their estimation algorithms are validated on both synthetic and real images showing competitive results regarding the quality of the inferences and the computational complexity when compared with the state of the art algorithms

properties of the michaelis menten mechanism in phase space abstract we study the two dimensional reduction of the michaelis menten reaction of enzyme kinetics first we prove the existence and uniqueness of a slow manifold between the horizontal and vertical isoclines second we determine the concavity of all solutions in the first quadrant third we establish the asymptotic behaviour of all solutions near the origin which generally is not given by a taylor series finally we determine the asymptotic behaviour of the slow manifold at infinity to this end we show that the slow manifold can be constructed as a centre manifold for a fixed point at infinity

electrical structures of interfaces a painleve ii model a painleve ii model derived out of the classical nernst planck system is applied in the context of boundary value problems that describe the electric field distribution in a region x 0 occupied by an electrolyte for privileged flux ratios of the ion concentrations the auto backlund transformation admitted by the painleve ii equation may be applied iteratively to construct exact solutions to classes of physically relevant boundary value problems these representations involve in turn either yablonskii vorob ev polynomials or classical airy functions the requirement that the electric field distribution and ion concentrations in these representations be non singular imposes constraints on the physical parameters these are investigated in detail along with asymptotic properties

the parameterization method for invariant manifolds iii overview and applications abstract we describe a method to establish existence and regularity of invariant manifolds and at the same time to find simple maps which are conjugated to the dynamics on them the method establishes several invariant manifold theorems for instance it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in banach spaces we also present several other applications of the method

multiple time scales in nonlinear flight mechanics diagnosis and modeling there are often disparate time scales in the dynamics of flight creating the potential for reduced order modeling to simplify simulation analysis and design there have been notable successes in developing reduced order models however in the case of nonlinear dynamics which one must typically deal with in guidance problems there has not been a systematic reliable means of diagnosing disparate time scales and developing reduced order models focusing on two time scale behavior in nonlinear dynamical systems we recall fenichel s characterization of the geometric structure of the flow and his theorem establishing the existence and properties of coordinates adapted to this structure adapted coordinates are difficult to construct directly without an appropriate singularly perturbed model of the dynamics we discuss the use of lyapunov exponents and vectors to diagnose two time scale behavior and to determine the corresponding tangent space structure for the linearized dynamics the structure of the linearized flow can then be translated into the manifold structure of the nonlinear flow we briefly mention the use of lyapunov vectors to locate a slow manifold and contrast this approach with two existing approaches the minimum time to climb problem provides an example of two time scale behavior and motivates the discussion

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

nonlinear unmixing of hyperspectral data with vector valued kernel functions this paper presents a kernel based nonlinear mixing model for hyperspectral data where the nonlinear function belongs to a hilbert space of vector valued functions the proposed model extends the existing ones by accounting for band dependent and neighboring nonlinear contributions the key idea is to work under the assumption that nonlinear contributions are dominant in some parts of the spectrum while they are less pronounced in other parts in addition to this we motivate the need for taking into account nonlinear contributions originating from the ground covers of neighboring pixels by practical considerations precisely the adjacency effect the relevance of the proposed model is that the nonlinear function is associated with a matrix valued kernel that allows to jointly model a wide range of nonlinearities and includes prior information regarding band dependences furthermore the choice of the nonlinear function input allows to incorporate neighboring effects the optimization problem is strictly convex and the corresponding iterative algorithm is based on the alternating direction method of multipliers finally experiments conducted using synthetic and real data demonstrate the effectiveness of the proposed approach

optimization problems with perturbations a guided tour this paper presents an overview of some recent and significant progress in the theory of optimization problems with perturbations we put the emphasis on methods based on upper and lower estimates of the objective function of the perturbed problems these methods allow one to compute expansions of the optimal value function and approximate optimal solutions in situations where the set of lagrange multipliers is not a singleton may be unbounded or is even empty we give rather complete results for nonlinear programming problems and describe some extensions of the method to more general problems we illustrate the results by computing the equilibrium position of a chain that is almost vertical or horizontal

series solution of painleve ii in electrodiffusion conjectured convergence a perturbation series solution is constructed with the use of airy functions for a nonlinear two point boundary value problem arising in an established model of steady electrodiffusion in one dimension with two ionic species carrying equal and opposite charges the solution includes a formal determination of the associated electric field which is known to satisfy a form of the painleve ii differential equation comparisons with the numerical solution of the boundary value problem show excellent agreement following termination of the series after a sufficient number of terms for a much wider range of values of the parameters in the model than suggested by previously presented analysis or admitted by previously presented approximation schemes these surprising results suggest that for a wide variety of cases a convergent series expansion for the painleve transcendent describing the electric field has been obtained a suitable weighting of error measures for the approximations to the field and its first derivative provides a monotonically decreasing overall measure of the error in a subset of these cases it is conjectured that the series does converge for this subset

quasi equilibrium grid algorithm geometric construction for model reduction the method of invariant grid mig is an iterative procedure for model reduction in chemical kinetics which is based on the notion of slow invariant manifold sim a n gorban i v karlin method of invariant manifold for chemical kinetics chem eng sci 58 2003 4751 4768 e chiavazzo a n gorban i v karlin comparison of invariant manifolds for model reduction in chemical kinetics commun comput phys 2 5 2007 964 992 a n gorban i v karlin a y zinovyev invariant grids for reaction kinetics physica a 333 2004 106 154 a n gorban i v karlin invariant manifolds for physical and chemical kinetics lecture notes physics 660 springer berlin heidelberg 2005 doi 10 1007 b98103 important role in that method is played by the initial grid which once refined gives a description of the invariant manifold the invariant grid a convenient way to get a first approximation of the sim is given by the spectral quasi equilibrium manifold sqem a n gorban i v karlin method of invariant manifold for chemical kinetics chem eng sci 58 2003 4751 4768 e chiavazzo a n gorban i v karlin comparison of invariant manifolds for model reduction in chemical kinetics commun comput phys 2 5 2007 964 992 in the present paper a flexible numerical method to construct the discrete analog of a quasi equilibrium manifold in any dimension is presented that object is named quasi equilibrium grid qeg while the procedure quasi equilibrium grid algorithm qega extensions of the qem notion are also suggested the qega is a numerical tool which can be used to find a grid based approximation for the locus of minima of a convex function under some linear constraints the method is validated by construction of one and two dimensional grids for a model of hydrogen oxidation reaction

exponential estimates of symplectic slow manifolds in this paper we prove the existence of an almost invariant symplectic slow manifold for analytic hamiltonian slow fast systems with finitely many slow degrees of freedom for which the error field is exponentially small we allow for infinitely many fast degrees of freedom the method we use is motivated by a paper of mackay from 2004 the method does not notice resonances and therefore we do not pose any restrictions on the motion normal to the slow manifold other than it being fast and analytic we also present a stability result and obtain a generalization of a result of gelfreich and lerman on an invariant slow manifold to finitely many fast degrees of freedom

canards in r3 abstract we give a geometric analysis of canard solutions in three dimensional singularly perturbed systems with a folded two dimensional critical manifold by analysing the reduced flow we obtain singular canard solutions passing through a singularity on the fold curve we classify these singularities called canard points as folded saddles folded nodes and folded saddle nodes we prove the existence of canard solutions in the case of the folded saddle we show the existence of canards in the folded node case provided a generic non resonance condition is satisfied and in a subcase of the folded saddle node the proof is based on the blow up method

higher order corrections in the approximation of low dimensional manifolds and the construction of simplified problems with the csp method in systems of stiff ordinary differential equations odes both fast and slow time scales are encountered the fast time scales are responsible for the development of low dimensional manifolds on which the solution moves according to the slow time scales in this paper methodologies for constructing highly accurate i expressions describing the manifold and ii simplified non stiff equations governing the slow evolution of the solution on the manifold are developed according to an iterative procedure proposed in the computational singular perturbation csp method it is shown that the increasing accuracy achieved with each iteration is directly related to the time rates of change of the csp vectors spanning the manifold along the solution trajectory here an algorithm is presented which implements these calculations and is validated on the basis of two simple examples

a maximum entropy approach to unsupervised mixed pixel decomposition due to the wide existence of mixed pixels the derivation of constituent components endmembers and their fractional proportions abundances at the subpixel scale has been given a lot of attention the entire process is often referred to as mixed pixel decomposition or spectral unmixing although various algorithms have been proposed to solve this problem two potential issues still need to be further investigated first assuming the endmembers are known the abundance estimation is commonly performed by employing a least squares error criterion which however makes the estimation sensitive to noise and outliers second the mathematical intractability of the abundance non negative constraint results in computationally expensive numerical approaches in this paper we propose an unsupervised decomposition method based on the classic maximum entropy principle termed the gradient descent maximum entropy gdme aiming at robust and effective estimates we address the importance of the maximum entropy principle for mixed pixel decomposition from a geometric point of view and demonstrate that when the given data present strong noise or when the endmember signatures are close to each other the proposed method has the potential of providing more accurate estimates than the popular least squares methods e g fully constrained least squares we apply the proposed gdme to the subject of unmixing multispectral and hyperspectral data the experimental results obtained from both simulated and real images show the effectiveness of the proposed method

joint bayesian endmember extraction and linear unmixing for hyperspectral imagery this paper studies a fully bayesian algorithm for endmember extraction and abundance estimation for hyperspectral imagery each pixel of the hyperspectral image is decomposed as a linear combination of pure endmember spectra following the linear mixing model the estimation of the unknown endmember spectra is conducted in a unified manner by generating the posterior distribution of abundances and endmember parameters under a hierarchical bayesian model this model assumes conjugate prior distributions for these parameters accounts for nonnegativity and full additivity constraints and exploits the fact that the endmember proportions lie on a lower dimensional simplex a gibbs sampler is proposed to overcome the complexity of evaluating the resulting posterior distribution this sampler generates samples distributed according to the posterior distribution and estimates the unknown parameters using these generated samples the accuracy of the joint bayesian estimator is illustrated by simulations conducted on synthetic and real aviris images

explicit time scale splitting algorithm for stiff problems auto ignition of gaseous mixtures behind a steady shock a new explicit algorithm based on the computational singular perturbation csp method is presented this algorithm is specifically designed to solve stiff problems and its performance increases with stiffness the key concept in its structure is the splitting of the fast from the slow time scales in the problem realized by embedding csp concepts into an explicit scheme in simple terms the algorithm marches in time with only the terms producing the slow time scales while the contribution of the terms producing the fast time scales is taken into account at the end of each integration step as a correction the new algorithm is designed for the integration of stiff systems of pdes by means of explicit schemes for simplicity in the presentation and discussion of the different features of the new algorithm a simple test case is considered involving the auto ignition of a methane air mixture behind a normal shock wave which is described by a system of odes the performance of the new algorithm accuracy and computational efficiency is then compared with the well known lsode package its merits when used for the solution of systems of pdes are discussed although when dealing with a stiff system of odes the new algorithm is shown to provide equal accuracy with that delivered by lsode at the cost of higher execution time the results indicate that its performance could be superior when facing a stiff system of pdes

slow integral manifolds for lagrangian fluid dynamics in unsteady geophysical flows abstract the authors consider lagrangian motion of fluid particles in unsteady gravity currents in geophysical flows the vertical motion of fluid particles especially the induced vertical mixing in these currents is partially responsible for the ocean thermohaline circulation and thus plays a role in the global climate dynamics first a reduced dynamic system for slow variables is derived for a nonautonomous multiscale system the reduced system still nonautonomous is the original system restricted to a centre like nonautonomous invariant manifold so called slow manifold which holds slow motions of the system an algorithm is also presented to obtain an approximation of the nonautonomous slow manifold a novelty here is that the reduction principle applies to nonautonomous multiscale systems which satisfy conditions that are true only locally in space as in many physical cases this makes the reduction principle applicable to real physical systems then this invariant manifold reduction principle is applied to an approximate conceptual lagrangian model of gravity currents and a reduced nonautonomous system for slow vertical motion is obtained this reduced system may be useful as a conceptual tractable tool for understanding some features of vertical mixing in unsteady gravity currents

periodic orbits and chaos in fast slow systems with bogdanov takens type fold points abstract the existence of stable periodic orbits and chaotic invariant sets of singularly perturbed problems of fast slow type having bogdanov takens bifurcation points in its fast subsystem is proved by means of the geometric singular perturbation method and the blow up method in particular the blow up method is effectively used for analyzing the flow near the bogdanov takens type fold point in order to show that a slow manifold near the fold point is extended along the boutroux s tritronquee solution of the first painleve equation in the blow up space

bounds for the chaotic region in the lorenz model abstract in a previous paper the authors made an extensive numerical study of the lorenz model changing all three parameters of the system we conjectured that the region of parameters where the lorenz model is chaotic is bounded for fixed r in this paper we give a theoretical proof of the conjecture by obtaining theoretical bounds for the chaotic region and by using fenichel theory the theoretical bounds are complemented with numerical studies performed using the maximum lyapunov exponent and ofli2 techniques and a comparison of both sets of results is shown finally we provide a complete three dimensional model of the chaotic regime depending on the three parameters

the first second and fourth painleve equations on weighted projective spaces abstract the first second and fourth painleve equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces c p 3 p q r s with suitable weights p q r s determined by the newton diagrams of the equations or the versal deformations of vector fields singular normal forms of the equations a simple proof of the painleve property and symplectic atlases of the spaces of initial conditions are given with the aid of the orbifold structure of c p 3 p q r s in particular for the first painleve equation a well known painleve s transformation is geometrically derived which proves to be the darboux coordinates of a certain algebraic surface with a holomorphic symplectic form the affine weyl group dynkin diagram and the boutroux coordinates are also studied from a view point of the weighted projective space

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

the use of algebraic sets in the approximation of inertial manifolds and lumping in chemical kinetic systems abstract the use of inertial manifolds for the derivation of reduced representations of complex chemical models is investigated using a method due to foias and temam physica d 32 1988 163 the method which is based on the approximation of inertial manifolds using algebraic sets is extended for use in pde systems and is shown to provide new and interesting results in reaction diffusion systems both the diffusion term and the initial fast transient behaviour are seen to have an effect on the long time dynamics of the remaining slow variable comparison with other methods for the reduction of complex chemical systems is made including some comments about the application of the quasi steady state approximation qssa in reaction diffusion systems the method is illustrated using as examples some systems which are of interest to applications in chemical and enzyme kinetics

some simple chaotic flows a systematic examination of general three dimensional autonomous ordinary differential equations with quadratic nonlinearities has uncovered 19 distinct simple examples of chaotic flows with either five terms and two nonlinearities or six terms and one nonlinearity the properties of these systems are described including their critical points lyapunov exponents and fractal dimensions

the role of slow system dynamics in predicting the degeneracy of slow invariant manifolds the case of vdp relaxation oscillations abstract the development of dissipative time scales which are much faster than the rest force the solution of a system of 1st order odes to land and then evolve on a slow invariant manifold the slow evolution on this manifold can be approximated by a reduced system which is constructed with asymptotic analysis and is free of the fast scales in this work the dynamics of the reduced system are compared to the fast and slow dynamics of the original system the comparison is based on the van der pol system which is typical of those exhibiting relaxation oscillation behavior and is characterized by successive dissipative and explosive phases special attention is given to the dissipative phase during which the solution evolves on a slow invariant manifold and the fast slow time scale gap tends to contract it is shown that one cannot predict from the dynamics of the van der pol system the approaching explosive phase and the associated degeneracy of the manifold such a prediction is possible only through the dynamics of the reduced system which involves an explosive component that is absent in the original system this explosive component accelerates as higher order effects are incorporated in the reduced system resulting in a smaller gap between the fast time scale of the original system and the fast time scale of the reduced one this feature sets a limit on the accuracy singular perturbation techniques can deliver even in the presence of a reasonably large fast slow time scale gap in the dynamics of the original system given a fast slow system predicting forthcoming changes in the dimensions of the manifold is very important when it is desired to identify the underlying major physical mechanisms or to control the process

the surface gradient method for the treatment of source terms in the shallow water equations a novel scheme has been developed for data reconstruction within a godunov type method for solving the shallow water equations with source terms in contrast to conventional data reconstruction methods based on conservative variables the water surface level is chosen as the basis for data reconstruction this provides accurate values of the conservative variables at cell interfaces so that the fluxes can be accurately calculated with a riemann solver the main advantages are 1 a simple centered discretization is used for the source terms 2 the scheme is no more complicated than the conventional method for the homogeneous terms 3 small perturbations in the water surface elevation can be accurately predicted and 4 the method is generally suitable for both steady and unsteady shallow water problems the accuracy of the scheme has been verified by recourse to both steady and unsteady flow problems excellent agreement has been obtained between the numerical predictions and analytical solutions the results indicate that the new scheme is accurate simple efficient and robust

an efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems the relation between the iterative algorithms based on the computational singular perturbation csp and the invariance equation ie methods is examined the success of the two methods is based on the appearance of fast and slow time scales in the dynamics of stiff systems both methods can identify the low dimensional surface in the phase space slow invariant manifold sim where the state vector is attracted under the action of fast dynamics it is shown that this equivalence of the two methods can be expressed by simple algebraic relations csp can also construct the simplified non stiff system that models the slow dynamics of the state vector on the sim an extended version of ie is presented which can also perform this task this new ie version is shown to be exactly similar to a modified version of csp which results in a very efficient algorithm especially in cases where the sim dimension is small so that significant model simplifications are possible

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed