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

generalizations of the fkkm theorem and the ky fan minimax inequality with applications to maximal elements price equilibrium and complementarity abstract this paper generalizes the fan knaster kuratowski mazurkiewicz fkkm theorem of ky fan game theory and related topics pp 151 156 north holland amsterdam 1979 and math ann 266 1984 519 537 and the ky fan minimax inequality by introducing a class of the generalized closedness and continuity conditions which are called the transfer closedness and transfer continuities we then apply these results to prove the existence of maximal elements of binary relations under very weak assumptions we also prove the existence of price equilibrium and the complementarity problem without the continuity assumptions thus our results generalize many of the existence theorems in the literature

involutive basic substructural core fuzzy logics involutive mianorm based logics abstract this paper deals with the standard completeness of involutive non associative non commutative substructural fuzzy logics and their axiomatic extensions first fuzzy systems based on involutively residuated mianorms binary monotonic identity aggregation operations on the real unit interval 0 1 their corresponding algebraic structures and their algebraic completeness results are discussed next completeness with respect to algebras whose lattice reduct is 0 1 known as standard completeness is established for these systems via a construction in the style of jenei montagna these standard completeness results resolve a problem left open by cintula horcik and noguera in the recent handbook of mathematical fuzzy logic and review of symbolic logic finally we briefly consider the similarities and differences between constructions of the author and wang s jenei montagna style

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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