minimal strongly eventually positive realization for a class of externally positive systems the minimal positive realization of externally positive systems is still an open problem due to various restrictions caused by the nonnegativity constraint on the state space matrices in this paper the minimal strongly eventually positive realization which relaxes the nonnegativity constraint on the state space model and only requires that the state trajectory is nonnegative after a certain number of steps is proposed for an externally positive system it is shown that a discrete time transfer function with a simple strictly dominant root is externally positive if and only if it has a minimal strongly eventually positive realization a constructive proof is provided the continuous time counterpart is also addressed by transforming it into a corresponding discrete time case

modulating traveling fronts for the swift hohenberg equation in the case of an additional conservation law abstract we consider the one dimensional swift hohenberg equation coupled to a conservation law as a parameter increases the system undergoes a turing bifurcation we study the dynamics near this bifurcation first we show that stationary periodic solutions bifurcate from a homogeneous ground state second we construct modulating traveling fronts which model an invasion of the unstable ground state by the periodic solutions this provides a mechanism of pattern formation for the studied system the existence proof uses center manifold theory for a reduction to a finite dimensional problem this is possible despite the presence of infinitely many imaginary eigenvalues for vanishing bifurcation parameter since the eigenvalues leave the imaginary axis with different velocities if the parameter increases furthermore compared to non conservative systems we address new difficulties arising from an additional neutral mode at fourier wave number k 0 by exploiting that the amplitude of the conserved variable is small compared to the other variables

nonnegative realization of autonomous systems in the behavioral approach nonnegative linear systems which have traditionally been investigated within the state space framework have been recently introduced and analyzed by means of the behavioral approach in a couple of recent papers j w nieuwenhuis linear algebra appl 281 1998 pp 43 58 m e valcher linear algebra appl 319 2000 pp 147 162 several general definitions and results about nonnegative behaviors as well as a complete analysis of nonnegativity property for autonomous behaviors have been presented in this contribution by focusing our interest again on autonomous behaviors we explore the nonnegative realization problem by deriving an extended set of necessary and sufficient geometric conditions for an autonomous behavior to be nonnegative realizable in the scalar case in particular necessary and sufficient conditions for nonnegative realizability which refer to the set of zeros of any polynomial involved in the kernel description of the behavior are provided finally a comparison between the nonnegative realizability property here investigated and k realizability addressed in h maeda and s kodama ieee trans ircuits systems i fund theory appl cas 281 1981 pp 39 47 is carried on

cellular automata as microcanonical simulators the deterministic q2r cellular automation has been tested by many authors as a fast algorithm to simulate the ising model in the microcanonical ensemble however the magnetic susceptibility curve measured from the fluctuations of the magnetization is found to be far below the expected results inside the ordered ising phase the non ergodicity degree of this automaton seems to be inadequate to simulate the ising dynamics in this work the authors introduce two modified automata and test their performance for the square lattice for the first one they found improved results concerning the ising transition for the second modified automaton the ising transition does not occur the magnetization vanishing even for energies far below the normal critical threshold on the other hand another transition appears at a different critical energy value that seems to be the same as that of the periodic chaotic transition already observed in the normal q2r dynamics the critical indices of this new transition are measured from finite size scaling the results indicating that it is at the same universality class as the ising transition

local and cluster critical dynamics of the 3d random site ising model abstract we present the results of monte carlo simulations for the critical dynamics of the three dimensional site diluted quenched ising model three different dynamics are considered these correspond to the local update metropolis scheme as well as to the swendsen wang and wolff cluster algorithms the lattice sizes of l 10 96 are analysed by a finite size scaling technique the site dilution concentration p 0 85 was chosen to minimize the correction to scaling effects we calculate numerical values of the dynamical critical exponents for the integrated and exponential autocorrelation times for energy and magnetization as expected cluster algorithms are characterized by lower values of dynamical critical exponent than the local one also in the case of dilution critical slowing down is more pronounced for the metropolis algorithm however the striking feature of our estimates is that they suggest that dilution leads to decrease of the dynamical critical exponent for the cluster algorithms this phenomenon is quite opposite to the local dynamics where dilution enhances critical slowing down

the gap lemma and geometric criteria for instability of viscous shock profiles an obstacle in the use of evans function theory for stability analysis of traveling waves occurs when the spectrum of the linearized operator about the wave accumulates at the imaginary axis since the evans function has in general been constructed only away from the essential spectrum a notable case in which this difficulty occurs is in the stability analysis of viscous shock profiles here we prove a general theorem the gap lemma concerning the analytic continuation of the evans function associated with the point spectrum of a traveling wave into the essential spectrum of the wave this allows geometric stability theory to be applied in many cases where it could not be applied previously we demonstrate the power of this method by analyzing the stability of certain undercompressive viscous shock waves a necessary geometric condition for stability is determined in terms of the sign of a certain melnikov integral of the associated viscous profile this sign can easily be evaluated numerically we also compute it analytically for solutions of several important classes of systems in particular we show for a wide class of systems that homoclinic solitary waves are linearly unstable confirming these as the first known examples of unstable viscous shock waves we also show that strong heteroclinic undercompressive waves are sometimes unstable similar stability conditions are also derived for lax and overcompressive shocks and for nn

behavior of periodic solutions of viscous conservation laws under localized and nonlocalized perturbations we establish nonlinear stability and asymptotic behavior of traveling periodic waves of viscous conservation laws under localized perturbations or nonlocalized perturbations asymptotic to constant shifts in phase showing that long time behavior is governed by an associated second order formal whitham modulation system a key point is to identify the way in which initial perturbations translate to initial data for this formal system a task accomplished by detailed estimates on the linearized solution operator about the background wave notably our approach gives both a common theoretical treatment and a complete classification in terms of phase coupling or decoupling of general systems of conservation or balance laws encompassing cases that had previously been studied separately or not at all at the same time our refined description of solutions gives the new result of nonlinear asymptotic stability with respect to localized perturbations in the phase decoupled case further distinguishing behavior in the different cases an interesting technical aspect of our analysis is that for systems of conservation laws the whitham modulation description is of system rather than scalar form as a consequence of which renormalization methods such as have been used to treat the reaction diffusion case in general do not seem to apply

reachability observability and realizability of continuous time positive systems this paper discusses reachability observability and realizability of single input single output linear time invariant systems in which state variables and or input output functions are restricted to be nonnegative to reflect physical constraints frequently encountered in real systems we define a set reachable from the origin with nonnegative inputs and also a set observable with nonnegative outputs we investigate geometrical structures of the sets through convex analysis and a duality relation between them is established next we consider positive realization of a given transfer function using the reachable set and the observable set we give a necessary and sufficient condition for positive realizability an example is given to demonstrate that a positive realizable transfer function does not in general have a jointly controllable and observable positive realization

stability of rolls and hexagonal patterns in non potential systems abstract the stability of rolls hexagons and non equilateral hexagonal patterns is considered in the framework of a system of amplitude equations containing non potential resonant quadratic terms the stability analysis of roll patterns reveals a new kind of instability which can generate steady non equilateral hexagons noise sensitive transitions between roll patterns or periodic oscillations the busse balloons for equilateral and non equilateral hexagons are calculated

non linear stability of modulated fronts for the swift hohenberg equation we consider front solutions of the swift hohenberg equation t u 1 x 2 2 u 2 u u 3 these are traveling waves which leave in their wake a periodic pattern in the laboratory frame using renormalization techniques and a decomposition into bloch waves we show the non linear stability of these solutions it turns out that this problem is closely related to the question of stability of the trivial solution for the model problem t u x t x 2 u x t 1 tanh x ct u x t u x t p with p 3 in particular we show that the instability of the perturbation ahead of the front is entirely compensated by a diffusive stabilization which sets in once the perturbation has hit the bulk behind the front

an example of how positivity may force realizations of large dimension abstract a standard result of linear system theory states that an siso proper rational transfer function of degree n always has a realization of dimension n in some applications one is interested in having a realization with nonnegative entries and it is known that when the dominant poles display a specific pattern forcing nonnegativity leads to a system which is not jointly reachable and observable in this paper we show that the minimal dimension of a positive realization may be large even in the case of a single dominant pole more precisely we provide a family of transfer functions each of which is of degree n 3 such that for any integer n 3 the corresponding member of the family admits a minimal positive realization of state space dimension not smaller than n

instability of spatially periodic patterns due to a zero mode in the phase field crystal equation abstract weakly nonlinear spatially periodic patterns coupled to a goldstone zero mode of the phase field crystal model are investigated rotationally invariant equations for the dynamics of the amplitudes of a hexagonal pattern are derived first which then allows us to determine stability regions for stripes and hexagons there are parameter regimes in which all periodic patterns become unstable as a result of long wavelength instabilities generated by the zero mode

on the definition of entropy production via examples we present a definition of entropy production rate for classes of deterministic and stochastic dynamics the point of departure is a gibbsian representation of the steady state path space measure for which the density is determined with respect to the time reversed process the gibbs formalism is used as a unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems our definition is motivated by recent work on the gallavotti cohen local fluctuation theorem and it is illustrated via a number of examples

linear stability of steady states for thin film and cahn hilliard type equations we study the linear stability of smooth steady states of the evolution equation h t f h xxx x g h h x x ah under both periodic and neumann boundary conditions if a 0 we assume f 1 in particular we consider positive periodic steady states of thin film equations where a 0 and f g might have degeneracies such as f 0 0 as well as singularities like g 0

stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions under natural spectral stability assumptions motivated by previous investigations of the associated spectral stability problem we determine sharp lp estimates on the linearized solution operator about a multidimensional planar periodic wave of a system of conservation laws with viscosity yielding linearized l1 lp lp stability for all p geqq 2 and dimensions d geqq 1 and nonlinear l1 hs lp hs stability and l2 asymptotic behavior for p geqq 2 and d geqq 3 the behavior can in general be rather complicated involving both convective that is wave like and diffusive effects

eigenvalue regions for positive systems positive systems are systems in which the state variables take only nonnegative values for all times in this paper we derive eigenvalue regions for discrete and continuous time positive linear systems by resorting to the immediately available information on the values of the main diagonal entries of the system matrix

information criteria for quantifying loss of reversibility in parallelized kmc parallel kinetic monte carlo kmc is a potent tool to simulate stochastic particle systems efficiently however despite literature on quantifying domain decomposition errors of the particle system for this class of algorithms in the short and in the long time regime no study yet explores and quantifies the loss of time reversibility in parallel kmc inspired by concepts from non equilibrium statistical mechanics we propose the entropy production per unit time or entropy production rate given in terms of an observable and a corresponding estimator as a metric that quantifies the loss of reversibility typically this is a quantity that cannot be computed explicitly for parallel kmc which is why we develop a posteriori estimators that have good scaling properties with respect to the size of the system through these estimators we can connect the different parameters of the scheme such as the communication time step of the parallelization the choice of the domain decomposition and the computational schedule with its performance in controlling the loss of reversibility from this point of view the entropy production rate can be seen both as an information criterion to compare the reversibility of different parallel schemes and as a tool to diagnose reversibility issues with a particular scheme as a demonstration we use sandia lab s spparks software to compare different parallelization schemes and different domain lattice decompositions

onset of wavy vortices in taylor couette flow with imperfect reflection symmetry we reveal experimentally a mechanism that alters the critical behavior of a hopf bifurcation substantially due to the presence of imperfections of the reflection symmetry in a hydrodynamic system the onset of rotating waves in taylor vortex flow which is widely considered as a classical example for a hopf bifurcation in hydrodynamics is investigated primarily by transient response experiments while wavy vortex flow is not influenced by such unavoidable experimental imperfections the critical behavior of the axially subharmonic rotating wave with wavy outflow boundaries also called the small jet mode is qualitatively altered experimental evidence is provided that the modified critical behavior at the hopf bifurcation is associated with imperfections of the reflection symmetry of the taylor couette setup the experimental results on hopf bifurcation are discussed in the context of a cusp hopf bifurcation model recently proposed by harlim and langford int j bifurcation chaos appl sci eng 17 2547 2007 and compared to experimental results on imperfect pitchfork bifurcation in small aspect ratio taylor couette flow

positive output state maps and quasi positive realization of mimo discrete systems the positive realization of externally positive systems described by their transfer matrix g z is a complex problem whose analysis has been completed only recently for the siso case on the basis of algebraic and geometric approaches that underline the many constraints that condition its solution these constraints concern the minimal order of the obtained models the minimality of their parameterizations and even their existence this paper considers the new category of quasi positive state space models introduced in guidorzi 2014 that limit the assumptions on the nonnegativeness of the state space trajectories to the only trajectories that can be actually generated by the system under nonnegative controls it is shown that all externally positive systems admit a quasi positive minimally parameterized state space realization whose existence is not conditioned by restrictions on the signs of the parameters of the polynomials in g z

nonlocalized modulation of periodic reaction diffusion waves the whitham equation in a companion paper we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized l1 perturbation here we determine time asymptotic behavior under such perturbations showing that solutions consist of a leading order of a modulation whose parameter evolution is governed by an associated whitham averaged equation

instability and localisation of patterns due to a conserved quantity abstract we describe the influence of a conserved quantity on the stability properties of roll square and hexagonal patterns near stationary onset the appropriate systems of amplitude equations are analysed they comprise equations for the amplitudes of the pattern modes together with an additional coupled equation governing the evolution of a large scale mode associated with the conserved quantity we find that the presence of this large scale mode may result in the destabilisation of all regular roll square or hexagonal patterns leading to amplitude modulation and strong localisation of the pattern our analytical results are complemented throughout by numerical simulations of a model partial differential equation to illustrate the modulational instabilities and their nonlinear development

brief paper positivity preserving h model reduction for positive systems this paper is concerned with the model reduction of positive systems for a given stable positive system our attention is focused on the construction of a reduced order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed h performance based upon a system augmentation approach a novel characterization on the stability with h performance of the error system is first obtained in terms of linear matrix inequality lmi then a necessary and sufficient condition for the existence of a desired reduced order model is derived accordingly furthermore iterative lmi approaches with primal and dual forms are developed to solve the positivity preserving h model reduction problem finally a compartmental network is provided to show the effectiveness of the proposed techniques

spatial two level interacting particle simulations and information theory based error quantification we propose a hierarchy of two level kinetic monte carlo methods for sampling high dimensional stochastic lattice particle dynamics with complex interactions the method is based on the efficient coupling of different spatial resolution levels taking advantage of the low sampling cost in a coarse space and developing local reconstruction strategies from coarse grained dynamics furthermore a natural extension to a multilevel kinetic coarse grained monte carlo is presented microscopic reconstruction corrects possibly significant errors introduced through coarse graining leading to the controlled error approximation of the sampled stochastic process in this manner the proposed algorithm overcomes known shortcomings of coarse graining of particle systems with complex interactions such as combined long and short range particle interactions and or complex lattice geometries specifically we provide error analysis for the approximation of long time stationary dynamics in terms of relative entropy measu

propagating fronts and the center manifold theorem we prove the existence of propagating front solutions for the swift hohenberg equation sh using the center manifold theorem we reduce the problem to a two dimensional system of ordinary differential equations they describe stationary solutions and front solutions of the partial differential equation sh we identify the well known amplitude equation as the lowest order approximation to the equation of motion on the center manifold

positive realizations of linear systems abstract this paper deals with the realization problem of linear systems in which a sign pattern is imposed on the realization a b c t this problem arises in the context of positive linear systems i e systems in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative using a unitary framework for continuous time and discrete time systems we derive necessary and sufficient conditions for positive realizability by means of convex analysis moreover the complete solution in the case of a transfer function of mcmillan degree n 3 is obtained

a tutorial on the positive realization problem this paper is a tutorial on the positive realization problem that is the problem of finding a positive state space representation of a given transfer function and characterizing existence and minimality of such representation this problem goes back to the 1950s and was first related to the identifiability problem for hidden markov models then to the determination of internal structures for compartmental systems and later embedded in the more general framework of positive systems theory within this framework developing some ideas sprang in the 1960s during the 1980s the positive realization problem was reformulated in terms of a geometric condition which was recently exploited as a tool for finding the solution to the existence problem and providing partial answers to the minimality problem in this paper the reader is carried through the key ideas which have proved to be useful in order to tackle this problem in order to illustrate the main results contributions and open problems several motivating examples and a comprehensive bibliography on positive systems organized by topics are provided

rooted tree analysis of runge kutta methods with exact treatment of linear terms we investigate a class of time discretization schemes called etd runge kutta methods where the linear terms of an ordinary differential equation are treated exactly while the other terms are numerically integrated by a one step method these schemes proposed by previous authors can be regarded as modified runge kutta methods whose coefficients are matrices instead of scalars from this viewpoint we reexamine the notion of consistency convergence and order to provide a mathematical foundation for new methods applying the rooted tree analysis expansion theorems of both the strict and numerical solutions are proved and two types of order conditions are defined several classes of formulas with up to four stages that satisfy the conditions are constructed and it is shown that the power series of matrices employed as their coefficients can be determined using the order conditions

nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case abstract extending previous results of oh zumbrun and johnson zumbrun we show that spectral stability implies linearized and nonlinear stability of spatially periodic traveling wave solutions of viscous systems of conservation laws for systems of generic type removing a restrictive assumption that wave speed be constant to first order along the manifold of nearby periodic solutions key to our analysis is a nonlinear cancellation estimate observed by johnson and zumbrun along with a detailed understanding of the whitham averaged system the latter motivates a careful analysis of the bloch perturbation expansion near zero frequency and suggests factoring out an appropriate translational modulation of the underlying wave allowing us to derive the sharpened low frequency estimates needed to close the nonlinear iteration arguments

numerical testing of the stability of viscous shock waves a new theoretical evans function condition is used as the basis of a numerical test of viscous shock wave stability accuracy of the method is demonstrated through comparison against exact solutions a convergence study and evaluation of approximate error equations robustness is demonstrated by applying the method to waves for which no current analytic results apply highly nonlinear waves from the cubic model and strong shocks from gas dynamics an interesting aspect of the analysis is the need to incorporate features from the analytic evans function theory for purposes of numerical stability for example we find it necessary for numerical accuracy to solve odes on the space of wedge products

parallelization processor communication and error analysis in lattice kinetic monte carlo in this paper we study from a numerical analysis perspective the fractional step kinetic monte carlo fs kmc algorithms proposed in 1 for the parallel simulation of spatially distributed particle systems on a lattice fs kmc are fractional step algorithms with a time stepping windowt and as such they are inherently partially asynchronous since there is no processor com munication during the periodt in this contribution we primarily focus on the error analysis of fs kmc algorithms as approximations of conventional serial kinetic monte carlo kmc a key as pect of our analysis relies on emphasizing a goal oriented approach for suitably defined macroscopic observables e g density energy correlations surface roughness rather than focusing on strong topology estimates for individual trajectories one of the key implications of our error analysis is that it allows us to address systematically the processor communication of different parallelization strategies for kmc by comparing their partial asynchrony which in turn is measured by their respective fractional time stept for a prescribed error tolerance

monomial patterns in the sequence akb abstract we consider the pattern of zero and nonzero elements in the sequence a k b where a is an n n nonnegative matrix and b is an n 1 nonnegative column vector we establish a tight bound of k n for the first occurrence of a given monomial pattern and we give a graph theoretic characterization of triples a b i such that there exists a k k n for which a k b is an i monomial the appearance of monomial patterns with a single nonzero entry is linked to controllability of discrete n dimensional linear dynamic systems with positivity constraints on the state and control

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

instability and stability of rolls in the swift hohenberg equation we develop a method for the stability analysis of bifurcating spatially periodic patterns under general nonperiodic perturbations in particular it enables us to detect sideband instabilities we treat in all detail the stability question of roll solutions in the two dimensional swift hohenberg equation and derive a condition on the amplitude and the wave number of the rolls which is necessary and sufficent for stability moreover we characterize the set of those wave vectors which give rise to unstable perturbations

spectral stability of ideal gas shock layers extending recent results in the isentropic case we use a combination of asymptotic ode estimates and numerical evans function computations to examine the spectral stability of shock wave solutions of the compressible navier stokes equations with ideal gas equation of state our main results are that in appropriately rescaled coordinates the evans function associated with the linearized operator about the wave i converges in the large amplitude limit to the evans function for a limiting shock profile of the same equations for which internal energy vanishes at one end state and ii has no unstable positive real part zeros outside a uniform ball thus the rescaled eigenvalue ode for the set of all shock waves augmented with the nonphysical limiting case form a compact family of boundary value problems that can be conveniently investigated numerically an extensive numerical evans function study yields one dimensional spectral stability independent of amplitude for gas constant in 1 2 3 and ratio of heat conduction to viscosity coefficient within 0 2 5 1 4 1 47 for air other values may be treated similarly but were not considered the method of analysis extends also to the multi dimensional case a direction that we shall pursue in a future work

long wavelength instabilities of square patterns the long wavelength instabilities of square and rectangular planforms are studied analytically and numerically using amplitude equations which describe the general interaction of two orthogonal coupled roll patterns the zigzag and two dimensional eckhaus instabilities are found and in addition it is discovered that the three dimensional equivalent of the eckhaus instability splits into two variants the square eckhaus instability is the direct equivalent of the two dimensional case whereas the rectangular eckhaus instability is truly three dimensional in character in the case of square patterns nonlinear phase diffusion equations are derived close to the onset of the instabilities a short wavelength cross square mode is also discussed briefly

feedback control of subcritical turing instability with zero mode a global feedback control of a system that exhibits a subcritical monotonic instability at a nonzero wave number short wave or turing instability in the presence of a zero mode is investigated using a ginzburg landau equation coupled to an equation for the zero mode this system is studied analytically and numerically it is shown that feedback control based on measuring the maximum of the pattern amplitude over the domain can stabilize the system and lead to the formation of localized unipulse stationary states or traveling solitary waves it is found that the unipulse traveling structures result from an instability of the stationary unipulse structures when one of the parameters characterizing the coupling between the periodic pattern and the zero mode exceeds a critical value that is determined by the zero mode damping coefficient

pattern selection in three dimensional magnetoconvection abstract weakly nonlinear boussinesq convection in an imposed vertical magnetic field is studied stress free boundary conditions are used at top and bottom of the layer where the magnetic field is assumed to remain vertical both steady and oscillatory convection are considered among all the spatially periodic patterns steady rolls are the preferred pattern when the onset of convection is steady state in the overstable regime a variety of spatially and temporally periodic patterns may be stable depending on the parameters these include in particular a pattern called alternating rolls the location of stable alternating rolls appears to be consistent with recent numerical simulations of compressible magnetoconvection where such patterns have been found in certain cases the conduction solution loses stability directly to spatially periodic patterns with nonperiodic time dependence the analysis of the possible patterns and their stability takes advantage of existing results from equivariant bifurcation theory and uses the critical wavenumber selected by the linear stability calculation as a function of the imposed magnetic field

realization of positive linear systems abstract positive linear systems are frequently used as mathematical models in research areas like biology and economics the problem of classifying all minimal realizations of these systems is treated in this paper extensive use is made of the theory of polyhedral cones sufficient and necessary conditions for the existence of a positive realization are given but the problem of minimality leads to the factorization problem for positive matrices ideas and results are given to come towards a solution of this factorization problem

stability of turing patterns in the brusselator model the selection and competition of turing patterns in the brusselator model are reviewed the stability of stripes and hexagons towards spatial perturbations is studied using the amplitude equation formalism for hexagonal patterns these equations include both linear and nonpotential spatial terms enabling distorted solutions the latter modify substantially the stability diagrams and select patterns with wave numbers quite different from the critical value the analytical results from the amplitude formalism agree with direct simulations of the model moreover we show that slightly squeezed hexagons are locally stable in a full range of distortion angles the stability regions resulting from the phase equation are similar to those obtained numerically by other authors and to those observed in experiments

the eckhaus instability in hexagonal patterns abstract the eckhaus instability of hexagonal patterns is studied within the model of three coupled envelope equations for the underlying roll systems the regions of instability in the parameter space are found analytically from both the phase approximation and a full system of amplitude equations beyond the stability limits of hexagons two different modes go unstable both provide symmetry breaking of an initially regular pattern via splitting of a triplet of rolls into two triplets of growing disturbances the parameters of fastest growing disturbances wavelength orientation growth rate are determined from the full set of linearized amplitude equations the nonlinear stage of the eckhaus instability is investigated numerically symmetry breaking due to the eckhaus instability indeed occurs within a certain range of parameters which for small supercriticality parameter leads to a metastable disordered hexagonal state with numerous line and point defects for larger the eckhaus instability triggers the transition of regular hexagonal pattern to disordered roll state the roll phase originates in the cores of defects and then spreads all over the pattern

self similar decay of localized perturbations in the integral boundary layer equation abstract the integral boundary layer equation ible arises as a long wave approximation for the flow of a viscous incompressible fluid down an inclined plane the trivial solution of the ible is linearly at best marginally stable i e it has essential spectrum at least up to the imaginary axis here we show that in the stable case this trivial solution is in fact nonlinearly stable with a burgers like self similar decay of localized perturbations the proof uses renormalization theory and the fact that in the stable case burgers equation is the amplitude equation for long small amplitude waves in the ible

invariants and canonical forms for systems structural and parametric identification the basic definitions regarding invariant functions and canonical forms for an equivalence relation on a generic set are first recalled with reference to observable state space models and to the equivalence relation induced by a change of basis it is then shown how the image of a complete set of independent invariants for the considered equivalence relation can be used to parametrize a subset of canonical forms in the given set then the set of polynomial input output models of the type p z y t q z u t and the equivalence relation induced by the premultiplication of p and q by a unimodular matrix are considered and canonical forms parametrized by a complete set of independent invariants introduced since the two sets of canonical forms share common sets of complete independent invariants very simple algebraical links between state space and input output canonical forms can be deduced the previous results are used to design efficient algorithms solving the problem of the canonical structural and parametric realization and identification of generic input output sequences generated by a linear discrete time invariant multivariable system the results obtained in the identification of a real process are then reported

modulated traveling fronts for a nonlocal fisher kpp equation a dynamical systems approach abstract we consider a nonlocal generalization of the fisher kpp equation in one spatial dimension as a parameter is varied the system undergoes a turing bifurcation we study the dynamics near this turing bifurcation our results are two fold first we prove the existence of a two parameter family of bifurcating stationary periodic solutions and derive a rigorous asymptotic approximation of these solutions we also study the spectral stability of the bifurcating stationary periodic solutions with respect to almost co periodic perturbations second we restrict to a specific class of exponential kernels for which the nonlocal problem is transformed into a higher order partial differential equation in this context we prove the existence of modulated traveling fronts near the turing bifurcation that describe the invasion of the turing unstable homogeneous state by the periodic pattern established in the first part both results rely on a center manifold reduction to a finite dimensional ordinary differential equation

the existence of dendritic fronts in this paper we study a fourth order semilinear parabolic equation on the infinite real line we show that in a certain parameter range this equation has propagating front solutions solutions tending to 0 at and advancing to the right with a speedc which leave behind them aperiodic pattern in the laboratory frame this is thus an example of spontaneous pattern formation

ginzburg landau equation coupled to a concentration field in binary mixture convection abstract localized travelling wave trains are investigated as they arise in binary mixture convection for free slip permeable boundary conditions a complex ginzburg landau equation is derived which is coupled to a mean concentration field this takes the slow mass diffusion into account numerical simulations show that in equations of that form the additional concentration field can lead to a substantial reduction of the drift velocity of localized wave trains moreover localized waves can be stable even if all the coefficients of the ginzburg landau equation are real and even if the bifurcation to extended waves is supercritical

on the modulation equations and stability of periodic generalized korteweg de vries waves via bloch decompositions in this paper we consider the relation between evans function based approaches to the stability of periodic travelling waves and other theories based on long wavelength asymptotics together with bloch wave expansions in previous work it was shown by rigorous evans function calculations that the formal slow modulation approximation resulting in the linearized whitham averaged system accurately describes the spectral stability to long wavelength perturbations to clarify the connection between bloch wave based expansions and evans function based approaches we reproduce this result without reference to the evans function by using direct bloch expansion methods and spectral perturbation analysis one of the novelties of this approach is that we are able to calculate the relevant bloch waves explicitly for arbitrary finite amplitude solutions furthermore this approach has the advantage of being applicable in the more general multi periodic setting where no conveniently computable evans function has yet been devised

stability of periodic solutions of conservation laws with viscosity pointwise bounds on the green function we investigate stability of periodic traveling wave solutions of systems of conservation laws with viscosity within the abstract evans function framework established by r a gardner our main result generalizing the work of zumbrun and howard in the traveling front or pulse setting is to establish sharp pointwise bounds on the green function for the linearized evolution equations provided that an appropriate evans function condition applies to the linearized operator about the wave this condition is equivalent to a spectral stability criterion introduced by schneider in the context of periodic reaction diffusion waves an immediate consequence is that strong spectral stability in the sense of schneider implies linearized l1 l p asymptotic stability for all p 1 on the other hand we show that the strict version of schneider s condition generically fails in the conservation law setting leading to complicated new metastable behavior reminiscent of that seen for degenerate neutrally stable families in the traveling front or pulse case our results apply also to the reaction diffusion setting sharpening at the linearized level results obtained by schneider using weighted norm and bloch decomposition methods

nonlinear modulational stability of periodic traveling wave solutions of the generalized kuramoto sivashinsky equation abstract in this paper we consider the spectral and nonlinear stabilities of periodic traveling wave solutions of a generalized kuramoto sivashinsky equation in particular we resolve the long standing question of nonlinear modulational stability by demonstrating that spectrally stable waves are nonlinearly stable when subject to small localized integrable perturbations our analysis is based upon detailed estimates of the linearized solution operator which are complicated by the fact that the necessarily essential spectrum of the associated linearization intersects the imaginary axis at the origin we carry out a numerical evans function study of the spectral problem and find bands of spectrally stable periodic traveling waves in close agreement with previous numerical studies of frisch she thual bar nepomnyashchy chang demekhin kopelevich and others carried out by other techniques we also compare predictions of the associated whitham modulation equations which formally describe the dynamics of weak large scale perturbations of a periodic wave train with numerical time evolution studies demonstrating their effectiveness at a practical level for the reader s convenience we include in an appendix the corresponding treatment of the swift hohenberg equation a nonconservative counterpart of the generalized kuramoto sivashinsky equation for which the nonlinear stability analysis is considerably simpler together with numerical evans function analyses extending spectral stability analyses of mielke and schneider

excitability stability and sign of equilibria in positive linear systems abstract the relationship between sign and stability of equilibria in positive linear systems is investigated in this paper in particular it is proved that the well known result stability and irreducibility imply strict positivity can be better specified by introducing the notion of excitability in fact stability excitability and strict positivity of the equilibria are three perfectly balanced properties since any one of them is implied by the two others

positive input reachability and controllability of positive systems abstract we study controllability and reachability for discrete linear control systems x k 1 ax k bu k in which the state vector x k the control vector u k the n n matrix a and the n m matrix b have nonnegative entries for an unconstrained system controllability is equivalent to reachability from the zero state in n steps furthermore a b is controllable if and only if the matrix c n b ab a 2 b a n 1 b has rank n we show that these properties are not equivalent for positive systems this has important implications for control strategy the timing of control inputs and the size and structure of the control matrix b are much more critical the zero nonzero pattern of c n plays a crucial role we illustrate these fundamental differences with models of a two species reversible chemical reaction and with two pharmacokinetic models of drug distribution in the human body

damage spreading and critical exponents for model a ising dynamics using damage spreading and heat bath dynamics we study the ising model in 2 and 3 dimensions with non conservative dynamics our algorithm differs in some important points from previous ones which makes it rather efficient we give estimates for the exponent z which seem to be the most precise published so far 2 172 0 006 for d 2 2 032 0 004 for d 3 we also give precise estimates of the exponent introduced by janssen et al z phys b 73 1989 539 and of analogous but in principle independent exponents we find surprisingly that some of the latter agree with and give an explanation for this

frequency limited mmb h _ infty model reduction for positive systems in this paper the problem of frequency limited mmb h infty model reduction for positive linear time invariant systems is investigated specifically our goal is to find a stable positive reduced order model for a given positive system such that the mmb h infty norm of the error system is bounded over a frequency interval of interest a new condition in terms of matrix inequality is developed for characterizing the frequency limited mmb h infty performance then an equivalent parametrization of a positive reduced order model is derived based on which an iterative algorithm is constructed for optimizing the reduced order model the algorithm utilizes coarse reduced order models resulting from generalized balanced truncation as the initial value both continuous and discrete time systems are considered in the same framework numerical examples clearly show the effectiveness and advantages of the proposed model reduction method

an efficient shooting algorithm for evans function calculations in large systems abstract in evans function computations of the spectra of asymptotically constant coefficient linear operators a basic issue is the efficient and numerically stable computation of subspaces evolving according to the associated eigenvalue ode for small systems a fast shooting algorithm may be obtained by representing subspaces as single exterior products j c alexander r sachs linear instability of solitary waves of a boussinesq type equation a computer assisted computation nonlinear world 2 4 1995 471 507 l q brin numerical testing of the stability of viscous shock waves ph d thesis indiana university bloomington 1998 l q brin numerical testing of the stability of viscous shock waves math comp 70 235 2001 1071 1088 l q brin k zumbrun analytically varying eigenvectors and the stability of viscous shock waves in seventh workshop on partial differential equations part i 2001 rio de janeiro mat contemp 22 2002 19 32 t j bridges g derks g gottwald stability and instability of solitary waves of the fifth order kdv equation a numerical framework physica d 172 1 4 2002 190 216 for large systems however the dimension of the exterior product space quickly becomes prohibitive growing as n k where n is the dimension of the system written as a first order ode and k typically n 2 is the dimension of the subspace we resolve this difficulty by the introduction of a simple polar coordinate algorithm representing pure monomial products as scalar multiples of orthonormal bases for which the angular equation is a numerically optimized version of the continuous orthogonalization method of drury davey a davey an automatic orthonormalization method for solving stiff boundary value problems j comput phys 51 2 1983 343 356 l o drury numerical solution of orr sommerfeld type equations j comput phys 37 1 1980 133 139 and the radial equation is evaluable by quadrature notably the polar coordinate method preserves the important property of analyticity with respect to parameters

controllability of positive linear discrete time systems the class of positive linear discrete time systems is considered reachability sets of such systems denned on cones are studied and some of their properties established reachability null controllability and controllability criteria are obtained in forms that are easier to implement than the corresponding criteria for linear discrete time systems of general type some classes of positive linear discrete time systems are investigated and their controllability properties are established

linear quadratic regulator problem with positive controls in this paper the linear quadratic regulator problem with a positivity constraint on the admissible control set is addressed necessary and sufficient conditions for optimality are presented in terms of inner products projections on closed convex sets pontryagin s maximum principle and dynamic programming the main results are concerned with smoothness of the optimal control and the value function the maximum principle will be extended to the infinite horizon case based on these analytical methods we propose a numerical algorithm for the computation of the optimal controls for the finite and infinite horizon problem the numerical methods will be justified by convergence properties between the finite and infinite horizon case on one side and discretized optimal controls and the true optimal control on the other

synchronous parallel kinetic monte carlo for continuum diffusion reaction systems a novel parallel kinetic monte carlo kmc algorithm formulated on the basis of perfect time synchronicity is presented the algorithm is intended as a generalization of the standard n fold kmc method and is trivially implemented in parallel architectures in its present form the algorithm is not rigorous in the sense that boundary conflicts are ignored we demonstrate however that in their absence or if they were correctly accounted for our algorithm solves the same master equation as the serial method we test the validity and parallel performance of the method by solving several pure diffusion problems i e with no particle interactions with known analytical solution we also study diffusion reaction systems with known asymptotic behavior and find that for large systems with interaction radii smaller than the typical diffusion length boundary conflicts are negligible and do not affect the global kinetic evolution which is seen to agree with the expected analytical behavior our method is a controlled approximation in the sense that the error incurred by ignoring boundary conflicts can be quantified intrinsically during the course of a simulation and decreased arbitrarily controlled by modifying a few problem dependent simulation parameters

controllability and reachability criteria for discrete time positive systems discrete time positive systems have been the object of widespread interest in the literature in the last decade particular attention has been devoted to the analysis and characterization of the different notions of reachability and controllability which in this context make their appearance in this paper the properties of ordinary and essential controllability and reachability are analysed based on a graph theoretic approach we provide practical criteria to verify whether a given positive system f g is endowed with these properties together with canonical forms for describing reachable controllable pairs

exponential splitting for unbounded operators we present a convergence analysis for exponential splitting methods applied to linear evolution equations our main result states that the classical order of the splitting method is retained in a setting of unbounded operators without requiring any additional order condition this is achieved by basing the analysis on the abstract framework of semi groups the convergence analysis also includes generalizations to splittings consisting of more then two operators and to variable time steps we conclude by illustrating that the abstract results are applicable in the context of the schrodinger equation with an external magnetic field or with an unbounded potential

the dynamic critical exponent of the three dimensional ising model we measure the dynamic exponent of the three dimensional ising model using a damage spreading monte carlo approach as described by macisaac and jan we simulate systems froml 5 tol 60 at the critical temperature tc 4 5115 we report a dynamic exponent z 2 35 0 05 a value much larger than the consensus value of 2 02 whereas if we assume logarithmic corrections we find thatz 2 05 0 05

billion atom synchronous parallel kinetic monte carlo simulations of critical 3d ising systems an extension of the synchronous parallel kinetic monte carlo spkmc algorithm developed by martinez et al j comp phys 227 2008 3804 to discrete lattices is presented the method solves the master equation synchronously by recourse to null events that keep all processors time clocks current in a global sense boundary conflicts are resolved by adopting a chessboard decomposition into non interacting sublattices we find that the bias introduced by the spatial correlations attendant to the sublattice decomposition is within the standard deviation of serial calculations which confirms the statistical validity of our algorithm we have analyzed the parallel efficiency of spkmc and find that it scales consistently with problem size and sublattice partition we apply the method to the calculation of scale dependent critical exponents in billion atom 3d ising systems with very good agreement with state of the art multispin simulations

attractivity of the ginzburg landau mode distribution for a pattern forming system with marginally stable long modes abstract the mathematical theory of the description of pattern forming systems close to the first instability via the ginzburg landau equation is based on approximation and attractivity results this theory allowed to prove global existence results and upper semicontinuity of attractors for classical hydrodynamical stability problems such as the couette taylor problem recently approximation results for the ginzburg landau approximation for pattern forming systems with marginally stable long modes such as the benard marangoni system have been shown it is the purpose of this paper to prove the second fundamental property namely the attractivity of the ginzburg landau mode distribution for such systems

positive linear observers for linear compartmental systems linear compartmental systems are mathematical systems that are frequently used in biology and mathematics the inputs states and outputs of such systems are positive because they denote amounts or concentrations of material for linear dynamic systems the observer problem has been solved the purpose of the observer problem is to determine a linear observer such that the state can be approximated the difference between the state and its estimate should converge to zero the interpretation in terms of a physical system requires that an estimate of the state be positive like the state itself in this paper conditions on the system matrices are presented that guarantee that there exists a positive linear observer such that both the error converges to zero and the estimate is positive

laminarizing effects of dispersion in an active dissipative nonlinear medium abstract the kuramoto sivashinsky equation has become a popular prototype for systems which exhibit spatial temporal chaos we show here that a linear dispersion term h xxx tends to arrest such irregular behavior in favor of spatially periodic cellular structures as is consistent with prior numerical and experimental observations the study includes a normal form analysis and a numerical investigation of periodic and solitary stationary wave solutions of the equation it is shown that by 1 1 the infinite families of stationary periodic waves of the kuramoto sivashinsky equation each ending in a solitary wave have been annihilated successively such that only a lone family of periodic waves consisting of one hump kdv pulses for 3 7 remains as the only periodic stationary wave attractors of the system these periodic waves have much larger domains of attraction than the strange attractors and hence tend to dominate spatial temporal chaos in an extended domain with significant dispersion

hexagonal patterns in finite domains in many mathematical models for pattern formation a regular hexagonal pattern is stable in an infinite region however laboratory and numerical experiments are carried out in finite domains and this imposes certain constraints on the possible patterns in finite rectangular domains it is shown that a regular hexagonal pattern cannot occur if the aspect ratio is rational in practice it is found experimentally that in a rectangular region patterns of irregular hexagons are often observed this work analyses the geometry and dynamics of irregular hexagonal patterns these patterns occur in two different symmetry types either with a reflection symmetry involving two wavenumbers or without symmetry involving three different wavenumbers the relevant amplitude equations are studied to investigate the detailed bifurcation structure in each case it is shown that hexagonal patterns can bifurcate subcritically either from the trivial solution or from a pattern of rolls numerical simulations of a model partial differential equation are also presented to illustrate the behaviour