convergence to equilibria in scalar nonquasimonotone functional differential equations abstract we consider a class of scalar functional differential equations generating a strongly order preserving semiflow with respect to the exponential ordering introduced by smith and thieme it is shown that the boundedness of all solutions and the stability properties of an equilibrium are exactly the same as for the ordinary differential equation which is obtained by ignoring the delays the result on the boundedness of the solutions combined with a convergence theorem due to smith and thieme leads to explicit necessary and sufficient conditions for the convergence of all solutions starting from a dense subset of initial data under stronger conditions guaranteeing that the functional differential equation is asymptotically equivalent to a scalar ordinary differential equation a similar result is proved for the convergence of all solutions

a comparison of estimators for 1 f noise abstract we use a monte carlo approach to investigate the performance of five different time series estimators of the exponent in 1 f noise we find that a maximum likelihood estimator is markedly superior to fourier regression methods and hurst exponent methods the results indicate that useful estimates of can be made from time series that are much shorter than generally presumed

retracted asymptotic constancy for a differential equation with multiple state dependent delays in this paper we investigate the asymptotic behavior of solutions to a differential equation with multiple state dependent delays it is shown that every bounded solution of such an equation tends to a constant as t our results improve and extend some corresponding ones already known

sharp conditions for existence of nontrivial invariant cones of nonnegative initial values of difference equations it is proved that an arbitrary scalar difference equation has a nontrivial invariant cone in the set of the nonnegative initial values if and only if the associated characteristic equation has a positive root by using this fact there is given a construction of the maximal invariant cone in the set of nonnegative initial values via a positive root of the associated characteristic equation finally the present results are compared with some earlier ones which were obtained for delay differential equations in the same direction

degenerate asymptotic perturbation theory asymptotic rayleigh schrodinger perturbation theory for discrete eigenvalues is developed systematically in the general degenerate case for this purpose we study the spectral properties ofm m matrix functionsa of a complex variable which have an asymptotic expansion ea k k as 0 we show that asymptotic expansions for groups of eigenvalues and for the corresponding spectral projections ofa can be obtained from the set a by analytic perturbation theory special attention is given to the case wherea is borel summable in some sector originating from 0 with opening angle here we prove that the asymptotic series describe individual eigenvalues and eigenprojections ofa which are shown to be holomorphic ins near 0 and borel summable ifa k a k for allk we then fit these results into the scheme of rayleigh schrodinger perturbation theory and we give some examples of asymptotic estimates for schrodinger operators

positive operators and elliptic eigenvalue problems on considere le probleme aux valeurs propres lu mu sur et u 0 sur dans un domaine borne lisse cr n l est une expression differentielle uniformement elliptique a coefficients a valeurs reels continus de holder le coefficient de u dans l est non negatif est nu x est une fonction continue a valeur reelle on utilise la theorie abstraite des operateurs positifs pour retrouver certains resultats

multiscaled randomness a possible source of 1 f noise in biology we evaluate the possibility that the 1 f fluctuations observed in many biological time series result simply from the fact that biological processes have many inputs with differing time scales we present a stochastic model whose output is the summation of multiple random inputs i e different regulatory mechanisms we derive the conditions under which the model reproduces the complex fluctuations and 1 f scaling observed in biological systems simulations demonstrate that if model parameters are unconstrained the likelihood of generating 1 f noise is quite small thus while the model can be used to generate 1 mathit f mathrm ensuremath beta noise with various scaling exponents it is unlikely that the 1 f behavior observed in many biological systems is due only to the fact that these systems are regulated by many different inputs acting on different time scales textcopyright 1996 the american physical society

retracted asymptotic behavior of solutions to a system of differential equations with state dependent delays in this paper we investigate the asymptotic behavior of solutions to a system of differential equations with state dependent delays it is shown that every bounded solution of such a system tends to a constant vector as t our results improve and extend some corresponding ones already known

global asymptotic behavior of a chemostat model with discrete delays this paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat the model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations in the case where only two species are engaged in competition it is shown that competitive exclusion holds for any monotone growth response functions sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n species case in regard to the delay effects on the qualitative outcome of competition it is demonstrated that when the delays are relatively small the predictions of the model are identical with the predictions given by corresponding models without time delays however including large delays in the model may alter the predicted outcome of competition the techniques used also work when different removal rates are permitted an

embedding as a modeling problem abstract standard approaches to time delay embedding will often fail to provide an embedding that is useful for many common applications this happens in particular when there are multiple timescales in the dynamics we present a modified procedure non uniform embedding which overcomes such problems in many cases for more complex nonlinear dynamics we introduce variable embedding where in a suitable sense the embedding changes with the state of the system we show how to implement these procedures by combining embedding and modeling into a single procedure with a single optimization goal

on selecting models for nonlinear time series abstract constructing models from time series with nontrivial dynamics involves the problem of how to choose the best model from within a class of models or to choose between competing classes this paper discusses a method of building nonlinear models of possibly chaotic systems from data while maintaining good robustness against noise the models that are built are close to the simplest possible according to a description length criterion the method will deliver a linear model if that has shorter description length than a nonlinear model we show how our models can be used for prediction smoothing and interpolation in the usual way we also show how to apply the results to identification of chaos by detecting the presence of homoclinic orbits directly from time series

spectral analysis and stability properties of a relativistic deformation of the harmonic oscillator the spectral properties are established and the stability of the eigenvalues under the singular perturbation representing the relativistic corrections are proven for a family of ordinary differential operators describing a relativistic deformation of the quantum harmonic oscillator

limit set dichotomy and multistability for a class of cooperative neural networks with delays recent papers have pointed out the interest to study convergence in the presence of multiple equilibrium points eps multistability for neural networks nns with nonsymmetric cooperative nonnegative interconnections and neuron activations modeled by piecewise linear pl functions one basic difficulty is that the semiflows generated by such nns are monotone but due to the horizontal segments in the pl functions are not eventually strongly monotone esm this notwithstanding it has been shown that there are subclasses of irreducible interconnection matrices for which the semiflows although they are not esm enjoy convergence properties similar to those of esm semiflows the results obtained so far concern the case of cooperative nns without delays the goal of this paper is to extend some of the existing results to the relevant case of nns with delays more specifically this paper considers a class of nns with pl neuron activations concentrated delays and a nonsymmetric cooperative interconnection matrix a and delay interconnection matrix a the main result is that when a a satisfies a full interconnection condition then the generated semiflows which are monotone but not esm satisfy a limit set dichotomy analogous to that valid for esm semiflows it follows that there is an open and dense set of initial conditions in the state space of continuous functions on a compact interval for which the solutions converge toward an ep the result holds in the general case where the nns possess multiple eps i e is a result on multistability and is valid for any constant value of the delays

exponentially accurate error estimates of quasiclassical eigenvalues ii several dimensions we study the behavior of truncated rayleigh schrodinger series for low lying eigenvalues of the time independent schrodinger equation in the semiclassical limit 0 in particular we prove that if the potential energy satisfies certain conditions there is an optimal truncation of the series for the eigenvalues in the sense that this truncation is exponentially close to the exact eigenvalue these results were already discussed for the one dimensional case in a previous article this time we consider the multi dimensional problem where degeneracy plays a central role

detecting periodicity in experimental data using linear modeling techniques fourier spectral estimates and to a lesser extent the autocorrelation function are the primary tools to detect periodicities in experimental data in the physical and biological sciences we propose a new method which is more reliable than traditional techniques and is able to make clear identification of periodic behavior when traditional techniques do not this technique is based on an information theoretic reduction of linear autoregressive models so that only the essential features of an autoregressive model are retained these models we call reduced autoregressive models rarm the essential features of reduced autoregressive models include any periodicity present in the data we provide theoretical and numerical evidence from both experimental and artificial data to demonstrate that this technique will reliably detect periodicities if and only if they are present in the data there are strong information theoretic arguments to support the statement that rarm detects periodicities if they are present surrogate data techniques are used to ensure the converse furthermore our calculations demonstrate that rarm is more robust more accurate and more sensitive than traditional spectral techniques

long range correlation properties of stationary linear models with mixed periodicities we consider the problem of stationary and linear source systems which generate time series data with long range correlations we use the discrete fourier transform dft and build stationary linear models using artificial time series data exhibiting a 1 f spectrum where the models can include only terms that contribute significantly to the model as assessed by information criteria the result is that the optimal best model is only composed of mixed periodicities that is the model does not include all continuous periodicities and the time series data generated by the model exhibit a clear 1 f spectrum in a wide frequency range it is considered that as the 1 f spectrum is a consequence of the contributions of all periods consecutive periods are indispensable to generate such data by stationary linear models however the results indicate that there are cases where this expectation is not always met these results also imply that although we can know linear features of time series data using the dft we always cannot substantially infer the type of the source system even if the system is stationary linear

quantum degeneracy and classical perturbation theory the degenerate rayleigh schrodinger perturbation theory for the eigenvalues of the class of schrodinger operators s e s0 ev where s0 is the laplace beltrami operator on the one dimensional torus t1 and v is an analytic potential perturbed rotator is explicitly constructed by means of the algorithm of the classical resonant birkhoff normal form as a consequence the degenerate quantum perturbation theory tends to the classical one in the classical limit

cyclic thermodynamic processes and entropy production we study the time evolution of a periodically driven quantum mechanical system coupled to several reservoirs of free fermions at different temperatures this is a paradigm of a cyclic thermodynamic process we introduce the notion of a floquet liouvillean as the generator of the dynamics of the coupled system on an extended hilbert space we show that the time periodic state which the state of the coupled system converges to after very many periods corresponds to a zero energy resonance of the floquet liouvillean we then show that the entropy production per cycle is strictly positive a property that implies carnot s formulation of the second law of thermodynamics

universal generation of 1 f noises this paper establishes a universal mechanism for the generation of 1 f noises the mechanism is based on a signal superposition model which superimposes signals transmitted from independent sources all sources transmit a statistically common yet arbitrary stochastic signal pattern each source has its own random transmission parameters amplitude frequency and initiation epoch we explore randomizations of the transmission parameters which render the power spectrum of the super imposed signal invariant with respect of the stochastic signal pattern transmitted analysis shows that the aforementioned randomizations yield superimposed signals which are 1 f noises the result obtained is in effect a randomized central limit theorem for 1 f noises

temporal exponential decay for the stark effect in atoms abstract let h 1 i 1 n i v x i 1 i x i x j 1 v x i n x y 1 y d y with a normalized gaussian suppose e 0 and that h h 1 e i 1 n x i has no eigenfunctions in l 2 r 3n if h 1 with ess h 1 then e ith decays exponentially at a rate governed by the positions of the resonances of h

on asymptotic perturbation theory for quantum mechanics almost invariant subspaces and gauge invariant magnetic perturbation theory singular perturbation theory for quantum mechanics is considered in a framework generalizing the spectral concentration theory under very general conditions asymptotic estimations on the rayleigh schrodinger expansions of the perturbed spectral projections are obtained as a consequence almost invariant subspaces of exponential order are constructed the results cover practically all singular perturbations considered in nonrelativistic quantum mechanics in the magnetic field case under the condition that the magnetic field does not increase at infinity a gauge invariant perturbation theory leading to convergent series with field dependent coefficients is developed

canonical expansion of symmetric operators and perturbation theory let h be any symmetric schrodinger operator of the type 2 x21 x2d igw x1 xd on where w is any odd homogeneous polynomial and it is proved that is self adjoint and that its eigenvalues coincide up to a sign with the singular values of h i e the eigenvalues of moreover we explicitly construct the canonical expansion of h and determine the singular values j of h through the borel summability of their divergent perturbation theory the singular values yield estimates of the location of the eigenvalues j of h by weyl s inequalities

delayed stochastic systems noise and time delay are two elements that are associated with many natural systems and often they are sources of complex behaviors understanding of this complexity is yet to be explored particularly when both elements are present as a step to gain insight into such complexity for a system with both noise and delay we investigate such delayed stochastic systems both in dynamical and probabilistic perspectives a langevin equation with delay and a random walk model whose transition probability depends on a fixed time interval past delayed random walk model are the subjects of in depth focus as well as considering relations between these two types of models we derive an approximate fokker planck equation for delayed stochastic systems and compare its solution with numerical results

new generic quasi convergence principles with applications in this paper essentially strongly order preserving and conditionally set condensing semiflows are considered obtained is a new type of generic quasi convergence principles implying the existence of an open and dense set of stable quasi convergent points when the state space is order bounded the generic quasi convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations giving some results of theoretical and practical significance

aging wiener khinchin theorem and critical exponents of 1 f β noise the power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the wiener khinchin theorem we here generalize the wiener khinchin theorem for nonstationary processes and introduce a time dependent power spectrum ensuremath langle s t m ensuremath omega ensuremath rangle where t m is the measurement time for processes with an aging autocorrelation function of the form ensuremath langle i t i t ensuremath tau ensuremath rangle t mathrm ensuremath upsilon ensuremath phi mathrm ea ensuremath tau t where ensuremath phi mathrm ea x is a nonanalytic function when x is small we find aging 1 f ensuremath beta noise aging 1 f ensuremath beta noise is characterized by five critical exponents we derive the relations between the scaled autocorrelation function and these exponents we show that our definition of the time dependent spectrum retains its interpretation as a density of fourier modes and discuss the relation to the apparent infrared divergence of 1 f ensuremath beta noise we illustrate our results for blinking quantum dot models single file diffusion and brownian motion in a logarithmic potential

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

the interaction of migratory birds and domestic poultry and its role in sustaining avian influenza we investigate the role of migratory birds in the spread of h5n1 avian influenza focusing on the interaction of a migratory bird species with nonmigratory poultry the model is of patch type and is derived with the aid of reaction advection equations for the migratory birds in the air along the flyways poultry may reside at some or all of the four patches of the model which consist of the breeding patch for the migratory birds their winter feeding patch and two stopover patches where birds rest and refuel on their migration outward and return migratory routes can be different the equations for the migratory birds contain time delays which represent the flight times for migratory birds along particular sectors delays also appear in the model coefficients via quantities which represent flight survival probabilities for the various sectors we establish results on positivity boundedness global asymptotic stability of the disease free equilibrium and the persistence of infection we also discuss extensions of the model to include the seasonality of the migration phenomenon numerical simulations support the analytical findings here we used data on h5n1 infected ducks in the poyang lake region of china

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

exponentially accurate error estimates of quasiclassical eigenvalues we study the behaviour of truncated rayleigh schrodinger series for the low lying eigenvalues of the one dimensional time independent schrodinger equation in the semiclassical limit 0 under certain hypotheses on the potential v x we prove that for any given small 0 there is an optimal truncation of the series for the approximate eigenvalues such that the difference between an approximate and exact eigenvalue is smaller than exp c for some positive constant c we also prove the analogous results concerning the eigenfunctions

stability of schrodinger eigenvalue problems we derive a general stability criterion for discrete eigenvalues of schrodinger operators such asa p 2 v x using only strong continuity ofa anda in the perturbation parameter the theory is developed for non selfadjoint operators and illustrated with examples like the anharmonic oscillator the stark and the zeeman effect the principal tools are weyl s criterion for the essential spectrum and a construction due to enss 5 they are also used to extend the classical invariance theorems for the essential spectrum to certain singular perturbations including some local perturbations of the laplacian by differential operators of arbitrary high order

on the periodic solutions of periodic multi species competitive systems with delays the paper considers general n species periodic kolmogorov competitive systems of functional differential equations with finite or infinite delays applying schauder s fixed point theorem a criterion on the existence of the positive periodic solutions is established as applications the existence and global asymptotic stability of the single species logistic periodic model and n species lotka volterra type periodic model with delays are discussed