dynamics of coarse grained variables in spatially extended systems abstract some new ways to probe the dynamics of spatially extended systems giving rise to spatio temporal chaos are explored coarse grained observables suitable for describing large scale behavior are introduced by locally averaging the original variables over a spatial region and their stability spatial correlation function and statistical properties are compared to those of the original variables in general these observables do not obey closed form equations truncation schemes in which high order fourier modes or variables other than the coarse grained ones are discarded are developed and their sensitivity to the parameters is assessed

comparison of systems with complex behavior abstract we present a formalism for comparing the asymptotic dynamics of dynamical systems with physical systems that they model based on the spectral properties of the koopman operator we first compare invariant measures and discuss this in terms of a statistical takens theorem proved here we also identify the need to go beyond comparing only invariant ergodic measures of systems and introduce an ergodic theoretic treatment of a class of spectral functionals that allow for this the formalism is extended for a class of stochastic systems discrete random dynamical systems the ideas introduced in this paper can be used for parameter identification and model validation of driven nonlinear models with complicated behavior as an illustration we provide an example in which we compare the asymptotic behavior of a combustion system measured experimentally with the asymptotic behavior of a class of models that have the form of a random dynamical system

exposure response relationships for transportation noise this article presents synthesis curves for the relationship between dnl and percentage highly annoyed for three transportation noise sources the results are based on all 21 datasets examined by schultz j acoust soc am 64 377 405 1978 and fidell et al j acoust soc am 89 221 233 1991 for which acceptable dnl and percentage highly annoyed measure could be derived augmented with 34 datasets separate nonidentical curves were found for aircraft road traffic and railway noise a difference between sources was found using data for all studies combined and for only those studies in which respondents evaluated two sources the latter outcome strengthens the conclusion that the differences between sources cannot be explained by differences in study methodology

community response to noise a review of factors influencing the relationship between noise exposure and reaction social surveys of the relationship between noise exposure and the subjective reactions to it were reviewed this review indicated that remarkably similar results have been obtained across different nationalities with different measurement techniques only a small percentage typically less than 20 of the variation in individual reaction is accounted for by noise exposure analysis of potential errors in both measurement of noise and reaction suggests that elimination of errors would only slightly increase the observed correlations variables such as attitude to the noise source and sensitivity to noise account for more variation in reaction than does noise exposure the weaker relationship between noise exposure and attitude than between reaction and attitude suggests that the attitude reaction relationship is not entirely due to noise exposure causing a change in attitude itself noise reaction correlations based on individual data are significantly lower in studies of impulsive noise than nonimpulsive noise this may be caused in part by the restricted range of noise exposure studied in some socioacoustic investigations of impulsive noise however the significantly higher correlations of attitude and reaction in impulsive noise studies suggest that attitude plays an even larger part while noise exposure plays a lesser part in determining reaction to impulsive noise relative to nonimpulsive noise

ordinary dichotomy and global linearization abstract suppose that f t x t t d t f t x 1 f t x 2 r t x 1 x 2 r t d t c c is some constant then if system x a t x has an ordinary dichotomy then x a t x f t x is topologically equivalent to x a t x if system x a t x has an ordinary dichotomy with asymptotically stable manifolds then x a t x f t x is strongly topologically equivalent to x a t x x f t x can be considered as a perturbation of the linear system x 0 which has an ordinary dichotomy the structure of the solution set of x f t x is clear since x f t x is strongly topologically equivalent to x 0

on the complete integrability and linearization of nonlinear ordinary differential equations ii third order equations we introduce a method for finding general solutions of third order nonlinear differential equations by extending the modified prelle singer method we describe a procedure to deduce all the integrals of motion associated with the given equation so that the general solution follows straightforwardly from these integrals the method is illustrated with several examples further we propose a powerful method of identifying linearizing transformations the proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation in addition to the above we provide an algorithm to invert the non local linearizing transformation through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation

non perturbative linearization of dynamical systems we consider the classical problem of linearizing a vector field x around a fixed point we adopt a non perturbative point of view based on the symmetry properties of linear vector fields

isostables isochrons and koopman spectrum for the action angle representation of stable fixed point dynamics abstract for asymptotically periodic systems a powerful phase reduction of the dynamics is obtained by computing the so called isochrons i e the sets of points that converge toward the same trajectory on the limit cycle motivated by the analysis of excitable systems a similar reduction has been attempted for non periodic systems admitting a stable fixed point in this case the isochrons can still be defined but they do not capture the asymptotic behavior of the trajectories instead the sets of interest that we call isostables are defined in the literature as the sets of points that converge toward the same trajectory on a stable slow manifold of the fixed point however it turns out that this definition of the isostables holds only for systems with slow fast dynamics also efficient methods for computing the isostables are missing the present paper provides a general framework for the definition and the computation of the isostables of stable fixed points which is based on the spectral properties of the so called koopman operator more precisely the isostables are defined as the level sets of a particular eigenfunction of the koopman operator through this approach the isostables are unique and well defined objects related to the asymptotic properties of the system also the framework reveals that the isostables and the isochrons are two different but complementary notions which define a set of action angle coordinates for the dynamics in addition an efficient algorithm for computing the isostables is obtained which relies on the evaluation of laplace averages along the trajectories the method is illustrated with the excitable fitzhugh nagumo model and with the lorenz model finally we discuss how these methods based on the koopman operator framework relate to the global linearization of the system and to the derivation of special lyapunov functions

separation of fast and slow variables for a linear system by the method of multiple time scales abstract a matrix method introduced by geigenmuller titulaer and felderhof for separating fast and slow motions of a system of first order linear differential equations is shown to arise in a natural way from a multiple time scales expansion the general method is then used to derive the smoluchowski equation and its initial data for brownian motion in a strong magnetic field from the corresponding kramers chandrasekhar initial value problem

the schultz curve 25 years later a research perspective the contemporary technical rationale for assessing effects impacts of transportation noise on communities rests in large part on a purely descriptive dosage effect relationship of the sort first synthesized by schultz j acoust soc am 64 377 405 1978 although u s federal adoption of an annoyance based rationale for regulatory policy has made this approach a familiar one it is only one of several historical perspectives and not necessarily the most useful for all purposes last reviewed by the u s federal interagency committee on noise ficon 10 years ago the accuracy and precision of estimates of the prevalence of a consequential degree of noise induced annoyance yielded by functions of noise exposure leave much to be desired this tutorial article traces the development of the dosage effect relationship on which ficon currently relies in a wider historical context of efforts to understand and predict community response to transportation noise it also identifies areas in which advances in genuine understanding might lead to improved means for predicting community response to transportation noise

approximation methods for nonlinearm accretive operator equations abstract suppose e is a real reflexive banach space and t d t e e is a locally lipschitzian m accretive operator where the domain of t d t is a proper subset of e for any f e approximation methods are constructed which converge strongly to the solution of the equation x tx f explicit error estimates are given and convergence is at least as fast a geometric progression

phase reduction theory for hybrid nonlinear oscillators hybrid dynamical systems characterized by discrete switching of smooth dynamics have been used to model various rhythmic phenomena however the phase reduction theory a fundamental framework for analyzing the synchronization of limit cycle oscillations in rhythmic systems has mostly been restricted to smooth dynamical systems here we develop a general phase reduction theory for weakly perturbed limit cycles in hybrid dynamical systems that facilitates analysis control and optimization of nonlinear oscillators whose smooth models are unavailable or intractable on the basis of the generalized theory we analyze injection locking of hybrid limit cycle oscillators by periodic forcing and reveal their characteristic synchronization properties such as ultrafast and robust entrainment to the periodic forcing and logarithmic scaling at the synchronization transition we also illustrate the theory by analyzing the synchronization dynamics of a simple physical model of biped locomotion

on iterations methods for zeros of accretive operators in banach spaces in this paper three iterations are designed to approach zeros of set valued accretive operators in banach spaces the first one is the continuous picard type iteration involving the resolvent the second one is the approximate picard type iteration involving the resolvent and the third one is the halpern type iteration involving the resolvent some strong convergence theorems for three iterations are proved

a centre manifold description of containment dispersion in channels with varying flow properties high order asymptotic approximations to the equation governing the longitudinal dispersion of a passive contaminant in poiseuille channel flow are derived and their validity discussed the derivation uses centre manifold theory which provides a systematic and near rigorous approach to calculating each successive approximation it also enables the derivation of the correct initial conditions for the taylor model of shear dispersion approximations that are valid when the channel is of a varying cross section are systematically derived via this approach and the effects of time dependent flow and variable diffusivity are also investigated the resultant modifications to the effective advection velocity and the effective dispersion coefficient are calculated and some general trends indicated

geometry of the ergodic quotient reveals coherent structures in flows abstract dynamical systems that exhibit diverse behaviors can rarely be completely understood using a single approach however by identifying coherent structures in their state spaces i e regions of uniform and simpler behavior we could hope to study each of the structures separately and then form the understanding of the system as a whole the method we present in this paper uses trajectory averages of scalar functions on the state space to a identify invariant sets in the state space and b to form coherent structures by aggregating invariant sets that are similar across multiple spatial scales first we construct the ergodic quotient the object obtained by mapping trajectories to the space of the trajectory averages of a function basis on the state space second we endow the ergodic quotient with a metric structure that successfully captures how similar the invariant sets are in the state space finally we parametrize the ergodic quotient using intrinsic diffusion modes on it by segmenting the ergodic quotient based on the diffusion modes we extract coherent features in the state space of the dynamical system the algorithm is validated by analyzing the arnold beltrami childress flow which was the test bed for alternative approaches the ulam s approximation of the transfer operator and the computation of lagrangian coherent structures furthermore we explain how the method extends the poincare map analysis for periodic flows as a demonstration we apply the method to a periodically driven three dimensional hill s vortex flow discovering unknown coherent structures in its state space finally we discuss differences between the ergodic quotient and alternatives propose a generalization to analysis of quasi periodic structures and lay out future research directions

an explicit iteration for zeros of accretive operators abstract in this paper for lipschitz accretive operator a an iteration scheme is defined as follows x n 1 1 n x n n u n ax n its strong convergence is established for finding some zero of a whenever n n 0 1 satisfying conditions lim n n 0 n 1 n lim n n 0 furthermore some applications for equilibrium problems are given also in particular the iteration coefficient is simpler and more general

kuramoto sivashinsky dynamics on the center unstable manifold this paper studies the dynamical behavior of solutions of the kuramoto sivashinsky partial differential equation with periodic boundary conditions on a spatial interval 0 h the length h is the bifurcation parameter and reduction is made to a two complex dimensional system on a local center unstable manifold near the second bifurcation point h 2 from the trivial solution the resulting o 2 equivariant system displays all the behavior found in high precision simulations of the partial differential equation near this bifurcation point in particular bifurcation sequences to stable traveling waves unstable modulated traveling waves and attracting heteroclinic cycles are reproduced qualitatively and quantitatively within 1 in the parameter range h 2 pm 20 a clear understanding of the global dynamical behavior in this region is thus obtained

strong convergence of approximants to fixed points of lipschitzian pseudocontractive maps abstract let e be a real q uniformly smooth banech space which is also uniformly convex for example l or 1 spaces 1 p 00 and k be a nonempty closed convex and bounded subset of e with 4 int k let t k be a lipschitzian pseudocontractive mapping such that for z e int k 1i z trll 112 tzll f or all i e a k then for zo e k arbitrary the iteration process z defined by zn l l l z l vn 1 cu z cr tz converges strongly to a fixed point of t provided that p and an satisfy certain conditions moreover if t is strictly pseudocontractive with a nonempty fixed point set then it is proved that the mann type iteration scheme converges strongly to a fixed point of t 2002 elsevier science ltd all rights reserved keywords accretive operators q uniformly smooth spaces duality maps uniformly continuous multivalued maps 1 introduction

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

spectral decomposition of a fokker planck equation at criticality the mean field for a complex network consisting of a large but finite number of random two state elements m has been shown to satisfy a nonlinear langevin equation the noise intensity is inversely proportional to sqrt m in the limiting case m infty the solution to the langevin equation exhibits a transition from exponential to inverse power law relaxation as criticality is approached from above or below the critical point when m infty the inverse power law is truncated by an exponential decay with rate vargamma the evaluation of which is the main purpose of this article an analytic numeric approach is used to obtain the lowest order eigenvalues in the spectral decomposition of the solution to the corresponding fokker planck equation and its equivalent schrodinger equation representation

closing the hierarchy of moment equations in nonlinear dynamical systems the moment equations associated with the evolution of the probability density are known to form an infinite hierarchy of coupled equations in nonlinear dynamical systems in the present paper a systematic approach for closing this hierarchy is proposed based on the ansatz that in the long time limit there exist groups of moments varying on the same time scale the method is applied to a one dimensional vector field in the presence of noise and to two prototypes of chaotic behavior excellent agreement with numerical results is obtained special emphasis is placed on the role of symmetries and on the origin of the composite oscillations found for certain types of moments in the chaotic systems

spectral decomposition of nonlinear systems with memory we present an alternative approach to the analysis of nonlinear systems with long term memory that is based on the koopman operator and a levy transformation in time memory effects are considered to be the result of interactions between a system and its surrounding environment the analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale on average the time evolution of a mode follows a mittag leffler function and the system can be described using the fractional calculus the general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation when analyzing data from an ill defined black box system the spectral decomposition in terms of mittag leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods consequently the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators or operators with long term memory in time when model details are unknown

a new renormalization method for the asymptotic solution of weakly nonlinear vector systems this paper considers the asymptotic integration of a special class of initial value problems involving a nonlinear regular perturbation scaled by a small parameter epsilon 0 for t mathcal o 1 epsilon these problems were classically solved using either the method of averaging or of multiple scales to remove secular terms that arise in the natural power series procedure our new ansatz is straightforward and effective moreover it indicates when problems might occur in providing the asymptotic solution on very long time intervals other closely related problems are also attacked using renormalization

asymptotic stability equals exponential stability and iss equals finite energy gain if you twist your eyes in this paper we show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables the same is shown for input to state stability and input to state exponential stability and for input to state exponential stability and a nonlinear h estimate

on the complete integrability and linearization of certain second order nonlinear ordinary differential equations a method for finding general solutions of second order nonlinear ordinary differential equations by extending the prelle singer ps method is briefly discussed we explore integrating factors integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the ps method we also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second order nonlinear ordinary differential equations to free particle equations we illustrate the theory with several potentially important examples and show that our procedure is widely applicable

approximation methods for nonlinear operator equations of the m accretive type abstract let e be a real banach space which is both uniformly convex and uniformly smooth let t d t e e be m accretive where the domain of t d t is a proper subset of e for any e approximation methods are constructed which converge strongly to a solution of the operator equation x tx explicit error estimates are obtained a related result deals with operator equations of the dissipative type

a holistic finite difference approach models linear dynamics consistently i prove that a centre manifold approach to creating finite difference models will consistently model linear dynamics as the grid spacing becomes small using such tools of dynamical systems theory gives new assurances about the quality of finite difference models under nonlinear and other perturbations on grids with finite spacing for example the linear advection diffusion equation is found to be stably modelled for all advection speeds and all grid spacings the theorems establish an extremely good form for the artificial internal boundary conditions that need to be introduced to apply centre manifold theory when numerically solving nonlinear partial differential equations this approach can be used to systematically derive finite difference models which automatically have excellent characteristics their good performance for finite grid spacing implies that fewer grid points may be used and consequently there will be less difficulties with stiff rapidly decaying modes in continuum problems

pulse based control using koopman operator under parametric uncertainty in applications such as biomedicine and systems synthetic biology technical limitations in actuation complicate implementation of time varying control signals in order to alleviate some of these limitations it may be desirable to derive simple control policies such as step functions with fixed magnitude and length or temporal pulses in this technical note we further develop a recently proposed pulse based solution to the convergence problem i e minimizing the convergence time to the target exponentially stable equilibrium for monotone systems in particular we extend this solution to monotone systems with parametric uncertainty our solutions also provide worst case estimates on convergence times furthermore we indicate how our tools can be used for a class of nonmonotone systems and more importantly how these tools can be extended to other control problems we illustrate our approach on switching under parametric uncertainty and regulation around a saddle point problems in a genetic toggle switch system

initial conditions for models of dynamical systems abstract the long time behaviour of many dynamical systems may be effectively predicted by a low dimensional model that describes the evolution of a reduced set of variables we consider the question of how to equip such a low dimensional model with appropriate initial conditions so that it faithfully reproduces the long term behaviour of the orginal high dimensional dynamical system our method involves putting the dynamical system into normal form which not only generates the low dimensional model but also provides the correct initital conditions for the model we illustrate the method with several examples

linearization in the large of nonlinear systems and koopman operator spectrum abstract according to the hartman grobman theorem a nonlinear system can be linearized in a neighborhood of a hyperbolic stationary point here we extend this linearization around stable unstable equilibria or periodic orbits to the whole basin of attraction for both discrete diffeomorphisms and flows we discuss the connection of the linearizing transformation to the spectrum of koopman operator

convergence properties for discrete time nonlinear systems three similar convergence notions are considered two of them are the long established notions of convergent dynamics and incremental stability the other is the more recent notion of contraction analysis all three convergence notions require that all solutions of a system converge to each other in this note we investigate the differences between these convergence properties for discrete time and time varying nonlinear systems by comparing the properties in pairs and using examples we also demonstrate a time varying smooth lyapunov function characterization for each of these convergence notions and with appropriate assumptions we provide several sufficient conditions to establish relationships between these properties in terms of lyapunov functions

synchronization in networks of general weakly nonlinear oscillators we present a general approach to the study of synchrony in networks of weakly nonlinear systems described by singularly perturbed equations of the type x x ef x x 0 by performing a perturbative calculation based on normal form theory we analytically obtain an o approximation to the floquet multipliers that determine the stability of the synchronous solution the technique allows us to prove and generalize recent results obtained using heuristic approaches as well as reveal the structure of the approximating equations we illustrate the results in several examples and discuss extensions to the analysis of stability of multisynchronous states in networks with complex architectures

resonances in a chaotic attractor crisis of the lorenz flow local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions lyapunov exponents and their associated covariant lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems however it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle this behavior is investigated here for a boundary crisis in the lorenz flow for which neither the lyapunov exponents nor the covariant lyapunov vectors provide a criterion for the crisis instead the convergence of the time evolution of probability densities to the invariant measure governed by the semigroup of transfer operators is expected to slow down at the approach of the crisis such convergence is described by the eigenvalues of the generator of this semigroup which can be divided into two families referred to as the stable and unstable ruelle pollicott resonances respectively the former describes the convergence of densities to the attractor or escape from a repeller and is estimated from many short time series sampling the state space the latter is responsible for the decay of correlations or mixing and can be estimated from a long times series invoking ergodicity it is found numerically for the lorenz flow that the stable resonances do approach the imaginary axis during the crisis as is indicative of the loss of global stability of the attractor on the other hand the unstable resonances and a fortiori the decay of correlations do not flag the proximity of the crisis thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises

the application of centre manifolds to amplitude expansions i ordinary differential equations abstract the theory of centre manifolds for a system of ordinary differential equations is summarized and its relationship to amplitude expansions based upon multiple time scales is discussed emphasis is placed upon the practical computational aspects of applying centre manifold theory to near critical problems and in particular to the computation of the centre manifold the calculations are illustrated by a detailed analysis of two problems in fluid mechanics

extending phase reduction to excitable media theory and applications phase reduction methods have been tremendously useful for understanding the dynamics of nonlinear oscillators but have been difficult to extend to systems with a stable fixed point such as an excitable system using the notion of isostables which measure the time it takes for a given initial condition in phase space to approach a stable fixed point we present a general method for isostable reduction for excitable systems we also devise an adjoint method for calculating infinitesimal isostable response curves which are analogous to infinitesimal phase response curves for oscillatory systems through isostable reduction we are able to implement sophisticated control strategies in a high dimensional model of cardiac activity for the termination of alternans a precursor to cardiac fibrillation

strongly topological linearization with generalized exponential dichotomy abstract it has been proved that if x a t x has a generalized exponential dichotomy and f t x satisfies certain conditions then the nonlinear system x a t x f t x is topologically equivalent to its linear system x a t x in this paper we prove that if the condition a t m a t is added then x a t x f t x is strongly topologically equivalent to x a t x where m is some positive number and a t is the eigenfunction of the generalized exponential dichotomy and therefore the corresponding solutions of x a t x f t x and x a t x have the same stability

classical nonlinear response of a chaotic system ii langevin dynamics and spectral decomposition the spectrum of a strongly chaotic system consists of discrete complex ruelle pollicott rp resonances we interpret the rp resonances as eigenstates and eigenvalues of the fokker planck operator obtained by adding an infinitesimal diffusion term to the first order liouville operator we demonstrate how the deterministic expression for the linear response is reproduced in the limit of vanishing noise for the second order response function we establish an equivalence of the spectral decomposition in the limit of vanishing noise and the long time asymptotic expansion in the deterministic case

global iterative schemes for accretive operators abstract let e be a real q uniformly smooth banach space and a e 2e be an m accretive operator which satisfies a linear growth condition of the form ax c 1 x for some constant c 0 and for all x e it is proved that if two real sequences n and n satisfy appropriate conditions the sequence xn generated from arbitrary x0 e by xn 1 xn n un n xn z un axn n 0 converges strongly to some x a 1 0 furthermore if e is a reflexive banach space with a uniformly g teaux differentiable norm and if every weakly compact convex subset of e has the fixed point property for nonexpansive mappings and a d a e 2e is m accretive then for arbitrary z x0 e the sequence xn defined by xn 1 n un 1 n xn 1 z xn en for un axn where en e is such that en

systematic elimination of fast variables in linear systems we consider linear systems in which some of the variables evolve on a much faster time scale than the remaining ones by means of a systematic perturbation theory in the time scale ratio we extract a reduced dynamics in terms of the slow variables only valid after an initial transient period the procedure provides a systematic extension of the usual adiabatic elimination scheme and gives corrections to it we also find an expression for the long time behavior of the correlation functions of the slow variables these asymptotic expressions do not extrapolate back towards the equal time correlations for t going to zero the reason for this initial slip is given and its magnitude calculated method and results are illustrated with a simple example of coupled oscillators

period adding bifurcations and chaos in a periodically stimulated excitable neural relaxation oscillator the response of an excitable neuron to trains of electrical spikes is relevant to the understanding of the neural code in this paper we study a neurobiologically motivated relaxation oscillator with appropriately identified fast and slow coordinates that admits an explicit mathematical analysis an application of geometric singular perturbation theory shows the existence of an attracting invariant manifold which is used to construct the fenichel normal form for the system this facilitates the calculation of the response of the system to pulsatile stimulation and allows the construction of a so called extended isochronal map the isochronal map is shown to have a single discontinuity and be of a type that can admit three types of response mode locked quasiperiodic and chaotic the bifurcation structure of the system is seen to be extremely rich and supports period adding bifurcations separated by windows of both chaos and periodicity a bifurcation analysis of the isochronal map is presented in conjunction with a description of the various routes to chaos in this system

spectral signature of the pitchfork bifurcation liouville equation approach the time evolution of probability densities of one dimensional nonlinear vector fields is studied using a liouville equation approach it is shown that the liouville operator admits a discrete spectrum of eigenvalues of decaying type if the vector field is far from bifurcation the associated right and left eigenvectors are explicitly constructed for simple models and shown to be distributions rather than regular functions on the other hand the spectrum of the liouville operator may become continuous at the bifurcation point a phenomenon illustrated explicitly in the paper in the case of the pitchfork bifurcation the relationship between the spectral decompositions of the liouville and of the fokker planck equations is discussed in particular the spectral decompositions constructed here for the liouville equation are obtained as the noiseless limit of the well known spectral decompositions of the fokker planck equation of the associated stochastic process

a spectral characterization of nonlinear normal modes belgian network dysco dynamical systems control and optimization funded by the interuniversity attraction poles programme initiated by the belgian science policy office

polymorphism in rapidly changing cyclic environment selection in a time periodic environment is modeled via the continuous time two player replicator dynamics which for symmetric payoffs reduces to the fisher equation of mathematical genetics for a sufficiently rapid and cyclic fine grained environment the time averaged population frequencies are shown to obey a replicator dynamics with a nonlinear fitness that is induced by environmental changes the nonlinear terms in the fitness emerge due to populations tracking their time dependent environment these terms can induce a stable polymorphism though they do not spoil the polymorphism that exists already without them in this sense polymorphic populations are more robust with respect to their time dependent environments the overall fitness of the problem is still given by its time averaged value but the emergence of polymorphism during genetic selection can be accompanied by decreasing mean fitness of the population the impact of the uncovered polymorphism scenario on the models of diversity is exemplified via the rock paper scissors dynamics and also via the prisoner s dilemma in a time periodic environment

normal hyperbolicity and linearisability it is well known that one can linearise a diffeomorphism near a compact invariant submanifold in the presence of 1 normal hyperbolicity in this note we give a counterexample to a statement suggested by c pugh and m shub that one can weaken this normal hyperbolicity assumption

the accurate dynamic modelling of contaminant dispersion in channels throw some dye into a flowing river how is the dye going to disperse along the river the dynamics may be modelled by taylor s one dimensional model of shear dispersion we discuss the creation and utility of models of the dispersion of the dye using invariant manifold theory the models are effective exponentially quickly we describe a one mode centre manifold and a two mode and a three mode invariant manifold model from the analysis we decide under what conditions these three models are valid using the geometric picture provided by the theory of invariant manifolds we find a complete set of initial conditions for the models so that the evolution of the full and approximate systems are identical after a short time throughout a comparison of the three models is made to identify the practical situations when each model will be preferred

normal form transforms separate slow and fast modes in stochastic dynamical systems modelling stochastic systems has many important applications normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics we explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes the thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems thus we not only must reduce the dimensionality of the dynamics but also endeavour to separate emph all slow processes from emph all fast time processes both deterministic and stochastic quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system the results will help us accurately model interpret and simulate multiscale stochastic systems

the mann process for perturbed m accretive operators in banach spaces this work was carried out while this author was visiting the university ofalabama in huntsville underthe nancial support by the lg yonam foundation 1998 and he would like to thank professor claudio h morales for his hospitality in the department of mathematics 0362 546x 01 see front matter 2001 elsevier science ltd all rights reserved pii s0362 546x 00 00115 2

reaction diffusion theory in the presence of an attractive harmonic potential problems involving the capture of a moving entity by a trap occur in a variety of physical situations the moving entity being an electron an excitation an atom a molecule a biological object such as a receptor cluster a cell or even an animal such as a mouse carrying an epidemic theoretical considerations have almost always assumed that the particle motion is translationally invariant we study here the case when that assumption is relaxed in that the particle is additionally subjected to a harmonic potential this tethering to a center modifies the reaction diffusion phenomenon using a smoluchowski equation to describe the system we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions interesting features emerge depending on the relative location of the trap the attractive center and the initial placement of the diffusing particle

a normal form of thin fluid film equations resolves the transient paradox abstract imagine two constant thickness thin films of fluid colliding together the transient flow forms a hump where they collide thereafter they slowly relax but apparently reliable lubrication models expressed only in the thickness of the fluid forecast that precisely nothing happens how can we resolve this paradox dynamical systems theory constructs a normal form of the navier stokes equations for the flow of a thin layer of fluid upon a solid substrate these normal form equations illuminate the fluid dynamics by decoupling the interesting long term lubrication flow from the rapid viscous decay of transient shear modes the normal form clearly shows the slow manifold of the lubrication model and demonstrates that the initial condition for the fluid thickness of the lubrication model is not the initial physical fluid thickness but instead is modified by any initial lateral shear flow with these initial conditions the lubrication model makes better forecasts this dynamical systems approach could similarly illuminate other models of complicated dynamics

a new class of linearizable equations using the symmetry approach we find a class of integrable nonlinear pdes with dispersion law all these equations turn out to be linearizable by means of a differential parametrization

spatiotemporal control to eliminate cardiac alternans using isostable reduction abstract cardiac alternans an arrhythmia characterized by a beat to beat alternation of cardiac action potential durations is widely believed to facilitate the transition from normal cardiac function to ventricular fibrillation and sudden cardiac death alternans arises due to an instability of a healthy period 1 rhythm and most dynamical control strategies either require extensive knowledge of the cardiac system making experimental validation difficult or are model independent and sacrifice important information about the specific system under study isostable reduction provides an alternative approach in which the response of a system to external perturbations can be used to reduce the complexity of a cardiac system making it easier to work with from an analytical perspective while retaining many of its important features here we use isostable reduction strategies to reduce the complexity of partial differential equation models of cardiac systems in order to develop energy optimal strategies for the elimination of alternans resulting control strategies require significantly less energy to terminate alternans than comparable strategies and do not require continuous state feedback

global stability analysis using the eigenfunctions of the koopman operator we propose a novel operator theoretic framework to study global stability of nonlinear systems based on the spectral properties of the so called koopman operator our approach can be regarded as a natural extension of classic linear stability analysis to nonlinear systems the main results establish the necessary and sufficient relationship between the existence of specific eigenfunctions of the koopman operator and the global stability property of hyperbolic fixed points and limit cycles these results are complemented with numerical methods which are used to estimate the region of attraction of the fixed point or to prove in a systematic way global stability of the attractor within a given region of the state space

hartman s linearization on nonautonomous unbounded system abstract we extend hartman s linearization theorem to the uniformly asymptotic stable and unbounded case we get the following conclusion there is a constant 0 such that the nonlinear system d x d t a t x f x t and its linear part d x d t a t x are topologically equivalent if the linear system is uniformly asymptotically stable and f x t satisfies lipschitz condition with constant

trace formula for noisy flows a trace formula is derived for the fokker planck equation associated with ito stochastic differential equations describing noisy time continuous nonlinear dynamical systems in the weak noise limit the trace formula provides estimations of the eigenvalues of the fokker planck operator on the basis of the pollicott ruelle resonances of the noiseless deterministic system which is assumed to be non bifurcating at first order in the noise amplitude the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo energy of the onsager machlup freidlin wentzell scheme

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

properties of center manifolds the center manifold has a number of puzzling properties associated with the basic questions of existence uniqueness differentiability and analyticity which may cloud its profitable application in e g bifurcation theory this paper aims to deal with some of these subtle properties regarding existence and uniqueness it is shown that the cut off function appear ing in the usual existence proof is responsible for the selection of a single center manifold thereby hiding the inherent nonuniqueness conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection this intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium in the case of nonuniqueness the difference between any two center manifolds will be less than el exp e2x l with el and e2 constants which explains why the formal taylor expansions of different center manifolds are the same while the expansions do not converge the differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost center manifolds are usually not analytic however an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant using this manifold a new and very simple proof of the lyapunov subcenter theorem is given

on the low dimensional modelling of stratonovich stochastic differential equations abstract we develop further ideas on how to construct low dimensional models of stochastic dynamical systems the aim is to derive a consistent and accurate model from the originally high dimensional system this is done with the support of centre manifold theory and techniques aspects of several previous approaches are combined and extended adiabatic elimination has previously been used but centre manifold techniques simplify the noise by removing memory effects and with less algebraic effort than normal forms analysis of associated fokker plank equations replace nonlinearly generated noise processes by their long term equivalent white noise the ideas are developed by examining a simple dynamical system which serves as a prototype of more interesting physical situations