invariant manifolds for impulsive equations and nonuniform polynomial dichotomies for impulsive differential equations in banach spaces we construct stable and unstable invariant manifolds for sufficiently small perturbations of a polynomial dichotomy we also consider the general case of nonuniform polynomial dichotomies moreover we introduce the notions of polynomial lyapunov exponent and of regularity coefficient for a linear impulsive differential equation and we show that when the lyapunov exponent never vanishes the linear equation admits a nonuniform polynomial dichotomy

nonconvex duality in optimal control we derive a representation of the minimum cost of an optimal control problem in terms of the upper envelope of generalized semisolutions of the hamilton jacobi equation

stable manifolds for delay equations and parameter dependence abstract we establish the existence of lipschitz stable invariant manifolds for semiflows generated by a delay equation x l t x t f t x t assuming that the linear equation x l t x t admits a nonuniform exponential dichotomy and that f is a sufficiently small lipschitz perturbation we also show that the stable invariant manifolds are lipschitz in the parameter

the synthesis of universal feedback pursuit strategies in differential games we show how any generalized supersolution of the hamilton jacobi equation can be used to construct a feedback pursuit strategy which guarantees to any given tolerance a capture time not exceeding the solution s value if the supersolution is the value function then a near optimal pursuit strategy is obtained in this way an important feature of the construction is its universal nature i e the fact that the feedback law is uniformly effective on compact sets of initial conditions this implies in particular that the feedback construction is one that exploits nonoptimal behavior on the part of the evader

dynamics of stochastic 2d navier stokes equations abstract in this paper we study the dynamics of a two dimensional stochastic navier stokes equation on a smooth domain driven by linear multiplicative white noise we show that solutions of the 2d navier stokes equation generate a perfect and locally compacting c 1 1 cocycle using multiplicative ergodic theory techniques we establish the existence of a discrete non random lyapunov spectrum for the cocycle the lyapunov spectrum characterizes the asymptotics of the cocycle near an equilibrium stationary solution we give sufficient conditions on the parameters of the navier stokes equation and the geometry of the planar domain for hyperbolicity of the zero equilibrium uniqueness of the stationary solution viz ergodicity local almost sure asymptotic stability of the cocycle and the existence of global invariant foliations of the energy space

existence of solutions to a global eikonal equation abstract we derive a geometric necessary and sufficient condition for the existence of solutions to a global eikonal equation we also study the existence of a minimal solution to this equation and its relation with the well known minimal time function

various lipschitz like properties for functions and sets i directional derivative and tangential characterizations in this work we introduce for extended real valued functions defined on a banach space x the concept of k directionally lipschitzian behavior where k is a bounded subset of x for different types of sets k e g zero singleton or compact the k directionally lipschitzian behavior recovers well known concepts in variational analysis locally lipschitzian directionally lipschitzian or compactly epi lipschitzian properties respectively characterizations of this notion are provided in terms of the lower dini subderivatives we also adapt the concept for sets and establish characterizations of the mentioned behavior in terms of the bouligand tangent cones the special case of convex functions and sets is also studied

coherent sets for nonautonomous dynamical systems we describe a mathematical formalism and numerical algorithms for identifying and tracking slowly mixing objects in nonautonomous dynamical systems in the autonomous setting such objects are variously known as almost invariant sets metastable sets persistent patterns or strange eigenmodes and have proved to be important in a variety of applications in this current work we explain how to extend existing autonomous approaches to the nonautonomous setting we call the new time dependent slowly mixing objects coherent sets as they represent regions of phase space that disperse very slowly and remain coherent the new methods are illustrated via detailed examples in both discrete and continuous time

entropy of coupled map lattices the entropy of coupled map lattices with respect to the group of space time translations is considered we use the notion of generalized lyapunov spectra to prove the analogue of the ruelle inequality and the pesin formula

a perron type theorem for nonautonomous differential equations abstract we show that the asymptotic exponential behavior of the solutions of a linear equation x a t x is reproduced by the solutions of the nonlinear equation x a t x f t x for any sufficiently small perturbation f more precisely we show that if the lyapunov exponents of the linear equation are limits then the same happens with the lyapunov exponents of the solutions of the nonlinear equations in fact without introducing new values our approach is based on lyapunov s theory of regularity including its generalization to infinite dimensional spaces

lyapunov type characterization of hyperbolic behavior abstract we give a complete characterization of the uniform hyperbolicity and nonuniform hyperbolicity of a cocycle with values in the space of bounded linear operators acting on a hilbert space in terms of the existence of appropriate quadratic forms our work unifies and extends many results in the literature by considering the general case of not necessarily invertible cocycles acting on a hilbert space over an arbitrary invertible dynamics as a nontrivial application of we study the persistence of hyperbolicity under small linear perturbations

smooth stable invariant manifolds and arbitrary growth rates abstract for nonautonomous linear equations v a t v with a generalized exponential dichotomy we show that there is a smooth stable invariant manifold for the perturbed equation v a t v f t v provided that f is sufficiently small the generalized exponential dichotomies may exhibit stable and unstable behaviors with respect to arbitrary growth rates e c t for some function t we consider the general case of nonuniform exponential dichotomies and the result is obtained in banach spaces moreover we show that for an equivariant system the dynamics on the stable manifold in a certain class of graphs is also equivariant we emphasize that this result cannot be obtained by averaging over the symmetry

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

on random linear dynamical systems in a banach space i multiplicative ergodic theorem and krein rutman type theorems abstract for linear random dynamical systems in a separable banach space x we derived a series of krein rutman type theorems with respect to co invariant cone family with rank k which present a quasi equivalence relation between the measurably co invariant cone family and the measurably dominated splitting of x moreover such quasi equivalence relation turns out to be an equivalence relation whenever i k 1 or ii in the frame of the multiplicative ergodic theorem with certain lyapunov exponent being greater than the negative infinity for the second case we thoroughly investigated the relations between the lyapunov exponents the co invariant cone family and the measurably dominated splitting for linear random dynamical systems in x

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

characterization of dominated splittings for operator cocycles acting on banach spaces abstract versions of the oseledets multiplicative ergodic theorem for cocycles acting on infinite dimensional banach spaces have been investigated since the pioneering work of ruelle in 1982 and are a topic of continuing research interest for a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in a complementary subbundle requires additional assumptions a necessary and sufficient condition for the existence of such a dominated splitting also known as exponential separation was recently given by j bochi and n gourmelon for invertible finite dimensional cocycles in discrete time we extend this result to cocycles of injective bounded and possibly noncompact linear maps acting on banach spaces in both discrete and continuous time using an essentially geometric approach based on a notion of approximate singular value decomposition in banach spaces our method is constructive and in the finite dimensional case yields explicit growth estimates on the dominated splitting which may be of independent interest

a note on linear rank tests and gill and schumacher s tests of proportionality summary gill schumacher 1987 assess proportionality of hazards in the two sample case by comparing two generalized linear rank tests with different weights they consider the difference of the two linear rank statistics under proportionality the expected value of this difference is zero and its variance may be estimated the purpose of this note is to show that one can perform a similar procedure within the traditional regression setting the procedure has the advantage of generalizability to multi sample or continuous data and is computable using existing regression programs

observing lyapunov exponents of infinite dimensional dynamical systems can lyapunov exponents of infinite dimensional dynamical systems be observed by projecting the dynamics into n using a typical nonlinear projection map we answer this question affirmatively by developing embedding theorems for compact invariant sets associated with c1 maps on hilbert spaces examples of such discrete time dynamical systems include time t maps and poincare return maps generated by the solution semigroups of evolution partial differential equations we make every effort to place hypotheses on the projected dynamics rather than on the underlying infinite dimensional dynamical system in so doing we adopt an empirical approach and formulate checkable conditions under which a lyapunov exponent computed from experimental data will be a lyapunov exponent of the infinite dimensional dynamical system under study provided the nonlinear projection map producing the data is typical in the sense of prevalence

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

entropy chaos and weak horseshoe for infinite dimensional random dynamical systems in this paper we study the complicated dynamics of infinite dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a polish space without assuming any hyperbolicity we prove if a continuous random map has a positive topological entropy then it contains a topological horseshoe we also show that the positive topological entropy implies the chaos in the sense of li yorke the complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system 2017 wiley periodicals inc

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

stability of nonautonomous differential equations in hilbert spaces abstract we introduce a large class of nonautonomous linear differential equations v a t v in hilbert spaces for which the asymptotic stability of the zero solution with all lyapunov exponents of the linear equation negative persists in v a t v f t v under sufficiently small perturbations f this class of equations which we call lyapunov regular is introduced here inspired in the classical regularity theory of lyapunov developed for finite dimensional spaces that is nowadays apparently overlooked in the theory of differential equations our study is based on a detailed analysis of the lyapunov exponents essentially the equation v a t v is lyapunov regular if for every k the limit of t 1 t as t exists where t is any k volume defined by solutions v 1 t v k t we note that the class of lyapunov regular linear equations is much larger than the class of uniformly asymptotically stable equations

the stable manifold theorem for non linear stochastic systems with memory i existence of the semiflow abstract we consider non linear stochastic functional differential equations sfde s on euclidean space we give sufficient conditions for the sfde to admit locally compact smooth cocycles on the underlying infinite dimensional state space our construction is based on the theory of finite dimensional stochastic flows and a non linear variational technique in part ii of this article the above result will be used to prove a stable manifold theorem for non linear sfde s

synchronization analysis of linearly coupled systems described by differential equations with a coupling delay when a transmission delay occurs in the interconnection of linearly coupled systems described by ordinary differential equations lcodes both synchronization and the final synchronized state will vary in this paper mathematical analysis is presented on the synchronization phenomena of lcodes with a single coupling delay criteria are derived for both local and global synchronization it is known that in addition to the dynamical behaviors of the underlying uncoupled system and the coupling configuration the coupling strength and the coupling delay also play key roles on the stability of synchronization both theoretical and numerical analysis indicate that under some conditions if the coupling strength is large enough the coupled system can be completely synchronized for any coupling delay on the other hand in some cases the coupled system can be synchronized if the coupling delay is small enough

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

zariski closure of subgroups of the symplectic group and lyapunov exponents of the schrodinger operator on the strip we consider the schrodinger equation with a random potential y n 1 q n ei y n y n 1 0 whereqn is a sequence of independent identically distributed random symmetricm m matrices with real valued elements yn m n e is the real parameter andi is the identity matrix we show that if the smallest jordan algebra of matrices containing the support of the distribution of matricesqn coincides with jordan algebra of all real valued symmetric matrices then for all but maybe a finite number of values ofe all the lyapunov exponents of our schrodinger equation are different and thus the spectrum of the corresponding schrodinger operator is pure point

attractor minimal sets for cooperative and strongly convex delay differential systems abstract we study the skew product semiflow induced by a family of convex and cooperative delay differential systems under some monotonicity assumptions we obtain an ergodic representation for the upper lyapunov exponent of a minimal subset in addition when eventually strong convexity at one point is assumed and there exist two completely strongly ordered minimal subsets k1 ck2 we show that k1 is an attractor subset which is a copy of the base the long time behaviour of every trajectory strongly ordered with k2 is then deduced some examples of application of the theory are shown

approximating lyapunov exponents and sacker sell spectrum for retarded functional differential equations we consider lyapunov exponents and sacker sell spectrum for linear nonautonomous retarded functional differential equations posed on an appropriate hilbert space a numerical method is proposed to approximate such quantities based on the reduction to finite dimension of the evolution family associated to the system to which a classic discrete qr method is then applied the discretization of the evolution family is accomplished by a combination of collocation and generalized fourier projection a rigorous error analysis is developed to bound the difference between the computed stability spectra and the exact stability spectra the efficacy of the results is illustrated with some numerical examples

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

large volume limit of the distribution of characteristic exponents in turbulence for spatially extended conservative or dissipative physical systems it appears natural that a density of characteristic exponents per unit volume should exist when the volume tends to infinity in the case of a turbulent viscous fluid however this simple idea is complicated by the phenomenon of intermittency in the present paper we obtain rigorous upper bounds on the distribution of characteristic exponents in terms of dissipation these bounds have a reasonable large volume behavior for two dimensional fluids a particularly striking result is obtained the total information creation is bounded above by a fixed multiple of the total energy dissipation at fixed viscosity the distribution of characteristic exponents is estimated in an intermittent model of turbulence see 7 and it is found that a change of behavior occurs at the valued 2 6 of the self similarity dimension

on the largest lyapunov exponent for products of gaussian matrices the paper provides a new integral formula for the largest lyapunov exponent of gaussian matrices which is valid in the real complex and quaternion valued cases this formula is applied to derive asymptotic expressions for the largest lyapunov exponent when the size of the matrix is large and compare the lyapunov exponents in models with a spike and no spikes

smooth invariant manifolds in banach spaces with nonuniform exponential dichotomy we establish the existence of smooth stable manifolds for semiflows defined by ordinary differential equations v a t v f t v in banach spaces assuming that the linear equation v a t v admits a nonuniform exponential dichotomy our proof of the ck smoothness of the manifolds uses a single fixed point problem in the unit ball of the space of ck functions with holder continuous kth derivative this is a closed subset of the space of continuous functions with the supremum norm by an apparently not so well known lemma of henry see proposition 3 the estimates showing that the functions maintain the original bounds when transformed under the fixed point operator are obtained through a careful application of the faa di bruno formula for the higher derivatives of the compositions see 31 and 35 as a consequence we obtain in a direct manner not only the exponential decay of solutions along the stable manifolds but also of their derivatives up to order k when the vector field is of class ck

smooth center manifolds for nonuniformly partially hyperbolic trajectories abstract we establish the existence of unique smooth center manifolds for ordinary differential equations v a t v f t v in banach spaces assuming that v a t v admits a nonuniform exponential trichotomy this allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories in addition we prove that the center manifolds are as regular as the vector field our proof of the c k smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space to the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting

numerical approximations to the stationary solutions of stochastic differential equations in this paper we investigate the possibility of approximating the stationary solution of a stochastic differential equation sde we start with the random dynamical system generated by the sde with the multiplicative noise we prove that the pullback flow has a stationary point however the stationary point is not constructible explicitly therefore we look at the numerical approximation we prove that the discrete time random dynamical system also has a stationary point finally we prove mean square convergence of the approximate stationary solution to the exact stationary solution as the time step diminishes as well as almost surely convergence when the time step is rational

lyapunov exponents of free operators abstract lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system this paper offers a definition of lyapunov exponents for a sequence of free linear operators the definition is based on the concept of the extended fuglede kadison determinant we establish the existence of lyapunov exponents derive formulas for their calculation and show that lyapunov exponents of free variables are additive with respect to operator product we illustrate these results using an example of free operators whose singular values are distributed by the marchenko pastur law and relate this example to c m newman s triangle law for the distribution of lyapunov exponents of large random matrices with independent gaussian entries as an interesting by product of our results we derive a relation between the extended fuglede kadison determinant and voiculescu s s transform

probability of local bifurcation type from a fixed point a random matrix perspective results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems real random matrices time delay dynamical systems companion matrices and a set of mappings known for their properties as universal approximators neural networks the eigenvalue spectrum is considered both numerically and analytically using previous work of edelman et al based upon the numerical evidence various conjectures are presented the conclusion is that in many circumstances most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues nevertheless surprising situations are presented for which the aforementioned conclusion does not hold e g real random matrices with gaussian elements with a large positive mean and finite variance

dualization of the euler and hamiltonian inclusions optimal control emerged as a distinct eld of research in the late 1950s with the derivation of the pontryagin maximum principle a fundamental set of necessary conditions on minimizing arcs which takes account of di erential constraints the pontryagin maximum principle directly generates the euler equation and other classical necessary conditions from the calculus of variations when specialised to problems without dynamic constraints however deeper connections between necessary conditions in optimal control and the calculus of variations were only subsequently revealed when clarke developed new analytic machinery now referred to as nonsmooth analysis a term introduced by clarke himself for extending classical necessary conditions to apply to the generalized

zariski closure and the dimension of the gaussian low of the product of random matrices i we prove the central limit theorem for products of i i d random matrices the main aim is to find the dimension of the corresponding gaussian law it turns out that ifg is the zariski closure of a group generated by the support of the distribution of our matrices and ifg is semi simple then the dimension of the gaussian law is equal to the dimension of the diagonal part of cartan decomposition ofg

pathwise random periodic solutions of stochastic differential equations abstract in this paper we study the existence of random periodic solutions for semilinear stochastic differential equations we identify these as the solutions of coupled forward backward infinite horizon stochastic integral equations in general cases we then use the argument of the relative compactness of wiener sobolev spaces in c 0 0 t l 2 and generalized schauder s fixed point theorem to prove the existence of a solution of the coupled stochastic forward backward infinite horizon integral equations the condition on f is then further weakened by applying the coupling method of forward and backward gronwall inequalities the results are also valid for stationary solutions as a special case when the period can be an arbitrary number

invariant manifolds for stochastic wave equations abstract in this paper we consider a class of stochastic wave equations with nonlinear multiplicative noise we first show that these stochastic wave equations generate random dynamical systems or stochastic flows by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism then we establish the existence of random invariant manifolds for the random wave equations due to the temperedness of the nonlinearity we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases based on these random dynamical systems we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium finally we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations

nonlinear perturbations of nonuniform exponential dichotomy on measure chains abstract this paper focuses on nonlinear perturbations of flows in banach spaces corresponding to a nonautonomous dynamical system on measure chains admitting a nonuniform exponential dichotomy we first define the nonuniform exponential dichotomy of linear nonuniformly hyperbolic systems on measure chains then establish a new version of the grobman hartman theorem for nonuniformly hyperbolic dynamics on measure chains with the help of nonuniform exponential dichotomies moreover we also construct stable invariant manifolds for sufficiently small nonlinear perturbations of a nonuniform exponential dichotomy in particular it is shown that the stable invariant manifolds are lipschitz in the initial values provided that the nonlinear perturbation is a sufficiently small lipschitz perturbation

proximal analysis and the minimal time function under general hypotheses on the target set s and the dynamics of the system we show that the minimal time function ts cdot is a proximal solution to the hamilton jacobi equation uniqueness results are obtained with two different kinds of boundary conditions a new propagation result is proven and as an application we give necessary and sufficient conditions for ts cdot to be lipschitz continuous near s a petrov type modulus condition is also shown to be sufficient for continuity of ts cdot near s

biometrika centenary survival analysis although papers relating to survival analysis have appeared in biometrika since the very first issue the current emphasis on the topic in this journal is fairly new especially as it relates to applications in biostatistics this survey traces the development of survival analysis throughout the twentieth century as reflected in the pages of biometrika focusing primarily on work published since 1980

smoothness of invariant manifolds for nonautonomous equations for semiflows generated by ordinary differential equations v a t v admitting a nonuniform exponential dichotomy we show that for any sufficiently small perturbation f there exist smooth stable and unstable manifolds for the perturbed equation v a t v f t v as an application we establish the existence of invariant manifolds for the nonuniformly hyperbolic trajectories of a semiflow in particular we obtain smooth invariant manifolds for a class of vector fields that need not be c1 for any 0 1 to the best of our knowledge no similar statement was obtained before in the nonuniformly hyperbolic setting we emphasize that we do not need to assume the existence of an exponential dichotomy but only the existence of a nonuniform exponential dichotomy with sufficiently small nonuniformity when compared to the lyapunov exponents of the original linear equation furthermore for example in the case of stable manifolds we only need to assume that there exist negative lyapunov exponents while we also allow zero exponents our proof of the smoothness of the invariant manifolds is based on the construction of an invariant family of cones

entropy volume growth and srb measures for banach space mappings we consider c 2 frechet differentiable mappings of banach spaces leaving invariant compactly supported borel probability measures and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces srb measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds equivalently the sum of positive lyapunov exponents of the map in addition to numerous difficulties incurred by our infinite dimensional setting a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions the results here generalize previously known results for diffeomorphisms of finite dimensional riemannian manifolds and are applicable to dynamical systems defined by large classes of dissipative parabolic pdes

necessary conditions in optimal control a new approach we present necessary conditions of optimality for a general control problem formulated in terms of a differential inclusion these conditions unify and substantially extend previous results in the literature and also incorporate a stratified feature of a novel nature when specialized to the calculus of variations the results yield necessary conditions and regularity theorems that go significantly beyond the previous standard they also give rise to new and stronger maximum principles of pontryagin type the principal focus here is the development of structural criteria on the problem which guarantee a priori that the necessary conditions apply to any local minimizer

proportional hazards tests and diagnostics based on weighted residuals summary nonproportional hazards can often be expressed by extending the cox model to include time varying coefficients e g for a single covariate the hazard function for subject i is modelled as exp fl t zi t a common example is a treatment effect that decreases with time we show that the function 3 t can be directly visualized by smoothing an appropriate residual plot also many tests of proportional hazards including those of cox 1972 gill schumacher 1987 harrell 1986 lin 1991 moreau o quigley mesbah 1985 nagelkerke oosting hart 1984 o quigley pessione 1989 schoenfeld 1980 and wei 1984 are related to time weighted score tests of the proportional hazards hypothesis and can be visualized as a weighted least squares line fitted to the residual plot

fine properties of lp cocycles which allow abundance of simple and trivial spectrum in this paper we generalize 3 and prove that the class of accessible and saddle con servative cocycles a wide class which includes cocycles evolving in glpd rq slpd rq and sppd rq l p densely have a simple spectrum we also generalize 3 1 and prove that for an l p residual subset of accessible cocycles we have a one point spectrum by using a different approach of the one given in 3 finally we show that the linear differential system versions of previous results also hold and give some applications

principal lyapunov exponents and principal floquet spaces of positive random dynamical systems i general theory this series of papers is concerned with principal lyapunov exponents and principal floquet subspaces of positive random dynamical systems in ordered banach spaces the current part of the series focuses on the development of general theory first the notions of generalized principal floquet subspaces generalized principal lyapunov exponents and generalized exponential separations for general positive random dynamical systems in ordered banach spaces are introduced which extend the classical notions of principal floquet subspaces principal lyapunov exponents and exponential separations for strongly positive deterministic systems in strongly ordered banach to general positive random dynamical systems in ordered banach spaces under some quite general assumptions it is then shown that a positive random dynamical system in an ordered banach space admits a family of generalized principal floquet subspaces a generalized principal lyapunov exponent and a generalized exponential separation we will consider in the forthcoming part s the applications of the general theory developed in this part to positive random dynamical systems arising from a variety of random mappings and differential equations including random leslie matrix models random cooperative systems of ordinary differential equations and random parabolic equations institute of mathematics and computer science wroc law university of technology wybrzeze wyspianskiego 27 pl 50 370 wroc law poland url http www im pwr wroc pl mierczyn index html e mail address mierczyn pwr wroc pl department of mathematics and statistics auburn university auburn university al 36849 usa e mail address wenxish auburn edu 2010 mathematics subject classification primary 37h15 37l55 37a30 secondary 15b52 34f05 35r60

robustness of nonuniform exponential dichotomies in banach spaces we give conditions for the robustness of nonuniform exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation x a t x persists under a sufficiently small linear perturbation we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the angles between the stable and unstable subspaces our proofs exhibit implicitly the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions we emphasize that we do not need the notion of admissibility of bounded nonlinear perturbations we also obtain related robustness results in the case of nonuniform exponential contractions in addition we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time

stable manifolds for nonautonomous equations without exponential dichotomy abstract we consider nonautonomous ordinary differential equations v a t v in banach spaces and under fairly general assumptions we show that for any sufficiently small perturbation f there exists a stable invariant manifold for the perturbed equation v a t v f t v which corresponds to the set of negative lyapunov exponents of the original linear equation the main assumption is the existence of a nonuniform exponential dichotomy with a small nonuniformity i e a small deviation from the classical notion of uniform exponential dichotomy in fact we showed that essentially any linear equation v a t v admits a nonuniform exponential dichotomy and thus the above assumption only concerns the smallness of the nonuniformity of the dichotomy this smallness is a rather common phenomenon at least from the point of view of ergodic theory almost all linear variational equations obtained from a measure preserving flow admit a nonuniform exponential dichotomy with arbitrarily small nonuniformity we emphasize that we do not need to assume the existence of a uniform exponential dichotomy and that we never require the nonuniformity to be arbitrarily small only sufficiently small our approach is related to the notion of lyapunov regularity which goes back to lyapunov himself although it is apparently somewhat forgotten today in the theory of differential equations

floquet bundles for scalar parabolic equations for linear scalar parabolic equations such as u t u xx a t x u x b t x u on a finite interval 0 x with various boundary conditions we obtain canonical floquet solutions u n t x these solutions are characterized by the property that z u n t x n for all te where z denotes the zero crossing lap number of matano the coefficients a t x and b t x are not assumed to be periodic in t but if they are the solutions u n t x reduce to the standard floquet solutions our results may naturally be expressed in the language of linear skew product flows in this context we obtain for each n 1 an exponential dichotomy between the bundles span u n 1 and overline span u n cdot cdot n n 1 infty

lyapunov exponents of linear stochastic functional differential equations ii examples and case studies we give several examples and examine case studies of linear stochastic functional differential equations the examples fall into two broad classes regular and singular according to whether an underlying stochastic semi flow exists or not in the singular case we obtain upper and lower bounds on the maximal exponential growth rate overline lambda 1 sigma of the trajectories expressed in terms of the noise variance roughly speaking we show that for small sigma overline lambda 1 sigma behaves like sigma 2 2 while for large sigma it grows like log sigma in the regular case it is shown that a discrete oseledec spectrum exists and upper estimates on the top exponent lambda 1 are provided these estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases

spatial estimates for stochastic flows in euclidean space we study the behavior for large x of kunita type stochastic flows t x on r d driven by continuous spatial semimartingales for this class of flows we prove new spatial estimates for large x under very mild regularity conditions on the driving semimartingale random field it is expected that the results would be of interest for the theory of stochastic flows on noncompact manifolds as well as in the study of nonlinear filtering stochastic functional and partial differential equations some examples and counterexamples are given

a multivariate faa di bruno formula with applications a multivariate faa di bruno formula for computing arbitrary partial derivatives of a function composition is presented it is shown by way of a general identity how such derivatives can also be expressed in the form of an infinite series applications to stochastic processes and multivariate cumulants are then delineated

invariant foliations near normally hyperbolic invariant manifolds for semiflows let m be a compact c1 manifold which is invariant and normally hyperbolic with respect to a c1 semiflow in a banach space then in an neighborhood of m there exist local c1 center stable and center unstable manifolds w cs and w cu respectively here we show that w cs and w cu may each be decomposed into the disjoint union of c1 submanifolds leaves in such a way that the semiflow takes leaves into leaves of the same collection furthermore each leaf intersects m in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time

links between directional derivatives through multidirectional mean value inequalities we prove in the general setting of lower semicontinuous functions on banach spaces the relation between the rockafellar directional derivative and the mixed lower limit of the lower dini derivatives as a byproduct we derive the famous inclusions of tangent cones of closed sets in banach spaces the results are established using as principal tool multidirectional mean value inequalities aussel et al siam j optim 9 3 690 706 1999

multiplicative ergodic theorem for a non irreducible random dynamical system we study the asymptotic properties of the trajectories of a discrete time random dynamical system in an infinite dimensional hilbert space under some natural assumptions on the model we establish a multiplica tive ergodic theorem with an exponential rate of convergence the assumptions are satisfied for a large class of parabolic pdes including the 2d navier stokes and complex ginzburg landau equations perturbed by a non degenerate bounded random kick force as a consequence of this er godic theorem we derive some new results on the statistical properties of the trajectories of the underlying random dynamical system in particular we obtain large deviations principle for the occupation measures and the analyticity of the pressure function in a setting where the system is not irreducible the proof relies on a refined version of the uniform feller property combined with some contraction and bootstrap arguments

nonsmooth constrained optimization and multidirectional mean value inequalities we establish a general fermat rule for the problem of minimizing a lower semicontinuous function on a convex subset of a banach space our basic tool is a constrained variational principle derived from the smooth variational principle of borwein and preiss specializing the fermat rule to the case when the convex set is a drop we obtain a multidirectional rolle type inequality from which in turn we deduce a multidirectional mean value inequality in the line of clarke and ledyaev we follow the abstract approach of our previous paper trans amer math soc 347 1995 pp 4147 4161 thus covering all standard situations met in applications while stressing the links between the results and the few key properties that are needed

the generalised berger wang formula and the spectral radius of linear cocycles using ergodic theory we prove two formulae describing the relationships between different notions of joint spectral radius for sets of bounded linear operators acting on a banach space the first formula was previously obtained by v s shulman and yu v turovski using operator theoretic ideas the second formula shows that the joint spectral radii corresponding to several standard measures of noncompactness share a common value when applied to a given precompact set of operators this result may be seen as an extension of classical formulae for the essential spectral radius given by r nussbaum a lebow and m schechter both results are obtained as a consequence of a more general theorem concerned with continuous operator cocycles defined over a compact dynamical system as a byproduct of our method we answer a question of j e cohen on the limiting behaviour of the spectral radius of a measurable matrix cocycle

ergodic theory of infinite dimensional systems with applications to dissipative parabolic pdes we consider a class of randomly perturbed dynamical systems satisfying conditions which reflect the properties of general nonlinear dissipative parabolic pdes results on invariant measures and their exponential mixing properties are proved and applications to 2d navier stokes systems are included

characteristic exponents for a viscous fluid subjected to time dependent forces we consider a viscous incompressible fluid enclosed in a bounded region of 2 or 3 and subjected to time dependent forces using bound state estimates for the schrodinger operator we obtain rigorous bounds for the characteristic exponents entropy kolmogorov sinai invariant and hausdorff dimension of attracting sets our methods are of potential use for more general time evolutions described by nonlinear partial differential equations

a proximal characterization of the reachable set we show that the graph of the reachable set of a control system given by a differential inclusion is uniquely characterized by a hamilton jacobi equation involving proximal normals

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

stable manifolds for nonuniform polynomial dichotomies we establish the existence of smooth stable manifolds in banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy this new dichotomy is more restrictive in the nonuniform part but allow the uniform part to obey a polynomial law instead of an exponential more restrictive law we consider two families of perturbations for one of the families we obtain local lipschitz stable manifolds and for the other family assuming more restrictive conditions on the perturbations and its derivatives we obtain c1 global stable manifolds finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family

the wong zakai approximations of invariant manifolds and foliations for stochastic evolution equations abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated dynamics of a class of stochastic evolution equations with a multiplicative white noise we prove that the solutions of wong zakai approximations almost surely converge to the solutions of the stratonovich stochastic evolution equation we also show that the invariant manifolds and stable foliations of the wong zakai approximations converge to the invariant manifolds and stable foliations of the stratonovich stochastic evolution equation respectively

parameter dependence of stable manifolds under nonuniform hyperbolicity abstract for a nonautonomous linear equation v a t v in a banach space with a nonuniform exponential dichotomy we show that the nonlinear equation v a t v f t v has stable invariant manifolds v which are lipschitz in the parameter provided that f is a sufficiently small lipschitz perturbation since any linear equation with nonzero lyapunov exponents has a nonuniform exponential dichotomy the above assumption is very general we emphasize that passing from a classical uniform exponential dichotomy to a general nonuniform exponential dichotomy requires a substantially new approach