invariant manifolds of partial functional differential equations this paper is concerned with the existence smoothness and attractivity of invariant manifolds for evolutionary processes on general banach spaces when the nonlinear perturbation has a small global lipschitz constant and locally ck smooth near the trivial solution such a nonlinear perturbation arises in many applications through the usual cut off procedure but the requirement in the existing literature that the nonlinear perturbation is globally ck smooth and has a globally small lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut off function our general results are illustrated by and applied to partial functional differential equations for which the phase space c r 0 x where r 0 and x being a banach space has no smooth inner product structure and for which the validity of variation of constants formula is still an interesting open problem

center manifolds and contractions on a scale of banach spaces we give a new proof for the existence of a ck center manifold at a nonhyperbolic equilibrium point of a finite dimensional vector field of class ck the problem is reduced to a fixed point problem on a scale of banach spaces these banach spaces consist of mappings with a certain maximal exponential growth at infinity we give conditions under which there is a unique fixed point depending differentiably on the parameters the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of banach spaces

center manifolds for invariant sets abstract we derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector field in r n by applying the hadamard graph transform technique it is shown that associated to a natural dynamical characteristic of the linearized flow along the invariant set there exists an invariant manifold called a center manifold of the invariant set which contains every locally bounded solution in particular contains the invariant set and is persistent under small perturbations

evolutionary induction of global model trees with specialized operators and memetic extensions metaheuristics such as evolutionary algorithms ea s have been successfully applied to the problem of decision tree induction recently an ea was proposed to evolve model trees which are a particular type of decision tree that is employed to solve regression problems however there is a need to specialize the ea s in order to exploit the full potential of evolutionary induction the main contribution of this paper is a set of solutions and techniques that incorporates knowledge about the inducing problem for the global model tree into the evolutionary search the objective of this paper is to demonstrate that specialized ea can find more accurate and less complex solutions to the traditional greedy induced counterparts and the straightforward application of ea this paper proposes a novel solution for each step of the evolutionary process and presents a new specialized ea for model tree induction called the global model tree gmt an empirical investigation shows that trees induced by the gmt are one order of magnitude less complex than trees induced by popular greedy algorithms and they are equivalent in terms of predictive accuracy with output models from straightforward implementations of evolutionary induction and state of the art methods

invariant manifolds of admissible classes for semi linear evolution equations abstract consider an evolution family u u t s t s 0 on a half line r and a semi linear integral equation u t u t s u s s t u t f u d we prove the existence of invariant manifolds of this equation these manifolds are constituted by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of l p type the lorentz spaces l p q and many other function spaces occurring in interpolation theory the existence of such manifolds is obtained in the case that u t s t s 0 has an exponential dichotomy and the nonlinear forcing term f t x satisfies the non uniform lipschitz conditions f t x 1 f t x 2 t x 1 x 2 for being a real and positive function which belongs to certain classes of admissible function spaces

a center manifold for second order semilinear differential equations on the real line and applications to the existence of wave trains for the gurtin mccamy 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

polynomial time verification of diagnosability of partially observed discrete event systems the problem of verifying the properties of diagnosability and i diagnosability is considered we present new polynomial time algorithms for deciding diagnosability and i diagnosability these algorithms are based on the construction of a nondeterministic automaton called a verifier

analysis of a singularly perturbed traveling wave problem a singularly perturbed traveling wave problem derived from the drift diffusion model describing electron transport in two valley semiconductor materials is analysed it is shown that the reduced problem defined on a submanifold of the phase space has homoclinic orbits for certain parameters by using methods from invariant manifold theory and methods from homoclinic continuation theory it is proved that the homoclinic orbits of the reduced problem persist in a global center manifold near the manifold of the reduced problem

centre manifolds for partial differential equations with delays a centre manifold theory for reaction diffusion equations with temporal delays is developed besides an existence proof we also show that the equation on the centre manifold is a coupled system of scalar ordinary differential equations of higher order as an illustration this reduction procedure is applied to the hutchinson equation with diffusion

admissibility and exponential trichotomy of dynamical systems described by skew product flows abstract the aim of this paper is to present a new and very general method for the detection of the uniform exponential trichotomy of dynamical systems the investigation is done in several constructive stages that correspond to three admissibility properties that are progressively introduced with respect to an associated input output system we prove that the uniform admissibility of the pair c b r x l 1 r x for the associated system is a sufficient condition for the existence of a uniform trichotomic behavior of the initial dynamical system if p 1 and the pair c b r x l 1 r x is uniformly p admissible then we obtain the uniform exponential trichotomy next we study whether the admissibility conditions are also necessary for the uniform exponential trichotomy supposing that a dynamical system has a uniform exponential trichotomy we prove that the associated input output system has unique bounded solutions in certain subspaces finally we obtain that the uniform p admissibility of the pair c b r x l 1 r x is a necessary and sufficient condition for uniform exponential trichotomy

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

hierarchical mixtures of experts and the em algorithm we present a tree structured architecture for supervised learning the statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models glim s learning is treated as a maximum likelihood problem in particular we present an expectation maximization em algorithm for adjusting the parameters of the architecture we also develop an on line learning algorithm in which the parameters are updated incrementally comparative simulation results are presented in the robot dynamics domain

orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine gordon equation abstract in this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two degree of freedom hamiltonian systems our methods are fundamentally different than other global perturbation methods e g standard melnikov theory in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation our methods combine the higher dimensional melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds we apply our methods to a modified model of the forced and damped sine gordon equation developed by bishop et al we give explicit conditions in terms of the system parameters for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle focus type chaotic dynamics follow from a theorem of silnikov this provides a mechanism for chaotic dynamics geometrically similar to that observed by bishop et al namely a random jumping between two spatially dependent states with an intermediate passage through a spatially independent state however in order for this type of silnikov dynamics to exist we require a different and unphysical type of damping compared to that used by bishop et al

asymptotic stability of a nonlinear korteweg de vries equation with critical lengths abstract we study an initial boundary value problem of a nonlinear korteweg de vries equation posed on the finite interval 0 2 k where k is a positive integer the whole system has dirichlet boundary condition at the left end point and both of dirichlet and neumann homogeneous boundary conditions at the right end point it is known that the origin is not asymptotically stable for the linearized system around the origin we prove that the origin is locally asymptotically stable for the nonlinear system if the integer k is such that the kernel of the linear korteweg de vries stationary equation is of dimension 1 this is for example the case if k 1

admissibly inertial manifolds for a class of semi linear evolution equations consider the semi linear evolution equation du t dt au t f t u we prove the existence of a new class of inertial manifolds called admissibly inertial manifolds for this equation these manifolds are constituted by trajectories of the solutions belonging to rescaledly admissible function spaces which contain wide classes of function spaces like weighted lp spaces the lorentz spaces lp q and many other rescaling function spaces occurring in interpolation theory the existence of these manifolds is obtained in the case that the partial differential operator a is positive definite and self adjoint with a discrete spectrum and the nonlinear forcing term f satisfies the lipschitz conditions on the domain d a 0 1 i e f t x f t y t a x y and f t x t 1 a x for being a real and positive function which belongs to certain classes of admissible function spaces

center manifolds for infinite dimensional nonautonomous differential equations abstract we study a nonlinear integral equation for a center manifold of a semilinear nonautonomous differential equation having mild solutions we assume that the linear part of the equation admits in a very general sense a decomposition into center and hyperbolic parts the center manifold is obtained directly as the graph of a fixed point for a lyapunov perron type integral operator we prove that this integral operator can be factorized as a composition of a nonlinear substitution operator and a linear integral operator the operator is formed by the green s function for the hyperbolic part and composition operators induced by a flow on the center part we formulate the usual gap condition in spectral terms and show that this condition is in fact a condition of boundedness of on corresponding spaces of differentiable functions this gives a direct proof of the existence of a smooth global center manifold

top down induction of model trees with regression and splitting nodes model trees are an extension of regression trees that associate leaves with multiple regression models in this paper a method for the data driven construction of model trees is presented namely the stepwise model tree induction smoti method its main characteristic is the induction of trees with two types of nodes regression nodes which perform only straight line regression and splitting nodes which partition the feature space the multiple linear model associated with each leaf is then built stepwise by combining straight line regressions reported along the path from the root to the leaf in this way internal regression nodes contribute to the definition of multiple models and have a global effect while straight line regressions at leaves have only local effects experimental results on artificially generated data sets show that smoti outperforms two model tree induction systems m5 and retis in accuracy results on benchmark data sets used for studies on both regression and model trees show that smoti performs better than retis in accuracy while it is not possible to draw statistically significant conclusions on the comparison with m5 model trees induced by smoti are generally simple and easily interpretable and their analysis often reveals interesting patterns

a variation of constants formula for abstract functional differential equations in the phase space abstract for linear functional differential equations with infinite delay in a banach space a variation of constants formula is established in the phase space as an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line

verification of the observer property in discrete event systems the observer property is an important condition to be satisfied by abstractions of discrete event system des models this technical note presents a new algorithm that tests if an abstraction of a des obtained through natural projection has the observer property the procedure called op verifier can be applied to potentially nondeterministic automata with no restriction on the existence of cycles of non relevant events this procedure has quadratic complexity in the number of states the performance of the algorithm is illustrated by a set of experiments

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

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

robust linear and support vector regression the robust huber m estimator a differentiable cost function that is quadratic for small errors and linear otherwise is modeled exactly in the original primal space of the problem by an easily solvable simple convex quadratic program for both linear and nonlinear support vector estimators previous models were significantly more complex or formulated in the dual space and most involved specialized numerical algorithms for solving the robust huber linear estimator numerical test comparisons with these algorithms indicate the computational effectiveness of the new quadratic programming model for both linear and nonlinear support vector problems results are shown on problems with as many as 20000 data points with considerably faster running times on larger problems

formulation of the normal form of turing hopf bifurcation in partial functional differential equations abstract turing hopf bifurcation is considered as an important mechanism for generating complex spatio temporal patterns in dynamical systems in this work the normal form up to the third order for the hopf steady state bifurcation which includes the turing hopf bifurcation of a general system of partial functional differential equations pfdes in a bounded spatial region of any dimension is derived based on the center manifold and the normal form theories of pfdes the explicit formula for the coefficients in the normal form of hopf steady state bifurcation are presented concisely in a matrix form which makes it more convenient in not only symbolic derivation but also numerical implementation this provides an approach of showing the existence and stability of stationary and time periodic solutions with spatial heterogeneity the derived formulas are also applicable to partial differential equations and functional differential equations

the effect of oscillations in the dynamics of differential equations with delay abstract the purpose of this paper is to study the dynamic behavior of delay differential equations of the form x t f x t 1 a e sin t e cos t e r provided that a and f meet some hypotheses by augmenting the above equation the explicit time dependent terms are replaced by state dependent terms the augmented system is autonomous and has a pair of purely imaginary and simple zero eigenvalues applying the center manifold reduction the existence of an attractive integral manifold with periodic structure for the original equation is shown furthermore we give a description of the flow on the obtained manifold this allows us to determine the sufficient conditions for existence of saddle node bifurcation to illustrate our results we consider an autonomous equation perturbed by a periodic function

optimal sensor selection for discrete event systems with partial observation for discrete event systems under partial observation we study the problem of selection of an optimal set of sensors that can provide sufficient yet minimal events observation information the sufficiency of the observed information is captured as the fulfillment of a desired formal property selection of sensors can be viewed as a selection of an observation mask and also of an equivalence class of events a sensor set is called optimal if any coarser selection of the corresponding equivalence class of events results in some significant loss of the events observation information we study an optimal selection of sensors over the set of general nonprojection observation masks we show that this problem is np hard in general for mask monotonic properties we present a top down and a bottom up algorithm each of polynomial complexity we show that observerness is not mask monotonic we show that the computational complexity can be further improved if the property is preserved under the projection via an intermediary observation mask that is an observer our results are obtained in a general setting so that they can be adapted for an optimal selection of sensors for a variety of applications

shadowing lemma and singularly perturbed boundary value problems a complete procedure is given to determine the outer and inner expansions of a singularly perturbed boundary value problem in mathbb r n the validity of such expansions is deduced from a gene

inertial manifolds for semi linear parabolic equations in admissible spaces we prove the existence of inertial manifolds for the solutions to the semi linear parabolic equation du t dt au t f t u when the partial differential operator a is positive definite and self adjoint with a discrete spectrum having a sufficiently large distance between some two successive points of the spectrum and the nonlinear forcing term f satisfies the lipschitz conditions on the domain d a 0 1 i e f t x f t y t a x y and f t x t 1 a x where t belongs to one of admissible function spaces containing wide classes of function spaces like lp spaces the lorentz spaces lp q and many other function spaces occurring in interpolation theory

behavior near hyperbolic stationary solutions for partial functional differential equations with infinite delay abstract the aim of this work is to investigate the asymptotic behavior of solutions near hyperbolic stationary solutions for partial functional differential equations with infinite delay we suppose that the linear part satisfies the hille yosida condition on a banach space and it is not necessarily densely defined firstly we establish a new variation of constants formula for the nonhomogeneous linear equations secondly we use this formula and the spectral decomposition of the phase space to show the existence of stable and unstable manifolds the estimations of solutions on these manifolds are obtained for illustration we propose to study the stability of stationary solutions for the lotka volterra model with diffusion

reduction principle and dynamic behaviors for a class of partial functional differential equations abstract we investigate the dynamic of solutions in the norm for some nonhomogeneous linear partial functional differential equations we suppose that the undelayed homogeneous part is the infinitesimal generator of an analytic semigroup and the delayed part is continuous with respect to fractional powers of the generator we establish a reduction principle for the infinite dimensional system in order to reduce its qualitative analysis to a finite dimensional one our reduction method is based on a new variation of constants formula in application the reduced system is used to prove the existence of almost automorphic almost periodic and periodic solutions for the whole infinite dimensional system

compositional verification in supervisory control this paper proposes a compositional approach to verifying whether a large discrete event system is nonblocking the new approach avoids computing the synchronous product of a large set of finite state machines instead the synchronous product is computed gradually and intermediate results are simplified using conflict preserving abstractions based on process algebraic results about fair testing heuristics are used to choose between different possible abstractions by translating the problem representation the same method can also be applied to verify safety properties in particular controllability experimental results show that the method is applicable to finite state machine models of industrial scale and brings considerable improvements in performance over other methods for nonblocking verification

persistent homoclinic orbits for a perturbed nonlinear schrodinger equation the persistence of homoclinic orbits for certain perturbations of the integrable nonlinear schrodinger equation under even periodic boundary conditions is established more specifically the existence of a symmetric pair of homoclinic orbits is established for the perturbed nls equation through an argument that combines melnikov analysis with a geometric singular perturbation theory for the pde 1996 john wiley sons inc

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

stable manifolds for semi linear evolution equations and admissibility of function spaces on a half line abstract consider an evolution family u u t s t s 0 on a half line r and a semi linear integral equation u t u t s u s s t u t f u d we prove the existence of stable manifolds of solutions to this equation in the case that u t s t s 0 has an exponential dichotomy and the nonlinear forcing term f t x satisfies the non uniform lipschitz conditions f t x 1 f t x 2 t x 1 x 2 for being a real and positive function which belongs to admissible function spaces which contain wide classes of function spaces like function spaces of l p type the lorentz spaces l p q and many other function spaces occurring in interpolation theory

normal forms and hopf bifurcation for partial differential equations with delays the paper addresses the computation of normal forms for some partial functional differential equations pfdes near equilibria the analysis is based on the theory previously developed for autonomous retarded functional differential equations and on the existence of center or other invariant manifolds as an illustration of this procedure two examples of pfdes where a hopf singularity occurs on the center manifold are considered

integral manifolds for partial functional differential equations in admissible spaces on a half line abstract in this paper we investigate the existence of stable and center stable manifolds for solutions to partial functional differential equations of the form u t a t u t f t u t t 0 when its linear part the family of operators a t t 0 generates the evolution family u t s t s 0 having an exponential dichotomy or trichotomy on the half line and the nonlinear forcing term f satisfies the lipschitz condition i e f t u t f t v t t u t v t c where u t v t c c r 0 x and t belongs to some admissible function space on the half line our main methods invoke lyapunov perron methods and the use of admissible function spaces

invariant manifolds for flows in banach spaces on presente une theorie des varietes invariantes lisses basee sur la methode classique de lyapunov penon pour des semi flots continus dans des espaces de banach

the sil nikov problem exponential expansion strong λ lemma c1 linearization and homoclinic bifurcation on introduit une nouvelle theorie appelee le developpement exponentiel et on explore ses interactions avec la theorie de la variete invariante locale les lemmes et la theorie de linerarisation reguliere