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

bicyclic graphs with small positive index of inertia abstract the positive index of inertia of a graph g denoted by i g is the number of the positive eigenvalues of the adjacency matrix of g in this paper we investigate the minimal positive index of inertia among all bicyclic graphs of order n with pendant vertices and characterize the bicyclic graphs with positive index 1 or 2 among all bicyclic graphs of order n

bounds for the matching number and cyclomatic number of a signed graph in terms of rank abstract a signed graph g is a graph with a sign attached to each of its edges where g is the underlying graph of g let m g c g and r g be the matching number the cyclomatic number and the rank of the adjacency matrix of g respectively in this paper we investigate the relation among the rank the matching number and the cyclomatic number of a signed graph and prove that 2 m g 2 c g r g 2 m g c g furthermore signed graphs reaching the lower bound or the upper bound are respectively characterized

on the relation between the positive inertia index and negative inertia index of weighted graphs abstract let g w be an edge weighted graph and let a g w be its adjacency matrix the positive inertia index respectively the negative inertia index of g w denoted by p g w resp p g w is defined to be the number of positive eigenvalues resp negative eigenvalues of a g w clearly p g w p g w is the rank of g w whereas p g w p g w is defined as the signature of g w in this paper some upper and lower bounds on p g w p g w are determined for the weighted chordal graphs resp bipartite graphs complete graphs cactus graphs join graphs bisplit graphs power graphs

jacobi ritz method for free vibration analysis of uniform and stepped circular cylindrical shells with arbitrary boundary conditions a unified formulation abstract a semi analytical approach is employed to analyze free vibration characteristics of uniform and stepped circular cylindrical shells subject to arbitrary boundary conditions the analytical model is established on the basis of multi segment partitioning strategy and flugge thin shell theory the admissible displacement functions are handled by unified jacobi polynomials and fourier series in order to obtain continuous conditions and satisfy arbitrary boundary conditions the penalty method about the spring technique is adopted the solutions about free vibration behavior of circular cylindrical shells were obtained by approach of rayleigh ritz to confirm the reliability and accuracy of this method convergence study and numerical verifications for circular cylindrical shells subject to different boundary conditions jacobi parameters spring parameters and maximum degree of permissible displacement function are carried out through comparative analyses it is obvious that the present method has a good stable and rapid convergence property and the results of this paper agree closely with the published literature in addition some interesting results about the geometric dimensions are investigated

inertia of complex unit gain graphs let t z c z 1 be a subgroup of the multiplicative group of all nonzero complex numbers c a t gain graph is a triple g t consisting of a graph g v e the circle group t and a gain function e t such that e i j e j i 1 e j i the adjacency matrix a of the t gain graph g of order n is an n n complex matrix aij where a i j e i j if v i is adjacent to v j 0 otherwise evidently this matrix is hermitian the inertia of is defined to be the triple i n i i i 0 where i i i 0 are numbers of the positive negative and zero eigenvalues of a including multiplicities respectively in this paper we investigate some properties of inertia of t gain graph

exponential splitting for nonautonomous linear discrete time systems in banach spaces in this paper we consider some concepts of exponential splitting for nonautonomous linear discrete time systems these concepts are generalizations of some well known concepts of uniform and nonuniform exponential dichotomies connections between these concepts are presented and some illustrating examples prove that these are distinct

which functions may be the upper bohl function of the diagonal discrete linear time varying systems abstract in this paper we give the necessary and sufficient conditions for a function f to be an upper bohl function of the diagonal discrete linear time varying systems

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

on the regularity of discrete linear systems in this paper we have introduced a new regularity coefficient of time varying discrete linear system on the base of this coefficient we have characterized the regularity of homogeneous discrete time varying linear systems by nonhomogeneous ones moreover we provided bounds for the regularity coefficient in terms of the existing in the literature regularity coefficients

robustness of discrete dynamics via lyapunov sequences for a nonautonomous dynamics with discrete time defined by a sequence of matrices we give a complete characterization of nonuniform exponential contractions and nonuniform exponential dichotomies in terms of lyapunov sequences we note that these include as very special cases uniform exponential contractions and uniform exponential dichotomies due to the central role played by these properties in a substantial part of the theory of dynamical systems in particular in connection with the study of stable and unstable invariant manifolds it is important to have available optimal characterizations that are more amenable to check whether a given dynamics has such a property we also obtain inverse theorems that give explicitly lyapunov sequences for a given contraction or dichotomy as a nontrivial application we establish the robustness under sufficiently small linear perturbations both of nonuniform exponential contractions and nonuniform exponential dichotomies we emphasize that when compared to former work our proof of the robustness property is much simpler

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

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

characterizing the mixed graphs with exactly one positive eigenvalue and its application to mixed graphs determined by their h spectra abstract a mixed graph is obtained by orienting some edges of a simple graph the positive inertia index of a mixed graph is defined as the number of positive eigenvalues of its hermitian adjacency matrix including multiplicities in this paper we study the positive inertia indices of mixed graphs our main results provide a characterization of mixed graphs with positive inertia index 1 as an application some classes of mixed graphs determined by their h spectra are fixed

complex unit gain graphs of rank 2 abstract let t be the group of all complex numbers z with z 1 a complex unit gain graph or simply a t gain graph is a triple g t consisting of a graph g v e the circle group t and a gain function e t such that v i v j v j v i 1 v j v i for any adjacent vertices v i and v j the adjacency matrix a of the t gain graph g t of order n is an n n complex matrix a i j where a i j a j i v i v j if v i is adjacent to v j and a i j 0 if otherwise the rank of a t gain graph denoted by r is the rank of the adjacency matrix of yu et al 24 obtained some properties of inertia indexes of a t gain graph and they characterized the t gain unicyclic graphs with small positive or negative index lu 8 characterized the t gain bicyclic graphs with rank 2 3 or 4 motivated by above we determine the rank of a t gain unicyclic graph and classify the t gain graphs with rank 2

robustness via lyapunov functions for nonautonomous linear equations x a t x we give a complete characterization of nonuniform exponential dichotomies in terms of strict quadratic lyapunov functions nonuniform exponential dichotomies include as a very special case uniform exponential dichotomies in particular we construct explicitly strict lyapunov functions for each exponential dichotomy as a nontrivial application we establish in a simple and direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations this represents a considerable simplification of former work

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

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

robustness of nonuniform μ ν dichotomies in banach spaces abstract this paper focuses on the linear perturbation of dynamical systems in banach spaces which corresponds to a nonautonomous dynamical system admitting a nonuniform dichotomy we establish the robustness or roughness of nonuniform dichotomies in the sense that the linear dynamical system persists under a sufficiently small linear perturbation

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

the inertia of weighted unicyclic graphs let gw be a weighted graph the inertia of gw is the triple in gw i gw i gw i0 gw where i gw i gw i0 gw are the numbers of the positive negative and zero eigenvalues of the adjacency matrix a gw of gw including their multiplicities respectively i gw i gw are called the positive negative indices of inertia of gw respectively in this paper we present a lower bound for the positive negative indices of weighted unicyclic graphs of order n with fixed girth and characterize all weighted unicyclic graphs attaining this lower bound moreover we characterize the weighted unicyclic graphs of order n with two positive two negative and at least n 6 zero eigenvalues respectively

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

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

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

on the sigma exponent of discrete linear systems in this technical note we introduce the concepts of sigma exponent of discrete linear time varying system it is shown that these exponents describe the possible changes in the lyapunov exponents under perturbation decreasing at infinity at exponential rate finally we present formulas for the exponential exponents in terms of transition matrix of the system

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

free vibration analysis of a circular cylindrical shell using the rayleigh ritz method and comparison of different shell theories abstract this study uses the rayleigh ritz method to derive a dynamic model for the free vibration analysis of a circular cylindrical shell in particular explicit expressions for the mass and stiffness matrices are obtained to easily implement a computer simulation under different shell theories and boundary conditions the dynamic model is constructed according to the donnell mushtari theory which is fully discussed herein and then dynamic models are constructed by using sanders theory love timoshenko theory reissner theory flugge theory and vlasov theory this paper also discusses the use of eigenfunctions of a uniform beam as admissible functions that produce compact expressions for the mass and stiffness matrices the numerical results indicate that the donnell mushtari theory is not sufficiently accurate to calculate the natural frequencies and that there is no discernible difference between the other shell theories considered in this study

ck centre unstable manifolds we consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps we show that every global centre unstable manifold can be expressed as a graph of a c k map provided that the nonlinearities are c k smooth the proofs are based on a lemma by d henry on a necessary and sufficient condition for a lipschitz map to be continuously differentiable

free vibration analysis of functionally graded conical cylindrical shell and annular plate structures with a four parameter power law distribution based on the first order shear deformation theory fsdt this paper focuses on the dynamic behavior of moderately thick functionally graded conical cylindrical shells and annular plates the last two structures are obtained as special cases of the conical shell formulation the treatment is developed within the theory of linear elasticity when materials are assumed to be isotropic and inhomogeneous through the thickness direction the two constituent functionally graded shell consists of ceramic and metal these constituents are graded through the thickness from one surface of the shell to the other a generalization of the power law distribution presented in literature is proposed two different four parameter power law distributions are considered for the ceramic volume fraction some material profiles through the functionally graded shell thickness are illustrated by varying the four parameters of power law distributions for the first power law distribution the bottom surface of the structure is ceramic rich whereas the top surface can be metal rich ceramic rich or made of a mixture of the two constituents and on the contrary for the second one symmetric and asymmetric volume fraction profiles are presented in this paper the homogeneous isotropic material can be inferred as a special case of functionally graded materials fgm the governing equations of motion are expressed as functions of five kinematic parameters by using the constitutive and kinematic relationships the solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell the discretization of the system equations by means of the generalized differential quadrature gdq method leads to a standard linear eigenvalue problem where two independent variables are involved without using the fourier modal expansion methodology numerical results concerning six types of shell structures illustrate the influence of the power law exponent of the power law distribution and of the choice of the four parameters on the mechanical behaviour of shell structures considered

on the exponential exponents of discrete linear systems abstract in this paper we introduce the concepts of exponential exponents of discrete linear time varying systems it is shown that these exponents describe the possible changes in the lyapunov exponents under perturbation decreasing at infinity at exponential rate finally we present formulas for the exponential exponents in terms of the transition matrix of the system

on the nullity of bicyclic graphs the nullity of a graph is the multiplicity of the eigenvalue zero in its spectrum in this paper we obtain the nullity set of n vertex unicyclic graphs and characterize the unicyclic graphs with extremal nullity

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

signed graphs with cut points whose positive inertia indexes are two abstract a signed graph g consists of an underlying graph g and a sign function which assigns each edge uv of g a sign u v either positive or negative the adjacency matrix of g is defined as a g a u v with a u v u v a u v where a u v 1 if u v v g are adjacent and a u v 0 otherwise the positive inertia index of g written as p g is defined to be the number of positive eigenvalues of a g recently yu et al 2016 12 characterized the signed graphs g with pendant vertices such that p g 2 in this paper we extend the above work to a more general case characterizing the signed graphs g with cut points whose positive inertia index is 2

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

stability by the linear approximation for discrete equations this paper is concerned with exponential stability of solutions of perturbed discrete equations for a given m 1 we will provide necessary and sufficient conditions for exponential stability of all perturbed systems with perturbation of order m under the assumption that the unperturbed linear system is exponentially stable basing on this result we obtained necessary and sufficient conditions for exponential stability of the perturbed system for all perturbations of order m 1 for regular systems our results are expressed in terms of regular coefficients of the unperturbed system

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

2 d differential quadrature solution for vibration analysis of functionally graded conical cylindrical shell and annular plate structures this paper focuses on the dynamic behavior of functionally graded conical cylindrical shells and annular plates the last two structures are obtained as special cases of the conical shell formulation the first order shear deformation theory fsdt is used to analyze the above moderately thick structural elements the treatment is developed within the theory of linear elasticity when materials are assumed to be isotropic and inhomogeneous through the thickness direction the two constituent functionally graded shell consists of ceramic and metal that are graded through the thickness from one surface of the shell to the other two different power law distributions are considered for the ceramic volume fraction the homogeneous isotropic material is inferred as a special case of functionally graded materials fgm the governing equations of motion expressed as functions of five kinematic parameters are discretized by means of the generalized differential quadrature gdq method the discretization of the system leads to a standard linear eigenvalue problem where two independent variables are involved without using the fourier modal expansion methodology for the homogeneous isotropic special case numerical solutions are compared with the ones obtained using commercial programs such as abaqus ansys nastran straus pro mechanica very good agreement is observed furthermore the convergence rate of natural frequencies is shown to be very fast and the stability of the numerical methodology is very good different typologies of non uniform grid point distributions are considered finally for the functionally graded material case numerical results illustrate the influence of the power law exponent and of the power law distribution choice on the mechanical behavior of shell structures

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

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

a semi analytical method for the free vibration of doubly curved shells of revolution abstract in this paper a semi analytical method is used to investigate the free vibration of doubly curved shells of revolution with arbitrary boundary conditions the doubly curved shells of revolution are divided into their segments in the meridional direction and the theoretical model for vibration analysis is formulated by applying flugge s thin shell theory regardless of the boundary conditions the displacement functions of shell segments are composed by the jacobi polynomials along the revolution axis direction and the standard fourier series along the circumferential direction the boundary conditions at the ends of the doubly curved shells of revolution and the continuous conditions at two adjacent segments were enforced by the penalty method then the natural frequencies of the doubly curved shells are obtained by using the rayleigh ritz method for arbitrary boundary conditions this method does not require any changes to the mathematical model or the displacement functions and it is very effective in the analysis of free vibration for doubly curved shells of revolution the credibility and exactness of proposed method are compared with the results of finite element method fem and some numerical results are reported for free vibration of the doubly curved shells of revolution under classical and elastic boundary conditions results of this paper can provide reference data for future studies in related field

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

proportional local assignability of lyapunov spectrum of linear discrete time varying systems we consider a local version of the pole assignment problem for linear discrete time varying systems our aim is to obtain sufficient conditions to place the lyapunov spectrum of the closed loop sys

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

nonuniform μ ν dichotomies and local dynamics of difference equations abstract we obtain a local stable manifold theorem for perturbations of nonautonomous linear difference equations possessing a very general type of nonuniform dichotomy possibly with different growth rates in the uniform and nonuniform parts note that we consider situations where the classical lyapunov exponents can be zero additionally we study how the manifolds decay along the orbit of a point as well as the behavior under perturbations and give examples of nonautonomous linear difference equations that admit the dichotomies considered

existence of nonuniform exponential dichotomies and a fredholm alternative abstract we consider linear equations x a t x in a banach space with nonuniform exponential dichotomies in r and r and we give an admissibility condition for the existence of nonuniform exponential dichotomies in the whole line under sufficiently small perturbations we also obtain a fredholm alternative for the existence of bounded solutions under nonlinear perturbations

stability theory and lyapunov regularity abstract we establish the stability under perturbations of the dynamics defined by a sequence of linear maps that may exhibit both nonuniform exponential contraction and expansion this means that the constants determining the exponential behavior may increase exponentially as time approaches infinity in particular we establish the stability under perturbations of a nonuniform exponential contraction under appropriate conditions that are much more general than uniform asymptotic stability the conditions are expressed in terms of the so called regularity coefficient which is an essential element of the theory of lyapunov regularity developed by lyapunov himself we also obtain sharp lower and upper bounds for the regularity coefficient thus allowing the application of our results to many concrete dynamics it turns out that using the theory of lyapunov regularity we can show that the nonuniform exponential behavior is ubiquitous contrarily to what happens with the uniform exponential behavior that although robust is much less common we also consider the case of infinite dimensional systems

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

exact solutions for vibration of stepped circular cylindrical shells abstract based on the flugge thin shell theory this paper presents exact solutions for the vibration of circular cylindrical shells with step wise thickness variations in the axial direction the shell is sub divided into multiple segments at the locations of thickness variations the state space technique is adopted to derive the homogenous differential equations for a shell segment and domain decomposition method is employed to impose the equilibrium and compatibility requirements along the interfaces of the shell segments to ensure the correctness of the present results comparisons are made with one paper available in the open literature based on the donnell mushtari theory shells with various combinations of end boundary conditions can be analyzed by the proposed method furthermore the influences of the shell thickness ratios locations of step wise thickness variations and step thickness ratios on the natural frequencies and mode shapes are examined the exact vibration results can serve as important benchmark values for researchers to validate their numerical methods for such circular cylindrical shells

nonuniform exponential contractions and lyapunov sequences abstract for a nonautonomous dynamics with discrete time obtained from the product of linear operators we show that a nonuniform exponential contraction can be completely characterized in terms of what we call strict lyapunov sequences we note that nonuniform exponential contractions include as a very particular case the uniform exponential contractions that correspond to have a uniform asymptotic stability of the dynamics we also obtain inverse theorems that give explicitly strict lyapunov sequences for each nonuniform exponential contraction essentially the lyapunov sequences are obtained in terms of what are usually called lyapunov norms that is norms with respect to which the behavior of a nonuniform exponential contraction becomes uniform we also show how the characterization of nonuniform exponential contractions in terms of quadratic lyapunov sequences can be used to establish in a very simple manner the persistence of the asymptotic stability of a nonuniform exponential contraction under sufficiently small linear or nonlinear perturbations moreover we describe an appropriate version of our results in the context of ergodic theory showing that the existence of an eventually strict lyapunov function implies that all lyapunov exponents are negative almost everywhere