discrete riemann surfaces and the ising model we define a new theory of discrete riemann surfaces and present its basic results the key idea is to consider not only a cellular decomposition of a surface but the union with its dual discrete holomorphy is defined by a straightforward discretisation of the cauchy riemann equation a lot of classical results in riemann theory have a discrete counterpart hodge star harmonicity hodge theorem weyl s lemma cauchy integral formula existence of holomorphic forms with prescribed holonomies giving a geometrical meaning to the construction on a riemann surface we define a notion of criticality on which we prove a continuous limit theorem we investigate its connection with criticality in the ising model we set up a dirac equation on a discrete universal spin structure and we prove that the existence of a dirac spinor is equivalent to criticality

moment lyapunov exponent and stability index for linear conservative system with small random perturbation asymptotic expansion series for the moment lyapunov exponent and stability index are constructed and justified for the two dimensional linear stochastic system close to a harmonic oscillator as an example a one degree of freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained

quadri tilings of the plane we introduce quadri tilings and show that they are in bijection with dimer models on a family of graphs r arising from rhombus tilings using two height functions we interpret a sub family of all quadri tilings called triangular quadri tilings as an interface model in dimension 2 2 assigning critical weights to edges of r we prove an explicit expression only depending on the local geometry of the graph r for the minimal free energy per fundamental domain gibbs measure this solves a conjecture of kenyon invent math 150 409 439 2002 we also show that when edges of r are asymptotically far apart the probability of their occurrence only depends on this set of edges finally we give an expression for a gibbs measure on the set of all triangular quadri tilings whose marginals are the above gibbs measures and conjecture it to be that of minimal free energy per fundamental domain

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

moment lyapunov exponents of the stochastic parametrical hill s equation abstract the lyapunov exponent and moment lyapunov exponents of hill s equation with frequency and damping coefficient fluctuated by white noise stochastic process are investigated a perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises the results are applied to the study of the almost sure and the moment stability of the stationary solutions of the thin simply supported beam subjected to axial compressions and time varying damping which are small intensity stochastic excitations

conformal invariance of domino tiling let u be a multiply connected region in r 2 with smooth boundary let p be a polyomino in z 2 approximating u as 0 we show that for certain boundary conditions on p the height distribution on a random domino tiling dimer covering of p is conformally invariant in the limit as tends to 0 in the sense that the distribution of heights of boundary components or rather the difference of the heights from their mean values only depends on the conformal type of u the mean height is not strictly conformally invariant but transforms analytically under conformal mappings in a simple way the mean height and all the moments are explicitly evaluated

new topologies on colombeau generalized numbers and the fermat reyes theorem abstract based on the theory of fermat reals we introduce new topologies on spaces of colombeau generalized points and derive some of their fundamental properties in particular we obtain metric topologies on the space of near standard generalized points that induce the standard euclidean topology on the reals we analyze the description of the sharp topology in terms of the natural extension of the absolute value or of the defining semi norms in the case of locally convex spaces thereby preserving a number of classical notions building on a new point value characterization of colombeau generalized functions we prove a fermat reyes theorem that forms the basis of an approach to differentiation on spaces of generalized functions close to the classical one

rotational invariance and discrete analyticity in the 2d dimer model we exploit the discrete analytic structure present in the 2d dimer model to rigorously compute the exponents of a class of two point functions in all directions this is a dimer analogue of the critical 2d ising spin spin correlation function

nonstandard principles for generalized functions abstract we show that principles from nonstandard analysis hold to some extent for nonlinear generalized functions the generalized functions under consideration are constructed as families of functions modulo a free filter as it is usually done in applied analysis in contrast with models of nonstandard analysis we do not require the filter to be an ultrafilter the principles are intended to be used as a tool for proving theorems which we illustrate by means of an automatic continuity result that was not suspected by experts in the field

height fluctuations in the honeycomb dimer model we study a model of random surfaces arising in the dimer model on the honeycomb lattice for a fixed wire frame boundary condition as the lattice spacing 0 cohn kenyon and propp 3 showed the almost sure convergence of a random surface to a non random limit shape 0 in 12 okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions in particular constructing a natural conformal structure on them we show here that when 0 has no facets for a family of boundary conditions approximating the wire frame the large scale surface fluctuations height fluctuations about 0 converge as 0 to a gaussian free field for the above conformal structure we also show that the local statistics of the fluctuations near a given point x are as conjectured in 3 given by the unique ergodic gibbs measure on plane configurations whose slope is the slope of the tangent plane of 0 at x

almost sure stability of linear stochastic differential equations with jumps under the nondegenerate condition as in the diffusion case see 14 21 6 the linear stochastic jump diffusion process projected on the unit sphere is a strong feller process and has a unique invariant measure which is also ergodic using the relation between the transition probabilities of jump diffusions and the corresponding diffusions due to 22 the unique deterministic lyapunov exponent can be represented by the furstenberg khas minskii formula as an integral over the sphere or the projective space with respect to the ergodic invariant measure so that the almost sure asymptotic stability of linear stochastic systems with jumps depends on its sign the critical case of zero lyapunov exponent is discussed and a large deviations result for asymptotically stable systems is further investigated several examples are treated for illustration

dimers in piecewise temperleyan domains we study the large scale behavior of the height function in the dimer model on the square lattice richard kenyon has shown that the fluctuations of the height function on temperleyan discretizations of a planar domain converge in the scaling limit as the mesh size tends to zero to the gaussian free field with dirichlet boundary conditions we extend kenyon s result to a more general class of discretizations moreover we introduce a new factorization of the coupling function of the double dimer model into two discrete holomorphic functions which are similar to discrete fermions defined in smirnov proceedings of the international congress of mathematicians icm madrid spain 2006 ann math 2 172 1435 1467 2010 for temperleyan discretizations with appropriate boundary modifications the results of kenyon imply that the expectation of the double dimer height function converges to a harmonic function in the scaling limit we use the above factorization to extend this result to the class of all polygonal discretizations that are not necessarily temperleyan furthermore we show that quite surprisingly the expectation of the double dimer height function in the temperleyan case is exactly discrete harmonic for an appropriate choice of laplacian even before taking the scaling limit

the emergence of the electrostatic field as a feynman sum in random tilings with holes we consider random lozenge tilings on the triangular lattice with holes q 1 q n in some fixed position for each unit triangle not in a hole consider the average orientation of the lozenge covering it we show that the scaling limit of this discrete field is the electrostatic field obtained when regarding each hole q i as an electrical charge of magnitude equal to the difference between the number of unit triangles of the two different orientations inside q i this is then restated in terms of random surfaces yielding the result that the average over surfaces with prescribed height at the union of the boundaries of the holes is in the scaling limit a sum of helicoids

moment lyapunov exponents of a two dimensional system under bounded noise parametric excitation the moment lyapunov exponents of a two dimensional system under bounded noise parametric excitation are studied in this paper the method of regular perturbation is applied to obtain weak noise expansions of the moment lyapunov exponent lyapunov exponent and stability index in terms of the small fluctuation parameter

conformal invariance in random cluster models i holmorphic fermions in the ising model we construct discrete holomorphic observables in the ising model at criticality and show that they have conformally covariant scaling limits as mesh of the lattice tends to zero in the sequel those observables are used to construct conformally invariant scaling limits of interfaces though ising model is often cited as a classical example of conformal invariance it seems that ours is the first paper where it is actually established

on the asymptotics of dimers on tori we study asymptotics of the dimer model on large toric graphs let mathbb l be a weighted mathbb z 2 periodic planar graph and let mathbb z 2 e be a large index sublattice of mathbb z 2 for mathbb l bipartite we show that the dimer partition function textsf textit z e on the quotient mathbb l mathbb z 2 e has the asymptotic expansion begin aligned textsf textit z exp a mathbf f 0 textsf fsc o 1 end aligned where a is the area of mathbb l mathbb z 2 e mathbf f 0 is the free energy density in the bulk and textsf fsc is a finite size correction term depending only on the conformal shape of the domain together with some parity type information assuming a conjectural condition on the zero locus of the dimer characteristic polynomial we show that an analogous expansion holds for mathbb l non bipartite the functional form of the finite size correction differs between the two classes but is universal within each class our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated double dimer models

conformal invariance of loops in the double dimer model the dimer model is the study of random dimer covers perfect matchings of a graph a double dimer configuration on a graph mathcal g is a union of two dimer covers of mathcal g we introduce quaternion weights in the dimer model and show how they can be used to study the homotopy classes relative to a fixed set of faces of loops in the double dimer model on a planar graph as an application we prove that in the scaling limit of the uniform double dimer model on mathbb z 2 or on any other bipartite planar graph conformally approximating mathbb c the loops are conformally invariant

internal sets and internal functions in colombeau theory this paper is devoted to the study of generalized functions as pointwise functions so called internal functions on certain sets of generalized points so called internal sets we treat the case of the colombeau algebras of generalized functions for which these notions have turned out to constitute a fundamental technical tool we provide general foundations for the notion of internal functions and internal sets and prove a saturation principle various applications to colombeau algebras are given

moment lyapunov exponents of a two dimensional system under combined harmonic and real noise excitations abstract the moment lyapunov exponents and the lyapunov exponents of a two dimensional system under combined excitations of harmonic and real noise which is modelled as an ornstein uhlenbeck process are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a regular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the real noise excitation on the parametric resonance due to the harmonic excitation is investigated

the moment lyapunov exponent of a co dimension two bifurcation system driven by non gaussian colored noise in the present paper based on the concept of the pth moment lyapunov exponent the stochastic stability of a typical co dimension two bifurcation system that is on a three dimensional center manifold and possesses two pure imaginary eigenvalues and one zero eigenvalue and is excited by a non gaussian colored noise is investigated the non gaussian colored noise is treated as an ornstein uhlenbeck process by means of the path integral approach based on the perturbation approach and the green s functions method the second differential eigenvalue equation which governing the moment lyapunov exponent is established by solving the eigenvalue problem the weak noise asymptotic expansions for the finite pth moment lyapunov exponent are obtained and which matches the approximation of the numerical monte carlo simulations finally the conclusions are given

moment lyapunov exponents of a two dimensional system under both harmonic and white noise parametric excitations the moment lyapunov exponents and the lyapunov exponents of a 2d system under both harmonic and white noise excitations are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a singular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the white noise excitation on the parametric resonance due to the harmonic excitation is investigated

large deviations and stochastic flows of diffeomorphisms previous results in the theory of large deviations for additive functionals of a diffusion process on a compact manifold m are extended and then applied to the analysis of the lyapunov exponents of a stochastic flow of diffeomorphisms of m an approximation argument relates these results to the behavior near the diagonal in m2 of the associated two point motion finally it is shown under appropriate non degeneracy conditions that the two point motion is ergodic on m2 if the top lyapunov exponent is positive

moment lyapunov exponents of a two dimensional system under real noise excitation abstract the moment lyapunov exponents of a two dimensional system under real noise excitation an ornstein uhlenbeck process are studied in this paper the method of regular perturbation is applied to obtain the weak noise expansions of the moment lyapunov exponent lyapunov exponent and stability index in terms of the small fluctuation parameter

almost sure and moment lyapunov exponents for a stochastic beam equation the transverse vibrations of an euler bernoulli beam with axial tension p and axial white noise forcing are given by mytt m yt eiyxxxx pyxx 0yxxw t 0 x l with hinged endpoints y 0 t yxx 0 t y l t yxx l t 0 the solution may be written y x t n 1 sin n xl cn t where the amplitude cn t of the nth mode of vibration is given by a two dimensional linear stochastic ordinary differential equation the main results give formulas for the almost sure and moment lyapunov exponents for the spde in terms of the almost sure and moment lyapunov exponents for each of the sdes corresponding to modes which are present in the initial condition it is shown that the almost sure and moment lyapunov exponents for the spde depend sensitively on the initial distribution of energy amongst the infinitely many modes of vibration and that they cannot be well approximated using only finitely many modes of vibration

loop statistics in the toroidal honeycomb dimer model the dimer model on a graph embedded in the torus can be interpreted as a collection of random self avoiding loops in this paper we consider the uniform toroidal honeycomb dimer model we prove that when the mesh of the graph tends to zero and the aspect of the torus is fixed the winding number of the collection of loops converges in law to a two dimensional discrete gaussian distribution this is known to physicists in more generality from their analysis of toroidal two dimensional critical loop models and their mapping to the massless free field on the torus this paper contains the first mathematical proof of this more general physics result in the specific case of the loop model induced by a toroidal dimer model

hilbert ℂ modules structural properties and applications to variational problems abstract we develop a theory of hilbert c over tilde modules which forms the core of a new functional analytic approach to algebras of generalized functions particular attention is given to nitely generated submodules projection operators representation theorems for c over tilde linear functionals and c over tilde sesquilinear forms we establish a generalized lax milgram theorem and use it to prove existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form

a characterisation of the gaussian free field we prove that a random distribution in two dimensions which is conformally invariant and satisfies a natural domain markov property is a multiple of the gaussian free field this result holds subject only to a fourth moment assumption

characterization of distributions having a value at a point in the sense of robinson abstract we characterize schwartz distributions having a value at a single point in the sense introduced by means of nonstandard analysis by a robinson they appear to be continuous functions in a neighborhood of the point this characterization improves a result by p loeb which assumes the everywhere existence of point values

a variational principle for domino tilings 1 1 description of results a domino is a 1 x 2 or 2 x 1 rectangle and a tiling of a region by dominos is a way of covering that region with dominos so that there are no gaps or overlaps in 1961 kasteleyn kal found a formula for the number of domino tilings of an m x n rectangle with mn even as shown in figure 1 for rm n 68 temperley and fisher tf used a differenit method and arrived at the same result at almost exactly the same time both lines of calculation showed that the logarithm of number of tilings divided by the number of dominos in a tiling that is mn 2 converges to 2g 7r 0 58 here g is catalan s constant on the other hand in 1992 elkies et al eklp studied domino tilings of regions they called aztec diamonds figure 2 shows an aztec diamond of order 48 and showed that the logarithm of the number of tilings divided by the number of dominos converges to the smaller number log 2 2 0 35 thus even though the region in figure 1 has slightly smaller area than the region in figure 2 the former has far more domino tilings for regions with other shapes neither of these asymptotic formulas may apply in the present paper we consider simply connected regions of arbitrary shape we give an exact formula for the limiting value of the logarithm of the number of tilings per unit area as a function of the shape of the boundary of the region as the size of the region goes to infinity in particular we show that computation of this limit is intimately linked with an understanding of long range variations in the local statistics of random domino tilings such variations can be seen by comparing figures 1 and 2 each of the two tilings is random in the sense that the algorithm pwi that was used to create it generates each of the possible tilings of the region being tiled with the same probability hence one can expect each tiling to be qualitatively typical of the overwhelming majority of tilings of the region in

dimers and amoebae we study random surfaces which arise as height functions of random perfect matchings a k a dimer configurations on a weighted bipartite doubly periodic graph g embedded in the plane we derive explicit formulas for the surface tension and local gibbs measure probabilities of these models the answers involve a certain plane algebraic curve which is the spectral curve of the kasteleyn operator of the graph for example the surface tension is the legendre dual of the ronkin function of the spectral curve the amoeba of the spectral curve represents the phase diagram of the dimer model further we prove that the spectral curve of a dimer model is always a real curve of special type namely it is a harnack curve this implies many qualitative and quantitative statement about the behavior of the dimer model such as existence of smooth phases decay rate of correlations growth rate of height function fluctuations etc

a pfaffian formula for monomer dimer partition functions we consider the monomer dimer partition function on arbitrary finite planar graphs and arbitrary monomer and dimer weights with the restriction that the only non zero monomer weights are those on the boundary we prove a pfaffian formula for the corresponding partition function as a consequence of this result multipoint boundary monomer correlation functions at close packing are shown to satisfy fermionic statistics our proof is based on the celebrated kasteleyn theorem combined with a theorem on pfaffians proved by one of the authors and a careful labeling and directing procedure of the vertices and edges of the graph

strongly internal sets and generalized smooth functions abstract based on a refinement of the notion of internal sets in colombeau s theory so called strongly internal sets we introduce the space of generalized smooth functions a maximal extension of colombeau generalized functions generalized smooth functions as morphisms between sets of generalized points form a sub category of the category of topological spaces in particular they can be composed unrestrictedly

moment lyapunov exponents and stochastic stability of a double beam system under compressive axial loading the lyapunov exponent and moment lyapunov exponents of two degrees of freedom linear systems subjected to white noise parametric excitation are investigated the method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises the lyapunov exponent and moment lyapunov exponents are important characteristics for determining the almost sure and moment stability of a stochastic dynamical system as an example we study the almost sure and moment stability of the double beam system under stochastic compressive axial loading the validity of the approximate results for moment lyapunov exponents is checked by the numerical monte carlo simulation method for this stochastic system

rhombic embeddings of planar quad graphs given a finite or infinite planar graph all of whose faces have degree 4 we study embeddings in the plane in which all edges have length 1 that is in which every face is a rhombus we give a necessary and sucient condition for the existence of such an embedding as well as a description of the set of all such embeddings r etant donne un graphe planaire fini ou infini dont toutes les faces sont de degre 4 onetudie ses plongements dans le plan dont toutes les aretes sont de longueur 1 c estdire dont toutes les faces sont des losanges on donne une condition necessaire et susante pour l existence d un tel plongement et on decrit l ensemble de ces plongements

algebraic theory of colombeauʼs generalized numbers abstract let k denote the commutative ring of colombeau s full generalized numbers endowed with scarpalezos sharp topology it becomes a topological ring we study the algebraic and topological properties of this topological ring in particular we prove that the group of units of k is dense in the sharp topology determine its boolean algebra show that it has minimal primes describe them completely which results in a complete classification of the maximal ideals from the description of the prime and maximal ideals it becomes clear that they should be determined by certain ultra filters

kam theorem and quantum field theory we give a new proof of the kam theorem for analytic hamiltonians the proof is inspired by a quantum field theory formulation of the problem and is based on a renormalization group argument treating the small denominators inductively scale by scale the crucial cancellations of resonances are shown to follow from the ward identities expressing the translation invariance of the corresponding field theory

lyapunov exponent and rotation number of two dimensional linear stochastic systems with small diffusion by means of a singular perturbation scheme and large deviation estimates we expand the lyapunov exponent and rotation number in terms of the small noise intensity varepsilon for all 2 times 2 systems dx t varepsilon ax t varepsilon dt sqrt varepsilon sum nolimits k 1 r b k x t varepsilon circ dw t k for which a is complex diagonalizable examples are given

bosonization on higher genus riemann surfaces we prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions the theories have fields of arbitrary spin on compact surfaces with any number of handles global considerations require that we add new topological terms to the bosonic action the proof that our prescription is correct relies on methods of complex algebraic geometry

dominos and the gaussian free field we define a scaling limit of the height function on the domino tiling model dimer model on simply connected regions in z 2 and show that it is the massless free field a gaussian process with independent coefficients when expanded in the eigenbasis of the laplacian

boundary monomers in the dimer model the correlation functions of an arbitrary number of boundary monomers in the system of close packed dimers on the square lattice are computed exactly in the scaling limit the equivalence of the 2n point correlation functions with those of a complex free fermion is proved thereby reinforcing the description of the monomer dimer model by a conformal free field theory with central charge c 1 comment 15 pages 2 figures

locally convex topological algebras of generalized functions compactness and nuclearity in a nonlinear context in this paper we introduce hausdorff locally convex algebra topologies on subalgebras of the whole algebra of nonlinear generalized functions these topologies are strong duals of frechet schwartz space topologies and even strong duals of nuclear frechet space topologies in particular any bounded set is relatively compact and one benefits from all deep properties of nuclearity these algebras of generalized functions contain most of the classical irregular functions and distributions they are obtained by replacing the mathematical tool of c functions in the original version of nonlinear generalized functions by the far more evolved tool of holomorphic functions this paper continues the nonlinear theory of generalized functions in which such locally convex topological properties were strongly lacking up to now

inhomogeneous bond percolation on square triangular and hexagonal lattices the star triangle transformation is used to obtain an equivalence extending over the set of all in homogeneous bond percolation models on the square triangular and hexagonal lattices among the consequences are box crossing rsw inequalities for such models with parameter values at which the transformation is valid this is a step toward proving the universality and conformality of these processes it implies criticality of such values thereby providing a new proof of the critical point of inhomogeneous systems the proofs extend to certain isoradial models to which previous methods do not apply

dimers on surface graphs and spin structures ii partition functions for dimers on closed oriented surfaces are known to be alternating sums of pfaffians of kasteleyn matrices in this paper we obtain the formula for the coefficients in terms of discrete spin structures

the laplacian and dirac operators on critical planar graphs on a periodic planar graph whose edge weights satisfy a certain simple geometric condition the discrete laplacian and dirac operators have the property that their determinants and inverses only depend on the local geometry of the graph we obtain explicit expressions for the logarithms of the normalized determinants as well as the inverses of these operators we relate the logarithm of the determinants to the volume plus mean curvature of an associated hyperbolic ideal polyhedron in the associated dimer and spanning tree models for which the determinants of the dirac operator and the laplacian respectively play the role of the partition function this allows us to compute the entropy and correlations in terms of the local geometry in addition we define a continuous family of special discrete holomorphic functions which via convolutions gives a general process for constructing discrete holomorphic functions and discrete harmonic functions on critical planar graphs

relations between the coulomb gas picture and conformal invariance of two dimensional critical models partition functions of critical 2d models on a torus can be derived from their microscopic formulation and their free field representation in the continuum limit this is worked out explicitly for theo n andq state potts model forn orq integer we recover results obtained from conformal invariance but our procedure also extends to nonintegral values in the latter case the expansion on characters of the virasoro algebra involves real coefficients of either sign the operator content of both models is discussed in detail

discrete complex analysis on isoradial graphs abstract we study discrete complex analysis and potential theory on a large family of planar graphs the so called isoradial ones along with discrete analogues of several classical results we prove uniform convergence of discrete harmonic measures green s functions and poisson kernels to their continuous counterparts among other applications the results can be used to establish universality of the critical ising and other lattice models

instability of the harmonic oscillator with small noise we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is small and is defined by a temporally homogeneous markov process with a finite number of states in the case of two states the telegraph process we obtain additional terms in the expansions affording comparison between the exact values and the asymptotic formulas

on the stability properties of a van der pol duffing oscillator that is driven by a real noise abstract in this paper we consider a van der pol duffing oscillator that is excited parametrically by a small intensity real noise which is assumed to be an integrable function of an n dimensional ornstein uhlenbeck vector process that is an output of a linear filter system the stability properties include the moment lyapunov exponent g p x 0 and the maximal lyapunov exponent and the stability in probability are examined to study a model of enhanced generality we remove both the detailed balance condition and the strong mixing condition in the case of an arbitrary finite real number p we employ the perturbation method and a spectrum representation of the fokker planck operator of the linear filter system to construct asymptotic expansions of the pth moment lyapunov exponent and the top lyapunov exponent the same methods are also used for a nonlinear stochastic system to obtain the fpk fokker planck kolmogonov equation for the amplitude process which is identical to the one that is derived from the stochastic averaging method in the case of a broadband noise excitation on the basis of this fpk equation we also examine the almost sure stability condition of the ito stochastic differential equation for the amplitude process which matches the result that is derived from the maximal lyapunov exponent finally the method proposed by lin and cai probabilistic structural dynamics advanced theory and application mcgraw hill new york 1995 is adopted to examine the stability in probability of the amplitude process for the nonlinear ito differential equation

mean square stability of a second order parametric linear system excited by a colored gaussian noise abstract the stochastic stability of a second order linear parametric oscillator whose stiffness is perturbed by a stationary zero mean colored gaussian process is investigated in this paper referring to the stability in the second response moments mean square stability the stochastic perturbation is the output of a linear filter excited by a stationary gaussian white noise stochastic process and the hypothesis of weak excitation is not formulated the first assumption allows using ito s stochastic differential calculus the moment equations of the response are written till the second order by means of ito s differential rule they form an infinite hierarchy that is the equations for the moments of order r contain moments of order larger than r in order to close the hierarchy the cumulant neglect closure method is applied herein the moment equation set is closed by neglecting the cumulants of third order nonlinear moment equations result they are linearized and the limit of stability is searched by studying the eigenvalues of the matrix of the coefficients of the linearized moment equations several numerical analyses are performed to evaluate the critical mean square value of the excitation when a system parameter is varied for the cases of both first and second order parametric excitation a first order excitation an ornstein uhlenbeck process does not cause stochastic resonance while a second order excitation does the first result was not known previously