subharmonic and homoclinic bifurcations in a parametrically forced pendulum abstract depending on the parameters of a parametrically forced pendulum system the boundaries of subharmonic and homoclinic bifurcations are calculated on the basis of the melnikov method and of averaging methods it is shown that as a parameter is varied repeated resonances of successively higher periods occur culminating in homoclinic orbits according to the theorem of smale homoclinic bifurcation is the source of the unstable chaotic motions observed for some selected parameter sets the theoretical predictions are tested by numerical calculations very good agreement is found between analytical and numerical results

comparisons between the pendulum with varying length and the pendulum with oscillating support we consider two forced dissipative pendulum systems the pendulum with vertically oscillating support and the pendulum with periodically varying length with a view to draw comparisons between their behaviour we study the two systems for values of the parameters for which the dynamics are non chaotic we focus our investigation on the persisting attractive periodic orbits and their basins of attraction utilising both analytical and numerical techniques although in some respect the two systems have similar behaviour we find that even within the perturbation regime they may exhibit different dynamics in particular for the same value of the amplitude of the forcing the pendulum with varying length turns out to be perturbed to a greater extent furthermore the periodic attractors persist under larger values of the damping coefficient in the pendulum with varying length finally unlike the pendulum with oscillating support the pendulum with varying length cannot be stabilised around the upward position for any values of the parameters

middle convolution and harnad duality dettweiler reiter s additive description of katz s middle convolution can be interpreted in terms of the harnad duality using this interpretation together with mumford s geometric invariant theory we generalize the additive middle convolution to have a multi parameter and act on systems of linear ordinary differential equations with irregular singularities we show that the generalized operation holds basic properties of the original one and additionally show that katz s algorithm involving our generalization works well in generic case

dynamics of the pendulum with periodically varying length dynamic behavior of a weightless rod with a point mass sliding along the rod axis according to periodic law is studied this is the pendulum with periodically varying length which is also treated as a simple model of a child s swing asymptotic expressions for boundaries of instability domains near resonance frequencies are derived domains for oscillation rotation and oscillation rotation motions in parameter space are found analytically and compared with a numerical study chaotic motions of the pendulum depending on problem parameters are investigated numerically

global attraction to the origin in a parametrically driven nonlinear oscillator we consider a parametrically driven nonlinear ode which encompasses a simple model of an electronic circuit known as a parametric amplifier whose linearisation has a zero eigenvalue by adopting two different approaches we obtain conditions for the origin to be a global attractor which is approached a non monotonically and b monotonically in case b we obtain an asymptotic expression for the convergence to the origin some further numerical results are reported

the melnikov theory for subharmonics and their bifurcations in forced oscillations the subharmonic melnikov theory for periodic perturbations of planar hamiltonian systems is improved an approximation to the associated poincare map in action angle coordinates is explicitly constructed and existence stability and bifurcation theorems for subharmonics are obtained in particular simple formulas for determining the stability of subharmonics and invariant circles bifurcating from them at hopf bifurcations are obtained and a degenerate resonance case which was not appropriately treated in previous references is discussed furthermore the weak nonlinearity case in which the unperturbed system is linear is studied the results are also useful to describe dynamics near the unperturbed centers in strongly nonlinear systems several examples are given to illustrate our theory

accessory parameters in conformal mapping exploiting the isomonodromic tau function for painleve vi we present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral the schwarz christoffel accessory parameter problem relevant when all sides have zero curvature is also captured within our approach the method exploits the isomonodromic tau function associated with the painleve vi equation recently these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the young diagrams after showing how to extract the monodromy data associated with the target domain we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem the viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides

semi discretization method for delayed systems summary the paper presents an ecient numerical method for the stability analysis of linear delayed systems the method is based on a special kind of discretization technique with respect to the past eect only the resulting approximate system is delayed and also time periodic but still it can be transformed analytically into a high dimensional linear discrete system the method is applied to determine the stability charts of the mathieu equation with continuous time delay copyright 2002 john wiley sons ltd

study of a class of subharmonic motions using a non smooth temporal transformation nstt abstract a nonlinear boundary value problem nbvp formulation for computing strongly nonlinear subharmonic orbits of a class of harmonically forced conservative systems is presented the formulation is based on a non smooth temporal transformation nstt to replace the independent temporal variable of the problem with two new piecewise linear periodic temporal variables as a result the problem of computing the subharmonic motions is reduced to a set of nbvps with homogeneous boundary conditions some numerical and analytical solutions of the reduced nbvps are given and the asymptotic behavior in the limit of large period of the derived analytical approximations is discussed in particular the formulation shows a definite connection between vibro impact oscillations considered in mechanics and strongly nonlinear oscillations close to a heteroclinic orbit an interesting feature of a family of subharmonic solutions analyzed in this work is that in the limit of large periods it degenerates to the perturbed stable and unstable manifolds of an unstable periodic orbit of the system

resurgence stokes phenomenon and alien derivatives for level one linear differential systems abstract a precise description of the singularities of the borel transform of solutions of a level one linear differential system is deduced from a proof of the summable resurgence of the solutions by the perturbative method of j ecalle then we compare the meromorphic classification stokes phenomenon from the viewpoint of the stokes cocycle and the viewpoint of alien derivatives we make explicit the stokes ramis matrices as functions of the connection constants in the borel plane and we develop two examples no assumption of genericity is made

multi resurgence and connection to stokes formulae for some linear meromorphic differential systems abstract in this article we consider a linear meromorphic differential system with several levels r 1 r p for any k we prove that the borel transforms of its r k reduced formal solutions are resurgent and we give the general form of all their singularities next under some convenient hypotheses on the geometric configuration of singular points we display exact formulae to express some stokes multipliers of level r k of initial system in terms of connection constants in the borel plane generalizing thus formulae already obtained by m loday richaud and the author for systems with a single level as an illustration we develop one numerical example

non commutative painleve equations and hermite type matrix orthogonal polynomials we study double integral representations of christoffel darboux kernels associated with two examples of hermite type matrix orthogonal polynomials we show that the fredholm determinants connected with these kernels are related through the its izergin korepin slavnov iiks theory with a certain riemann hilbert problem using this riemann hilbert problem we obtain a lax pair whose compatibility conditions lead to a non commutative version of the painleve iv differential equation for each family

integrity measures quantifying the erosion of smooth and fractal basins of attraction abstract the phase portraits of a dissipative dynamical system are characterized by the existence of one or more stable attractors which typically include point equilibria periodic oscillations harmonic and subharmonic quasi periodic solutions and chaotic attractors each attractor is embedded in its own domain or basin of attraction bounded by a separatrix associated with an unstable saddle solution under the variation of a control parameter as the attractors move and bifurcate the basins also undergo corresponding changes and metamorphoses these changes in size and shape are usually continuous but can be discontinuous as when an attractor vanishes along with its basin at a saddle node bifurcation associated with the homoclinic tangling of the invariant manifolds of the saddle solution basin boundaries can also change in nature from smooth to fractal in this paper the escape of a driven oscillator from a cubic potential well as an archetypal example is used to explore the engineering significance of the basin erosions that occur under increased forcing various measures of engineering integrity of the constrained attractor are introduced a global measure assesses the overall basin area a local measure assesses the distance from the attractor to the basin boundary and a velocity measure is related to the size of impulse that could be sustained without failure since engineering systems may be subjected to pulse loads of finite duration attention is given to both the absolute and transient basin boudaries the significant erosion of these at homoclinic tangencies is particularly highlighted in the present study the fractal basins having a severely reduced integrity under all three criteria

asymptotic equivalence of density estimation and gaussian white noise signal recovery in gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation having all the essential traits in a pure form the equivalence has mostly been stated informally but an approximation in the sense of le cam s deficiency distance would make it precise the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss in nonparametrics a first result of this kind has recently been established for gaussian regression we consider the analogous problem for the experiment given by n i i d observations having density f on the unit interval our basic result concerns the parameter space of densities which are in a holder ball with exponent 1 2 and which are uniformly bounded away from zero we show that an i i d sample of size n with density f is globally asymptotically equivalent to a white noise experiment with drift f l 2 and variance 1 4n l this represents a nonparametric analog of le cam s heteroscedastic gaussian approximation in the finite dimensional case the proof utilizes empirical process techniques related to the hungarian construction white noise models on f and log f are also considered allowing for various automatic asymptotic risk bounds in the i i d model from white noise

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

indeterminate sub critical bifurcations in parametric resonance in this paper we show that an indeterminate sub critical bifurcation with an unpredictable outcome arises in typical parametrically excited systems this occurs when the trivial solution is located on a fractal boundary as it loses its stability at a sub critical bifurcation here in the presence of even infinitesimal noise the outcome is indeterminate in the sense that we cannot predict on to which co existing attractor the system will settle

second order linear differential equations with two irregular singular points of rank three the characteristic exponent abstract for a second order linear differential equation with two irregular singular points of rank three multiple laplace type contour integral solutions are considered an explicit formula in terms of the stokes multipliers is derived for the characteristic exponent of the multiplicative solutions the stokes multipliers are represented by converging series with terms for which limit formulas as well as more detailed asymptotic expansions are available here certain new recursively known coefficients enter which are closely related to but different from the coefficients of the formal solutions at one of the irregular singular points of the differential equation the coefficients of the formal solutions then appear as finite sums over subsets of the new coefficients as a by product the leading exponential terms of the asymptotic behaviour of the late coefficients of the formal solutions are given and this is a concrete example of the structural results obtained by immink in a more general setting the formulas displayed in this paper are not of merely theoretical interest but they also are complete in the sense that they could be and have been implemented for computing accurate numerical values of the characteristic exponent although the computational load is not small and increases with the rank of the singular point under consideration

rotating solutions and stability of parametric pendulum by perturbation method abstract rotating solutions of a parametrically driven pendulum are studied via a perturbation method by assuming the undamped unforced hamiltonian case as basic solution and damping and excitation as small perturbations the existence and stability of the harmonic solution are determined analytically first order terms are mainly considered but the extensions to higher order terms as well as to subharmonic rotations are straightforward setting up the analysis in the phase space instead of in the physical space has facilitated development of simple but comprehensive formulas comparison with numerical simulations shows that the analytical results are very effective in accurate predictions even for fairly large amplitudes and in detecting the saddle node bifurcation where the rotating solution is born a limited accuracy is observed to first order in detecting the period doubling bifurcation where it loses stability and an explanation is proposed

the asymptotics of monotone subsequences of involutions we compute the limiting distributions of the lengths of the longest monotone subsequences of random signed involutions with or without conditions on the number of fixed points and negated points as the sizes of the involutions tend to infinity the resulting distributions are depending on the number of fixed points 1 the tracy widom distributions for the largest eigenvalues of random goe gue gse matrices 2 the normal distribution or 3 new classes of distributions which interpolate between pairs of the tracy widom distributions we also consider the second rows of the corresponding young diagrams in each case the convergence of moments is also shown the proof is based on the algebraic work of j baik and e rains in 7 which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the riemann hilbert analysis for the orthogonal polynomials by p deift k johansson and baik in 3

dynamics of a parametrically excited double pendulum abstract the 2 2 mode interaction of a parametrically excited double pendulum is explored in the excitation frequency excitation amplitude plane to determine the bifurcation structure at small amplitudes of oscillation the method of averaging combined with centre manifold reduction is used the full equations are solved numerically to extend the bifurcation set to larger amplitudes of response numerical centre manifold reduction is employed to derive two maps valid near two multiple bifurcation points which organise the dynamical phenomena of the mode interaction region iteration of these maps shows the existence of global bifurcations these results are discussed in the light of numerical integrations which show that a whole range of interesting behaviour occurs including torus doubling torus gluing and chaos

experimental versus theoretical robustness of rotating solutions in a parametrically excited pendulum a dynamical integrity perspective the objective of this paper is showing how global safety arguments can be fruitfully used to interpret experimental results of a pendulum parametrically excited by wave motion in fact the results of an experimental campaign developed with the aim of simulating sea waves energy production by a parametric pendulum show that rotations exist in a region which is smaller than the theoretical one this discrepancy can be partially attributed to the experimental approximations and constraints but it has a deeper theoretical motivation by comparing the experimental results with the dynamical integrity profiles we have found that experimental rotations exist only where a measure of dynamical integrity accounting for both attractor robustness and basin compactness is large enough so that they can support experimental imperfections leading to changes in initial conditions

equivalence theory for density estimation poisson processes and gaussian white noise with drift this paper establishes the global asymptotic equivalence between a poisson process with variable intensity and white noise with drift under sharp smoothness conditions on the unknown function this equivalence is also extended to density estimation models by poissonization the asymptotic equivalences are established by constructing explicit equivalence mappings the impact of such asymptotic equivalence results is that an investigation in one of these nonparametric models automatically yields asymptotically analogous results in the other models

stability of rank 2 vector bundles along isomonodromic deformations we are interested in the stability of holomorphic rank 2 vector bundles of degree 0 over compact riemann surfaces which are provided with irreducible meromophic tracefree connections in the case of a logarithmic connection on the riemann sphere such a vector bundle will be trivial up to the isomonodromic deformation associated to a small move of the poles according to a result of a bolibruch in the general case of meromorphic connections over riemann surfaces of arbitrary genus we prove that the vector bundle will be semi stable up to a small isomonodromic deformation more precisely the vector bundle underlying the universal isomonodromic deformation is generically semi stable along the deformation and even maximally stable for curves of genus g 2 this result is non trivial even in the case of non singular connections

characterization in bi parameter space of a non ideal oscillator abstract we investigate the dynamical behavior of a non ideal duffing oscillator a system composed of a mass spring pendulum driven by a dc motor with limited power supply to identify new features on duffing oscillator parameter space due to the limited power supply we provide an extensive numerical characterization in the bi parameter space by using lyapunov exponents following this procedure we identify remarkable new organized distribution of periodic windows the ones known as arnold tongues and also shrimp shaped structures in addition we also identify intertwined basins of attraction for coexisting multiple attractors connected with tongues

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

identifying evaluating and controlling dynamical integrity measures in non linear mechanical oscillators the erosion curves for the helmholtz the duffing and rigid block oscillators are constructed to investigate the loss of integrity in different mechanical systems various definitions of safe basin and integrity measures are reviewed compared with each other and applied to the considered cases a control method aimed at shifting the erosion profiles towards larger excitation amplitudes is considered too and its main features are discussed in depth

material symmetry and singularities in solids we examine the relations between the symmetry group of a crystalline or solid body and the line singularities in the deformation or microstructural field which the material can support a method is developed which in some relevant cases allows us to describe a singularity by means of a suitable derham current supported on the defect submanifold this current measures the obstruction to the extension of the local deformation or microstructure to a globally defined field the procedure described here yields the basis of a formulation of weak field equations for a defective material and clarifies the structure of the compatibility relations for virtually any class of material symmetry

initiation and directional control of period 1 rotation for a parametric pendulum we study a time delayed feedback control for initiating period 1 rotations of a vertically excited parametric pendulum from arbitrary initial conditions the possibility of controlling the direction of rotation has also been explored we start with a simple linear time delayed control for which the control gain corresponding to the most stable period 1 rotation has been obtained using the floquet theory this control increases the basins of attraction of rotations but they do not encompass the full initial condition space we modify our control law by using a switched control gain that destabilizes all the oscillatory solutions and the entire initial condition space becomes the basin of attraction of either the clockwise or the anticlockwise rotation by a suitable modification of the switching condition we can choose a preferential stable direction of rotation hence we can initiate either clockwise or anticlockwise rotation for a parametric pendulum from arbitrary initial conditions performance of our controller in achieving this objective has been demonstrated for different sets of parameters to establish its effectiveness

non periodic driving of coupled oscillators a spherical swing abstract nonlinearly coupled damped oscillators at 1 1 frequency ratio one oscillator being driven coherently for efficient excitation are exemplified by a spherical swing with some phase mismatch between drive and response for certain damping range excitation is found to succeed if it lags behind but to produce a chaotic attractor if it leads the response although a period doubling sequence for damping increasing leads to the attractor this is actually born as a hard as regards amplitude bifurcation at a zero growth rate parametric line as damping decreases an unstable fixed point crosses an invariant plane to enter as saddle focus a phase space domain of physical solutions a second hard bifurcation occurs at the zero mismatch line the saddle focus leaving that domain times on the attractor diverge when approaching either line leading to exactly one dimensional and noninvertible limit maps which are analytically determined

opers with irregular singularity and spectra of the shift of argument subalgebra joint work with boris feigin and edward frenkel math qa 0712 1183 let g be a semisimple complex lie algebra and u g its universal enveloping algebra the algebra u g bears a natural filtration by the degree with respect to the generators the associated graded algebra is the symmetric algebra s g c g by the poincare birkhoff witt theorem the commutator on u g defines the poisson bracket on s g let zs g s g be the center of s g with respect to the poisson bracket and let g g be a regular semisimple element due to the result of mischenko and fomenko 1978 the algebra a s g generated by the elements where zs g is commutative with respect to the poisson bracket and has maximal possible transcendence degree equal to 12 dim g rk g if is a regular element contained in the cartan subalgebra h g then h a and if f is the principal nilpotent element then the subalgebra af contains the principal nilpotent subalgebra zg f it was recently shown that the shift of argument subalgebras can be quantized

computational techniques for nonlinear dynamics in multiple friction oscillators this paper presents some numerical techniques used to investigate the nonlinear dynamics of a class of mechanical systems affected by dry friction forces they allow for the exact detection of the points which separate slip phases from stick phases in this way the correct dynamic equations are integrated during the different motion phases finally a method is described to perform the numerical computation of the lyapunov exponents in such non smooth systems

jacobi elliptic functions a review of nonlinear oscillatory application problems abstract this review paper is concerned with the applications of jacobi elliptic functions to nonlinear oscillators whose restoring force has a monomial or binomial form that involves cubic and or quadratic nonlinearity first geometric interpretations of three basic jacobi elliptic functions are given and their characteristics are discussed it is shown then how their different forms can be utilized to express exact solutions for the response of certain free conservative oscillators these forms are subsequently used as a starting point for a presentation of different quantitative techniques for obtaining an approximate response for free perturbed nonlinear oscillators an illustrative example is provided further two types of externally forced nonlinear oscillators are reviewed i those that are excited by elliptic type excitations with different exact and approximate solutions ii those that are damped and excited by harmonic excitations but their approximate response is expressed in terms of jacobi elliptic functions characteristics of the steady state response are discussed and certain qualitative differences with respect to the classical duffing oscillator excited harmonically are pointed out parametric oscillations of the oscillators excited by an elliptic type forcing are considered as well and the differences with respect to the stability chart of the classical mathieu equation are emphasized the adjustment of the melnikov method to derive the general condition for the onset of homoclinic bifurcations in a system parametrically excited by an elliptic type forcing is provided and compared with those corresponding to harmonic excitations advantages and disadvantages of the use of jacobi elliptic functions in nonlinear oscillatory application problems are discussed and some suggestions for future work are given

inverted oscillations of a driven pendulum inverted oscillations of a parametrically driven planar pendulum are considered together with their relationship to the inverted solution in particular a horseshoe structure of the associated manifolds is identified which explains the similarity between the bifurcations of the inverted position and the hanging position this allows us to apply a large body of existing knowledge to the dynamics enabling a lower bound on the forcing required to achieve inverted oscillations to be established

indeterminate trans critical bifurcations in parametrically excited systems this paper may be regarded as a companion paper to a recent article m s soliman j m t thompson proc r soc lond a 438 511 518 1992 that addressed the phenomena of indeterminate sub critical bifurcations in typical parametrically excited systems this earlier work focused on the principal subharmonic resonance which occurs when the excitation frequency is almost twice the linear natural frequency in contrast we here focus on the fundamental harmonic resonance that arises when the two frequencies are almost the same we show that under this condition indeterminate trans critical bifurcations with unpredictable outcome can arise when the trivial solution is located on a fractal boundary as it loses its stability at the bifurcation the outcome is then unpredictable even under infinitesimal perturbations from the trivial solution

subharmonic and homoclinic bifurcations in the driven and damped sine gordon system abstract chaotic responses induced by an applied biharmonic driven signal on the sine gordon sg system influenced by a constant dc driven and the damping fields are investigated using a collective coordinate approach for the motion of the breather in the system for this biharmonic signal one term has a large amplitude at low frequency thus the classical melnikov method does not apply to such a system however we use the modified version of the melnikov method to homoclinic bifurcations of the perturbed sg system additionally resonant breathers are studied using the modified subharmonic melnikov theory this dynamic behavior is illustrated by some numerical computations

algebraic aspects of increasing subsequences we present a number of results relating partial cauchy littlewood sums integrals over the compact classical groups and increasing subsequences of permutations these include integral formulae for the distribution of the longest increasing subsequence of a random involution with constrained number of fixed points new formulae for partial cauchy littlewood sums as well as new proofs of old formulae relations of these expressions to orthogonal polynomials on the unit circle and explicit bases for invariant spaces of the classical groups together with appropriate generalizations of the straightening algorithm

stokes matrices poisson lie groups and frobenius manifolds the purpose of this paper is to point out and then draw some consequences of the fact that the poisson lie group g dual to g gln c may be identified with a certain moduli space of meromorphic connections over the unit disc having an irregular singularity at the origin g will be fully described in sect 2 the key feature of this point of view is that there is a holomorphic map g g from the dual of the lie algebra to the group g for each choice of diagonal matrix a0 with distinct eigenvalues the irregular type this map is essentially the riemann hilbert map or de rham morphism for such connections we will call it the monodromy map it is generically a local analytic isomorphism the main result is

zeros of the jimbo miwa ueno tau function we introduce a family of local deformations for meromorphic connections on p1 in the neighborhood of a higher rank simple singularity following the scheme in malgrange mathematique et physique progress in mathematics birkhauser boston 1983 vol 37 pp 381 400 ibid pp 401 426 ibid pp 427 438 we use these local models to prove that the zeros of the tau function introduced by jimbo miwa and ueno in their pioneering work on birkhoff deformations at irregular singular points physica d 2 306 352 2 407 448 1981 4 26 46 1983 publ rims kyoto univ 17 2 703 721 1981 occur at precisely those points in the deformation space at which a certain birkhoff riemann hilbert problem fails to have a solution

tusnady s inequality revisited tusnady s inequality is the key ingredient in the kmt hungarian coupling of the empirical distribution function with a brownian bridge we present an elementary proof of a result that sharpens the tusnady inequality modulo constants our method uses the beta integral representation of binomial tails simple taylor expansion and some novel bounds for the ratios of normal tail probabilities

matrix models and stochastic growth in donaldson thomas theory we show that the partition functions which enumerate donaldson thomas invariants of local toric calabi yau threefolds without compact divisors can be expressed in terms of specializations of the schur measure we also discuss the relevance of the hall littlewood and jack measures in the context of bps state counting and study the partition functions at arbitrary points of the kahler moduli space this rewriting in terms of symmetric functions leads to a unitary one matrix model representation for donaldson thomas theory we describe explicitly how this result is related to the unitary matrix model description of chern simons gauge theory this representation is used to show that the generating functions for donaldson thomas invariants are related to tau functions of the integrable toda and toeplitz lattice hierarchies the matrix model also leads to an interpretation of donaldson thomas theory in terms of non intersecting paths in the lock step model of vicious walkers we further show that these generati

the dependence on the monodromy data of the isomonodromic tau function the isomonodromic tau function defined by jimbo miwa ueno vanishes on the malgrange s divisor of generalized monodromy data for which a vector bundle is nontrivial or which is the same a certain riemann hilbert problem has no solution in their original work jimbo miwa ueno provided an algebraic construction of its derivatives with respect to isomonodromic times however the dependence on the generalized monodromy data i e monodromy representation and stokes parameters was not derived we fill the gap by providing a simpler and more general description in which all the parameters of the problem monodromy changing and monodromy preserving are dealt with at the same level we thus provide variational formulae for the isomonodromic tau function with respect to the generalized monodromy data the construction applies more generally given any sufficiently well behaved family of riemann hilbert problems rhp where the jump matrices depend arbitrarily on deformation parameters we can construct a one form not necessarily closed on the deformation space malgrange s differential defined off malgrange s divisor we then introduce the notion of discrete schlesinger transformation it means that we allow the solution of the rhp to have poles or zeros at prescribed point s even if is not closed its difference evaluated along the original solution and the transformed one is shown to be the logarithmic differential on the deformation space of a function as a function of the position of the points of the schlesinger transformation it yields a natural generalization of the sato formula for the baker akhiezer vector even in the absence of a tau function and it realizes the solution of the rhp as such ba vector some exemplifications in the setting of the painleve ii equation and finite toplitz hankel determinants are provided

random words toeplitz determinants and integrable systems ii this paper connects the analysis of the length of the longest weakly increasing subsequence of inhomogeneous random words to a riemnn hilbert problem and an associated system of integrable partial differential equations in particular we show that the poissonization of the distribution function of this length can be identified as the jimbo miwa ueno tau function

duality of 2d gravity as a local fourier duality the p q duality is a relation between the p q model and the q p model of two dimensional quantum gravity geometrically this duality corresponds to a relation between the two relevant points of the sato grassmannian kharchev and marshakov have expressed such a relation in terms of matrix integrals some explicit formulas for small p and q have been given in the work of fukuma kawai nakayama already in the duality between the 2 3 model and the 3 2 model the formulas are long in this work a new approach to p q duality is given it can be realized in a precise sense as a local fourier duality of d modules this result is obtained as a special case of a local fourier duality between irregular connections associated to kac schwarz operators therefore since these operators correspond to virasoro constraints this allows us to view the p q duality as a consequence of the duality of the relevant virasoro constraints

homoclinic motions and chaos in the quasiperiodically forced van der pol duffing oscillator with single well potential we study a two frequency quasiperiodically forced oscillator with single well po tential which reduces to the van der pol and duffing oscillators in certain special cases the unperturbed system without damping and forcing terms has a one parameter family of periodic orbits we concentrate on the dynamics near the unperturbed resonant periodic orbits using the second order averaging method and a version of melnikov s method we show that when double resonance occurs the stable and unstable manifolds of normally hyperbolic invariant tori intersect transversely i e transverse homoclinic motions exist near the unperturbed reso nant periodic orbits in certain parameter regions such homoclinic motions yield chaotic dynamics characterized by a generalization of the bernoulli shift numer ical simulation results are also given to demonstrate the theoretical results

a partial differential equation with infinitely many periodic orbits chaotic oscillations of a forced beam this paper delineates a class of time periodically perturved evolution equations in a banach space whose associated poincar e map contains a smale horseshoe this implies that such systems possess periodic orbits with arbitrarily high period the method uses techniques originally due to melnikov and applies to systems of the form x f0 x f1 x t where x f0 x is hamiltonian and has a homoclinic orbit we give an example from structural mechanics sinusoidally forced vibrations of a buckled beam

monodromy dependence and connection formulae for isomonodromic tau functions we discuss an extension of the jimbo miwa ueno differential 1 form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients this extension is based on the results of m bertola generalizing a previous construction by b malgrange we show how this 1 form can be used to solve a long standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors we explain how this scheme works for fuchsian systems and in particular calculate the connection constant for the generic painleve vi tau function the result proves the conjectural formula for this constant proposed by iorgov lisovyy and tykhyy we also apply the method to non fuchsian systems and evaluate constant factors in the asymptotics of the painleve ii tau function

a co rotational element time integration strategy for non linear dynamics new procedures are proposed for implicit dynamic analysis using the finite element method the main aim is to give stable solutions with significant rigid body motions in particular rotations in contrast to most conventional approaches the time integration strategy is closely linked to the element technologies with the latter involving a form of co rotational procedure for the undamped situation one of the solution procedures leads to an algorithm that exactly conserves energy when constant external forces are applied i e with gravity loading

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

geometry and braiding of stokes data fission and wild character varieties a family of new algebraic poisson varieties will be constructed generalising the complex character varieties of riemann surfaces then the well known poisson mapping class group actions on the character varieties will be generalised

monodromy for the hypergeometric function nfn 1 la fonction hypergeometrique le groupe hypergeometrique la forme hermitienne invariante le cas imprimitif theorie de galois differentielle fonctions hypergeometriques algebriques

fredholm determinants and pole free solutions to the noncommutative painleve ii equation we extend the formalism of integrable operators a la its izergin korepin slavnov to matrix valued convolution operators on a semi infinite interval and to matrix integral operators with a kernel of the form frac e 1 t lambda e 2 mu lambda mu thus proving that their resolvent operators can be expressed in terms of solutions of some specific riemann hilbert problems we also describe some applications mainly to a noncommutative version of painleve ii recently introduced by retakh and rubtsov and a related noncommutative equation of painleve type we construct a particular family of solutions of the noncommutative painleve ii that are pole free for real values of the variables and hence analogous to the hastings mcleod solution of commutative painleve ii such a solution plays the same role as its commutative counterpart relative to the tracy widom theorem but for the computation of the fredholm determinant of a matrix version of the airy kernel

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

on the asymptotic and numerical solution of linear ordinary differential equations an investigation is made of the nature of asymptotic solutions of homogeneous linear differential equations of arbitrary order in the neighborhood of a singularity of unit rank we introduce a classification of the solutions into two types explicit and implicit for the former there exists a sector on which the solution is dominated by all independent solutions as the singularity is approached no such sector exists for implicit solutions in consequence the two types of solution have different uniqueness properties another difference is that error bounds for the asymptotic expansions of explicit solutions are generally stronger than those for implicit solutions we also investigate the computation of the solutions by numerical integration of the differential equation it is shown that it is the exception rather than the rule for the integration process to be stable particularly so for implicit solutions to overcome these instabilities we develop boundary value methods complete with error analysis numerical examples illustrate the computation of both explicit and implicit solutions and also the associated stokes multipliers

symplectic manifolds and isomonodromic deformations abstract we study moduli spaces of meromorphic connections with arbitrary order poles over riemann surfaces together with the corresponding spaces of monodromy data involving stokes matrices natural symplectic structures are found and described both explicitly and from an infinite dimensional viewpoint generalising the atiyah bott approach this enables us to give an intrinsic symplectic description of the isomonodromic deformation equations of jimbo miwa and ueno thereby putting the existing results for the six painleve equations and schlesinger s equations into a uniform framework

a pendulum theorem we consider n linked pendulums which are inverted and balanced on top of one another and establish a general theorem which shows how they may be stabilized by small vertical oscillations of the support

globally and locally attractive solutions for quasi periodically forced systems abstract we consider a class of differential equations x x g x f t with r d describing one dimensional dissipative systems subject to a periodic or quasi periodic diophantine forcing we study existence and properties of trajectories with the same quasi periodicity as the forcing for g x x 2 p 1 p n we show that when the dissipation coefficient is large enough there is only one such trajectory and that it describes a global attractor in the case of more general nonlinearities including g x x 2 describing the varactor equation we find that there is at least one trajectory which describes a local attractor

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

the indian wire trick via parametric excitation a comparison between theory and experiment a seemingly paradoxical experiment is described whereby a length of wire is stabilized upside down by vertical periodic oscillation of its support the experimental results reveal an upper and a lower bound on the excitation frequency for stability the results of recent theories are presented and used to explain the essential details of the observations the theory relies on a novel phenomenon of so called resonancetongue interaction the result is verified via asymptotic calculations based on a one dimensional rod model and numerical results on a spatially discretized system of links this gravity defying effect has potential application to the stabilization of other spatially extended systems via parametric excitation

the 192 solutions of the heun equation a machine generated list of 192 local solutions of the heun equation is given they are analogous to kummer s 24 solutions of the gauss hypergeometric equation since the two equations are canonical fuchsian differential equations on the riemann sphere with four and three singular points respectively tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the coxeter group d n each of the 192 expressions is labeled by an element of d 4 of the 192 24 are equivalent expressions for the local heun function hl and it is shown that the resulting order 24 group of transformations of hl is isomorphic to the symmetric group s 4 the isomorphism encodes each transformation as a permutation of an abstract four element set not identical to the set of singular points

on the analysis of incomplete spectra in random matrix theory through an extension of the jimbo miwa ueno differential abstract several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by jimbo miwa and ueno jmu in the early 1980s 45 recent advances in the theory of tau functions 41 based on earlier works of b malgrange and m bertola have allowed to extend the original jimbo miwa ueno differential form to a 1 form closed on the full space of extended monodromy data of the underlying lax pairs this in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions including the exact computation of all relevant constant factors we use this method to efficiently compute the tail asymptotics of soft edge hard edge and bulk scaled distribution and gap functions in the complex wishart ensemble provided each eigenvalue particle has been removed independently with probability 1 0 1

asymptotics of plancherel measures for symmetric groups 1 1 plancherel measures given a finite group g by the corresponding plancherel measure we mean the probability measure on the set g of irreducible representations of g which assigns to a representation g the weight dim g for the symmetric group s n the set s n is the set of partitions of the number n which we shall identify with young diagrams with n squares throughout this paper the plancherel measure on partitions arises naturally in representation theoretic combinatorial and probabilistic problems for example the plancherel distribution of the first part of a partition coincides with the distribution of the longest increasing subsequence of a uniformly distributed random permutation 31 we denote the plancherel measure on partitions of n by mn

random unitary matrices permutations and painleve this paper is concerned with certain connections between the ensemble of n n unitary matrices specifically the characteristic function of the random variable tr u and combinatorics specifically ulam s problem concerning the distribution of the length of the longest increasing subsequence in permutation groups and the appearance of painleve functions in the answers to apparently unrelated questions among the results is a representation in terms of a painleve v function for the characteristic function of tr u and using recent results of baik deift and johansson an expression in terms of a painleve ii function for the limiting distribution of the length of the longest increasing subsequence in the hyperoctahedral groups