a general method for the existence of periodic solutions of differential systems in the plane abstract we propose a general method to prove the existence of periodic solutions for planar systems of ordinary differential equations which can be used in many different circumstances applications are given to some nonresonant cases even for systems with superlinear growth in some direction or with a singularity systems at resonance are also considered provided a landesman lazer type of condition is assumed

rotation numbers eigenvalues and the poincare birkhoff theorem using the relation between rotation numbers and eigenvalues we prove the existence of nontrivial t periodic solutions having precise oscillatory properties for a class of asymptotically linear second order differential equations

periodic solutions of weakly coupled superlinear systems abstract by the use of a higher dimensional version of the poincare birkhoff theorem we are able to generalize a result of jacobowitz 15 and hartman 14 thus proving the existence of infinitely many periodic solutions for a weakly coupled superlinear system

convergence rates and decoupling in linear stochastic approximation algorithms almost sure convergence rates for linear algorithms h k 1 h k frac 1 k chi b k a kh k are studied where chi in 0 1 a k k 1 infty are symmetric positive semidefinite random matrices and b k k 1 infty are random vectors it is shown that n gamma h n a 1 b rightarrow 0 a s for the gamma in 0 chi positive definite a and vector b such that frac 1 n chi gamma sum k 1 n a k a to 0 and frac 1 n chi gamma sum k 1 n b k b to 0 a s when chi gamma in left frac12 1 right these assumptions are implied by the marcinkiewicz strong law of large numbers which allows the a k and b k to have heavy tails long range dependence or both finally corroborating experimental outcomes and decreasing gain design considerations are provided

periodic solutions of second order equations via rotation numbers abstract we consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems we show how a rotation number approach together with the poincare birkhoff theorem and the phase plane analysis of the spiral properties allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity these systems may also be resonant in particular our approach can be used to deal with the problems without both global existence of the cauchy problems associated to the equation and the sign assumption on f the typical example is a partially superlinear second order equation our main result generalizes some classical results of jacobowitz and hartman among others

forced vibrations of superquadratic hamiltonian systems on etudie l existence des solutions t periodiques ter t 0 du systeme hamiltonien suivant z gh z f t ou g est la matrice asymetrique standard

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

infinity of subharmonics for asymmetric duffing equations with the lazer leach dancer condition abstract in this paper based on a generalized version of the poincare birkhoff twist theorem by franks we establish the existence of infinitely many subharmonics for the asymmetric duffing equation with the classical lazer leach dancer condition as a consequence of our result we obtain a sufficient and necessary condition for existence of arbitrarily large amplitude periodic solutions for a class of asymmetric duffing equations at resonance

applications of a kushner and clark lemma to general classes of stochastic algorithms two general classes of stochastic algorithms are considered including algorithms considered by ljung as well as algorithms of the form theta n 1 theta n gamma n 1 v n 1 theta n z where z is a stationary ergodic process it is shown how one can apply a lemma of kushner and clark to obtain properties of these algorithms this is done by using in particular martingale arguments in the generalized ljung case in these various situations the convergence is obtained by the method of the associated ordinary differential equation under the classical boundedness assumptions in the case of linear algorithms the boundedness assumptions are dropped

noise conditions for prespecified convergence rates of stochastic approximation algorithms we develop deterministic necessary and sufficient conditions on individual noise sequences of a stochastic approximation algorithm for the error of the iterates to converge at a given rate specifically suppose spl rho sub n is a given positive sequence converging monotonically to zero consider a stochastic approximation algorithm x sub n 1 x sub n a sub n a sub n x sub n b sub n a sub n e sub n where x sub n is the iterate sequence a sub n is the step size sequence e sub n is the noise sequence and x is the desired zero of the function f x ax b then under appropriate assumptions we show that x sub n x o spl rho sub n if and only if the sequence e sub n satisfies one of five equivalent conditions these conditions are based on well known formulas for noise sequences kushner and clark s 1978 condition chen s see proc ifac world congr p 375 80 1996 condition kulkarni and horn s see ieee trails automat contr vol 41 p 419 24 1996 condition a decomposition condition and a weighted averaging condition our necessary and sufficient condition on e sub n to achieve a convergence rate of spl rho sub n is basically that the sequence e sub n spl rho sub n satisfies any one of the above five well known conditions we provide examples to illustrate our result in particular we easily recover the familiar result that if a sub n a n and e sub n is a martingale difference process with bounded variance then x sub n x o n sup 1 2 log n sup spl beta for any spl beta 1 2

a continuation approach to superlinear periodic boundary value problems this paper deals with the problem of the existence of t periodic solutions for the first order differential system x i t x 1 1 where f 0 t x r r is a caratheodory function in what follows we prove some results for the solvability of the periodic bvp in the case when the dimension of the space is even such a limitation is motivated by our interest in applications to the second order equation d g t u u o 1 2 which takes the form of 1 1 when it is written as the equivalent system u v 0 g t u v 1 3

strong laws of large numbers under weak assumptions with application the employment of strong laws of large numbers is instrumental to the analysis of system estimation and identification strategies however the vast bulk of such laws as presented in the wider literature assume independence or at least uncorrelatedness of random components and these assumptions are quite restrictive from an engineering point of view by way of contrast the paper shows how to establish strong laws for possibly nonstationary random processes with very general dependence structure a brief example is provided that illustrates the utility of the strong law of large numbers presented

reducing the backscattering via complex impedance coating a method for reducing the back scattering from bodies with complex impedance boundaries is developed a nonlinear integral equation system providing zero backscattering is obtained for determining the impedance distribution on the boundaries of infinite cylinders numerical results for circular and elliptic cylinders are given frequency dependence of backscattering at obtained optimal impedance is discussed on the example of circular cylinder

periodic solutions of duffing s equations with superquadratic potential abstract this paper is devoted to the study of harmonic and subharmonic solutions for the second order scalar nonlinear duffing s equation x g x p t x x where g and p are continuous functions with p bounded and periodic in the first variable and g satisfying the assumption g x sign x as x among other results we prove the existence of infinitely many harmonic and subharmonic solutions of any order p p t and if the potential g x of g x satisfies certain conditions of superquadratic growth at the new existence results can be applied to situations in which the more classical superlinear growth condition g x x as x is not satisfied in this manner various preceding theorems are improved and sharpened see the introduction for more details proofs are based on a generalized version of the poincare birkhoff twist theorem due to w ding

on the convergence of linear stochastic approximation procedures many stochastic approximation procedures result in a stochastic algorithm of the form h sub k 1 h sub k 1 k b sub k a sub k h sub k for all k 1 2 3 here b sub k k 1 2 3 is a r sup d valued process a sub k k 1 2 3 is a symmetric positive semidefinite re sup d spl times d valued process and h sub k k 1 2 3 is a sequence of stochastic estimates which hopefully converges to h sup spl delta lim sub n spl rarr spl infin 1 n spl sigma sub k 1 sup n ea sub k sup 1 lim sub n spl rarr spl infin 1 n spl sigma sub k 1 sup n eb sub k assuming everything here is well defined we give an elementary proof which relates the almost sure convergence of h sub k k 1 2 3 to strong laws of large numbers for b sub k k 1 2 3 and a sub k k 1 2 3

analysis of stochastic gradient algorithms for linear regression problems parameter estimation problems that can be formulated as linear regressions are quite common in many applications recursive on line sequential estimation of such parameters can be performed using the recursive least squares rls algorithm or a stochastic gradient version of this algorithm in this paper the convergence properties of the gradient algorithm are analyzed under the assumption that the gain tends to zero the technique is the same as the so called ordinary differential equation approach but the treatment here is self contained and includes a proof of the boundedness of the estimates a main result is that the convergence conditions for the gradient algorithm are the same as those for the recursive least squares algorithm

generalizations of the poincare birkhoff theorem in this article we prove some generalizations of the classical poincar birkhoff theorem on area preserving homeomorphisms of the annulus which satisfy a boundary twist condition the work of g d birkhoff on this theorem and its applications can be found in b1 b2 and chapter v of b3 a more modern treatment can be found in b n we prove a theorem for the open annulus a s1 x 0 1 since as we will see the theorem for the closed annulus can easily be obtained from this result for the open annulus however it is not immediately obvious what should be the analogue of the twist condition it turns out that the most general hypothesis and the most natural from the point of view of our proof involves the notion of positively and negatively returning disks for some lift of f to the covering space a r x 0 1 more precisely if f a a is a lift of f a a we will say that the e is a positively returning disk for f if there is an open disk u c a such that f u n u 0 and fp u n u k 0 forsome n k 0 here u k denotes the set x k t i x t e u thus u is disjoint from its image but under iteration by f returns to a positive translate of itself a negatively returning disk is defined similarly but with k 0 such that fl u n u 0 in section 2 we prove the following

a new fixed point theorem and periodic solutions of nonconservative weakly coupled systems abstract in this paper we prove a new fixed point theorem for the coupling of twist conditions and poincare bohl type conditions as its applications we prove the existence of periodic solutions for various nonconservative mixed type weakly coupled systems

efficient computation of nonparaxial surface fields excited on an electrically large circular cylinder with an impedance boundary condition an alternative numerical approach is presented for the evaluation of the fock type integrals that exist in the uniform geometrical theory of diffraction utd based asymptotic solution for the nonparaxial surface fields excited by a magnetic or an electric source located on the surface of an electrically large circular cylinder with an impedance boundary condition ibc this alternative approach is based on performing numerical integration of the fock type integrals on a deformed path on which the integrands are nonoscillatory and rapidly decaying comparison of this approach with the previously developed one presented in which is based on invoking the cauchy s residue theorem by finding the pole singularities numerically reveals that the alternative approach is considerably more efficient

efficient computation of surface fields excited on a dielectric coated circular cylinder an efficient method to evaluate the surface fields excited on an electrically large dielectric coated circular cylinder is presented the efficiency of the method results from the circumferentially propagating representation of the green s function as well as its efficient numerical evaluation along a steepest descent path the circumferentially propagating series representation of the appropriate green s function is obtained from its radially propagating counterpart via watson s transformation and then the path of integration is deformed to the steepest descent path on which the integrand decays most rapidly numerical results are presented that indicate that the representations obtained here are very efficient and valid even for arbitrary small separations of the source and field points this work is especially useful in the moment method analysis of conformal microstrip antennas where the mutual coupling effects are important

on the almost sure rate of convergence of linear stochastic approximation algorithms the almost sure rate of convergence of linear stochastic approximation algorithms is analyzed as the main result it is demonstrated that their almost sure rate of convergence is equivalent to the almost sure rate of convergence of the averages of their input data sequences as opposed to most of the existing results on the rate of convergence of stochastic approximation which cover only algorithms with the noise decomposable as the sum of a martingale difference vanishing and telescoping sequence the main results of this paper hold under assumptions not requiring the input data sequences to admit any particular decomposition although no decomposition of the input data sequences is required the results on the almost sure rate of convergence of linear stochastic approximation algorithms obtained in this correspondence are as tight as the rate of convergence in the law of iterated logarithm moreover the main result of this correspondence yields the law of iterated logarithm for linear stochastic approximation if the law of iterated logarithm holds for the input data sequences the obtained general results are illustrated with two nontrivial examples where the input data sequences are strongly mixing strictly stationary random processes or functions of a uniformly ergodic markov chain these results are also applied to the analysis of least mean square lms algorithms

subharmonic solutions with prescribed minimal period for nonautonomous hamiltonian systems on etablit un resultat de multiplicite pour une certaine classe de systemes hamiltoniens periodiques en temps

stochastic approximation with correlated data new almost sure convergence results for a special form of the multidimensional robbins monro stochastic approximation procedure are developed the results are applicable to cases where the training data is heavily correlated no conditional expectation properties or boundedness assumptions are required to apply the new results for example when the data sequence is normal and i m dependent ii autoregressive moving average or iii band limited the results can be used to establish the almost sure convergence of each algorithm treated the special form of the robbins monro procedure considered is motivated by a consideration of several algorithms that have been proposed for discrete time adaptive signal processing applications most of these algorithms can also be viewed as stochastic gradient following algorithm the ease with which the new results can be applied is illustrated

convergence of an adaptive linear estimation algorithm in this work we prove the almost sure convergence of an adaptive linear estimator governed by a stochastic gradient algorithm with decreasing step size in the presence of correlated observations two complementary contributions are added to the famous 1977 ljung theorem first we drop the condition of nondivergence of the algorithm assumed by ljung while that condition can be ensured by adding a barrier the convergence of the suitably bounded algorithm itself is not established even on the basis of ljung theorem here the barrier problem is overcome by proving that it is not necessary for the convergence our second contribution is to generalize the model describing the correlated observations no state space model is used and no linear relationship between the observations and the signal to be estimated needs to be assumed instead we use a decreasing covariance model that agrees with a very wide class of practical applications

existence of forced oscillations for some nonlinear differential equations abstract this article studies the existence of t periodic solutions for systems of nonlinear second order ordinary differential equations assuming function v to be superquadratic it is shown that this system possesses infinitely many t periodic solutions the proof of this result rests on showing that certain homotopy groups of level sets of the functional associated with the system are not trivial some more general results concerning systems of another type are also presented

periodic solutions of superlinear impulsive differential equations a geometric approach abstract a geometric method is introduced to study superlinear second order differential equations with impulsive effects basing on a reference continuous polar lifting of a planar orientation preserving homeomorphism we prove via the poincare birkhoff twist theorem the existence of infinitely many periodic solutions of conservative superlinear second order equations with the finite twist impulsive terms we also prove by developing a new twist fixed point theorem the existence of periodic solutions for non conservative superlinear second order equations with degenerate impulsive terms

positively homogeneous hamiltonian systems in the plane i try to give a general description of the dynamics of the solutions for a planar hamiltonian system with positively homogeneous hamiltonian function and periodic forcing term most of the results obtained are already known in the special case of a scalar second order differential equation with asymmetric nonlinearity

almost sure approximations to the robbins monro and kiefer wolfowitz processes with dependent noise we study a recursive algorithm which includes the multidimensional robbins monro and kiefer wolfowitz processes the assumptions on the disturbances are weaker than the usual assumption that they be a martingale difference sequence it is shown that the algorithm can be represented as a weighted average of the disturbances this representation can be used to prove asymptotic results for stochastic approximation procedures as an example we approximate the one dimensional kiefer wolfowitz process almost surely by brownian motion and as a byproduct obtain a law of the iterated logarithm

on the use of morse index and rotation numbers for multiplicity results of resonant bvps abstract motivated by a recent series of papers by k li and co workers 14 15 16 17 18 we consider the problem of the existence and multiplicity of solutions for the neumann or periodic bvps associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behavior of its solutions near zero and near infinity may be compared with the behavior of the solutions of two suitable linear systems one considered near zero and the other near infinity we show how a rotation number approach together with the poincare birkhoff theorem and a recent variant of it allows to obtain multiplicity results in terms of the gap between the morse indexes of the referred linear systems at zero and at infinity these systems may also be resonant when the gaps are sufficiently large our multiplicity results improve the ones obtained by variational methods in the quoted papers also our approach allows a description of the solutions obtained in terms of their nodal properties

the exact number of solutions for a class of ordinary differential equations through morse index computation we consider a class of second order ordinary differential equations with jumping nonlinearities that cross the first k eigenvalues under zero boundary conditions and by means of a morse index computation show the existence of exactly 2 k solutions

realization of impedance boundary impedance boundary is generally considered as an approximate model for material interfaces considering an electromagnetic field in a certain wave guiding anisotropic material it is shown that a slab of such a material backed by a perfect electric conductor pec plane can be exactly represented by impedance boundary conditions as a result of the analysis a novel explicit relation is derived between the surface admittance dyadic of the impedance boundary and the material and geometric parameters of the anisotropic slab which can be used to realize a given admittance dyadic the relation is verified with results known for the perfect electromagnetic pemc boundary and bi axial material considered as special cases of the theory

multiplicity results for an ode problem with even nonlinearity on considere le probleme u g u h dans u 0 sur ou r n est un ouvert borne a frontiere lisse en se restreignant au cas u u 2 t sin x dans 0 u 0 u 0 etant donne n 0 n il existe t nd 0 tel que pour tout t t nd 0 l equation consideree a au moins n 0 solutions

existence and multiplicity results for periodic solutions of semilinear duffing equations we obtain existence an multiplicity of harmonic and subharmonic solutions to the periodically forced nonlinear duffing s equation x g x p t in the semi linear case 0 a g x x b for x large we introduce new conditions of nonresonance with respect to the spectrum of the associated linear differential operator x x and show by some nonlinear counterexamples that the results are sharp

maslov index poincare birkhoff theorem and periodic solutions of asymptotically linear planar hamiltonian systems abstract we study the relationship between the twist condition in the poincare birkhoff fixed point theorem and the assumptions on the maslov index for asymptotically linear planar hamiltonian systems for this aim we develop a variant of the poincare birkhoff theorem which together with its classical version allows to obtain a lower bound for the number of nontrivial periodic solutions in terms of the gap between the maslov indexes associated to the linearizations of the planar system at zero and at infinity