weak focus limit cycles and bifurcations for bounded quadratic systems abstract we study the bounded quadratic systems with either two weak foci or a weak focus of order 2 from the first case we obtain 1 1 configuration of limit cycles and in the second case we prove that there is no limit cycle surrounding the weak focus of order 2 also we unfold the bounded quadratic system with a center and the maximum number of limit cycles in this case is two

linear estimate for the number of limit cycles of a perturbed cubic polynomial differential system abstract perturbing the cubic polynomial differential systems x y a 1 x a 0 b 1 y b 0 y x a 1 x a 0 b 1 y b 0 having a center at the origin inside the class of all polynomial differential systems of degree n we obtain using the averaging theory of second order that at most 17 n 15 limit cycles can bifurcate from the periodic orbits of the center

global bifurcation of limit cycles in a family of polynomial systems in this paper we study the number of limit cycles in a family of polynomial systems using bifurcation methods and melnikov functions we obtain the maximal number of limit cycles in global bifurcation

performance of a distributed stochastic approximation algorithm in this paper a distributed stochastic approximation algorithm is studied applications of such algorithms include decentralized estimation optimization control or computing the algorithm consists in two steps a local step where each node in a network updates a local estimate using a stochastic approximation algorithm with decreasing step size and a gossip step where a node computes a local weighted average between its estimates and those of its neighbors convergence of the estimates toward a consensus is established under weak assumptions the approach relies on two main ingredients the existence of a lyapunov function for the mean field in the agreement subspace and a contraction property of the random matrices of weights in the subspace orthogonal to the agreement subspace a second order analysis of the algorithm is also performed under the form of a central limit theorem the polyak averaged version of the algorithm is also considered

on hopf cyclicity of planar systems we investigate the maximal number of limit cycles which appear under perturbations in hopf bifurcations by using the first order melnikov function with multiple parameters

the number of limit cycles from a cubic center by the melnikov function of any order abstract in this paper we consider the system x y 1 x 2 p x y y x 1 x 2 q x y where p x y and q x y are arbitrary quadratic polynomials we study the maximum number of limit cycles bifurcating from the periodic orbits by using the melnikov function of any order we prove that the upper bound for the number of limit cycles is 3 and reached

on the limit cycles of planar polynomial system with non rational first integral via averaging method at any order in this paper we consider a class of cubic planar polynomial differential system with non rational first integral using the averaging method at any order we bound the maximum number of limit cycles which bifurcate from the periodic annulus of the origin when we perturb them inside the class of all polynomial systems of degree n

on the chebyshev property for a new family of functions abstract we analyze whether a given set of analytic functions is an extended chebyshev system this family of functions appears studying the number of limit cycles bifurcating from some nonlinear vector field in the plane our approach is mainly based on the so called derivation division algorithm we prove that under some natural hypotheses our family is an extended chebyshev system and when some of them are not fulfilled then the set of functions is not necessarily an extended chebyshev system one of these examples constitutes an extended chebyshev system with high accuracy

simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli abstract we study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations the cubic planar system x y y x 1 2 y 2 x x 1 2 y 2 has simultaneously a center at the origin and at infinity we study up to first order averaging analysis the bifurcation of periodic orbits from the two period annuli first separately and second simultaneously this problem is a generalization of 24 to the piecewise systems class when the polynomial perturbation has degree n we prove that the inner and outer abelian integrals are rational functions and we provide an upper bound for the number of zeros when the perturbation is cubic the same degree as the unperturbed vector field the maximum number of limit cycles up to first order perturbation from the inner and outer annuli is 9 and 8 respectively when the simultaneous bifurcation problem is considered 12 limit cycles exist these limit cycles appear in three types of configurations 9 3 6 6 and 4 8 in the non piecewise scenario only 5 limit cycles were found

on the number of limit cycles near a homoclinic loop with a nilpotent singular point abstract in this article we study the expansion of the first melnikov function appearing by perturbing an integrable and reversible system with a homoclinic loop passing through a nilpotent singular point and obtain formulas for computing the first coefficients of the expansion based on these coefficients we obtain a lower bound for the maximal number of limit cycles near the homoclinic loop moreover as an application of our main results we consider a type of integrable and reversible polynomial systems obtaining at least 3 4 or 5 limit cycles respectively

a linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system abstract in this paper we study a class of planar piecewise smooth quadratic integrable non hamiltonian systems which have a center by using the averaging method we give an estimation of the number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n our estimation is linear depending on n and it is at least twice the associated estimation of smooth systems

bifurcation of critical periods from the reversible rigidly isochronous centers abstract in this paper we study bifurcation of critical periods by perturbing a nonlinear vector field containing m th degree homogeneous terms with a rigidly isochronous center at the origin first we give expressions of period bifurcation functions in the form of integrals and then study the number of critical periods for cases of m 1 2 3 respectively

on the number of limit cycles of a class of polynomial differential systems we study the number of limit cycles of polynomial differential systems of the form where g1 f1 g2 and f2 are polynomials of a given degree note that when g1 x f1 x 0 we obtain the generalized pol

the number of limit cycles of a quintic polynomial system in this paper we consider the bifurcation of limit cycles of the system email protected y x 2 a 2 y 2 b 2 email protected x y email protected x x 2 a 2 y 2 b 2 email protected x y for e sufficiently small where a email protected r 0 and p q are polynomials of degree n we obtain that up to first order in e the upper bounds for the number of limit cycles that bifurcate from the period annulus of the quintic center given by e 0 are 3 2 n sin 2 email protected 2 1 if a b and n 1 if a b moreover there are systems with at least 3 2 n sin 2 email protected 2 1 if a b and n 1 limit cycles if a b

nested generalized cyclic row column designs summary a nested row column design has v treatments arranged in b blocks each comprising pq units grouped into p rows and q columns a class of designs based on a cyclical method of construction is defined the canonical efficiency factors of such nested generalized cyclic row column designs are examined efficient designs are tabulated for 5 v 15 p 3 q 7 and r v where r bpq v is the number of replications of each treatment row column designs are used to eliminate heterogeneity in two directions more generally an experiment may have several blocks each containing pq plots arranged in p rows and q columns where the blocks may represent a further blocking factor or provide replications of the basic row column design these designs are called nested row column designs since rows and columns are nested within blocks the lattice square designs introduced by yates 1940 are examples of such designs a general analysis and the concept of balance have been considered by singh dey 1979 several series of balanced nested row column designs have been given by singh dey 1979 street 1981 and agrawal prasad 1982a preece 1967 provides a table of nested balanced incomplete block designs and a number of these designs can also be used as balanced nested row column designs other series based on partially balanced designs have been given by street 1981 and agrawal prasad 1982b jarrett hall 1982 gave some examples of small designs based on cyclic and generalized cyclic methods of con struction which they suggested could provide useful alternatives to nearest neighbour designs and methods as an example of a balanced nested row column design consider the following design given by singh dey 1979 for five treatments 0 1 2 3 4 arranged in five blocks of two rows and two columns

limit cycles appearing from the perturbation of a system with a multiple line of critical points abstract consider planar ordinary differential equations of the form x y c x y y x c x y where c x y is an algebraic curve we are interested in knowing whether the existence of multiple factors for c is important or not when we study the maximum number of zeros of the abelian integral m that controls the limit cycles that bifurcate from the period annulus of the origin when we perturb it with an arbitrary polynomial vector field with this aim we study in detail the case c x y 1 y m where m is a positive integer number and prove that m has essentially no impact on the number of zeros of m this result improves the known studies on m one of the key points of our approach is that we obtain a simple expression of m based on some successive reductions of the integrals appearing during the procedure

bifurcations and distribution of limit cycles which appear from two nests of periodic orbits abstract in this paper we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation z i z z 3 after a small polynomial perturbation we first show that under small perturbations of the form e p 2 m 1 z z where p 2 m 1 z z is a polynomial of degree 2 m 1 in which the power of z is odd and the power of z is even the only possible distribution of limit cycles is u u for all values of u 0 1 2 m 3 hence the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of z i z z 3 is m 3 for m 4 then we consider a perturbation of the form e p m z z where p m z z is a polynomial of degree m in which the power of z is odd and obtain the upper bound m 5 for m 6 moreover we show that the distribution u v of limit cycles is possible for 0 u m 5 0 v m 5 with u v m 2 and m 9

bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems this paper presents a survey of results on the bifurcation of limit cycles from centers and separatrix cycles of perturbed planar analytic systems and contributes some new results on the bifurcation of multiple limit cycles from centers and on the multiplicity of separatrix cycles of such systems the basic theme throughout the paper is that the number positions and multiplicities of the limit cycles that bifurcate under perturbations are related to the number positions and multiplicities of the zeros of the melnikov function for the system the general theory is illustrated by a number of examples from the literature some of which are extended to include new results

limit cycles of cubic polynomial vector fields via the averaging theory abstract in this paper we study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging method more precisely we prove that the perturbations of the period annulus of the center located at the origin of the cubic polynomial differential system x y f x y y x f x y where f x y 0 is a conic such that f 0 0 0 by arbitrary cubic polynomial differential systems provide at least six limit cycles bifurcating from the periodic orbits of the period annulus using only the first order averaging method

bifurcation of periodic orbits of periodic equations with multiple parameters by averaging method abstract we investigate a class of n dimensional periodic differential equations with multiple small parameters by averaging method and obtain sufficient conditions for the existence of multiple periodic orbits as applications of the main results a cubic system is studied

averaging theory at any order for computing periodic orbits abstract we provide a recurrence formula for the coefficients of the powers of e in the series expansion of the solutions around e 0 of the perturbed first order differential equations using it we give an averaging theory at any order in e for the following two kinds of analytic differential equation d x d k 1 e k f k x d x d k 0 e k f k x a planar polynomial differential system with a singular point at the origin can be transformed using polar coordinates to an equation of the previous form thus we apply our results for studying the limit cycles of a planar polynomial differential systems

bifurcation of limit cycles from a two dimensional center inside rn abstract we study the bifurcation of limit cycles from the periodic orbits of a linear differential system in r n perturbed inside a class of piecewise linear differential systems which appears in a natural way in control theory our main result shows that at most one limit cycle can bifurcate up to first order expansion of the displacement function with respect to the small parameter this upper bound is reached for proving this result we use the averaging theory in a form where the differentiability of the system is not needed

abelian integrals and limit cycles for a class of cubic polynomial vector fields of lotka volterra type with a rational first integral of degree two abstract in this paper we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of lotka volterra type having a rational first integral of degree 2 under polynomial perturbations of degree n the analysis is carried out by estimating the number of zeros of the corresponding abelian integrals moreover using chebyshev criterion we show that the sharp upper bound for the number of zeros of the abelian integrals defined on each period annulus is 3 for n 3 the simultaneous bifurcation and distribution of limit cycles for the system with two period annuli under cubic polynomial perturbations are considered all configurations u v with 0 u v 3 u v 5 are realizable

some constructions for balanced incomplete block designs with nested rows and columns summary a method of construction of balanced incomplete block designs with nested rows and columns is developed using difference techniques from which many infinite series of designs are obtained

information exchange and learning dynamics over weakly connected adaptive networks this paper examines the learning mechanism of adaptive agents over weakly connected graphs and reveals an interesting behavior on how information flows through such topologies the results clarify how asymmetries in the exchange of data can mask local information at certain agents and make them totally dependent on other agents a leader follower relationship develops with the performance of some agents being fully determined by the performance of other agents that are outside their domain of influence this scenario can arise for example due to intruder attacks by malicious agents or as the result of failures by some critical links the findings in this paper help explain why strong connectivity of the network topology adaptation of the combination weights and clustering of agents are important ingredients to equalize the learning abilities of all agents against such disturbances the results also clarify how weak connectivity can be helpful in reducing the effect of outlier data on learning performance

averaging methods of arbitrary order periodic solutions and integrability in this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems this result extends and improves results on averaging theory of periodic analytic differential systems and it unifies many different kinds of averaging methods applying our theory to autonomous analytic differential systems we obtain some conditions on the existence of limit cycles and integrability for polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues we prove that there always exists a positive number n such that if its first n averaging functions vanish then all averaging functions vanish and consequently there exists a neighborhood of the origin filled with periodic orbits consequently if all averaging functions vanish the origin is a center for n 2 furthermore in a punctured neighborhood of the origin the system is c infty completely integrable for n 2 provided that each periodic orbit has a trivial holonomy finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities

bifurcation of limit cycles from quadratic isochrones abstract for a one parameter family of plane quadratic vector fields x e depending analytically on a small real parameter e we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous or linearizable center techniques are developed for treating the bifurcations of all orders and these are applied to prove the following results for the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three for one class of nonlinear isochrones at most one continuous family of limit cycles can emerge whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge moreover for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center

the number of limit cycles in perturbations of polynomial systems with multiple circles of critical points abstract this paper investigates the problem for limit cycle bifurcations of system x y f x y e p x y y x f x y e q x y where f x y consists of multiple circles and p x y q x y are polynomials of degree n the upper bound for the maximal number of limit cycles emerging from the period annulus surrounding the origin is provided in terms of n and the involved multiplicities of circles by using the first order melnikov function furthermore hopf bifurcation for a cubic system of this type is discussed

melnikov theory for a class of planar hybrid systems abstract in this paper we extend the well known melnikov theory for smooth systems to a class of two zonal planar hybrid piecewise smooth systems in this class each zone of differentiability is separated by a straight line and in these zones the dynamics are governed by a smooth system when an orbit reaches the separation line then a reset map applies before entering the orbit in the other zone the family of non generic systems of this class which has a continuum of periodic orbits is considered then we study the periodic orbits of the continuum that persist under a perturbation to achieve this objective first we build an appropriate poincare map after that we introduce a function of one variable whose zeros provide us the periodic orbits that remain after the perturbation this function will be called the melnikov function because it is very close to the so called melnikov function for smooth systems finally we apply the obtained results to some piecewise linear systems continuous as well as discontinuous ones

the center problem via averaging method the averaging method has been used to study the problem of the determination of the number of hyperbolic limit cycles that can bifurcate from the period annulus of a center in this paper we use the averaging method up to any order in the perturbation parameter to determine the center conditions of monodromic singular points of analytic planar vector fields

cyclicity of a family of vector fields abstract we study the cyclicity and the center problem for a special family of planar differential equations we also relate this cyclicity with the maximum number of limit cycles that bifurcate from a hamiltonian vector field afterwards we apply the result obtained to several known types of differential equations

the cyclicity of period annuli for a class of cubic hamiltonian systems with nilpotent singular points abstract this paper deals with the limit cycles of a class of cubic hamiltonian systems under polynomial perturbations we suppose that the corresponding hamiltonian system which has at least one center has finite singular points and is symmetrical with respect to both x axis and y axis and also the origin is a nilpotent singular point hence the hamiltonian h x y can be written as h x y x 2 a x 4 b x 2 y 2 c y 4 a b c r 3 c 0 for the above h x y the corresponding vector field is x 2 y b x 2 2 c y 2 y 2 x 1 2 a x 2 b y 2 we first obtain that the above vector fields with at least one center can be divided into 8 classes by its topological phase portraits for the following perturbed system x 2 y b x 2 2 c y 2 e f x y y 2 x 1 2 a x 2 b y 2 e g x y where 0 e 1 f x y and g x y are polynomials in x y with degree n we give an estimation of the number of isolated zeros of the corresponding abelian integral the number of limit cycles follows from poincare pontryagin theorem

adaptive robust distributed learning in diffusion sensor networks in this paper the problem of adaptive distributed learning in diffusion networks is considered the algorithms are developed within the convex set theoretic framework more specifically they are based on computationally simple geometric projections onto closed convex sets the paper suggests a novel combine project adapt protocol for cooperation among the nodes of the network such a protocol fits naturally with the philosophy that underlies the projection based rationale moreover the possibility that some of the nodes may fail is also considered and it is addressed by employing robust statistics loss functions such loss functions can easily be accommodated in the adopted algorithmic framework all that is required from a loss function is convexity under some mild assumptions the proposed algorithms enjoy monotonicity asymptotic optimality asymptotic consensus strong convergence and linear complexity with respect to the number of unknown parameters finally experiments in the context of the system identification task verify the validity of the proposed algorithmic schemes which are compared to other recent algorithms that have been developed for adaptive distributed learning

on the abelian integrals of quadratic reversible centers with orbits formed by genus one curves of higher degree abstract in this study we determine the number of zeros for abelian integrals in four cases of quadratic reversible centers of genus one based on the results of li et al we prove that the upper bound of the associated number of zeros for abelian integrals with orbits formed by quartics quintics sextics and eight degree curves under polynomial perturbations of arbitrary degree n depends linearly on n

limit cycles bifurcating from piecewise quadratic systems separated by a straight line abstract in this paper we are concerned with limit cycles bifurcating from piecewise quadratic systems separated by a straight line by means of the first order averaging theory of discontinuous differential systems and the argument principle we present an estimate of the maximum number of limit cycles bifurcating from the period annulus around the center of piecewise smooth systems

periodic orbits for perturbations of piecewise linear systems abstract we consider the existence of periodic orbits in a class of three dimensional piecewise linear systems firstly we describe the dynamical behavior of a non generic piecewise linear system which has two equilibria and one two dimensional invariant manifold foliated by periodic orbits the aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system in order to analyze this situation we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation by using this function we state some results of existence and stability of limit cycles in the perturbed system as well as results of bifurcations of limit cycles the techniques presented are similar to the melnikov theory for smooth systems and the method of averaging

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

limit cycle bifurcations in perturbations of planar piecewise smooth systems with multiply lines of critical points abstract this paper investigates the limit cycle bifurcation problem of planar piecewise polynomial systems whose unperturbed system has a center at the origin and may have invariant multiply lines by melnikov function method this paper provides the upper and lower bounds for the poincare cyclicity in particular the exact cyclicities of two specific cases are given using analyzing method one of which has been discussed in the paper 7

periodic orbits from second order perturbation via rational trigonometric integrals abstract the second order poincare pontryagin melnikov perturbation theory is used in this paper to study the number of bifurcated periodic orbits from certain centers this approach also allows us to give the shape and the period up to the first order we address these problems for some classes of abel differential equations and quadratic isochronous vector fields in the plane we prove that two is the maximum number of hyperbolic periodic orbits bifurcating from the isochronous quadratic centers with a birational linearization under quadratic perturbations of second order in particular the configurations 2 0 and 1 1 are realizable when two centers are perturbed simultaneously the required computations show that all the considered families share the same iterated rational trigonometric integrals

distributed consensus algorithms in sensor networks with imperfect communication link failures and channel noise the paper studies average consensus with random topologies intermittent links and noisy channels consensus with noise in the network links leads to the bias variance dilemma running consensus for long reduces the bias of the final average estimate but increases its variance we present two different compromises to this tradeoff the a nd algorithm modifies conventional consensus by forcing the weights to satisfy a persistence condition slowly decaying to zero and the a nc algorithm where the weights are constant but consensus is run for a fixed number of iterations iota then it is restarted and rerun for a total of p runs and at the end averages the final states of the p runs monte carlo averaging we use controlled markov processes and stochastic approximation arguments to prove almost sure convergence of a nd to a finite consensus limit and compute explicitly the mean square error mse variance of the consensus limit we show that a nd represents the best of both worlds zero bias and low variance at the cost of a slow convergence rate rescaling the weights balances the variance versus the rate of bias reduction convergence rate in contrast a nc because of its constant weights converges fast but presents a different bias variance tradeoff for the same number of iterations iota p shorter runs smaller iota lead to high bias but smaller variance larger number p of runs to average over for a static nonrandom network with gaussian noise we compute the optimal gain for a nc to reach in the shortest number of iterations iota p with high probability 1 delta epsiv delta consensus epsiv residual bias our results hold under fairly general assumptions on the random link failures and communication noise

distributed linear parameter estimation asymptotically efficient adaptive strategies this paper considers the problem of distributed adaptive linear parameter estimation in multiagent inference networks local sensing model information is only partially available at the agents and interagent communication is assumed to be unpredictable the paper develops a generic mixed time scale stochastic procedure consisting of simultaneous distributed learning and estimation in which the agents adaptively assess their relative observation quality over time and fuse the innovations accordingly under rather weak assumptions on the statistical model and the interagent communication it is shown that by properly tuning the consensus potential with respect to the innovation potential the asymptotic information rate loss incurred in the learning process may be made negligible as such it is shown that the agent estimates are asymptotically efficient in that their asymptotic covariance coincides with that of a centralized estimator the inverse of the centralized fisher information rate for gaussian

the number of limit cycles of a class of polynomial differential systems abstract in this paper we consider the bifurcation of limit cycles of a class of polynomial differential systems of the form x y 1 x 4 e p n x y y x 1 x 4 e q n x y where p n q n are arbitrary polynomials of degree n we prove that there is a system of the above form having at least 3 n 1 2 2 limit cycles in hopf bifurcation then applying the argument principle we obtain that up to first order in e the number of limit cycles that bifurcate from period annulus surrounding the origin of this system is at most 5 n 1 4 n 1 2 for n 5

linear estimate of the number of zeros of abelian integrals for a kind of quartic hamiltonians abstract an upper bound b n 7n 5 is derived for the number of zeros of abelian integrals i h h g x y dy f x y dx on the open interval where h is an oval lying on the algebraic curve h x y 1 2 y2 u x h deg u x 4 and is the maximal interval of existence of h f x y g x y are polynomials of x and y and n max deg f x y deg g x y

on the number of limit cycles bifurcating from a non global degenerated center abstract we give an upper bound for the number of zeros of an abelian integral this integral controls the number of limit cycles that bifurcate by a polynomial perturbation of arbitrary degree n from the periodic orbits of the integrable system 1 x d h 0 where h is the quasi homogeneous hamiltonian h x y x 2 k 2 k y 2 2 the tools used in our proofs are the argument principle applied to a suitable complex extension of the abelian integral and some techniques in real analysis

on the number of limit cycles surrounding a unique singular point for polynomial differential systems of arbitrary degree abstract we study the number of limit cycles that bifurcate from the periodic orbits of the center x y r x y y x r x y where r is a convenient polynomial of degree 2 when we perturb it inside the class of all polynomial differential systems of degree n we use averaging theory for computing this number as a consequence of our study we provide the biggest number of limit cycles surrounding a unique singular point in terms of the degree of the system known up to now

on the number of limit cycles of a perturbed cubic polynomial differential center abstract for e sufficiently small we consider the planar polynomial differential system x y y 2 a x 2 b x c e p x y y x y 2 a x 2 b x c e q x y where p x y and q x y are polynomials of degree n by using the averaging method of first order we bound the number of limit cycles that can bifurcate from period annulus of the unperturbed system

limit cycles of polynomial differential equations with quintic homogeneous nonlinearities abstract in this paper we mainly study the number of limit cycles which can bifurcate from the periodic orbits of the two centers x y y x x y 1 x 2 y 2 2 y x 1 x 2 y 2 2 when they are perturbed inside the class of all polynomial differential systems with quintic homogeneous nonlinearities we do this study using the averaging theory of first second and third orders

upper bounds for the number of zeroes for some abelian integrals consider the vector field x yg x y y xg x y where the set of critical points g x y 0 is formed by k straight lines not passing through the origin and parallel to one or two orthogonal directions we perturb it with a general polynomial perturbation of degree n and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of k and n our approach is based on the explicit computation of the abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions when we apply our results for k 4 we recover or improve some results obtained in several previous works

limit cycles near a homoclinic loop by perturbing a class of integrable systems abstract in this paper the problem of limit cycle bifurcation is investigated by perturbing a class of integrable systems with a homoclinic loop under the assumption that the homoclinic loop passes through a degenerate singular point at the origin the asymptotic expansion of the melnikov function along the level curves of the first integral inside the homoclinic loop is studied near the loop meanwhile the formulas for the first coefficients in the expansion are given which can be used to study the number of limit cycles near the homoclinic loop finally an example is provided to demonstrate the obtained results

linear estimation of the number of zeros of abelian integrals for some cubic isochronous centers abstract this paper consists of two parts in the first part we study the relationship between conic centers all orbits near a singular point of center type are conics and isochronous centers of polynomial systems in the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics when we perturb such systems inside the class of all polynomial systems of degree n

a collaborative training algorithm for distributed learning in this paper an algorithm is developed for collaboratively training networks of kernel linear least squares regression estimators the algorithm is shown to distributively solve a relaxation of the classical centralized least squares regression problem a statistical analysis shows that the generalization error afforded agents by the collaborative training algorithm can be bounded in terms of the relationship between the network topology and the representational capacity of the relevant reproducing kernel hilbert space numerical experiments suggest that the algorithm is effective at reducing noise the algorithm is relevant to the problem of distributed learning in wireless sensor networks by virtue of its exploitation of local communication several new questions for statistical learning theory are proposed

some series constructions for minimal size equineighboured balanced incomplete block designs with nested rows and columns summary three infinite series for minimal size equineighboured balanced incomplete block designs with nested rows and columns are constructed for many parameter combinations the proposed equineighboured designs do not require more replicates and blocks than ordinary balanced incomplete block designs with nested rows and columns in the absence of neighbour properties available in the literature the designs can also be used as nested balanced incomplete block designs taking either the rows or the columns as subblocks with neighbour balance at all levels within the subblocks

higher dimensional consensus learning in large scale networks the paper considers higher dimensional consensus hdc hdc is a general class of linear distributed algorithms for large scale networks that generalizes average consensus and includes other interesting distributed algorithms like sensor localization leader follower algorithms in multiagent systems or distributed jacobi algorithm in hdc the network nodes are partitioned into anchors nodes whose states are fixed over the hdc iterations and sensors nodes whose states are updated by the algorithm the paper starts by briefly considering what we call the forward problem by presenting the conditions for hdc to converge the limiting state to which it converges and what is its convergence rate the main focus of the paper is the inverse or design problem i e learning the weights or parameters of the hdc so that the algorithm converges to a desired prespecified state this generalizes the well known problem of designing the weights in average consensus we pose learning as a constrained nonconvex optimization problem that we cast in the framework of multiobjective optimization mop and to which we apply pareto optimality we derive the solution to the learning problem by proving relevant properties satisfied by the mop solutions and by the pareto front finally the paper shows how the mop approach leads to interesting tradeoffs speed of convergence versus performance arising in resource constrained networks simulation studies illustrate our approach for a leader follower architecture in multiagent systems

bifurcation of a piecewise smooth cubic system via expansion of melnikov function abstract asymptotic expansion formulae of the first order melnikov function near the perturbed center of a cubic system are investigated studied through a suitable transformation by expanding the formulae we obtain that the given cubic system can have at least ten limit cycles

the number of limit cycles of a quintic polynomial system with center abstract in this paper we study the global bifurcation of limit cycles for the system x y 1 x 4 y x 1 x 4 e y 2 m 1 j 0 l a j x j for e sufficiently small where l 2 n 2 or 2 n 3 m n are arbitrary positive integers and a 0 a 1 a l are real we have used argument principle to give estimate of an upper bound for the number of limit cycles that can bifurcate from period annulus of this system for e 0 furthermore we have shown that there exist a set of constant a 0 a 1 a l where the related abelian integral of this system has at least 3 m n 2 isolated zeros we have in order to prove our result applied the argument principle to a complex extension of the abelian integral

limit cycles of polynomial differential systems with homogeneous nonlinearities of degree 4 via the averaging method by using the averaging method we study the limit cycles for a class of quartic polynomial differential systems as well as their global shape in the plane more specifically we analyze the global shape of limit cycles bifurcating from a hopf bifurcation and also from periodic orbits with linear center x y y x the perturbation of these systems is made inside the class of quartic polynomial differential systems without quadratic and cubic terms

some methods of construction of balanced incomplete block designs with nested rows and columns summary a number of methods of construction of balanced incomplete block designs with nested rows and columns are developed using the method of differences

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

decentralized parameter estimation by consensus based stochastic approximation in this paper an algorithm for decentralized multi agent estimation of parameters in linear discrete time regression models is proposed in the form of a combination of local stochastic approximation algorithms and a global consensus strategy an analysis of the asymptotic properties of the proposed algorithm is presented taking into account both the multi agent network structure and the probabilities of getting local measurements and implementing exchange of inter agent messages in the case of non vanishing gains in the stochastic approximation algorithms an asymptotic estimation error covariance matrix bound is defined as the solution of a lyapunov like matrix equation in the case of asymptotically vanishing gains the mean square convergence is proved and the rate of convergence estimated in the discussion the problem of additive communication noise is treated in a methodologically consistent way it is also demonstrated how the consensus scheme in the algorithm can contribute to the overall reduction of measurement noise influence some simulation results illustrate the obtained theoretical results

diffusion strategies outperform consensus strategies for distributed estimation over adaptive networks adaptive networks consist of a collection of nodes with adaptation and learning abilities the nodes interact with each other on a local level and diffuse information across the network to solve estimation and inference tasks in a distributed manner in this work we compare the mean square performance of two main strategies for distributed estimation over networks consensus strategies and diffusion strategies the analysis in the paper confirms that under constant step sizes diffusion strategies allow information to diffuse more thoroughly through the network and this property has a favorable effect on the evolution of the network diffusion networks are shown to converge faster and reach lower mean square deviation than consensus networks and their mean square stability is insensitive to the choice of the combination weights in contrast and surprisingly it is shown that consensus networks can become unstable even if all the individual nodes are stable and able to solve the estimation task on their own when this occurs cooperation over the network leads to a catastrophic failure of the estimation task this phenomenon does not occur for diffusion networks we show that stability of the individual nodes always ensures stability of the diffusion network irrespective of the combination topology simulation results support the theoretical findings