principal poincare pontryagin function associated to polynomial perturbations of a product of d 1 straight lines abstract in this paper we study small polynomial perturbations of a hamiltonian vector field with hamiltonian f formed by a product of d 1 real linear functions in two variables we assume that the corresponding lines are in a general position in r 2 that is the lines are distinct non parallel no three of them have a common point and all critical values not corresponding to intersections of lines are distinct we prove in this paper that the principal poincare pontryagin function m k t associated to such a perturbation and to any family of ovals surrounding a singular point of center type belongs to the c t 1 t module generated by abelian integrals and some integrals i i j t with 1 i j d defined in the paper moreover i i j t are not abelian integrals they are iterated integrals of length two

on the critical periods of perturbed isochronous centers abstract consider a family of planar systems x x x e having a center at the origin and assume that for e 0 they have an isochronous center firstly we give an explicit formula for the first order term in e of the derivative of the period function we apply this formula to prove that up to first order in e at most one critical period bifurcates from the periodic orbits of isochronous quadratic systems when we perturb them inside the class of quadratic reversible centers moreover necessary and sufficient conditions for the existence of this critical period are explicitly given from the tools developed in this paper we also provide a new characterization of planar isochronous centers

quadratic systems with center and their perturbations abstract we study the quadratic systems on a plane z i z az2 b z 2 cz2 z x iy we give a simple algebraic proof of the center conditions 0 and b 0 or a 1 2 b 1 or a b 1 c c or a 2 b 1 c 1 and of the theorem of bautin that the number of small amplitude limit cycles bifurcating from a center or a focus is not greater than 3 we present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case here one mistake in bautin s work is revealed and some questions concerning focus numbers are answered we study also global not small limit cycles for a small perturbation of the center case with two invariant lines the lotka volterra system using abelian integrals we show that the number of zeroes of the corresponding abelian integral is 0 1 or 2 some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two fold center when two center conditions hold the problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients

the monotonicity of the period function for planar hamiltonian vector fields if the hamiltonian system with hamiltonian h x y 12y2 v x has a center at the origin it is shown that the period function for the family of periodic trajectories surrounding the center is monotone when the function v v 2 is convex this theorem is applied to determine existence and uniqueness of solutions for the neumann boundary value problem x a x 2 bx c x 0 x 1 0

random field reconstruction with quantization in wireless sensor networks we develop new and novel algorithms for spatial field reconstruction and spatial exceedance level estimation we consider spatial physical phenomena which are partially observed by a wireless sensor network we focus on the practical scenario in which due to bandwidth and power constraints the observations at the sensors are quantized and transmitted over imperfect wireless channels we first develop the spatial best linear unbiased estimator s blue algorithm for the spatial field reconstruction estimation we then develop an algorithm that is based on a multivariate series expansion approach resulting in a saddle point type approximation to solve both problems of spatial field reconstruction and exceedance level estimation we derive the posterior cramer rao lower bound pcrlb and quantify the achievable mse in the estimation the estimation accuracy of the proposed algorithms is quantified via numerical comparisons between the two algorithms and the pcrlb

nonzero bound on fiedler eigenvalue causes exponential growth of h infinity norm of vehicular platoon we consider platoons composed of identical vehicles and controlled in a distributed way that is each vehicle has its own onboard controller the regulation errors in spacing to the immediately preceeding and following vehicles are weighted differently by the onboard controller which thus implements an asymmetric bidirectional control scheme the weights can vary along the platoon we prove that such platoons have a nonzero uniform bound on the second smallest eigenvalue of the graph laplacian matrix the fiedler eigenvalue furthermore it is shown that existence of this bound always signals undesirable scaling properties of the platoon namely the h infinity norm of the transfer function of the platoon grows exponentially with the number of vehicles regardless of the controllers used hence the benefits of a uniform gap in the spectrum of a laplacian with an asymetric distributed controller are paid for by poor scaling as the number of vehicles grows

distributed control for identical dynamically coupled systems a decomposition approach we consider the problem of designing distributed controllers for a class of systems which can be obtained from the interconnection of a number of identical subsystems if the state space matrices of these systems satisfy a certain structural property then it is possible to derive a procedure for designing a distributed controller which has the same interconnection pattern as the plant this procedure is basically a multiobjective optimization under linear matrix inequality constraints with system norms as performance indices the explicit expressions for computing these controllers are given for both h infin or h 2 performance and both for static state feedback and dynamic output feedback in discrete time at the end of the paper two application examples illustrate the effectiveness of the approach

abelian integrals in unfoldings of codimension 3 singularities with nilpotent linear parts abstract this paper is concerned with the upper bound of the number of limit cycles in unfolding of codimension 3 planar singularities with nilpotent linear parts after making a central rescaling the problem reduces to a perturbation problem of a one parameter family of quadratic reversible systems as the parameter a 1 1 0 is rational except the case a 2 3 based on the chebyshev criterion for abelian integrals and a rationalizing transformation the problem could be solved theoretically to illustrate our approaches two particular cases corresponding to nilpotent codimension 3 saddle and elliptic case respectively are proved where the upper bound of the number of limit cycles is two

a class of hamiltonian systems with increasing periods on considere un systeme hamiltonien de la forme x g y y f x ou h f et h g satisfont certaines hypotheses on montre qu il y a un intervalle maximal 0 a a 0 tel que chaque solution x t y t avec x 0 a 0 a y 0 0 est periodique avec son orbite enfermant 0 0 soit p a la plus petite periode de cette solution alors p 0 a r est differentiable avec p a 0

the period functionsʼ higher order derivatives abstract we prove a formula for the n th derivative of the period function t in a period annulus of a planar differential system for n 1 we obtain freire gasull and guillamon formula for the period s first derivative chicone and dumortier 1993 17 we apply such a result to hamiltonian systems with separable variables and other systems we give some sufficient conditions for the period function of conservative second order ode s to be convex

mistuning based control design to improve closed loop stability margin of vehicular platoons we consider a decentralized bidirectional control of a platoon of n identical vehicles moving in a straight line the control objective is for each vehicle to maintain a constant velocity and inter vehicular separation using only the local information from itself and its two nearest neighbors each vehicle is modeled as a double integrator to aid the analysis we use continuous approximation to derive a partial differential equation pde approximation of the discrete platoon dynamics the pde model is used to explain the progressive loss of closed loop stability with increasing number of vehicles and to devise ways to combat this loss of stability if every vehicle uses the same controller we show that the least stable closed loop eigenvalue approaches zero as o 1 n2 in the limit of a large number n of vehicles we then show how to ameliorate this loss of stability by small amounts of mistuning i e changing the controller gains from their nominal values we prove that with arbitrary small amounts of mistuning the asymptotic behavior of the least stable closed loop eigenvalue can be improved to o 1 n all the conclusions drawn from analysis of the pde model are corroborated via numerical calculations of the state space platoon model

distributed control of spatially invariant systems we consider distributed parameter systems where the underlying dynamics are spatially invariant and where the controls and measurements are spatially distributed these systems arise in many applications such as the control of vehicular platoons flow control microelectromechanical systems mems smart structures and systems described by partial differential equations with constant coefficients and distributed controls and measurements for fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator lqr h sub 2 and h sub spl infin optimal controllers can be obtained by solving a parameterized family of standard finite dimensional problems we show that optimal controllers have an inherent degree of decentralization and this provides a practical distributed controller architecture we also prove a general result that applies to partially distributed control and a variety of performance criteria stating that optimal controllers inherit the spatial invariance structure of the plant connections of this work to that on systems over rings and systems with dynamical symmetries are discussed

the period function of reversible quadratic centers abstract in this paper we investigate the bifurcation diagram of the period function associated to a family of reversible quadratic centers namely the dehomogenized loud s systems the local bifurcation diagram of the period function at the center is fully understood using the results of chicone and jacobs c chicone m jacobs bifurcation of critical periods for plane vector fields trans amer math soc 312 1989 433 486 most of the present paper deals with the local bifurcation diagram at the polycycle that bounds the period annulus of the center the techniques that we use here are different from the ones in c chicone m jacobs bifurcation of critical periods for plane vector fields trans amer math soc 312 1989 433 486 because while the period function extends analytically at the center it has no smooth extension to the polycycle at best one can hope that it has some asymptotic expansion another major difficulty is that the asymptotic development has to be uniform with respect to the parameters in order to prove that a parameter is not a bifurcation value we study also the bifurcations in the interior of the period annulus and we show that there exist three germs of curves in the parameter space that correspond to this type of bifurcation moreover we determine some regions in the parameter space for which the corresponding period function has at least one or two critical periods finally we propose a complete conjectural bifurcation diagram of the period function of the dehomogenized loud s systems our results can also be viewed as a contribution to the proof of chicone s conjecture c chicone review in mathscinet ref 94h 58072

the cyclicity of a cubic system with nonradical bautin ideal abstract we present a method for investigating the cyclicity of an elementary focus or center of a polynomial system of differential equations by means of complexification of the system and application of algorithms of computational algebra showing an approach to treating the case that the bautin ideal b of focus quantities is not a radical ideal more precisely when the ideal b k is not radical where b k is the ideal generated by the shortest initial string of focus quantities that like the bautin ideal determines the center variety we illustrate the method with a family of cubic 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

limit cycles bifurcated from a class of quadratic reversible center of genus one abstract we investigate the bifurcation of limit cycles in a class of planar quadratic reversible non hamiltonian systems this systems has a center of genus one the exact upper bound of the number of limit cycles is given

period function for a class of hamiltonian systems abstract this paper studies the period function of the class of hamiltonian systems x h y y h x where h x y has the special form h x y f x g y and the origin is a non degenerate center more concretely if t h denotes the period of the periodic orbit contained in h x y h we solve the inverse problem of characterizing all systems with a given function t h we also characterize the limiting behaviour of t at infinity when the origin is a global center and apply this result to prove among other results that there are no nonlinear polynomial isochronous centers in this family

on the centers of the weight homogeneous polynomial vector fields on the plane we classify all centers of a planar weight homogeneous polynomial vector field of weight degree 1 2 3 and 4

information based objective functions for active data selection learning can be made more efficient if we can actively select particularly salient data points within a bayesian learning framework objective functions are discussed that measure the expected informativeness of candidate measurements three alternative specifications of what we want to gain information about lead to three different criteria for data selection all these criteria depend on the assumption that the hypothesis space is correct which may prove to be their main weakness

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

classification of the centers their cyclicity and isochronicity for the generalized quadratic polynomial differential systems in this paper we classify the centers the cyclicity of their hopf bifurcation and the isochronicity of the polynomial differential systems in r2 of degree d that in complex notation z x iy can be written as z i z zz d 22 az2 bzz cz 2 where d 2 is an arbitrary even positive integer r and a b c c note that if d 2 we obtain the well known class of quadratic polynomial differential systems which can have a center at the origin

transition time analysis in singularly perturbed boundary value problems the paper deals with the boundary value problem x xj x2 0 with x 0 a x t b for a b t 0 and e 0 close to zero it is shown that for t sufficiently big the problem has exactly three solutions two of which reach negative values solutions reaching negative values occur for t t e 0 and we show that asymptotically for e 0 t e ln e i e lim6 o t e 1 the main tools are transit time analysis in the lienard plane and normal form techniques as such the methods are rather qualitative and useful in other similar problems introduction in this paper the authors will apply desingularization techniques to a singularly perturbed boundary value problem by doing so we will be able to explain apparent transition phenomena in a geometric way the equation we study is given by

the infinitesimal 16th hilbert problem in the quadratic case let h x y be a real cubic polynomial with four distinct critical values in a complex domain and let x h h y frac partial partial x h x frac partial partial y be the corresponding hamiltonian vector field we show that there is a neighborhood of x h in the space of all quadratic plane vector fields such that any x has at most two limit cycles

asymptotic expansion of the period function abstract let p be a not necessarily bounded polycycle of an analytic vector field on an open set of the plane suppose that the singularities which appear after desingularization of the vertices of p are formally linearizable consider the function t defined by the return time near p it is shown that the function t and its derivative t have asymptotic expansions in x r and x log x r it is also shown that under some other conditions imposed on the polycycle vertices the asymptotic expansions of t and t converge absolutely and uniformly to these functions respectively these results are applied to the polycycles of the analytic vector fields which have a darboux first integral in particular it is obtained that if p is a polycycle of a hamiltonian vector field with an analytic polynomial if p is unbounded hamiltonian function t is a nonoscillating function another application concerns the nilpotent centers or focus since the singularities which appear after desingularization of such a singularity have analytic first integrals

geometric methods for two point nonlinear boundary value problems on considere une classe d equations differentielles de la forme x f x g x avec des conditions aux limites soit de dirichlet soit de neumann on etudie l existence de la solution triviale

optimal design for synchronization of cooperative systems state feedback observer and output feedback this technical note studies synchronization of identical general linear systems on a digraph containing a spanning tree a leader node or command generator is considered which generates the desired tracking trajectory a framework for cooperative tracking control is proposed including full state feedback control observer design and dynamic output feedback control the classical system theory notion of duality is extended to networked systems it is shown that unbounded synchronization regions that achieve synchronization on arbitrary digraphs containing a spanning tree can be guaranteed by using linear quadratic regulator based optimal control and observer design methods at each node

on achieving size independent stability margin of vehicular lattice formations with distributed control we study the stability margin of a vehicular formation with distributed control in which the control at each vehicle only depends on the information from its neighbors in an information graph we consider a d dimensional lattice as information graph of which the 1 d platoon is a special case the stability margin is measured by the real part of the least stable eigenvalue of the closed loop state matrix which quantifies the rate of decay of initial errors in 1 it was shown that with symmetric control in which two neighbors put equal weight on information received from each other the stability margin of a 1 d vehicular platoon decays to 0 as 0 1 n2 where n is the number of vehicles moreover a perturbation analysis was used to show that with vanishingly small amount of asymmetry in the control gains the stability margin scaling can be improved to 0 1 n in this technical note we show that with judicious choice of nonvanishing asymmetry in control the stability margin of the closed loop can be bounded away from zero uniformly in n asymmetry in control gains thus makes the control architecture highly scalable the results are also generalized to d dimensional lattice information graphs that were studied in 2 and the correspondingly stronger conclusions than those derived in 2 are obtained in addition we show that the size independent stability margin can be achieved with relative position and relative velocity rprv feedback as well as relative position and absolute velocity rpav feedback while the analysis in 1 2 was only for the rpav case

mixed discrete continuous bayesian inference censored measurements of sparse signals this paper addresses bayesian inference of sparse signals by censored measurements to reduce the sensor s duty cycle signals below a threshold are censored i e are set to zero sparse signals are random vectors that or whose elements are zero with given probabilities the corresponding probabilistic model induces random measurement and signal vectors of mixed absolute continuous discrete and singular continuous nature therefore mixed probability densities the expectation regarding these densities and a generalized bayes rule are constructively derived for the inference proper a posteriori expected loss functions are defined their derivative free minimizations gives bayesian inferrers similar to traditional minimum mean square error mmse maximum a posteriori map and median estimators and detectors the result provides a unified bayesian inference framework eventually this leads to closed form solutions for the inference problem numerical results and the analysis of the probability of censorship

limit cycle bifurcations by perturbing a quadratic integrable system with a triangle abstract finding cyclicity of homoclinic or heteroclinic loops in quadratic systems is an open problem which is related to the second part of hilbert s 16th problem this paper mainly considers the problem of limit cycle bifurcation for quadratic polynomial systems near a triangle it is proved that the cyclicity of the triangle in quadratic systems is at least three

the monotonicity of period function for codimension four quadratic system q4 abstract in this paper we study the codimension four quadratic system q 4 z iz 4 z 2 2 z 2 b ic z 2 where b and c are real constants i 2 1 z x iy b ic 2 it is proved that the period function of periodic trajectories for q 4 is monotone the content of this paper can be viewed as a contribution to the proof of chicone s conjecture cf mr 94h 58072

asymptotic expansion of the dulac map and time for unfoldings of hyperbolic saddles local setting abstract in this paper we study unfoldings of planar vector fields in a neighbourhood of a hyperbolic resonant saddle we give a structure theorem for the asymptotic expansion of the local dulac time as well as the local dulac map with the remainder uniformly flat with respect to the unfolding parameters here local means close enough to the saddle in order that the normalizing coordinates provided by a suitable normal form can be used the principal part of the asymptotic expansion is given in a monomial scale containing a deformation of the logarithm the so called roussarie ecalle compensator especial attention is paid to the remainder s properties concerning the derivation with respect to the unfolding parameters

bernstein inequalities and applications to analytic geometry and differential equations we introduce new methods of complex analysis inequalities of bernstein type to study projections of analytic sets these techniques are applied to problems of bifurcations of periodic orbits of differential equations such as the local hilbert s 16th problem

coherence in large scale networks dimension dependent limitations of local feedback we consider distributed consensus and vehicular formation control problems specifically we address the question of whether local feedback is sufficient to maintain coherence in large scale networks subject to stochastic disturbances we define macroscopic performance measures which are global quantities that capture the notion of coherence a notion of global order that quantifies how closely the formation resembles a solid object we consider how these measures scale asymptotically with network size in the topologies of regular lattices in 1 2 and higher dimensions with vehicular platoons corresponding to the 1 d case a common phenomenon appears where a higher spatial dimension implies a more favorable scaling of coherence measures with a dimensions of 3 being necessary to achieve coherence in consensus and vehicular formations under certain conditions in particular we show that it is impossible to have large coherent 1 d vehicular platoons with only local feedback we analyze these effects in terms of the underlying energetic modes of motion showing that they take the form of large temporal and spatial scales resulting in an accordion like motion of formations a conclusion can be drawn that in low spatial dimensions local feedback is unable to regulate large scale disturbances but it can in higher spatial dimensions this phenomenon is distinct from and unrelated to string instability issues which are commonly encountered in control problems for automated highways

sensor centric data reduction for estimation with wsns via censoring and quantization consider a wireless sensor network wsn with a fusion center fc deployed to estimate signal parameters from noisy sensor measurements if the wsn has a large number of low cost battery operated sensor nodes with limited transmission bandwidth then conservation of transmission resources power and bandwidth is paramount to this end the present paper develops a novel data reduction method which requires no inter sensor collaboration and results in only a subset of the sensor measurements transmitted to the fc using interval censoring as a data reduction method each sensor decides separately whether to censor its acquired measurements based on a rule that promotes censoring of measurements with least impact on the estimator mean square error mse leveraging the statistical distribution of sensor data the censoring mechanism and the received uncensored data fc based estimators are derived for both deterministic via maximum likelihood estimation and random parameters via maximum a posteriori probability estimation for a linear gaussian model quantization of the uncensored measurements at the sensor nodes offers an additional degree of freedom in the resource conservation versus estimator mse reduction tradeoff cramer rao bound analysis for the different censor estimators and censor quantizer estimators is also provided to benchmark and facilitate mse based performance comparisons numerical simulations corroborate the analytical findings and demonstrate that the proposed censoring estimation approach performs competitively with alternative methods under different sensing conditions while having lower computational complexity

reachability of maximal number of critical periods without independence abstract the reachability of the maximal number of critical periods bifurcated from a weak center is judged with independence of period quantities since the independence is a sufficient condition in this paper we give a method to judge the reachability without the independence our main idea is to consider how the number of zeros for derivative of the period function changes as a curve in the parameter space scans near a specific point and choose an appropriate curve to find the largest number of zeros we apply this method to two polynomial differential systems having zero dimensional isochronicity variety and nonzero dimensional one separately to show that the maximal number is reached and obtain greater numbers than those obtained by classical independence separately with our method the two examples give the possibility that more critical periods arise from an isochronous center than a finite order weak center an opposite fact to many known results

classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree abstract in this paper we classify the centers localized at the origin of coordinates and their isochronicity for the polynomial differential systems in r 2 of degree d that in complex notation z x i y can be written as z i z a z d n 1 2 z d n 1 2 b z d n 1 2 z d n 1 2 c z d 1 2 z d 1 2 d z d 2 j n 1 2 z d 2 j n 1 2 where j is either 0 or 1 if j 0 then d 5 is an odd integer and n is an even integer satisfying 2 n d 1 2 if j 1 then d 3 is an integer and n is an integer with converse parity with d and satisfying 0 n d 1 3 where denotes the integer part function furthermore r and a b c d c note that if d 3 and j 0 we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in k e malkin 1964 17 n i vulpe and k s sibirskii 1988 25 j llibre and c valls 2009 15 and if d 2 j 1 and c 0 we are also obtaining as a particular case the quadratic polynomial differential systems studied in n n bautin 1952 2 h zoladek 1994 26 so the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations

brief paper an internal model principle is necessary and sufficient for linear output synchronization output synchronization of a network of heterogeneous linear state space models under time varying and directed interconnection structures is investigated it is shown that assuming stabilizability and detectability of the individual systems and imposing very mild connectedness assumptions on the interconnection structure an internal model requirement is necessary and sufficient for synchronizability of the network to polynomially bounded trajectories the resulting dynamic feedback couplings can be interpreted as a generalization of existing methods for identical linear systems

optimal dimensionality reduction of sensor data in multisensor estimation fusion when there exists the limitation of communication bandwidth between sensors and a fusion center one needs to optimally precompress sensor outputs sensor observations or estimates before the sensors transmission in order to obtain a constrained optimal estimation at the fusion center in terms of the linear minimum error variance criterion or when an allowed performance loss constraint exists one needs to design the minimum dimension of sensor data this paper will answer the above questions by using the matrix decomposition pseudo inverse and eigenvalue techniques

bi center problem and bifurcation of limit cycles from nilpotent singular points in z2 equivariant cubic vector fields abstract in this paper bi center problem and bifurcation of limit cycles from nilpotent singular points in z 2 equivariant cubic vector fields are studied first the system is simplified by using some proper transformations and the first five lyapunov constants at a nilpotent singular point are calculated by applying the inverse integrating factor method then sufficient and necessary conditions are obtained for two nilpotent singular points of the system being centers a new perturbation scheme is present to prove the existence of 12 small amplitude limit cycles in cubic z 2 equivariant vector fields which bifurcate from two nilpotent singular points this is a new lower bound of the number of limit cycles bifurcating in such systems

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

the period function of the generalized lotka volterra centers abstract the present paper deals with the period function of the quadratic centers in the literature different terminologies are used to classify these centers but essentially there are four families hamiltonian reversible q 3 r codimension four q 4 and generalized lotka volterra systems q 3 lv chicone c chicone review in mathscinet ref 94h 58072 conjectured that the reversible centers have at most two critical periods and that the centers of the three other families have a monotonic period function with regard to the second part of this conjecture only the monotonicity of the hamiltonian and q 4 families w a coppel l gavrilov the period function of a hamiltonian quadratic system differential integral equations 6 1993 1357 1365 y zhao the monotonicity of period function for codimension four quadratic system q 4 j differential equations 185 2002 370 387 has been proved concerning the q 3 lv family no substantial progress has been made since the middle 80s when several authors showed independently the monotonicity of the classical lotka volterra centers f rothe the periods of the volterra lokta system j reine angew math 355 1985 129 138 r schaaf global behaviour of solution branches for some neumann problems depending on one or several parameters j reine angew math 346 1984 1 31 j waldvogel the period in the lotka volterra system is monotonic j math anal appl 114 1986 178 184 by means of the first period constant one can easily conclude that the period function of the centers in the q 3 lv family is monotone increasing near the inner boundary of its period annulus i e the center itself thus according to chicone s conjecture it should be also monotone increasing near the outer boundary which in the poincare disc is a polycycle in this paper we show that this is true in addition we prove that except for a zero measure subset of the parameter plane there is no bifurcation of critical periods from the outer boundary finally we show that the period function is globally i e in the whole period annulus monotone increasing in two other cases different from the classical one

bandwidth constrained distributed estimation for wireless sensor networks part i gaussian case we study deterministic mean location parameter estimation when only quantized versions of the original observations are available due to bandwidth constraints when the dynamic range of the parameter is small or comparable with the noise variance we introduce a class of maximum likelihood estimators that require transmitting just one bit per sensor to achieve an estimation variance close to that of the clairvoyant sample mean estimator when the dynamic range is comparable or larger than the noise standard deviation we show that an optimum quantization step exists to achieve the best possible variance for a given bandwidth constraint we will also establish that in certain cases the sample mean estimator formed by quantized observations is preferable for complexity reasons we finally touch upon algorithm implementation issues and guarantee that all the numerical maximizations required by the proposed estimators are concave

spectra of certain large tridiagonal matrices abstract we characterize the eigenvalues and eigenvectors of a class of complex valued tridiagonal n by n matrices subject to arbitrary boundary conditions i e with arbitrary elements on the first and last rows of the matrix for large n we show there are up to 4 eigenvalues the so called special eigenvalues whose behavior depends sensitively on the boundary conditions the other eigenvalues the so called regular eigenvalues vary very little as function of the boundary conditions for large n we determine the regular eigenvalues up to o n 2 and the special eigenvalues up to o n for some 0 1 the components of the eigenvectors are determined up to o n 1 the matrices we study have important applications throughout the sciences among the most common ones are arrays of linear dynamical systems with nearest neighbor coupling and discretizations of second order linear partial differential equations in both cases we give examples where specific choices of boundary conditions substantially influence leading eigenvalues and therefore the global dynamics of the system

on the number of critical periods for planar polynomial systems abstract in this paper we get some lower bounds for the number of critical periods of families of centers which are perturbations of the linear one we give a method which lets us prove that there are planar polynomial centers of degree l with at least 2 l 2 2 critical periods as well as study concrete families of potential reversible and lienard centers this last case is studied in more detail and we prove that the number of critical periods obtained with our approach does not increases with the order of the perturbation

critical periods of perturbations of reversible rigidly isochronous centers abstract in this paper we discuss bifurcation of critical periods in an m th degree time reversible system which is a perturbation of an n th degree homogeneous vector field with a rigidly isochronous center at the origin we present period bifurcation functions as integrals of analytic functions which depend on perturbation coefficients and reduce the problem of critical periods to finding zeros of a judging function this procedure gives not only the number of critical periods bifurcating from the period annulus but also the location of these critical periods applying our procedure to the case n m 2 we determine the maximum number of critical periods and their location to the case n m 3 we investigate the bifurcation of critical periods up to the first order in e and obtain the expression of the second period bifurcation function when the first one vanishes

on the number of critical points of the period in the study of subharmonic bifurcations from a planar hamiltonian system it is often assumed that the period of a one parameter family of periodic orbits of the unperturbed system is a monotone function of the parameter see e g hale and taboas s chow and hale es however this hypothesis is in many instances difficult to verify for example it is well known that every orbit in the first quadrant of the vollterra lotka equation is periodic it is only in recent years that the monotonicity of the periods of these periodic orbits has been shown hsu 9 and waldvogel 151 such problems also occur in the study of bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation u f u o uer 1 1 on some interval in 14 smoller and wasserman showed that if f u u u u u cc

on the number of critical periods for planar polynomial systems of arbitrary degree we construct a class of planar systems of arbitrary degree n having a reversible center at the origin and such that the number of critical periods on its period annulus grows quadratically with n as far as we know the previous results on this subject gave systems having linear growth

sequential decentralized parameter estimation under randomly observed fisher information we consider the problem of decentralized scalar parameter estimation using wireless sensor networks with gaussian noise specifically we propose a novel framework based on level triggered sampling a non uniform sampling strategy and sequential estimation the proposed estimator can be used as an asymptotically optimal fixed sample size decentralized estimator when the observed fisher information i e fisher information without expectation is deterministic as an alternative to the one shot estimators commonly found in the literature it can also be used as an asymptotically optimal sequential decentralized estimator when the observed fisher information is random we show that the optimal centralized estimator under gaussian noise which is the maximum likelihood estimator is characterized by two processes namely the observed fisher information ut and the observed correlation vt it is noted that vt is always random even when ut is not in the proposed scheme each sensor computes its local random processes and sends a single bit to the fusion center fc whenever the local random processes passes certain predefined levels the fc upon receiving a bit from a sensor updates its approximation to the corresponding global random process and accordingly its estimate the sequential estimation process terminates when ut or the approximation to it reaches a target value we provide an asymptotic analysis for the proposed estimator and the one based on conventional uniform in time sampling under both deterministic and random ut and determine the conditions under which they are asymptotically optimal consistent and asymptotically unbiased analytical results together with simulation results demonstrate the superiority of the proposed estimator based on level triggered sampling over the traditional decentralized estimator based on uniform sampling

distributed control of networked dynamical systems static feedback integral action and consensus this paper analyzes distributed control protocols for first and second order networked dynamical systems we propose a class of nonlinear consensus controllers where the input of each agent can be written as a product of a nonlinear gain and a sum of nonlinear interaction functions by using integral lyapunov functions we prove the stability of the proposed control protocols and explicitly characterize the equilibrium set we also propose a distributed proportional integral pi controller for networked dynamical systems the pi controllers successfully attenuate constant disturbances in the network we prove that agents with single integrator dynamics are stable for any integral gain and give an explicit tight upper bound on the integral gain for when the system is stable for agents with double integrator dynamics throughout the paper we highlight some possible applications of the proposed controllers by realistic simulations of autonomous satellites power systems and building temperature control

perturbed rigidly isochronous centers and their critical periods abstract this paper investigates the bifurcation of critical periods from a cubic rigidly isochronous center under any small polynomial perturbations of degree n it proves that for n 3 4 and 5 there are at most 2 and 4 critical periods induced by periodic orbits of the unperturbed cubic system respectively and in each case this upper bound is sharp moreover for any n 5 there are at most n 1 2 critical periods induced by periodic orbits of the unperturbed cubic system an example is given to show that the upper bound in the case of n 11 can be reached

the period function for quadratic integrable systems with cubic orbits this paper is concerned with the monotonicity of the period function for families of quadratic systems with centers whose orbits lie on cubic planar curves it is proved that each such system has a period function with at most one critical point

bifurcation of critical periods of polynomial systems abstract we describe a general approach to studying bifurcations of critical periods based on a complexification of the system and algorithms of computational algebra using this approach we obtain upper bounds on the number of critical periods of several families of cubic systems in some cases we overcome the problem of nonradicality of a relevant ideal by moving it to a subalgebra generated by invariants of a group of linear transformations

the problem of center for resonant singular points of polynomial vector fields we generalize the notion of center to the case of a p q resonant singular point of a polynomial vector field in 2 and to some other situations resonant node saddle node non elementary singular point of vector field and resonant fixed point of one dimensional complex diffeomorphism we show some similarities and differences with the classical case in particular the analogue of bautin s theorem does not hold four small amplitude limit cycles can bifurcate from the center after quadratic perturbation of a quadratic vector field with resonant center

parametric frugal sensing of power spectra for moving average models wideband spectrum sensing is a fundamental component of cognitive radio and other applications a novel frugal sensing scheme was recently proposed as a means of crowdsourcing the task of spectrum sensing using a network of scattered low end sensors transmitting randomly filtered power measurement bits to a fusion center a non parametric approach to spectral estimation was adopted to estimate the ambient power spectrum here a parametric spectral estimation approach is considered within the context of frugal sensing assuming a moving average ma representation for the signal of interest the problem of estimating admissible ma parameters and thus the ma power spectrum from single bit quantized data is formulated this turns out being a non convex quadratically constrained quadratic program qcqp which is np hard in general approximate solutions can be obtained via semi definite relaxation sdr followed by randomization but this rarely produces a feasible solution for this particular kind of qcqp a new sequential parametric convex approximation spca method is proposed for this purpose which can be initialized from an infeasible starting point and yet still produce a feasible point for the qcqp when one exists with high probability simulations not only reveal the superior performance of the parametric techniques over the globally optimum solutions obtained from the non parametric formulation but also the better performance of the spca algorithm over the sdr technique

the criticality of centers of potential systems at the outer boundary abstract the number of critical periodic orbits that bifurcate from the outer boundary of a potential center is studied we call this number the criticality at the outer boundary our main results provide sufficient conditions in order to ensure that this number is exactly 0 and 1 we apply them to study the bifurcation diagram of the period function of x y x x 1 p x 1 q y with q p this family was previously studied for q 1 by y miyamoto and k yagasaki

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

bifurcation of critical periods for plane vector fields on etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l origine pour toutes les valeurs d un parametre r n on definit la fonction periode p on determine les points critiques de cette fonction pres du centre avec comme parametre de bifurcation

finiteness for critical periods of planar analytic vector fields limburgs univ centrum dept math b 3610 diepenbeek belgium chicone c univ missouri dept math columbia mo 65211

the period function of classical lienard equations in this paper we study the number of critical points that the period function of a center of a classical lienard equation can have centers of classical lienard equations are related to scalar differential equations x x f x x 0 with f an odd polynomial let us say of degree 2 1 we show that the existence of a finite upperbound on the number of critical periods only depending on the value of can be reduced to the study of slow fast lienard equations close to their limiting layer equations we show that near the central system of degree 2 1 the number of critical periods is at most 2 2 we show the occurrence of slow fast lienard systems exhibiting 2 2 critical periods elucidating a qualitative process behind the occurrence of critical periods it all provides evidence for conjecturing that 2 2 is a sharp upperbound on the number of critical periods we also show that the number of critical periods multiplicity taken into account is always even

transients of platoons with asymmetric and different laplacians abstract we consider an asymmetric control of platoons of identical vehicles with nearest neighbor interaction recent results show that if the vehicle uses different asymmetries for position and velocity errors the platoon has a short transient and low overshoots in this paper we investigate the properties of vehicles with friction to achieve consensus an integral part is added to the controller making the vehicle a third order system we show that the parameters can be chosen so that the platoon behaves as a wave equation with different wave velocities simulations suggest that our system has a better performance than other nearest neighbor scenarios moreover an optimization based procedure is used to find the controller properties

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

centers with equal period functions abstract we give a sufficient condition for systems with symmetries to have periodic solutions with equal periods we show that the main result can be applied both to hamiltonian and to non hamiltonian systems we apply the main results to produce planar centers with equal period functions

distributed state estimation with dimension reduction preprocessing system state estimation relies heavily on the measurements with the advance of sensing technology the ability to measure is no longer a bottleneck in many systems and more and more researchers now focus on the rich information setting i e big data however although information never hurts it does not help unconditionally how to make the most of it depends on whether we can process the data efficiently in some systems the inherent constraint such as the bandwidth and cost makes it necessary to reduce the dimension of the measurement before further processing the problem that the raw measurements are first preprocessed to reduce size and then used for estimation is addressed in this paper it is shown that there is a lower bound on the size of the preprocessed data such that if the size is beyond the bound there exists a closed form estimator design that the linear minimum mean square estimation can be obtained moreover we propose an algorithm that is guaranteed to converge to a stationary point to design an estimator in the conditions that the lower bound cannot be reached besides convergence the proposed algorithm guarantees bounded performance loss compared with the global optimal solution under some additional conditions finally simulation results in three different applications are shown to demonstrate the effectiveness of the proposed algorithm

passivity as a design tool for group coordination we pursue a group coordination problem where the objective is to steer the differences between output variables of the group members to a prescribed compact set to stabilize this set we study a class of feedback rules that are implementable with local information available to each member when the information flow between neighboring members is bidirectional we show that the closed loop system exhibits a special interconnection structure which inherits the passivity properties of its components by exploiting this structure we develop a passivity based design framework which results in a broad class of feedback rules that encompass as special cases some of the existing formation stabilization and group agreement designs in the literature the passivity approach offers additional design flexibility compared to these special cases and systematically constructs a lurie type lyapunov function for the closed loop system we further study the robustness of these feedback laws in the presence of a time varying communication topology and present a persistency of excitation condition which allows the interconnection graph to lose connectivity pointwise in time as long as it is established in an integral sense

first derivative of the period function with applications abstract given a centre of a planar differential system we extend the use of the lie bracket to the determination of the monotonicity character of the period function as far as we know there are no general methods to study this function and the use of commutators and lie bracket was restricted to prove isochronicity we give several examples and a special method which simplifies the computations when a first integral is known

distributed estimation and quantization an algorithm is developed for the design of a nonlinear n sensor distributed estimation system subject to communication and computation constraints the algorithm uses only bivariate probability distributions and yields locally optimal estimators that satisfy the required system constraints it is shown that the algorithm is a generalization of the classical lloyd max results

distributed estimation using reduced dimensionality sensor observations we derive linear estimators of stationary random signals based on reduced dimensionality observations collected at distributed sensors and communicated to a fusion center over wireless links dimensionality reduction compresses sensor data to meet low power and bandwidth constraints while linearity in compression and estimation are well motivated by the limited computing capabilities wireless sensor networks are envisioned to operate with and by the desire to estimate random signals from observations with unknown probability density functions in the absence of fading and fusion center noise ideal links we cast this intertwined compression estimation problem in a canonical correlation analysis framework and derive closed form mean square error mse optimal estimators along with coordinate descent suboptimal alternatives that guarantee convergence at least to a stationary point likewise we develop estimators based on reduced dimensionality sensor observations in the presence of fading and additive noise at the fusion center nonideal links performance analysis and corroborating simulations demonstrate the merits of the novel distributed estimators relative to existing alternatives

exponential rate of convergence for lloyd s method i for a random variable with finite second moment and log concave density a unique quantizer exists which produces the minimum expected encoding error using squared error distortion an algorithm given by lloyd lloyd s method i yields a sequence of quantizers which converges to the optimum quantizer using results of fleischer it is shown that the convergence takes place exponentially fast if the logarithm of the density is not piecewise affine as a consequence the number of iterations of lloyd s algorithm needed to obtain the optimum distortion correct to n decimal places approaches infinity no faster than linearly in n another consequence is that if the output of a stationary information source at each time is distributed according to the given density the source output can be encoded at each time i using the quantizer obtained on the i th iteration of lloyd s method obtaining the same asymptotic behavior one would have obtained if the optimum quantizer had been used to encode at each time

the period in the lotka volterra system is monotonic abstract periodic solutions in a class of hamiltonian systems with one degree of freedom containing the lotka volterra system are considered their period is represented as an integral appropriate for numerical evaluation as well as for theoretical investigations the monotonicity of the period in the lotka volterra system follows as a simple consequence

distributed control design for spatially interconnected systems this paper deals with analysis synthesis and implementation of distributed controllers designed for spatially interconnected systems we develop a state space framework for posing problems of this type and focus on systems whose model is spatially discrete in this paper analysis and synthesis results are developed for this class of systems using the l sub 2 induced norm as the performance criterion the results are stated in terms of linear matrix inequalities and are thus readily amenable to computation a special implementation of the resulting controllers is presented which is particularly attractive for distributed operation of the controller several examples are provided to further illustrate the application of the results

non uniform continuity of periodic holm staley b family of equations abstract we consider a family of non evolutionary partial differential equations known as the holm staley b family which includes the integrable camassa holm and degasperis procesi equations we show that the solution map is not uniformly continuous the proof relies on a construction of smooth periodic traveling waves with small amplitude

estimating the number of limit cycles in polynomials systems abstract we describe a method based on algorithms of computational algebra for obtaining an upper bound for the number of limit cycles bifurcating from a center or a focus of polynomial vector field we apply it to a cubic system depending on six parameters and prove that in the generic case at most six limit cycles can bifurcate from any center or focus at the origin of the system

the critical points of the period function of x x2 x α x 1 0 0 α 1 on etudie les points critiques de la fonction periode des solutions periodiques de l equation x x 2 x x 1 0 0 2 1 on montre que le nombre de ces points critiques est borne

the cyclicity of quadratic reversible systems with a center of genus one and non morsean point in this paper we investigate a quadratic reversible system with non morsean point it is proved that the cyclicity of the period annulus under quadratic perturbations is less or equal to four

quadratic perturbations of quadratic codimension four centers abstract we study the stratum in the set of all quadratic differential systems x p 2 x y y q 2 x y with a center known as the codimension four case q 4 it has a center and a node and a rational first integral the limit cycles under small quadratic perturbations in the system are determined by the zeros of the first poincare pontryagin melnikov integral i we show that the orbits of the unperturbed system are elliptic curves and i is a complete elliptic integral then using picard fuchs equations and the petrov s method based on the argument principle we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center

low complexity methods for estimation after parameter selection statistical inference of multiple parameters often involves a preliminary parameter selection stage the selection stage has an impact on subsequent estimation for example by introducing a selection bias the post selection maximum likelihood psml estimator is shown to reduce the selection bias and the post selection mean squared error psmse compared with conventional estimators such as the maximum likelihood ml estimator however the computational complexity of the psml is usually high due to the multi dimensional exhaustive search for a global maximum of the post selection log likelihood psll function moreover the psll involves the probability of selection that in general does not have an analytical form in this paper we develop new low complexity post selection estimation methods for a two stage estimation after parameter selection architecture the methods are based on implementing the iterative maximization by parts mbp approach which is based on the decomposition of the psll function into easily optimized and complicated parts the proposed second best psml method applies the mbp psml algorithm with a pairwise probability of selection between the two highest ranked parameters w r t the selection rule the proposed sa psml method is based on using stochastic approximation sa and monte carlo integrations to obtain a non parametric estimation of the gradient of the probability of selection and then applying the mbp psml algorithm on this approximation for low complexity performance analysis we develop the empirical post selection cram acute text e r rao type lower bound simulations demonstrate that the proposed post selection estimation methods are tractable and reduce both the bias and the psmse compared with the ml estimator while only requiring moderate computational complexity