bifurcations of subharmonic solutions in reversible systems soit f r r n r r n une application c periodique cut on dit que l equation differentielle x f t x est reversible s il existe un operateur symetrique s sur r n tel que s 2 i de telles equations differentielles sont aussi appelees systemes avec la propriete e par rapport a s on etudie les bifurcations des solutions sous harmoniques de tels systemes reversibles

infinite period bifurcation and global bifurcation branches branches of periodic solutions which exhibit the alternatives of the global hopf bifurcation theorem are calculated for two general systems of differential equations in the first a branch of solu

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

the hyperelliptic limit cycles of the lienard systems abstract for lienard systems x y y f m x y g n x with f m and g n real polynomials of degree m and n respectively in h zoladek algebraic invariant curves for the lienard equation trans amer math soc 350 1998 1681 1701 the author showed that if m 3 and m 1 n 2 m there always exist lienard systems which have a hyperelliptic limit cycle llibre and zhang j llibre xiang zhang on the algebraic limit cycles of lienard systems nonlinearity 21 2008 2011 2022 proved that the lienard systems with m 3 and n 5 have no hyperelliptic limit cycles and that there exist lienard systems with m 4 and 5 n 8 which do have hyperelliptic limit cycles so it is still an open problem to characterize the lienard systems which have an algebraic limit cycle in cases m 4 and m 1 n 2 m in this paper we will prove that there exist lienard systems with m 5 and m 1 n 2 m which have hyperelliptic limit cycles

the cyclicity of period annuli of a class of quintic hamiltonian systems abstract in this paper we prove that the cyclicity of period annuli for system d x d t y d y d t a x b x 3 c x 5 under perturbations of polynomials with degree n is not more than 54 n 13 taking into account the multiplicity where a b c r with c 0 such that the unperturbed systems have at least one center

global hopf bifurcation of two parameter flows the behavior of center indices as introduced by j mallet paret j yorke is analyzed for two parameter flows the integer sum of center indices along a one dimensional curve in parameter space is called the h index a nonzero h index implies global hopf bifurcation the index h is not a homotopy invariant this fact is due to the occurrence of stationary points with an algebraically double eigenvalue zero which we call b points to each b point we assign an integer b index such that the h index relates to the b indices by a formula such as occurs in the calculus of residues

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

on polynomial hamiltonian planar vector fields abstract we study the dynamical behaviour of polynomial hamiltonian planar vector fields particularly we analyze the structure of finite and infinite critical points and we obtain the best upper bound of the number of centers and of the number of saddles that a system of this type can exhibit depending on its degree

perturbation from an elliptic hamiltonian of degree four iii global centre abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a global centre it is proven that the least upper bound of the number of zeros of the related elliptic integral is four and this is a sharp one this result permits to prove the existence of lienard equations of type 3 2 with a quadruple limit cycle with both a triple and a simple limit cycle with two semistable limit cycles with one semistable and two simple limit cycles or with four simple limit cycles

the period function for hamiltonian systems with homogeneous nonlinearities abstract the paper deals with hamiltonian systems with homogeneous nonlinearities we prove that such systems have no isochronous centers that the period annulus of any of its centres is either bounded or the whole plane and that the period function associated to the origin has at most one critical point

perturbations from an elliptic hamiltonian of degree four abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector field with an elliptic hamiltonian of degree 4 exhibiting a cuspidal loop it is proven that the least upper bound for the number of zeros of the related elliptic integral is four and this upper bound is a sharp one this permits to prove the existence of lienard equations of type 3 2 with at least four limit cycles the paper also contains a complete result on the respective number of small and large limit cycles

a chebyshev criterion for abelian integrals we present a criterion that provides an easy sufficient condition in order for a collection of abelian integrals to have the chebyshev property this condition involves the functions in the integrand of the abelian integrals and can be checked in many cases in a purely algebraic way by using this criterion several known results are obtained in a shorter way and some new results which could not be tackled by the known standard methods can also be deduced

bounding the number of zeros of certain abelian integrals abstract in this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n abelian integrals to have at most n k 1 zeros counted with multiplicities this condition involves the functions in the integrand of the abelian integrals and it can be checked in many cases in a purely algebraic way

robust localization of nodes and time recursive tracking in sensor networks using noisy range measurements simultaneous localization and tracking slat in sensor networks aims to determine the positions of sensor nodes and a moving target in a network given incomplete and inaccurate range measurements between the target and each of the sensors one of the established methods for achieving this is to iteratively maximize a likelihood function ml of positions given the observed ranges which requires initialization with an approximate solution to avoid convergence towards local extrema this paper develops methods for handling both gaussian and laplacian noise the latter modeling the presence of outliers in some practical ranging systems that adversely affect the performance of localization algorithms designed for gaussian noise a modified euclidean distance matrix edm completion problem is solved for a block of target range measurements to approximately set up initial sensor target positions and the likelihood function is then iteratively refined through majorization minimization mm to avoid the computational burden of repeatedly solving increasingly large edm problems in time recursive operation an incremental scheme is exploited whereby a new target node position is estimated from previously available node target locations to set up the iterative ml initial point for the full spatial configuration the above methods are first derived under gaussian noise assumptions and modifications for laplacian noise are then considered analytically the main challenges to overcome in the laplacian case stem from the non differentiability of l1 norms that arise in the various cost functions simulation results show that the proposed algorithms significantly outperform existing localization methods in the presence of outliers while providing comparable performance for gaussian noise

cyclicity of periodic annulus and hopf cyclicity in perturbing a hyper elliptic hamiltonian system with a degenerate heteroclinic loop abstract in this paper we study the cyclicity of periodic annulus and hopf cyclicity in perturbing a quintic hamiltonian system the undamped system is hyper elliptic non symmetric with a degenerate heteroclinic loop which connects a hyperbolic saddle to a nilpotent saddle we rigorously prove that the cyclicity is 3 for periodic annulus when the weak damping term has the same degree as that of the associated hamiltonian system this result provides a positive answer to the open question whether the annulus cyclicity is 3 or 4 when the smooth polynomial damping term has degree n first a transformation based on the involution of the hamiltonian is introduced and then we analyze the coefficients involved in the bifurcation function to show that the hopf cyclicity is 2 n 1 3 further for piecewise smooth polynomial damping with a switching manifold at the y axis we consider the damping terms to have degrees l and n respectively and prove that the hopf cyclicity of the origin is 3 l 2 n 4 3 3 n 2 l 4 3 when l n n l

a unified proof on the weak hilbert 16th problem for n 2 abstract the weak hilbert 16th problem for n 2 was solved by horozov and iliev proc london math soc 69 1994 198 244 zhang and li adv in math 26 1997 445 460 gavrilov invent math 143 2001 449 497 and li and zhang nonlinearity 15 2002 1775 1992 by using different methods for different cases the aim of this paper is to give a unified and easier proof for all cases the proof is restricted to the real domain combines geometric and analytical methods and uses deformation arguments

limit cycles of some polynomial lienard systems abstract in this paper we study the number of limit cycles of some polynomial lienard systems by the melnikov functions and the methods of hopf homoclinic and heteroclinic bifurcation theory we prove that h 2 5 3 h 4 5 5 h 6 5 10 h 8 5 10

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

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

explicit non algebraic limit cycles for polynomial systems we consider a system of the form email protected p n x y xr m x y email protected q n x y yr m x y where p n x y q n x y and r m x y are homogeneous polynomials of degrees n n and m respectively with n

bifurcation in classical bipolar transistor oscillator circuits the results of a systematic asymptotic analysis of two classical oscillator circuits are presented in this paper both circuits are composed of passive components and a single bipolar transistor they are each described by a third order system of nonlinear ordinary differential equations that is singularly perturbed in the limit varepsilon to 0 here varepsilon is the inverse of the circuit s q the two timing method is used to construct the asymptotic behavior of the solutions as varepsilon to 0 these solutions are shown to depend upon amplitudes that evolve on a slow time scale according to amplitude equations these amplitude equations are analyzed and the results are physically interpreted

on the persistence of quasi periodic invariant tori for double hopf bifurcation of vector fields abstract we analyze the persistence of quasi periodic invariant 2 and 3 tori for the double hopf hopf hopf bifurcation by using the kam method we prove that in a sufficiently small neighborhood of the bifurcation point the full system has quasi periodic 2 tori for most of the parameter sets where its truncated normal form possesses 2 tori under appropriate conditions we obtain that the full system also has quasi periodic 3 tori for most parameters near the hopf bifurcation curve of its truncated normal form and along the direction of the bifurcation and these 3 tori bifurcate from invariant 2 tori we also give concrete formulas on the existence of quasi periodic invariant 2 and 3 tori which are based on coefficients of the truncated normal form

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

oscillatory coexistence in the chemostat a codimension two unfolding a nontrivial branch of stable oscillatory solutions of a two predator one prey competition model is found in the positive octant by a multiparameter bifurcation analysis the branch of solutions is global in that it is a continuous branch connecting the oscillations in each of the one predator one prey planes this result gives analytical verification of the numerical observations of other authors that a branch of nonplanar oscillations crosses the positive octant in certain parameter ranges

algebraic limit cycles in polynomial systems of differential equations using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4 5 and 6 we also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3 moreover we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3 as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4

on the period function of reversible quadratic centers with their orbits inside quartics abstract this paper is concerned with the monotonicity of the period function for a class of reversible quadratic centers with their orbits inside quartics it is proved that such a system has a period function with at most one critical point

gaussian process regression for sensor networks under localization uncertainty in this paper we formulate gaussian process regression with observations under the localization uncertainty due to the resource constrained sensor networks in our formulation effects of observations measurement noise localization uncertainty and prior distributions are all correctly incorporated in the posterior predictive statistics the analytically intractable posterior predictive statistics are proposed to be approximated by two techniques viz monte carlo sampling and laplace s method such approximation techniques have been carefully tailored to our problems and their approximation error and complexity are analyzed simulation study demonstrates that the proposed approaches perform much better than approaches without considering the localization uncertainty properly finally we have applied the proposed approaches on the experimentally collected real data from a dye concentration field over a section of a river and a temperature field of an outdoor swimming pool to provide proof of concept tests and evaluate the proposed schemes in real situations in both simulation and experimental results the proposed methods outperform the quick and dirty solutions often used in practice

limit cycle bifurcations near a double homoclinic loop with a nilpotent saddle of order m abstract in this paper we study the explicit expansion of the first order melnikov function near a double homoclinic loop passing through a nilpotent saddle of order m in a near hamiltonian system for any positive integer m m 1 we derive the formulas of the coefficients in the expansion which can be used to study the limit cycle bifurcations for near hamiltonian systems in particular for m 2 we use the coefficients to consider the limit cycle bifurcations of general near hamiltonian systems and give the existence conditions for 10 11 13 15 and 16 11 13 and 16 respectively limit cycles in the case that the homoclinic loop is of cuspidal type smooth type respectively and their distributions as an application we consider a near hamiltonian system with a nilpotent saddle of order 2 and obtain the lower bounds of the maximal number of limit cycles

merging of resonance tongues abstract this paper considers the resonance behavior found near a saddle node hopf bifurcation in a flow the global structure of resonance tongues is examined and it is shown that in the absence of certain symmetries different resonance tongues can merge with one another the position and shape of the resonance tongues are seen to be related to the presence of homoclinic and heteroclinic bifurcations of two saddle foci the implications of these results for the normal form analysis of the saddle node hopf bifurcation are discussed

isochronicity into a family of time reversible cubic vector fields in this work we study necessary and sufficient conditions for the existence of isochronous centers into a family of cubic time reversible systems this class of reversible systems is characterized by the existence of an inverse integrating factor which is a certain power of an invariant straight line

exact and approximate solutions of source localization problems we consider least squares ls approaches for locating a radiating source from range measurements which we call r ls or from range difference measurements rd ls collected using an array of passive sensors we also consider ls approaches based on squared range observations sr ls and based on squared range difference measurements srd ls despite the fact that the resulting optimization problems are nonconvex we provide exact solution procedures for efficiently computing the sr ls and srd ls estimates numerical simulations suggest that the exact sr ls and srd ls estimates outperform existing approximations of the sr ls and srd ls solutions as well as approximations of the r ls and rd ls solutions which are based on a semidefinite relaxation

perturbation from an elliptic hamiltonian of degree four iv figure eight loop the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a figure eight loop it is proved that the least upper bound of the number of zeros of the related elliptic integral is five and this is a sharp bound multiplicity taken into account moreover if restricting to the level curves inside a saddle loop or outside the figure eight loop the sharp upper bound is respectively two or four also the multiplicity of the zeros is at most four this is the last one in a series of papers on this subject the results of this paper together with j differential equations 176 2001 114 j differential equations 175 2001 209 j differential equations to be published largely finish the study of the cubic perturbations of the elliptic hamiltonians of degree four and presumably provide a complete description of the number and the possible configurations of limit cycles for cubic lienard equations with small quadratic damping as a special case we obtain a configuration of four limit cycles surrounding three singularities together with a small limit cycle which surrounds one of the singularities

perturbations of a class of hyper elliptic hamiltonian systems of degree seven with nilpotent singular points in this paper we study the number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 2 under the perturbations of 0 1 x 2 2 x 4 3 x 6 y y where 0 1 and i r i 0 1 2 3 we find a special transformation such that four generating elements are reduced to special three ones and then the algebraic method can be used the bounds and sharp bounds are obtained

perturbations from an elliptic hamiltonian of degree four i saddle loop and two saddle cycle abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree 4 and especially to the study of the related elliptic integrals besides some general results the paper contains a complete treatment of the saddle loop case and the two saddle cycle case it is proven that the related elliptic integrals have at most two zeros respectively one zero the multiplicity taken into account the bifurcation diagram of the zeros is also obtained

synchronization and fluctuations coupling a finite number of stochastic units it is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit infinite number of elements in fact since the early work of kuramoto mean field theory has been used to analyze this transition in contrast work that directly deals with finite arrays is relatively scarce in the context of synchronization and yet it is worth noting that finite number effects should be seriously taken into account since in general the limits n where n is the number of units and t where t is time do not commute mean field theory implements the particular choice first n and then t here we analyze an ensemble of three state coupled stochastic units which has been widely studied in the thermodynamic limit we formally address the finite n problem by deducing a fokker planck equation that describes the system we compute the steady state solution of this fokker planck equation that is finite n but t we use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions

on melnikov functions of a homoclinic loop through a nilpotent saddle for planar near hamiltonian systems abstract the first order melnikov function of a homoclinic loop through a nilpotent saddle for general planar near hamiltonian systems is considered the asymptotic expansion of this melnikov function and formulas for its first coefficients are given the number of limit cycles which appear near the homoclinic loop is discussed by using the asymptotic expansion of the first order melnikov function an example is presented as an application of the main results

on the 16th hilbert problem for limit cycles on nonsingular algebraic curves abstract we give an upper bound for the maximum number n of algebraic limit cycles that a planar polynomial vector field of degree n can exhibit if the vector field has exactly k nonsingular irreducible invariant algebraic curves additionally we provide sufficient conditions in order that all the algebraic limit cycles are hyperbolic we also provide lower bounds for n

applications of second order cone programming in a second order cone program socp a linear function is minimized over the intersection of an affine set and the product of second order quadratic cones socps are nonlinear convex problems that include linear and convex quadratic programs as special cases but are less general than semidefinite programs sdps several efficient primaldual interior point methods for socp have been developed in the last few years after reviewing the basic theory of socps we describe general families of problems that tan be recast as socps these include robust linear programming and robust leastsquares problems problems involving sums or maxima of norms or with convex hyperbolic constraints we discuss a variety of engineering applications such as filter design antenna array weight design truss design and grasping forte optimization in robotics we describe an efficient primaldual interior point method for solving socps which shares many of the features of primaldual interior point methods for linear program

isochronicity for several classes of hamiltonian systems abstract in this paper we study isochronous centers of analytic hamiltonian systems giving special attention to the polynomial case we first revisit the potential systems and we show the connection between isochronicity and involutions we then study a more general system namely the ones associated to hamiltonians of the form h x y a x b x y c x y 2 as an application we classify the cubic polynomial hamiltonian isochronous centers and we give examples of nontrivial and nonglobal polynomial hamiltonian isochronous centers

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x is 2 taking into account their multiplicities for h 0 1 6 and this upper bound is a sharp one this implies that the number of limit cycles bifurcated from periodic orbits in the vicinity of the center of the unperturbed system for e 0 inside an eye figure loop is less than or equal to 2

bifurcations of limit cycles for a quintic hamiltonian system with a double cuspidal loop in this work we consider the number of limit cycles that can bifurcate from periodic orbits located inside a double cuspidal loop of the quintic hamiltonian vector field x h y x x 3 x 2 1 y under small perturbations of the form e a bx 2 cx 4 y y where 0 e 1 and a b c are real constants using picard fuchs equations for related abelian integrals asymptotic expansion of these integrals about critical level curves of h and some geometric properties of the curves defined by ratios of two especial integrals we show that the least upper bound for the number of limit cycles appeared in this bifurcation is two

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

bifurcations of limit cycles in a z4 equivariant planar polynomial vector field of degree 7 the present paper intends to explore the number and distribution of limit cycles for a class of perturbed z 4 equivariant planar hamiltonian system of degree 7 the bifurcation theory of planar dynamical systems and the method of detection functions are applied to study the given perturbed system by controlling the perturbation parameters an example of a special perturbed z 4 equivariant vector field having 41 limit cycles is found the distribution of the above 41 limit cycles is also given the results are significant to the study of weaken hilbert s 16th problem and to the acquaintance of the characteristics of higher order equivariant vector fields

exact bound on the number of zeros of abelian integrals for two hyper elliptic hamiltonian systems of degree 4 abstract this paper concerns the exact bound on the number of zeros of abelian integrals associated with two hyper elliptic hamiltonian systems of degree 4 the upper and lower bounds for the two systems have been obtained in several previous works however the sharp bounds are still unknown in this paper we provide a proof to show that the exact bound is 3 for both systems the basic idea of our method is to bound the parameter space in r 3 to obtain two parameter sets which might yield maximal 4 zeros of abelian integrals corresponding to the two sets further we show that the existence of 4 zeros on the two sets is not possible and thus the sharp bound is 3

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

uniqueness of algebraic limit cycles for quadratic systems abstract we know five different families of algebraic limit cycles in quadratic systems one of degree 2 and four of degree 4 moreover if there are other families of algebraic limit cycles for quadratic systems then their degrees must be larger than 4 it is known that if a quadratic system has an algebraic limit cycle of degree 2 then this is the unique limit cycle of the system the first main goal of this paper is to prove that if a quadratic system has an algebraic limit cycle of degree 4 then this is the unique limit cycle of the system for the yablonskii and filiptsov algebraic limit cycles the problem of proving their uniqueness has been open since 1966 and 1973 respectively the second main goal is to provide sufficient conditions in order that all the limit cycles of a quadratic system to be algebraic

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

planar polynomial vector fields having first integrals and algebraic limit cycles with the help of abel differential equations we obtain a new class of darboux integrable planar polynomial differential systems which have degenerate infinity moreover such integrable systems may have algebraic limit cycles also we present the explicit expressions of these algebraic limit cycles for quintic systems

global hopf bifurcation in a simple climate model the mathematical structure of a simple climate model is investigated the model is governed by a system of two nonlinear autonomous differential equations for the evolution in time of global temperature t and meridional ice sheet extent l the system s solutions are studied by a combination of qualitative reasoning with explicit calculations both analytical and numerical for plausible values of the physical parameters a branch of periodic solutions obtains which is both orbitally and structurally stable the amplitude of the stable periodic solutions in t and l correspond roughly to that obtained from proxy records of quaternary glaciation cycles the period of these solutions increases along the branch until it becomes infinite while the amplitude of the limiting solution is finite the limiting solution is a homoclinic orbit formed by the reconnecting separatrix of a saddle the exchange of stability between the branch of periodic solutions and the steady solution from which it arises is studied

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

distributed sensor localization in random environments using minimal number of anchor nodes the paper introduces diloc a distributed iterative algorithm to locate m sensors with unknown locations in rm m ges 1 with respect to a minimal number of m 1 anchors with known locations the sensors and anchors nodes in the network exchange data with their neighbors only no centralized data processing or communication occurs nor is there a centralized fusion center to compute the sensors locations diloc uses the barycentric coordinates of a node with respect to its neighbors these coordinates are computed using the cayley menger determinants i e the determinants of matrices of internode distances we show convergence of diloc by associating with it an absorbing markov chain whose absorbing states are the states of the anchors we introduce a stochastic approximation version extending diloc to random environments i e when the communications among nodes is noisy the communication links among neighbors may fail at random times and the internodes distances are subject to errors we show a s convergence of the modified diloc and characterize the error between the true values of the sensors locations and their final estimates given by diloc numerical studies illustrate diloc under a variety of deterministic and random operating conditions

bifurcations of mode equations spin waves abstract we study the bifurcations of the spin wave equations that describe the parametric pumping of collective modes in magnetic media we discuss the following dynamical phenomena i sequential excitation of modes via zero eigenvalue bifurcations ii hopf bifurcations followed or not by feigenbaum cascades of period doublings iii local and global homoclinic phenomena two new organizing centers for routes to chaos are identified in the classification given by guckenheimer and holmes one is the codimension two local bifurcation with one pair of imaginary eigenvalues and a zero eigenvalue to which many dynamical consequences are known the second global homoclinic bifurcations associated to splitting of separatrices in the limit where the system can be considered a hamiltonian subjected to weak dissipation and forcing

a codimension two bifurcation with a third order picard fuchs equation abstract using a third order picard fuchs equation we show that a certain two parameter family of planar vectorfields for parameter values in a certain cone has a unique limit cycle which is born from a hopf bifurcation and dies in a saddle connection this removes a superfluous hypothesis in theorem 3 2 chapter 13 of s n chow and j k hale methods of bifurcation theory springer verlag new york 1982

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

the perturbations of a class of hyper elliptic hamilton systems with a double homoclinic loop through a nilpotent saddle abstract in this paper we consider the upper bounds of a number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 under the perturbations of 0 1 x 2 x 2 3 x 3 4 x 4 y y where 0 1 and i r i 0 4 the unperturbed system has a double homoclinic loop with a nilpotent saddle of order 1 the sharp upper bounds are obtained for each of the cases of 1 4 0 1 3 0 2 3 0 and 3 4 0 when abelian integrals are defined in the bounded period annuli

relationships between limit cycles and algebraic invariant curves for quadratic systems abstract algebraic limit cycles for quadratic systems started to be studied in 1958 up to now we know 7 families of quadratic systems having algebraic limit cycles of degree 2 4 5 and 6 here we present some new results on the limit cycles and algebraic limit cycles of quadratic systems these results provide sometimes necessary conditions and other times sufficient conditions on the cofactor of the invariant algebraic curve for the existence or nonexistence of limit cycles or algebraic limit cycles in particular it follows from them that for all known examples of algebraic limit cycles for quadratic systems those cycles are unique limit cycles of the system

majorization minimization algorithms in signal processing communications and machine learning this paper gives an overview of the majorization minimization mm algorithmic framework which can provide guidance in deriving problem driven algorithms with low computational cost a general introduction of mm is presented including a description of the basic principle and its convergence results the extensions acceleration schemes and connection to other algorithmic frameworks are also covered to bridge the gap between theory and practice upperbounds for a large number of basic functions derived based on the taylor expansion convexity and special inequalities are provided as ingredients for constructing surrogate functions with the pre requisites established the way of applying mm to solving specific problems is elaborated by a wide range of applications in signal processing communications and machine learning

a criterion for determining the monotonicity of the ratio of two abelian integrals abstract a new method to determine the monotonicity of the ratio of two abelian integrals is given we find two criterion functions defined directly by the functions which appear in the abelian integrals the monotonicities of the criterion functions imply the monotonicity of the ratio of the abelian integrals by using this criterion several known results can be deduced easier and some new results are obtained

on the number of limit cycles in small perturbations of a class of hyper elliptic hamiltonian systems with one nilpotent saddle abstract in this paper we make a complete study on small perturbations of hamiltonian vector field with a hyper elliptic hamiltonian of degree five which is a lienard system of the form x y y q 1 x e y q 2 x with q 1 and q 2 polynomials of degree respectively 4 and 3 it is shown that this system can undergo degenerated hopf bifurcation and poincare bifurcation which emerges at most three limit cycles in the plane for sufficiently small positive e and the limit cycles can encompass only an equilibrium inside i e the configuration 3 0 of limit cycles can appear for some values of parameters where 3 0 stands for three limit cycles surrounding an equilibrium and no limit cycles surrounding two equilibria

local birth of homoclinic chaos abstract local generation of chaotic behaviours associated with a sil nikov homoclinic orbit is studied near the codimension two bifurcation at the double stationary oscillatory instability of eigenvalues 0 we develop a perturbative calculation of the sil nikov homoclinic orbit and of its bifurcation locus starting from a conservative vector field flat nonanalytic behaviours due to non axisymmetric perturbations of the normal form are discussed and we show that the codimension one locus of sil nikov homoclinicity is unavoidable near bicriticality under certain conditions the results are applied to rossler s model quasiperiodic regimes born at the double instability are also studied and illustrated in the rossler model

non nested configuration of algebraic limit cycles in quadratic systems this work deals with algebraic limit cycles of planar polynomial differential systems of degree two more concretely we show among other facts that a quadratic vector field cannot possess two non nested algebraic limit cycles contained in different irreducible invariant algebraic curves

multiple limit cycles of some strongly nonlinear lienard van der pol oscillator in this paper we investigate limit cycles of some lienard van der pol oscillator the system has a double homoclinic loop and two cuspidal loops if the damping effect has vanished by the related melnikov function theory and bifurcation theories the limit cycles near the singular circle and center with their distribution are found the number of limit cycles obtained also reveals that some recent results on the lower bounds of the maximal number of limit cycles bifurcated from this kind of systems can be improved

simple and fast convex relaxation method for cooperative localization in sensor networks using range measurements we address the sensor network localization problem given noisy range measurements between pairs of nodes we approach the nonconvex maximum likelihood formulation via a known simple convex relaxation we exploit its favorable optimization properties to the full to obtain an approach that is completely distributed has a simple implementation at each node and capitalizes on an optimal gradient method to attain fast convergence we offer a parallel but also an asynchronous flavor both with theoretical convergence guarantees and iteration complexity analysis experimental results establish leading performance our algorithms top the accuracy of a comparable state of the art method by one order of magnitude using one order of magnitude fewer communications

interaction of steady and periodic bifurcating modes with imperfection effects in reaction diffusion systems asymptotic solutions for two dimensional reaction diffusion systems are obtained under suitable conditions a trivial solution independent of both time and space is assumed to exist and the linearized system about this solution is considered the analysis is restricted to the case where this linearized system possesses one real eigenvalue and a complex pair of eigenvalues all with real part approximately equal to zero steady periodic and doubly periodic bifurcating solutions are found with special attention given to the effects of imperfection in one of the parameters

on the 16th hilbert problem for algebraic limit cycles abstract for a polynomial planar vector field of degree n 2 with generic invariant algebraic curves we show that the maximum number of algebraic limit cycles is 1 n 1 n 2 2 when n is even and n 1 n 2 2 when n is odd furthermore these upper bounds are reached

stability and transitions of the second grade poiseuille flow abstract in this study we consider the stability and transitions for the poiseuille flow of a second grade fluid which is a model for non newtonian fluids we restrict our attention to perturbation flows in an infinite pipe with circular cross section that are independent of the axial coordinate we show that unlike the newtonian 0 case in the second grade model 0 case the time independent base flow exhibits transitions as the reynolds number r exceeds the critical threshold r c 8 505 1 2 where is a material constant measuring the relative strength of second order viscous effects compared to inertial effects at r r c we find that the transition is either continuous or catastrophic and a small amplitude time periodic flow with 3 fold azimuthal symmetry bifurcates the time period of the bifurcated solution tends to infinity as r tends to r c our numerical calculations suggest that for low values the system prefers a catastrophic transition where the bifurcation is subcritical we also show that there is a reynolds number r e with r e r c such that for r r e the base flow is globally stable and attracts any initial disturbance with at least exponential speed we show that the gap between r e and r c vanishes quickly as increases

distributed maximum likelihood sensor network localization we propose a class of convex relaxations to solve the sensor network localization problem based on a maximum likelihood ml formulation this class as well as the tightness of the relaxations depends on the noise probability density function pdf of the collected measurements we derive a computational efficient edge based version of this ml convex relaxation class and we design a distributed algorithm that enables the sensor nodes to solve these edge based convex programs locally by communicating only with their close neighbors this algorithm relies on the alternating direction method of multipliers admm it converges to the centralized solution it can run asynchronously and it is computation error resilient finally we compare our proposed distributed scheme with other available methods both analytically and numerically and we argue the added value of admm especially for large scale networks

classification and unfoldings of 1 2 resonant hopf bifurcation in this paper we study the bifurcations of periodic solutions from an equilibrium point of a differential equation whose linearization has two pairs of simple pure imaginary complex conjugate eigenvalues which are in 1 2 ratio this corresponds to a hopf hopf mode interaction with 1 2 resonance as occurs in the context of dissipative mechanical systems using an approach based on liapunov schmidt reduction and singularity theory we give a framework in which to study these problems and their perturbations in two cases no distinguished parameter and one distinguished bifurcation parameter we give a complete classification of the generic cases and their unfoldings

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop ii abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x for h 1 6 is three and for h 0 is four counted with multiplicity for all parameters and this implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for e 0 outside an eye figure loop is less than or equal to three

sungular hopf bifurcation to relaxation oscillations this paper is a continuation of our study of singular hopf bifurcations to relaxation oscillations siam journal on applied mathematics 46 1986 pp 721 739 in our previous analysis a system of two first order nonlinear equatins is investigated and it is shown how the bifurcation problem can be reduced to a weakly perturbed conservative system of two first order equations in this paper a method based on analyzing nearly conservative quantities was developed to find the bifurcation diagram of the periodic solutions emerging from singular hopf bifurcation points specifically the subcritical hopf bifurcation of the fitzhugh nagumo equations is analyzed the extension from the earlier bifurcation analysis is achieved by performing a perturbation analysis from the first integral of the leading order nonlinear oscillator at a critical value of a parameter the first integral exists up to an additional order in the perturbed equation this condition permits the resolution of features determined by h