centers and isochronous centers for generalized quintic systems in this paper we classify the centers and the isochronous centers of certain polynomial differential systems in r 2 of degree d 5 odd that in complex notation are z i z z z d 5 2 a z 5 b z 4 z c z 3 z 2 d z 2 z 3 e z z 4 f z 5 where z x i y r and a b c d e f c note that if d 5 we obtain the class of polynomial differential systems in the form of a linear system with homogeneous polynomial nonlinearities of degree 5 due to the huge computations required for computing the necessary and sufficient conditions for the characterization of the centers and isochronous centers our study uses algorithms of computational algebra based on the grobner basis theory and on modular arithmetics

performance bounds for noncoherent detection under brownian phase noise the performance of noncoherent detection of orthogonal phase noise impaired signals in the presence of additive white gaussian noise is considered the authors present a novel class of upper and lower bounds on the error probability of a binary hypothesis test comprising quadratic forms of the additive noise and of the filtered noisy phase signal plus noise filtering the noisy phase signal gives rise to bounded nonlinear functionals of the brownian motion sample path the exact statistics of which are unknown the classical theory of chebyshev systems is utilized to solve the limiting values of the required stochastic expectations based on the availability of the corresponding generalized moments the resulting multidimensional moment bounds constitute the tightest possible error bounds for the given set of generalized moments and require only modest computational efforts the theory is applicable to assess the design and performance of optical heterodyne systems and is most suitable for coded systems employing hard decisions for which the obtained bounds are remarkably tight

noncoherent detection of coherent lightwave signals corrupted by phase noise waves are treated that modulate by either on off keying ook or binary frequency shift keying fsk and are further impaired by additive gaussian noise heterodyne detection of such a waveform produces an electronic bandpass signal which to ease demodulation in the presence of phase noise is noncoherently demodulated to extract the baseband pulse stream the treatment goes beyond previous bit error rate ber analyses of optical heterodyne receivers for ook and fsk first there is full adherence to the standard brownian motion model of phase noise also the receiver structure is formulated in such a way that the probability density function of the receiver output samples can be accurately determined this permits calculations of the additive noise and phase noise tolerable when achieving bit error rates as small as 10 sup 9 finally the study is comprehensive regarding the range of parameters explored filtering at an intermediate frequency if alone as well as if filtering plus postdetection low pass filtering is considered when the receiver parameters decision threshold for ook and if filter bandwidth are optimized large amounts of phase noise can be accommodated with only minor increases in required signal to noise ratio this is especially important when the bit rate is moderate compared to the laser linewidth

a class of polynomial planar vector fields with polynomial first integral abstract we give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity in the affirmative case our algorithm computes a minimal first integral in addition we solve the poincare problem for the class of systems which admit a polynomial first integral as above in the sense that the degree of the minimal first integral can be computed from the reduction of singularities of the corresponding vector field

the cyclicity of period annuli of a class of quadratic reversible systems with two centers abstract quadratic perturbations of a class of quadratic reversible systems are studied the associated systems can have at most four limit cycles in any compact region of the period annuli and if four limit cycles exist then their distribution is 3 1 in other words three of them are nested and surrounding one focus the other limit cycle is surrounding the different focus

calculation of mutual information for partially coherent gaussian channels with applications to fiber optics the mutual information between a complex valued channel input and its complex valued output is decomposed into four parts based on polar coordinates an amplitude term a phase term and two mixed terms numerical results for the additive white gaussian noise awgn channel with various inputs show that at high signal to noise ratio snr the amplitude and phase terms dominate the mixed terms for the awgn channel with a gaussian input analytical expressions are derived for high snr the decomposition method is applied to partially coherent channels and a property of such channels called spectral loss is developed spectral loss occurs in nonlinear fiber optic channels and it may be one effect that needs to be taken into account to explain the behavior of the capacity of nonlinear fiber optic channels

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

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

an algebraic approach to the classification of centers in polynomial lienard systems abstract we consider the second order lienard system x f x x g x 0 where f x and g x are polynomials which we rewrite in the equivalent two dimensional form x y y g x yf x we show that the local question of whether a critical point of this system is a center can be expressed in terms of global conditions on f and g using these results we give a simple classification of all such centers we also address the problem of the coexistence of centers foci and limit cycles for these systems systems with degenerate centers are also considered

darboux integrability and the inverse integrating factor abstract we mainly study polynomial differential systems of the form dx dt p x y dy dt q x y where p and q are complex polynomials in the dependent complex variables x and y and the independent variable t is either real or complex we assume that the polynomials p and q are relatively prime and that the differential system has a darboux first integral of the form h f 1 1 f p p exp h 1 g 1 n 1 1 exp h q g q n q q where the polynomials f i and g j are irreducible the polynomials g j and h j are coprime and the i and j are complex numbers when i 1 p and j 1 q prelle and singer proved that these systems have a rational integrating factor we improve this result as follows assume that h is a rational function which is not polynomial following to poincare we define the critical remarkable values of h then we prove that the system has a polynomial inverse integrating factor if and only if h has at most two critical remarkable values under some assumptions over the darboux first integral h we show first that the system has a polynomial inverse integrating factor and secondly that if the degree of the system is m the homogeneous part of highest degree of h is a multi valued function and the functions exp h j g j are exponential factors for j 1 q then the system has a polynomial inverse integrating factor of degree m 1 we also present versions of these results for real polynomial differential systems finally we apply these results to real polynomial differential systems having a darboux first integral and limit cycles or foci

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

classification of the centers their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities abstract in this paper we classify the centers the cyclicity of its hopf bifurcation and their isochronicity for the polynomial differential systems in r 2 of arbitrary degree d 3 odd that in complex notation z x i y can be written as z i z z z d 3 2 a z 3 b z 2 z c z z 2 d z 3 where r and a b c d c if d 3 we obtain the well known class of all polynomial differential systems of the form a linear system with cubic homogeneous nonlinearities

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

darboux integrating factors inverse problems abstract we discuss planar polynomial vector fields with prescribed darboux integrating factors in a nondegenerate affine geometric setting we establish a reduction principle which transfers the problem to polynomial solutions of certain meromorphic linear systems and show that the space of vector fields with a given integrating factor modulo a subspace of explicitly known standard vector fields has finite dimension for several classes of examples we determine this space explicitly

coding for dpsk noisy phase channels low to moderate complexity coding schemes are suggested and examined for binary dpsk modulation the model of the communication channel consists of both a brownian motion phase noise and an additive white gaussian noise decoding utilizes a mismatched soft decision metric which comprises the prefiltering of the phase noise impaired signals followed by differential demodulation the resulting equivalent binary input output symmetric discrete time coding channel is stationary and memoryless it accounts as well for an underlying time diversity thus providing an effective simple means of boosting the capabilities of less powerful codes to cope with the existing phase noise levels an analytical upper bound on the coded error probability is derived which strictly addresses the brownian motion phase model it features a convenient decoupling of the code structure from the coding channel the latter is completely specified via a single scalar generic parameter which relies on the univariate moments admitted by certain exponential functionals of the brownian motion sample path the exact statistics of which is intractable the theory is illustrated while studying the performance of low constraint length rate 1 n binary convolutional codes which are optimally combined with time diversity the advantages obtained by less trivial forms of code concatenation are examined the approach treated is to use a two level concatenation scheme for which the binary inner code dimension matches the alphabet size of an outer non binary code

darboux theory of integrability in cn taking into account the multiplicity abstract darboux in 1878 provided a theory on the existence of first integrals of polynomial systems based on the existence of sufficient invariant algebraic hypersurfaces called now the darboux theory of integrability in 1979 jouanolou successfully improved the darboux theory of integrability characterizing the existence of rational first integrals for this he used sophisticated tools of algebraic geometry the aim of this paper is to improve the classical result of darboux and the new one of jouanolou taking into account the multiplicity of the invariant algebraic hypersurfaces additionally our proof of the improved result of jouanolou is much simpler and elementary than the original one some examples show that the improved darboux theory of integrability with multiplicity is much useful than the classical one

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

poincare s reversibility condition we consider a real planar analytic vector field x such that the origin o is a centre for the linearization of x poincare s condition of reversibility with respect to a line passing through o is then a sufficient condition for o to be a centre for the vector field x we provide necessary and sufficient conditions involving the vanishing of certain polynomials in the coefficients in the expansion of x for reversibility we also show that if the linearization l x of the divergence of x is non trivial then the only possible reversibility line is given by l x 0 in such cases this provides the basis for a simple test of reversibility we examine the consequences of our various tests for quadratic and cubic vector fields all non hamiltonian cases are discussed when l x 0 in cubic systems it is possible for the reversibility line if it exists to be unique but it is also possible for there to be two such lines these possibilities are characterized algebraically and a prescription is provided for determining the reversibility line s in each case

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

envelope statistics for filtered optical signals corrupted by phase noise consideration is given to the case of an optical pulse containing phase noise which is passed through either an optical filter or following heterodyne lightwave detection an electrical filter because of the phase noise the envelope of the filter output at any instant is a random variable an analytical method is developed for estimating the probability density function pdf of this envelope for different kinds of filter responses and for realistic combinations of phase noise severity and filter bandwidth obvious applications are to detection analyses of coherent lightwave systems wherein finite laser linewidths constitute an important source of impairment for each of the several types of filters considered the envelope pdf can be accurately fitted by an exponential function approximation where the decay constant is related in a simple way to known system parameters

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

clipped diversity combining for channels with partial band interference ii ratio statistic combining for pt i see ibid vol com 3 no 12 p 1320 1987 ratio statistic combining is proposed for mitigating partial band interference in systems with diversity transmission and frequency hop signaling systems with noncoherent demodulation and binary orthogonal signaling are covered the partial band interference is gaussian and gaussian quiescent noise is included in the analysis to account for wideband noise sources the exact probability of error is found for a receiver using ratio statistic combining and this is compared to the exact error probabilities for receivers with optimum combining with perfect side information clipped linear combining the ratio threshold test with majority logic decoding and self normalization diversity combining numerical results are also given for a frequency hop system which uses ratio statistic combining for channels with rayleigh fading and partial band interference it is determined that ratio statistic combining is an excellent diversity combining scheme for systems with partial band interference and fading

performance of reed solomon coded frequency hop spread spectrum communications in partial band interference this paper is concerned with the performance of a communications system which utilizes frequency hop spread spectrum diversity transmission reed solomon coding and parallel error correction and erasure correction decoding both binary signaling and m ary orthogonal signaling are considered the goals are twofold first it is desirable to provide good performance in partial band gaussian noise interference by use of coding and diversity with an efficient error correction algorithm second it is necessary to totally neutralize narrow band interference regardless of the power level or statistical distribution of the interference in order to have an effective spread spectrum system through an analysis of the effects of partial band interference on a frequency hop spread spectrum system with diversity it is shown that the use of reedsolomon coding with a parallel errors and erasures decoding algorithm accomplishes these goals the paper also investigates the accuracy of the chernoff bound as an approximation to the true performance of a frequency hop spreadspectrum communication system with diversity side information m ary orthogonal signaling and reed solomon coding the performance results presented in the paper are based on analysis and computer evaluation approximate results based on the chernoff bound are also given it is shown that the chernoff bound for m ary orthogonal signaling gives a very poor approximation for many cases of interest this is largely due to the looseness of the union bound

models and information rates for wiener phase noise channels a waveform channel is considered where the transmitted signal is corrupted by wiener phase noise and additive white gaussian noise a discrete time channel model that considers the effect of filtering on the phase noise is developed the model is based on a multi sample receiver i e an integrate and dump filter whose output is sampled at a rate higher than the signaling rate it is shown that at high signal to noise ratio snr the multi sample receiver achieves a rate that grows logarithmically with the snr if the number of samples per symbol oversampling factor grows with the cubic root of the snr moreover the pre log factor is at least 1 2 and can be achieved by amplitude modulation for an approximate discrete time model of the multi sample receiver the capacity pre log at high snr is shown to be at least 3 4 if the number of samples per symbol grows with the square root of the snr the analysis shows that phase modulation achieves a pre log of at least 1 4 while amplitude modulation still achieves a pre log of 1 2 this is strictly greater than the capacity pre log of the approximate discrete time wiener phase noise channel with only one sample per symbol which is 1 2 numerical simulations are used to compute lower bounds on the information rates achieved by the multi sample receiver the simulations show that oversampling is beneficial for both strong and weak phase noise at high snr in fact the information rates are sometimes substantially larger than when using commonly used approximate discrete time models

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

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

planar quadratic differential systems with invariant straight lines of total multiplicity four in this article we prove a classification theorem of real planar quadratic vector fields which possess four invariant straight lines including the line at infinity and including their multiplicities this classification which is taken modulo the action of the group of real affine transformations and time rescaling is given in terms of algebraic invariants and comitants and also geometrically using cycles on the complex projective plane and on the line at infinity the algebraic invariants and comitants allow us to verify for any given real quadratic system whether or not it has invariant lines of total multiplicity four and to specify its configuration of lines endowed with their corresponding real singularities of this system the calculations can be implemented on computer this classification is instrumental in studying the problem of darboux integrability of such systems work which is in progress

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

darboux linearization and isochronous centers with a rational first integral abstract in this paper we study isochronous centers of polynomial systems it is known that a center is isochronous if and only if it is linearizable we introduce the notion of darboux linearizability of a center and give an effective criterion for verifying darboux linearizability if a center is darboux linearizable the method produces a linearizing change of coordinates most of the known polynomial isochronous centers are darboux linearizable moreover using this criterion we find a new two parameter family of cubic isochronous centers and give the linearizing changes of coordinates for centers belonging to that family we also determine all hamiltonian cubic systems which are darboux linearizable in the second part of this work we restrict to the study of isochronous centers having a rational first integral we prove that under certain conditions the cycle vanishing at the isochronous center is either zero homologous in the closure of a generic fiber or the function obtained from the first integral by eliminating the indeterminacy points has several critical points in the singular fiber passing through the isochronous center

the poincare problem algebraic integrability and dicritical divisors abstract we solve the poincare problem for plane foliations with only one dicritical divisor moreover in this case we give a simple algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case we also provide an algorithm to compute a rational first integral of prefixed genus g 1 of any type of plane foliation f when the number of dicritical divisors dic f is larger than 2 this algorithm depends on suitable families of invariant curves when dic f 2 it proves that the degree of the rational first integral can be bounded only in terms of g the degree of f and the local analytic type of the dicritical singularities of f

hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields we provide normal forms and the global phase portraits in the poincare disk for all hamiltonian nilpotent centers of linear plus cubic homogeneous planar polynomial vector fields

the performance of noncoherent orthogonal m fsk in the presence of timing and frequency errors practical m fsk systems experience a combination of time and frequency offsets errors this paper assesses the deleterious effect of these offsets first individually and then combined on the average bit error probability performance of the system exact expressions for these various error probability performances are derived and evaluated numerically for system parameters of interest also presented are upper bounds on average symbol error probability for the case of frequency error alone which are useful in assessing the absolute and relative performance of the system both continuous and discontinuous phase m fsk cases are considered when timing error is present the latter being much less robust to this type of offset

a method of constructing cycles without contact around a weak focus has a weak focus then limit cycles can be made to appear in its neighborhood by varying slightly the right sides 11 frommer 2 bautin 3 chin et al 4 and tung 5 are examples of excellent expositions of poincare s idea it is necessary to go from numerical examples to general theory in the first part of this paper we present a method of constructing cycles without contact around a weak focus which makes clear poincart s idea used in the above work 2 51 in the second part another method to analyze global phase portraits is used quadratic systems with at least four limit cycles are produced by means of both the above methods this contradicts the results of petrovskii and landis 6

center conditions at infinity for abel differential equations an abel differential equation y p x y 2 q x y 3 is said to have a center at a set a a 1 a r of complex numbers if y a 1 y a 2 y a r for any solution y x with the initial value y a 1 small enough the polynomials p q are said to satisfy the polynomial composition condition on a if there exist polynomials p q and w such that p p and q q are representable as p x p w x q x q w x and w a 1 w a 2 w a r we show that for wide ranges of degrees of p and q restricted only by certain assumptions on the common divisors of these degrees the composition condition provides a very accurate approximation of the center one up to a finite number of configurations not accounted for to our best knowledge this is the first general i e not restricted to small degrees of p and q or to a very special form of these polynomials result in the center problem for abel equations as an important intermediate result we show that at infinity according to an appropriate projectivization of the parameter space the center conditions are given by a system of the moment equations of the form as a1 p k q 0 s 2 r k 0 1

geometry of quadratic differential systems in the neighborhood of infinity in this article we give a complete global classification of the class qsess of planar essentially quadratic differential systems i e defined by relatively prime polynomials and whose points at infinity are not all singular according to their topological behavior in the vicinity of infinity this class depends on 12 parameters but due to the action of the affine group and re scaling of time the family actually depends on five parameters our classification theorem theorem 7 1 gives us a complete dictionary connecting very simple integer valued invariants which encode the geometry of the systems in the vicinity of infinity with algebraic invariants and comitants which are a powerful tool for computer algebra computations helpful in the route to obtain the full topological classification of the class qs of all quadratic differential systems

local bifurcations and a survey of bounded quadratic systems abstract this paper presents a survey of the known results for bounded quadratic systems as well as a study of the local bifurcations that occur at critical points of such systems it is shown that the only finite codimension bifurcations that occur at a critical point of a bounded quadratic system are the saddle node and the hopf takens bifurcations of codimensions 1 and 2 and the bogdanov takens bifurcations of codimensions 2 and 3 furthermore it is shown that whenever a bounded quadratic system has one of these critical points then a full generic unfolding of the critical point exists in the class of bounded quadratic systems finally we give a complete list of those limit periodic sets whose finite cyclicity still needs to be established in order to obtain the existence of a finite upper bound for the number of limit cycles that can occur in a bounded quadratic system

quadratic differential systems with complex conjugate invariant lines meeting at a finite point abstract in this article we study the class qs2cil of quadratic differential systems with two complex conjugate invariant lines meeting at a finite point from the literature we know that quadratic systems with invariant lines of total multiplicity at least four or with the line at infinity filled up with singularities are integrable via the method of darboux and hence they have no limit cycles these could only occur if we have only the two complex lines and the line at infinity all simple we first find all integrable systems in qs2cil due to the presence of invariant lines we next indicate a gap in the 1986 proof of suo and chen that systems in qs2cil have at most one limit cycle and we give a complete proof of this result finally we give the topological classification of qs2cil yielding 22 phase portraits three of which with a limit cycle

the centre and isochronicity problems for some cubic systems we present an efficient method for computing focus and linearizability quantities of polynomial differential equation systems we apply the method to computing these quantities for ten eight parametric cubic systems and obtain the necessary and sufficient conditions of linearizability isochronicity of these systems we also show that there is a kind of duality between the problem of constructing algebraic invariant curves first integrals and linearizing transformations on one side and the problem of solving some first order linear partial differential equations on the other side

on the poincare problem abstract we derive some restrictions on the possible degrees of algebraic invariant curves and on the possible form of algebraic integrating factors for plane polynomial vector fields whose stationary points at infinity satisfy a certain genericity condition the method is elementary and we show by example that it also yields strong results for certain non generic vector fields

an inverse approach to the center focus problem for polynomial differential system with homogenous nonlinearities abstract we consider polynomial vector fields x with a linear type and with homogenous nonlinearities it is well known that x has a center at the origin if and only if x has an analytic first integral of the form h 1 2 x 2 y 2 j 3 h j where h j h j x y is a homogenous polynomial of degree j the classical center focus problem already studied by h poincare consists in distinguishing when the origin of x is either a center or a focus in this paper we study the inverse center focus problem in particular for a given analytic function h defined in a neighborhood of the origin we want to determine the homogenous polynomials in such a way that h is a first integral of x and consequently the origin of x will be a center we study the particular case of centers which have a local analytic first integral of the form h 1 2 x 2 y 2 1 j 1 j in a neighborhood of the origin where j is a convenient homogenous polynomial of degree j for j 1 these centers are called weak centers they contain the class of center studied by alwash and lloyd the uniform isochronous centers and the isochronous holomorphic centers but they do not coincide with the class of isochronous centers we give a classification of the weak centers for quadratic and cubic vector fields with homogenous nonlinearities

a dynamical systems approach to the fourth painleve equation we use methods from dynamical systems to study the fourth painleve equation our starting point is the symmetric form of to which the poincare compactification is applied the motion on the sphere at infinity can be completely characterized there are fourteen fixed points which are classified into three different types generic orbits of the full system are curves from one of four asymptotically unstable points to one of four asymptotically stable points with the set of allowed transitions depending on the values of the parameters this allows us to give a qualitative description of a generic real solution of

bifurcation of limit cycles and integrability conditions for 6 parameter families of polynomial vector fields of arbitrary degree abstract we characterize the center variety and the maximum order of any focus of the 6 parameter family of real planar polynomial vectors given in complex notation by z i z a z n 1 z j 1 b z n 2 z j 2 c z n 3 z j 3 where a b c c 0 n 1 j 1 n 2 j 2 n 3 j 3 n k j k 1 for k 1 2 3 n 1 j 1 n 2 j 2 n 3 j 3 1 n 3 j 3 1 n 2 j 2 1 n 1 j 1 and 1 n 1 j 1 1 n 2 j 2 0

darboux theory of integrability in the sparse case darboux s theorem and jouanolou s theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field these results are given in terms of the degree of the polynomial vector field here we show that we can get the same kind of results if we consider the size of a newton polytope associated to the vector field furthermore we show that in this context the bound is optimal

coding for frequency hopped spread spectrum communication with partial band interference part i capacity and cutoff rate the performance of optimal codes on frequency hopped channels with partial band interference is investigated the performance measures considered are channel capacity and cutoff rate worst case partial band gaussian noise interference is assumed with the interference independent of the transmitted signal the capacity and cutoff rate are calculated as a function of the signal to noise ratio we consider soft decision receivers and hard decision receivers with and without side information optimal code rates are found for each of the above cases the required signal to noise ratio for reliable communication when codes are used is determined as a function of the code rate

algebraic particular integrals integrability and the problem of the center in this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields we first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center these curves control the changes in the systems as parameters vary the bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered namely the real four dimensional projective space next we consider the known four algebraic conditions for the center for quadratic vector fields one of them says that the system is hamiltonian a condition which has a clear geometric meaning we determine the geometric meaning of the remaining other three algebraic conditions i ii iii

characterizing filtered light waves corrupted by phase noise the phase noise associated with single mode semiconductor lasers must be accounted for in performance studies of lightwave communication systems the standard phase noise model is a brownian motion stochastic process although many analyses of lightwave communication systems have been published none to the authors knowledge has fully adhered to the standard model the reason is that a proper characterization of filtered lightwave signal had not been achieved such a characterization along with theoretical approaches to obtaining it is detailed the authors show for example how to generate probability density functions pdfs of the magnitude of a filtered laser tone with special attention to the tail region and how to analytically represent the characteristic function of the pdf in closed form in the small phase noise realm with the characterization in place the stage is now set for determining the bit error rate performance of advanced detection techniques which seek to mitigate the phase noise impairment

inverse jacobi multiplier as a link between conservative systems and poisson structures some aspects of the relationship between conservativeness of a dynamical system namely the preservation of a finite measure and the existence of a poisson structure for that system are analyzed from the local point of view due to the flow box theorem we restrict ourselves to neighborhoods of singularities in this sense we characterize poisson structures around the typical zero hopf singularity in dimension 3 under the assumption of having a local analytic first integral with non vanishing first jet by connecting with the classical poincar e center problem from the global point of view we connect the property of being strictly conservative the invariant measure must be positive with the existence of a poisson structure depending on the phase space dimension finally weak conservativeness in dimension two is introduced by the extension of inverse jacobi multipliers as weak solutions of its defining partial differential equation and some of its applications are developed examples including lotka volterra systems quadratic isochronous centers and non smooth oscillators are provided

algebraic integrability of foliations of the plane we give an algorithm to decide whether an algebraic plane foliation f has a rational first integral and to compute it in the affirmative case the algorithm runs whenever we assume the polyhedrality of the cone of curves of the surface obtained after blowing up the set bf of infinitely near points needed to get the dicritical exceptional divisors of a minimal resolution of the singularities of f this condition can be detected in several ways one of them from the proximity relations in bf and as a particular case it holds when the cardinality of bf is less than 9

the problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems abstract in this work we study the centers of planar analytic vector fields which are limit of linear type centers it is proved that all the nilpotent centers are limit of linear type centers and consequently the poincare liapunov method to find linear type centers can be also used to find the nilpotent centers moreover we show that the degenerate centers which are limit of linear type centers are also detectable with the poincare liapunov method

centers for a class of generalized quintic polynomial differential systems abstract we characterize the centers of the planar polynomial differential systems of degree d 5 odd that in complex notation writes as z iz z z d 5 2 az 5 bz 4 z cz 3 z 2 dz 2 z 3 ez z 4 f z 5 where a b c d e f c and either a re e 0 or a im e 0 or re a e 0 or im a e 0

probability of error analyses of a bfsk frequency hopping system with diversity under partial band jamming interference part ii performance of square law nonlinear combining soft decision receivers in this paper error probability analyses are performed for a binary frequency shift keying bfsk system employing l hop bit frequency hopping fh spread spectrum waveforms transmitted over a partial band gaussian noise jamming channel the error probabilities for the l hop bit bfsk fh systems are obtained as the performance measure of the square law linear combining soft decision receiver under the assumption of the worst case partial band jamming the receiver in our analysis assumes no knowledge of jamming state side information both exact and approximate multiple bound parameter chernoff bound solutions are obtained under two separate assumptions with and without the system s thermal noise in the analyses numerical results of the error rates are graphically displayed as a function of signal to jamming power ratio with l and signal to noise ratio as parameters all of our results exact and approximate indicated that the higher number of hops per bit produced higher error probabilities as a result of increased combining losses when the square law linear combining soft decision receiver is employed in demodulating the multihop per bit waveform

bifurcation diagrams for hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields as a natural continuation of the work done in 7 we provide the bifurcation diagrams for the global phase portraits in the poincare disk of all the hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields

hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields abstract we provide normal forms and the global phase portraits in the poincare disk for all the hamiltonian linear type centers of linear plus cubic homogeneous planar polynomial vector fields

an upper bound on the error probability of quadratic detection in noisy phase channels a rigorous method to find the upper bound of the error probability of noncoherent quadratically detected signals in the presence of both additive white gaussian noise and brownian carrier phase noise is presented an analytical upper bound on the bit error probability is derived relying on a bivariate moment generating function of two bounded exponential functionals of the brownian phase path these functionals whose exact statistics are unknown yield the two basic impairments arising due to phase noise in band signal suppression and intersymbol crosstalk the classical theory of chebyshev systems is applied to obtain the limiting values of the involved generating function utilizing a multidimensional moment characterization of the involved functionals the impact of the incomplete statistical characterization used on the resultant upper bound tightness is addressed assessing the design and performance of lightwave heterodyned systems using this method is discussed the method enables the analysis of interchannel crosstalk effects in frequency division multiplexing systems

power moment derivation for noisy phase lightwave systems the distribution law of the brownian motion exponential functional spl epsiv f sub 0 1 exp j spl radic 2 spl beta w sub t dt sup 2 where w sub t t spl isin spl rscr sup is a standard brownian motion modeling the laser s phase noise plays a key role in many of the proposed heterodyne lightwave systems the exact analytic derivation of these statistics appears to be however a formidable mathematical task we present a closed form formula for the kth power moment e spl epsiv sup k induced by the unknown distribution law the results are useful in assessing the performance of noisy phase lightwave systems

optical power requirements for detecting ook and fsk signals corrupted by phase noise the decision of on off keying ook and frequency shift keying fsk is considered for coherent lightwave communications in which the optical pulses are corrupted by the phase noise associated with nonzero laser linewidths heterodyne lightwave detection is assumed followed by intermediate frequency filtering envelope detection and postdetection low pass filtering using analytical results published previously theoretical minimum values of the required average power are obtained in photons bit for achieving a bit error rate of 10 sup 9 numerical results are given as functions of laser linewidth to bit rate ratio for ook binary fsk or 2 fsk and 8 fsk it is shown that heterodyne detection for lightwave signals of this kind can be made highly robust to phase noise in contrast to the case of binary phase shift keying

an explicit expression for the first return map in the center problem abstract the classical center focus problem posed by h poincare in 1880 s asks about the characterization of planar polynomial vector fields such that all their integral trajectories are closed curves whose interiors contain a fixed point a center in this paper we present a method allowing for the first time to obtain an explicit expression for the first return map in the center problem

darboux theory of integrability for a class of nonautonomous vector fields the goal of this paper is to extend the classical darboux theory of integrability from autonomous polynomial vector fields to a class of nonautonomous vector fields we also provide sufficient conditions for applying this theory of integrability and we illustrate this theory in several examples

reversible nilpotent centers with cubic homogeneous nonlinearities abstract we provide 13 non topological equivalent classes of global phase portraits in the poincare disk of reversible cubic homogeneous systems with a nilpotent center at origin which complete the classification of the phase portraits of the nilpotent centers with cubic homogeneous nonlinearities