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

time correlation statistics of the lms adaptive algorithm weights the transient and steady state weight correlation statistics of both the real and complex lms adaptive filters are obtained when the inputs are independent samples from real and circularly gaussian processes respectively a matrix relationship is derived between the covariance matrix of the weight vector at two different times and the covariance matrix of the weights at one time these expressions show that the weight fluctuations have the same time constants as the mean behavior of the lms algorithm itself i e the weights are correlated over the same number of iterations that it takes for the algorithm to converge to the wiener weights for stationary inputs

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

the complex lms adaptive algorithm transient weight mean and covariance with applications to the ale the transient and steady state mean and covariance of the complex valued lms adaptive element weights are investigated when the inputs are samples from circularly normal processes it is shown that the data covariance diagonalizing matrix also diagonalizes the weight covariance matrix this result permits describing the complete transient behavior of the adaptive line enhancer ale weight covariance matrix in closed form for the case of multiple equal power narrow band statistically independent orthogonal rayleigh fading sinusoids in broad band noise based on these transient covariance results it is shown that for any stage of adaptation there exists an optimum ale gain constant that minimizes the excess mean squared prediction error this result is of particular significance when processing time limited data

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

low rank second order splitting of large scale differential riccati equations we apply first and second order splitting schemes to the differential riccati equation such equations are very important in e g linear quadratic regulator lqr problems where they provide a link between the state of the system and the optimal input the methods can also be extended to generalized riccati equations e g arising from lqr problems given in implicit form in contrast to previously proposed schemes such as bdf or rosenbrock methods the splitting schemes exploit the fact that the nonlinear and affine parts of the problem when considered in isolation have closed form solutions we show that if the solution possesses low rank structure which is frequently the case then this is preserved by the method this feature is used to implement the methods efficiently for large scale problems the proposed methods are expected to be competitive as they at most require the solution of a small number of linear equation systems per time step finally we apply our low rank implementations to the riccati equations arising from two lqr problems the results show that the rank of the solutions stay low and the expected orders of convergence are observed

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

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

convergence analysis for splitting of the abstract differential riccati equation we consider a splitting based approximation of the abstract differential riccati equation in the setting of hilbert schmidt operators the riccati equation arises in many different areas and is important within the field of optimal control in this paper we conduct a temporal error analysis and prove that the splitting method converges with the same order as the implicit euler scheme under the same low regularity requirements on the initial values for a subsequent spatial discretization the abstract setting also yields uniform temporal error bounds with respect to the spatial discretization parameter the spatial discretizations commonly lead to large scale problems where the use of structural properties of the solution is essential we therefore conclude by proving that the splitting method preserves low rank structure in the matrix valued case numerical results demonstrate the validity of the convergence analysis less

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

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

galerkin approximation of optimal linear filtering of infinite dimensional linear systems in this paper an implementable approximation of the infinite dimensional kalman filter is proposed for linear distributed systems corrupted by white noise it relies on a galerkin type treatment of both the riccati and the filter equations and it is shown to converge to the best linear estimate of the state for each sample path of the noise a basic tool is the study of the riccati equation on the hilbert space of hilbert schmidt operators on the state space numerical results are given for a case concerning delay differential equation

error analysis for implicit approximations to solutions to cauchy problems exact error analyses for two implicit approximation schemes to the cauchy problem du t dt mathcal a u t mathrel backepsilon 0 u 0 u 0 in d mathcal a are presented the operator mathcal a is required to be maximal monotone on a hilbert space mathcal h the approximations considered are u h kh i h mathcal a k u 0 and solutions u lambda to the approximating problems du lambda t dt mathcal a lambda u lambda 0 u lambda 0 u 0 where a lambda is the yosida approximation of mathcal a the special case of a being a subgradient is considered and tight estimates of the convergence of both u h and mathcal a h u h or u h and a h u h are given the new results in this case are the linear rates of convergence of u h in l infty 0 infty mathcal h and in the energy norm and the sublinear rate of convergence of a h u h in l 2 0 infty mathcal h applications to both linear and

on the benefits of the ldlt factorization for large scale differential matrix equation solvers abstract we propose efficient algorithms for solving large scale matrix differential equations in particular we deal with the differential riccati equations dre and state the applicability to the differential lyapunov equations dle we focus on methods based on standard versions of ordinary differential equations in the matrix setting the application of these methods yields algebraic lyapunov equations ales with a certain structure to be solved in every step the alternating direction implicit adi algorithm and krylov subspace based methods allow to exploit this special structure however a direct application of classic low rank formulations requires the use of complex arithmetic using an l d l t type decomposition of both the right hand side and the solution of the equation we avoid this problem thus the proposed methods are a more practical alternative for large scale problems arising in applications also they make the application of higher order methods feasible the numerical results show the better performance of the proposed methods compared to earlier formulations

correlation function for the weight sequence of the frequency domain adaptive filter the statistical behavior of the single complex adaptive weight in the frequency domain adaptive filter is further investigated in a previous note 8 an expression was evaluated for the expected value of the magnitude square of the weight sequence at any time in this note the covariance of the weight at different times is evaluated and in steady state shown to be geometric with the same time constant as the frequency domain adaptive filter algorithm itself using these results the reduction in the fluctuations of the weight values are computed for an exponentially decaying post algorithm filter

inhomogeneous fuchs equations and the limit cycles in a class of near integrable quadratic systems we study the bifurcation of limit cycles in general quadratic perturbations of the particular quadratic system which represents one of two codimension five components in the intersection of two strata in the centre manifold and q 4 the study of limit cycles for this degenerate case requires us to investigate not the first but the second variation m 2 of the displacement function we prove that up to three limit cycles can emerge from the period annulus surrounding the centre this implies that the cyclicity of period annuli of nearby systems in and q 4 is at most three as well our approach relies upon the possibility of deriving appropriate picard fuchs equations satisfied by the four independent integrals included in m 2

the cyclicity and period function of a class of quadratic reversible lotka volterra system of genus one abstract in this paper we investigate a class of quadratic reversible lotka volterra system of genus one z i z z 2 b z 2 with b 3 5 it is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two moreover we prove that the period function of its period trajectories is monotone increasing

convergence of galerkin approximations for operator riccati equations a nonlinear evolution equation approach abstract we develop an approximation and convergence theory for galerkin approximations to infinite dimensional operator riccati differential equations formulated in the space of hilbert schmidt operators on a separable hilbert space we treat the riccati equation as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator we prove a generic approximation result for quasi autonomous nonlinear evolution systems involving accretive operators which we then use to demonstrate the hilbert schmidt norm convergence of galerkin approximations to the solution of the riccati equation we illustrate the application of our results in the context of a linear quadratic optimal control problem for a one dimensional heat equation

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

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

unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops abstract in this paper we present a complete study of quadratic 3 parameter unfoldings of some integrable system belonging to the classqr3 and having two centers and two unbounded heteroclinic loops we restrict to unfoldings that are transverse toqr3 obtain a versal bifurcation diagram and all global phase portraits including the precise number and configuration of the limit cycles it is proved that 3 is the maximal number of limit cycles surrounding a single focus and only the 1 1 configuration can occur in case of simultaneous nests of limit cycles essentially the proof relies on a careful analysis of a related non conservative abelian integral

on the decay rate of hankel singular values and related issues this paper investigates the decay rate of the hankel singular values of linear dynamical systems this issue is of considerable interest in model reduction by means of balanced truncation for instance since the sum of the neglected singular values provides an upper bound for an appropriate norm of the approximation error the decay rate involves a new set of invariants associated with a linear system which are obtained by evaluating a modified transfer function at the poles of the system these considerations are equivalent to studying the decay rate of the eigenvalues of the product of the solutions of two lyapunov equations the related problem of determining the decay rate of the eigenvalues of the solution to one lyapunov equation will also be addressed very often these eigenvalues like the hankel singular values are rapidly decaying this fact has motivated the development of several algorithms for computing low rank approximate solutions to lyapunov equations however until now conditions assuring rapid decay have not been well understood such conditions are derived here by relating the solution to a numerically low rank cauchy matrix determined by the poles of the system bounds explaining rapid decay rates are obtained under some mild conditions

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

quadratic centers defining elliptic surfaces abstract let x be a quadratic vector field with a center whose generic orbits are algebraic curves of genus one to each x we associate an elliptic surface a smooth complex compact surface which is a genus one fibration we give the list of all such vector fields and determine the corresponding elliptic surfaces

eigenvalue decay bounds for solutions of lyapunov equations the symmetric case abstract we present two new bounds for the eigenvalues of the solutions to a class of continuous and discrete time lyapunov equations these bounds hold for lyapunov equations with symmetric coefficient matrices and right hand side matrices of low rank they reflect the fast decay of the nonincreasingly ordered eigenvalues of the solution matrix

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

the cyclicity of the period annulus of the quadratic hamiltonian triangle abstract the paper studies quadratic hamiltonian centers surrounded by a separatrix contour having a form of a triangle it is proved that in this situation the cyclicity of the period annulus under quadratic perturbations is equal to three