the greatest number of limit cycles of the generalized rayleigh lienard oscillator abstract the limit cycles of the generalized rayleigh lienard oscillatorx ax 2bx3 z3 z2x2 z1x4 z4x2 x 0 fora 0 b 0 anda 0 are studied by using the jacobian elliptic functions with the generalized harmonic balance method for given values of the parameterszi 0 i 1 2 3 4 the values ofaandbfor which limit cycles exist are found as functions of a single parameter there is one limit cycle in the region where the hamiltonian esay is positive i e a solution of type cn and six limit cycles three double values in the region whereeis negative solution type dn

on the finite section method for computing exponentials of doubly infinite skew hermitian matrices abstract computing the exponential of large scale skew hermitian matrices or parts thereof is frequently required in applications in this work we consider the task of extracting finite diagonal blocks from a doubly infinite skew hermitian matrix these matrices usually have unbounded entries which impede the application of many classical techniques from approximation theory we analyze the decay property of matrix exponentials for several classes of banded skew hermitian matrices then finite section methods based on the decay property are presented we use several examples to demonstrate the effectiveness of these methods

bifurcation at infinity in polynomial vector fields abstract we study here the appearance of limit cycles from the equator in polynomial vector fields with no singular points at infinity this bifurcation is a generalized hopf bifurcation from the point at infinity we start with the general theory and then specialize to the particular case of cubic polynomial systems for which we study the simultaneous bifurcation of limit cycles from the origin and from the equator we finally discuss the transformation of a weak focus at infinity into a finite weak focus

number of limit cycles of the lienard equation in this paper we study a lienard system of the form x 5y2f x y52x where f x is an odd polyno mial we introduce a method that gives a sequence of algebraic approximations to the equation of each limit cycle of the system this sequence seems to converge to the exact equation of each limit cycle we obtain also a sequence of polynomials rn x whose roots of odd multiplicity are related to the number and location of the limit cycles of the system s1063 651x 97 02809 2

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

decay properties for functions of matrices over c algebras abstract we extend previous results on the exponential off diagonal decay of the entries of analytic functions of banded and sparse matrices to the case where the matrix entries are elements of a c algebra

applications of linking to hammerstein integral equations in this paper we establish some results for asymptotic linear hammerstein integral equations using the notion of linking both homotopical and co homological

quadratic lienard equations with quadratic damping beijing univ dept math beijing 100871 peoples r china beijing univ math inst beijing 100871 peoples r china dumortier f limburgs univ ctr univ campus b 3590 diepenbeek belgium

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

decomposition of time ordered products and path ordered exponentials we present a decomposition formula for un an integral of time ordered products of operators in terms of sums of products of the more primitive quantities cm which are the integrals of time ordered commutators of the same operators the resulting factorization enables a summation over n to be carried out to yield an explicit expression for the time ordered exponential an expression which turns out to be an exponential function of cm the campbell baker hausdorff formula and the non abelian eikonal formula obtained previously are both special cases of this result

small amplitude limit cycles of some lienard type systems abstract as we know the lienard system and its generalized forms are classical and important models of nonlinear oscillators and have been widely studied by mathematicians and scientists the main problem considered by most people is the number of limit cycles in this paper we investigate two kinds of lienard systems and obtain the maximal number i e the least upper bound of limit cycles appearing in hopf bifurcations by applying some known bifurcation theorems with technical analysis

limit cycles in highly non linear differential equations abstract this paper is concerned with both small amplitude and large amplitude limit cycle bifurcations of planar differential systems the analysis is not restricted to minimal models with few non linear terms in fact the novel approach adopted here is to consider differential equations containing highly non linear terms in both the damping and restoring coefficients the maximum number of limit cycles which may be bifurcated in a small region of the origin is given for certain classes of the more generalised mixed rayleigh lienard oscillator equations of the form f x h g x 0 certain mechanical systems are investigated

the limit cycles of the generalized rayleigh lienard oscillator abstract the limit cycles of the generalized rayleigh and mixed rayleigh lienard oscillator x ax 2bx 3 e z 3 z 2 x 2 z 1 x 4 z 4 dot x 2 dot x 0 for a and b 0 are studied by using the jacobian elliptic functions with the generalized harmonic balance method the transitory motion and consequently the limit cycles and their stability are also studied quantitatively with a generalized approximation of the krylov bogoliubov slowly varying amplitude and phase type giving the radius frequency and energy of the limit cycles whether there are zero one or two limit cycles and their stability depends on the values of the parameters z i i 1 2 3 4 these solutions are interesting because they do not depend on the value of e for the simple cases of only two non zero z i parameters plots of the universal functions are given as well as the limit cycle radii approximate solutions in the general case are found to coincide with certain exact solutions of the mixed type oscillators

application of secondary bifurcations to large amplitude limit cycles in mechanical systems large amplitude limit cycles have been described in terms of primary hopf secondary fold bifurcation pairs minimal models for two cases of large amplitude limit cycles have been developed for mechanical systems the mechanism responsible for large amplitude limit cycles in these models is non linear damping the theory is applied to the problems of surge oscillations in axial flow compressors and wing rock oscillations in aircraft flight dynamics to derive new and useful results

small amplitude limit cycles of certain lienard systems the paper is concerned with the number of limit cycles of systems of the form y f x g x where f and g are polynomials for several classes of such systems the maximum number of limit cycles that can bifurcate out of a critical point under perturbation of the coefficients in f and g is obtained in terms of the degree of f and g

small amplitude limit cycles in lienard systems via multiplicity abstract a recent paper of christopher and lloyd reduces the computation of the order of degeneracy of a weak focus for a polynomial lienard system x y f x y g x to the computation of the multiplicity of a polynomial map in this paper we first take advantage of that approach to obtain new lower and upper bounds for the maximum order of degeneracy of the origin in terms of the degrees of f and g later on we implement an algorithm to compute this maximum order for concrete values of these degrees as far as we know we enlarge the set of degrees for which this maximum order was known finally we extend christopher and lloyd s result to analytic degenerate or not lienard equations

compactness of the set of trajectories of the controllable system described by an affine integral equation in this paper the control system with integral constraint on the controls is studied it is assumed that the behavior of the system is described by an affine volterra integral equation which is nonlinear with respect to the state vector and is linear with respect to the control vector the closed ball of the space l p p 1 with radius m 0 and centered at the origin is chosen as the set of admissible control functions it is proved that the set of trajectories is a bounded closed precompact and hence compact subset of the space of continuous functions

polynomial systems a lower bound for the hilbert numbers let h n be the maximum possible number of limit cycles of systems dot x p x y dot y q x y where p and q are polynomials of degree at most n we are concerned with the rate of growth of h n as n increases it is known that h n geq kn 2 for some constant k in this paper we show that h n grows at least as rapidly as n 2 log n

an exact formulation of the time ordered exponential using path sums we present the path sum formulation for the time ordered exponential of a time dependent matrix the path sum formulation gives the time ordered exponential as a branched continued fraction of finite depth and breadth the terms of the path sum have an elementary interpretation as self avoiding walks and self avoiding polygons on a graph our result is based on a representation of the time ordered exponential as the inverse of an operator the mapping of this inverse to sums of walks on a graphs and the algebraic structure of sets of walks we give examples demonstrating our approach we establish a super exponential decay bound for the magnitude of the entries of the time ordered exponential of sparse matrices we give explicit results for matrices with commonly encountered sparse structures

the number of limit cycle bifurcation diagrams for the generalized mixed rayleigh lienard oscillator abstract this paper investigates the generalized mixed rayleigh lienard oscillator with highly nonlinear terms not restrict to the number of limit cycles this analysis considers mainly the number of limit cycle bifurcation diagrams of the system first the singularity theory approach is applied to the first order averaged approximation of the system with lower order nonlinear terms to reveal all possible bifurcation diagrams by summarizing the generating rule and structural distinction of different bifurcation diagrams a numerical procedure is then developed calculation suggests that the number of bifurcation diagrams increase very fast as the order of nonlinear terms lastly numerical simulations are adopted to approve the analytical results

small amplitude limit cycles of the generalized mixed rayleigh lienard oscillator abstract this paper concerned with the generalized mixed rayleigh lienard oscillator equations of the form x b30x3 a1 b21x2 b41x4 b03 2 0 the problem is to obtain the maximum possible number of limit cycles under perturbation of the coefficients arising in these equations an algorithm is implemented on a computer to determine a so called focal basis estimates can then be obtained for the number of limit cycles which may be bifurcated in a small region of the origin it is shown that at most three small amplitude limit cycles may be bifurcated

approximation of the integral funnel of the urysohn type integral operator abstract the integral funnel of the closed ball of the space lp p 1 with radius r and centered at the origin under urysohn type integral operator is defined as the set of graphs of the images of all functions from given ball approximation of the integral funnel is considered the closed ball of the space lp p 1 with radius r is replaced by the set consisting of a finite number of functions the hausdorff distance between integral funnel and the set consisting of the sections of the graphs of images of a finite number of functions is evaluated it is proved that in the case of appropriate choosing of the discretization parameters the approximating sets converges to the integral funnel

the mckean s caricature of the fitzhugh nagumo model i the space clamped system within the context of lienard equations we present the fitzhugh nagumo model with an idealized nonlinearity we give an analytical expression i for the transient regime corresponding to the emission of a finite number of action potentials or spikes and ii for the asymptotic regime corresponding to the existence of a limit cycle we carry out a global analysis to study periodic solutions the existence of which is linked to the solutions of a system of transcendental equations the periodic solutions are obtained with the help of the harmonic balance method or as limit behavior of the transient regime we show how the appearance of periodic solutions corresponds either to a fold limit cycle bifurcation or to a hopf bifurcation at infinity the resultsobtained are in agreement with local analysis methods i e the melnikov method and the averaging method the generalization of the model leads us to formulate two conjectures concerning the number of limit cycles for the piecewise linear lienard equ

improving a method for the study of limit cycles of the lienard equation in recent papers we have introduced a method for the study of limit cycles of the li enard system mathrm x ifmmode dot else fi y ensuremath f x mathrm y ifmmode dot else fi ensuremath x where f x is an odd polynomial the method gives a sequence of polynomials r n x whose roots are related to the number and location of the limit cycles and a sequence of algebraic approximations to the bifurcation set of the system in this paper we present a variant of the method that gives very important qualitative and quantitative improvements

a new method for optimal control of volterra integral equations abstract we formulate and analyze a new method for solving optimal control problems for systems governed by volterra integral equations our method utilizes discretization of the original volterra controlled system and a novel type of dynamic programming in which the hamilton jacobi function is parametrized by the control function rather than the state as in the case of ordinary dynamic programming we also derive estimates for the computational cost of our method