vanishing viscosity solutions of nonlinear hyperbolic systems we consider the cauchy problem for a strictly hyperbolic n n system in one space dimension ut a u ux 0 assuming that the initial data have small total variation we show that the solutions of the viscous approximations ut a u ux euxx are defined globally in time and satisfy uniform bv estimates independent of e moreover they depend continuously on the initial data in the l 1 distance with a lipschitz constant independent of t e letting e 0 these viscous solutions converge to a unique limit depending lipschitz continuously on the initial data in the conservative case where a df is the jacobian

one parameter lie groups and inverse integrating factors of n th order autonomous systems abstract the method of obtaining inverse integrating factors of n th order autonomous systems using one parameter lie groups admitted by the systems is given by describing n th order autonomous systems with differential form systems the properties of inverse integrating factors of the n th order autonomous systems are presented

on the limit cycles of planar polynomial system with non rational first integral via averaging method at any order in this paper we consider a class of cubic planar polynomial differential system with non rational first integral using the averaging method at any order we bound the maximum number of limit cycles which bifurcate from the periodic annulus of the origin when we perturb them inside the class of all polynomial systems of degree n

singular initial data and uniform global bounds for the hyper viscous navier stokes equation with periodic boundary conditions abstract in the hyper viscous navier stokes equations of incompressible flow the operator a is replaced by a a b aa ba for real numbers a b with 1 and b 0 we treat here the case a 0 and equip a and hence a a b with periodic boundary conditions over a rectangular solid r n for initial data in l p with n 2p 1 2 we establish local existence and uniqueness of strong solutions generalizing a result of giga miyakawa for 1 and b 0 specializing to the case p 2 which holds a particular physical relevance in terms of the total energy of the system it is somewhat interesting to note that the condition n 4 1 2 is sufficient also to establish global existence of these unique regular solutions and uniform higher order bounds for the borderline case n 4 1 2 we generalize standard existing for n 3 folklore results and use energy techniques and gronwall s inequality to obtain first a time dependent h bound and then convert to a time independent global exponential h bound this is to be expected given that uniform bounds already exist for n 2 1 6 pp 78 79 and the folklore bounds already suggest that the n 4 1 2 cases for n 3 should behave as well as the n 2 case what is slightly less expected is that the n 3 cases are easier to prove and give better bounds e g the uniform bound for n 3 depends on the square of the data in the exponential rather than the fourth power for n 2 more significantly for n 4 1 2 we use our own entirely semigroup techniques to obtain uniform global bounds which bootstrap directly from the uniform l2 estimate and are algebraic in terms of the uniform l2 bounds on the initial and forcing data the integer powers on the square of the data increase without bound as n 4 1 2 thus anticipating the exponential bound in the borderline case n 4 1 2 we prove our results for the case a 1 and b 0 the general case with a 0 and b 0 can be recovered by using norm equivalence we note that the hyperviscous navier stokes equations have both physical and numerical application

comparison of numerical methods for the calculation of two dimensional turbulence the estimate derived by henshaw kreiss and reyna for the smallest scale present in solutions of the two dimensional incompressible navier stokes equations is employed to obtain convergent pseudospectral approximations these solutions are then compared with those obtained by a number of commonly used numerical methods if the viscosity term is deleted and the energy in high wave numbers removed by setting the amplitudes of all wave numbers above a certain point in the spectrum to zero the chopped solution differs considerably from the convergent solution even at early times in the case that the regular viscosity is replaced by a hyperviscosity term i e the square of the laplacian we also derive an estimate for the smallest scale present if the coefficient of hyperviscosity is chosen so that the spectrum of the hyperviscosity solution disappears at the same point as for the regular viscosity solution the hyperviscosity solution is also completely different from the convergent solution if we tune the hyperviscosity coefficient then the solutions are similar in amplitude or phase but not both the solution obtained by a second order difference method with twice the number of points as the pseudospectral model or a fourth order difference method with the same number of points as the pseudospectral model is essentially identical to the convergent solution this is reasonable since most of the energy of the solution is contained in the lower part of the spectrum

scalar conservation laws with moving constraints arising in traffic flow modeling an existence result abstract we consider a strongly coupled pde ode system that describes the influence of a slow and large vehicle on road traffic the model consists of a scalar conservation law accounting for the main traffic evolution while the trajectory of the slower vehicle is given by an ode depending on the downstream traffic density the moving constraint is expressed by an inequality on the flux which models the bottleneck created in the road by the presence of the slower vehicle we prove the existence of solutions to the cauchy problem for initial data of bounded variation

l 2 stability estimates for shock solutions of scalar conservation laws using the relative entropy method we consider scalar nonviscous conservation laws with strictly convex flux in one spatial dimension and we investigate the behavior of bounded l2 perturbations of shock wave solutions to the riemann problem using the relative entropy method we show that up to a time dependent translation of the shock the l2 norm of a perturbed solution relative to the shock wave is bounded above by the l2 norm of the initial perturbation

limit cycles of cubic polynomial differential systems with rational first integrals of degree 2 the main goal of this paper is to study the maximum number of limit cycles that bifurcate from the period annulus of the cubic centers that have a rational first integral of degree 2 when they are perturbed inside the class of all cubic polynomial differential systems using the averaging theory the computations of this work have been made with mathematica and maple

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

inertial manifolds for the hyperviscous navier stokes equations abstract we prove the existence of inertial manifolds for the incompressible hyperviscous navier stokes equations on the two or three dimensional torus u t u u u p f t x r t d div u 0 where d 2 or 3 and 3 2 since the spectral gap condition is not necessarily satisfied for the aforementioned problem in three dimensions we employ the spatial averaging method introduced by mallet paret and sell in 26

coexistence of limit cycles and invariant algebraic curves for a kukles system abstract we consider a kukles system of the form x y y f x y where f x y is a polynomial with real coefficients of degree d without y as a divisor we study the maximum number of small amplitude limit cycles for these kind of systems which can coexist with invariant algebraic curves we give all the possible distributions of invariant straight lines for a kukles system and we give some bounds for the number of limit cycles we also give some necessary conditions for the existence of an invariant algebraic curve of degree 2 and we study the possible coexistence of this curve and a limit cycle finally we give two examples of cubic kukles systems both with an invariant hyperbola in the first example the hyperbola coexists with a center and in the second one it coexists with two small amplitude limit cycles these two examples contradict a previous result given in ann differential equations 7 3 1991 323

multiscale fractional step schemes for the numerical simulation of the rotating shallow water flows with complex periodic topography in this paper we study several multiscale fractional step schemes for the numerical solution of the rotating shallow water equations with complex topography we consider the case of periodic boundary conditions f plane model spatial discretization is obtained using a fourier spectral galerkin method for the schemes presented in this paper we consider two approaches the first approach multiscale schemes is based on topography scale separation and the numerical time integration is function of the scales the second approach is based on a splitting of the operators and the time integration method is function of the operator considered fractional step schemes the numerical results obtained are compared with the explicit reference scheme leap frog scheme with these multiscale fractional step schemes the objective is to propose new schemes giving numerical results similar to those obtained using only one uniform fine grid n nn n and a time step t but with a cpu time near the cpu time needed when using only one coarse grid n1 n1n1 n1 n1 t t t

towards large eddy and direct simulation of complex turbulent flows abstract recent advances in the methodology for direct numerical simulation of turbulent flows and some of the current applications are reviewed it is argued that high order finite difference schemes yield solutions with comparable accuracy to the spectral methods with the same number of degrees of freedom the effects of random inflow conditions on the downstream evolution of turbulence are discussed

limit cycles of cubic polynomial vector fields via the averaging theory abstract in this paper we study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging method more precisely we prove that the perturbations of the period annulus of the center located at the origin of the cubic polynomial differential system x y f x y y x f x y where f x y 0 is a conic such that f 0 0 0 by arbitrary cubic polynomial differential systems provide at least six limit cycles bifurcating from the periodic orbits of the period annulus using only the first order averaging method

existence and uniqueness of solutions for the navier stokes equations with hyperdissipation abstract the navier stokes equations with hyperdissipation are used to numerically simulate turbulent flows while numerical aspects of this model are well understood its analytic properties have not been thoroughly explored in this article we prove the existence and uniqueness of strong solutions to these equations on t 2 with periodic boundary conditions our proof extends the methods of cannone 1995 and cannone and meyer 1995 to the discrete setting and relies on littlewood paley theory and techniques from multiple fourier series

the perturbed riemann problem and delta contact discontinuity in chromatography equations abstract in this paper we discuss the perturbed riemann problem for the nonlinear chromatography equations the solutions to the problem are constructed with the method of the splitting delta function in solutions we find another kind of delta contact discontinuity on which both state variables simultaneously contain the dirac delta functions it is quite different from the previous ones on which only one state variable contains the dirac delta function this phenomenon is also predicted theoretically by mazzotti 2009 and mazzotti et al 2010 moreover by studying the limits of the solutions as the perturbed parameter e 0 it can be found that the riemann solutions are stable for such perturbations with the initial data

bifurcation of periodic orbits of periodic equations with multiple parameters by averaging method abstract we investigate a class of n dimensional periodic differential equations with multiple small parameters by averaging method and obtain sufficient conditions for the existence of multiple periodic orbits as applications of the main results a cubic system is studied

limit cycles in kukles systems of arbitrary degree with invariant ellipse abstract in this paper a normal form is obtained for the kukles polynomial systems of arbitrary degree with an invariant ellipse we study some existence and uniqueness results for limit cycles of that family moreover the tangential sixteenth hilbert problem is discussed for those polynomial systems showing an upper bound for the number of bifurcated limit cycles depending on the degree of the system we conclude with some numerical simulations of the results obtained

existence of solutions for a class of discontinuous differential equations in rn in this paper we extend the caratheodory existence theorem for a class of discontinuous differential equations with right hand sides in a special form but with no continuity or monotonicity assumptions a new existence result for a scalar discontinuous differential equation is also established

averaging theory at any order for computing periodic orbits abstract we provide a recurrence formula for the coefficients of the powers of e in the series expansion of the solutions around e 0 of the perturbed first order differential equations using it we give an averaging theory at any order in e for the following two kinds of analytic differential equation d x d k 1 e k f k x d x d k 0 e k f k x a planar polynomial differential system with a singular point at the origin can be transformed using polar coordinates to an equation of the previous form thus we apply our results for studying the limit cycles of a planar polynomial differential systems

bifurcations of small limit cycles in lienard systems with cubic restoring terms abstract in this paper we study bifurcations of small amplitude limit cycles of lienard systems of the form x y f x y g x where g x is a cubic polynomial and f x is a smooth or piecewise smooth polynomial of degree n by using involutions we obtain sharp upper bounds of the number of small amplitude limit cycles produced around a singular point for some systems of this type

on the limit cycles of a class of kukles type differential systems abstract in this paper we study the limit cycles of two families of differential systems in the plane these systems are obtained by polynomial perturbations with arbitrary degree on the second component of the standard linear center the classes under consideration are polynomial generalizations of certain canonical form of a kukles system with an invariant ellipse previously studied in the literature we provide in both cases an accurate upper bound of the maximum number of limit cycles that the perturbed system can have bifurcating from the periodic orbits of the linear center using the averaging theory of first second and third order these upper bounds are presented in terms of the degree of the respective systems moreover the existence of a weak focus with the highest order is also studied

solutions a variations bornees pour certains systemes hyperboliques de lois de conservation abstract blake temple trans amer math soc 280 1983 781 795 has described the hyperbolic systems of two conservation laws whose shock and rarefaction curves coincide in this note we prove the global existence of weak solutions for such systems with any bounded variation initial condition the proof is based upon standard numerical schemes as well as upon parabolic regularization the key is that the total variation of the riemann invariants is decreasing in time at least in the case of the initial condition with compact support we prove by using the glimm scheme that the system is decoupling in two conservation laws in one unknown in finite time

abelian integrals and limit cycles for a class of cubic polynomial vector fields of lotka volterra type with a rational first integral of degree two abstract in this paper we study the number of limit cycles which bifurcate from the periodic orbits of cubic polynomial vector fields of lotka volterra type having a rational first integral of degree 2 under polynomial perturbations of degree n the analysis is carried out by estimating the number of zeros of the corresponding abelian integrals moreover using chebyshev criterion we show that the sharp upper bound for the number of zeros of the abelian integrals defined on each period annulus is 3 for n 3 the simultaneous bifurcation and distribution of limit cycles for the system with two period annuli under cubic polynomial perturbations are considered all configurations u v with 0 u v 3 u v 5 are realizable

computing centre conditions for certain cubic systems abstract we present necessary and sufficient conditions for a critical point of certain two dimensional cubic differential systems to be a centre extensive use of the computer algebra system reduce is involved the search for necessary and sufficient conditions for a centre has long been of considerable interest in the theory of nonlinear differential equations it has proved to be a difficult problem and full conditions are known for very few classes of systems such conditions are also required in the investigation of hilbert s sixteenth problem concerning the number of limit cycles of polynomial systems

averaging methods of arbitrary order periodic solutions and integrability in this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems this result extends and improves results on averaging theory of periodic analytic differential systems and it unifies many different kinds of averaging methods applying our theory to autonomous analytic differential systems we obtain some conditions on the existence of limit cycles and integrability for polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues we prove that there always exists a positive number n such that if its first n averaging functions vanish then all averaging functions vanish and consequently there exists a neighborhood of the origin filled with periodic orbits consequently if all averaging functions vanish the origin is a center for n 2 furthermore in a punctured neighborhood of the origin the system is c infty completely integrable for n 2 provided that each periodic orbit has a trivial holonomy finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities

a class of non integrable systems admitting an inverse integrating factor abstract we study the existence of an inverse integrating factor for a class of systems in general non integrable whose lowest degree quasi homogeneous term is a hamiltonian system and its hamiltonian function only has simple factors over c x y

the center problem via averaging method the averaging method has been used to study the problem of the determination of the number of hyperbolic limit cycles that can bifurcate from the period annulus of a center in this paper we use the averaging method up to any order in the perturbation parameter to determine the center conditions of monodromic singular points of analytic planar vector fields

shape and period of limit cycles bifurcating from a class of hamiltonian period annulus abstract in this work we are concerned with the problem of shape and period of isolated periodic solutions of perturbed analytic radial hamiltonian vector fields in the plane francoise developed a method to obtain the first non vanishing poincare pontryagin melnikov function we generalize this technique and we apply it to know up to any order the shape of the limit cycles bifurcating from the period annulus of the class of radial hamiltonians we write any solution in polar coordinates as a power series expansion in terms of the small parameter this expansion is also used to give the period of the bifurcated periodic solutions we present the concrete expression of the solutions up to third order of perturbation of hamiltonians of the form h h r necessary and sufficient conditions that show if a solution is simple or double are also presented

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

multilevel schemes for the shallow water equations in this paper we study a number of multilevel schemes for the numerical solution of the shallow water equations new schemes and new perspectives of known schemes are examined we consider the case of periodic boundary conditions spatial discretization is obtained using a fourier spectral galerkin method for the time integration two strategies are studied the first one is based on scale separation and we choose the time scheme explicit or semi implicit as a function of the spatial scales multilevel schemes the second approach is based on a splitting of the operators and we choose the time integration method as a function of the operator considered multistep or fractional schemes the numerical results obtained are compared with the explicit reference scheme leap frog scheme and with the semi implicit scheme leap frog scheme with crank nicholson scheme for the gravity terms both computed with a similar mesh the drawback of the explicit reference scheme being the numerical stability constraint on the time step and the drawback of the semi implicit scheme being the dispersive error the aim with the new schemes is to obtain schemes with less dispersive error than the semi implicit scheme and with better stability properties than the explicit reference scheme the numerical results obtained show that the schemes proposed allow one to reduce the dispersive error and to increase the numerical stability at reduced cost

polynomial inverse integrating factors for quadratic differential systems we characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals we also prove that these families of quadratic systems have no limit cycles

unique conservative solutions to a variational wave equation relying on the analysis of characteristics we prove the uniqueness of conservative solutions to the variational wave equation u tt c u c u u x x 0 given a solution u t x even if the wave speed c u is only hor continuous in the t x plane one can still define forward and backward characteristics in a unique way using a new set of independent variables x y constant along characteristics we prove that t x u together with other variables satisfy a semilinear system with smooth coefficients from the uniqueness of the solution to this semilinear system one obtains the uniqueness of conservative solutions to the cauchy problem for the wave equation with general initial data u 0 cdot in h 1 i r u t 0 cdot in l 2 i r

periodic orbits from second order perturbation via rational trigonometric integrals abstract the second order poincare pontryagin melnikov perturbation theory is used in this paper to study the number of bifurcated periodic orbits from certain centers this approach also allows us to give the shape and the period up to the first order we address these problems for some classes of abel differential equations and quadratic isochronous vector fields in the plane we prove that two is the maximum number of hyperbolic periodic orbits bifurcating from the isochronous quadratic centers with a birational linearization under quadratic perturbations of second order in particular the configurations 2 0 and 1 1 are realizable when two centers are perturbed simultaneously the required computations show that all the considered families share the same iterated rational trigonometric integrals

the semigroup generated by 2 2 conservation laws consider the cauchy problem for a strictly hyperbolic 2 2 system of conservation laws in one space dimension ie1 01 assuming that each characteristic field is either linearly degenerate or genuinely nonlinear this paper develops a new algorithm based on wave front tracking which yields a cauchy sequence of approximate solutions converging to a unique limit depending continuously on the initial data the solutions that we obtain constitute a semigroup s defined on a set ie1 02 of integrable functions with small total variation for some lipschitz constant l we have the estimate ie1 03

wave interactions and stability of the riemann solutions for the chromatography equations we prove that the riemann solutions are stable for the chromatography system under the local small perturbations of the riemann initial data the proof is based on the detailed analysis of the wave interactions by applying the method of characteristic analysis it is noteworthy that both the propagation directions of the shock wave s and rarefaction wave r are unchanged when they interact with the contact discontinuity j moreover the global structures and large time asymptotic behaviors of the perturbed riemann solutions are constructed and analyzed case by case c 2009 elsevier inc all rights reserved

on the integrability of two dimensional flows this paper deals with the notion of integrability of flows or vector fields on two dimensional manifolds we consider the following two key points about first integrals 1 they must be defined on the whole domain of definition of the flow or vector field or defined on the complement of some special orbits of the system 2 how are they computed we prove that every local flow on a two dimensional manifold m always has a continuous first integral on each component of m where is the set of all separatrices of we consider the inverse integrating factor and we show that it is better to work with it instead of working directly with a first integral or an integrating factor for studying the integrability of a given two dimensional flow or vector field finally we prove the existence and uniqueness of an analytic inverse integrating factor in a neighborhood of a strong focus of a non resonant hyperbolic node and of a siegel hyperbolic saddle

high resolution conjugate filters for the simulation of flows this paper proposes a hermite kernel realization of the conjugate filter oscillation reduction cfor scheme for the simulation of fluid flows the hermite kernel is constructed by using the discrete singular convolution dsc algorithm which provides a systematic generation of low pass filter and its conjugate high pass filters the high pass filters are utilized for approximating spatial derivatives in solving flow equations while the conjugate low pass filter is activated to eliminate spurious oscillations accumulated during the time evolution of a flow as both low pass and high pass filters are derived from the hermite kernel they have similar regularity time frequency localization effective frequency band and compact support fourier analysis indicates that the cfor hermite scheme yields a nearly optimal resolution and has a better approximation to the ideal low pass filter than previous cfor schemes thus it has better potential for resolving natural high frequency oscillations from a shock extensive one and two dimensional numerical examples including both incompressible and compressible flows with or without shocks are employed to explore the utility test the resolution and examine the stability of the present cfor hermite scheme small ratio of point perwavelength ppw is achieved in advancing a wavepacket and resolving a shock entropy wave interaction the present results for the advection of an isentropic vortex compare very favorably to those in the literature

darboux integrability and invariant algebraic curves for planar polynomial systems in this paper we study the normal forms of polynomial systems having a set of given generic invariant algebraic curves

on the number of limit cycles surrounding a unique singular point for polynomial differential systems of arbitrary degree abstract we study the number of limit cycles that bifurcate from the periodic orbits of the center x y r x y y x r x y where r is a convenient polynomial of degree 2 when we perturb it inside the class of all polynomial differential systems of degree n we use averaging theory for computing this number as a consequence of our study we provide the biggest number of limit cycles surrounding a unique singular point in terms of the degree of the system known up to now

on the number of limit cycles of a perturbed cubic polynomial differential center abstract for e sufficiently small we consider the planar polynomial differential system x y y 2 a x 2 b x c e p x y y x y 2 a x 2 b x c e q x y where p x y and q x y are polynomials of degree n by using the averaging method of first order we bound the number of limit cycles that can bifurcate from period annulus of the unperturbed system

limit cycles for generalized kukles polynomial differential systems abstract we study the limit cycles of generalized kukles polynomial differential systems using averaging theory of first and second order

linear estimation of the number of zeros of abelian integrals for some cubic isochronous centers abstract this paper consists of two parts in the first part we study the relationship between conic centers all orbits near a singular point of center type are conics and isochronous centers of polynomial systems in the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics when we perturb such systems inside the class of all polynomial systems of degree n

conditions for the existence of a center for the kukles homogeneous systems abstract in this work we study necessary and sufficient conditions for the existence of centers in two families of linear centers with homogeneous quartic and quintic nonlinearities systems of this class are called kukles homogeneous systems systems of this type were studied for the first time by kukles who studied a linear center with cubic nonhomogeneous nonlinearities

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

linearization of isochronous centers abstract in this paper we study isochronous centers of polynomial systems we first discuss isochronous centers of quadratic systems cubic symmetric systems and reduced kukles system all these systems have rational first integrals we give a unified proof of the isochronicity of these systems by constructing algebraic linearizing changes of coordinates we then study two other classes of systems with isochronous centers namely the class of complex systems z ip z and the class of cubic systems symmetric with respect to a line and satisfying 1 both classes consist of darboux integrable systems we discuss their geometric properties and construct the linearizing changes of coordinates we show that the class of polynomial isochronous systems carries a very rich geometry finally we discuss the geometry of the linearizing changes of coordinates in the complex plane

bifurcations and distribution of limit cycles which appear from two nests of periodic orbits abstract in this paper we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation z i z z 3 after a small polynomial perturbation we first show that under small perturbations of the form e p 2 m 1 z z where p 2 m 1 z z is a polynomial of degree 2 m 1 in which the power of z is odd and the power of z is even the only possible distribution of limit cycles is u u for all values of u 0 1 2 m 3 hence the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of z i z z 3 is m 3 for m 4 then we consider a perturbation of the form e p m z z where p m z z is a polynomial of degree m in which the power of z is odd and obtain the upper bound m 5 for m 6 moreover we show that the distribution u v of limit cycles is possible for 0 u m 5 0 v m 5 with u v m 2 and m 9

limit cycles bifurcating from planar polynomial quasi homogeneous centers in this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi homogeneous center which can be obtained using first order averaging method this result improves the upper bounds given in 7

a second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter abstract we deal with nonlinear t periodic differential systems depending on a small parameter the unperturbed system has an invariant manifold of periodic solutions we provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation in the end two applications are done the key tool for proving the main result is the lyapunov schmidt reduction method applied to the t poincare andronov mapping

limit cycles of polynomial differential systems with homogeneous nonlinearities of degree 4 via the averaging method by using the averaging method we study the limit cycles for a class of quartic polynomial differential systems as well as their global shape in the plane more specifically we analyze the global shape of limit cycles bifurcating from a hopf bifurcation and also from periodic orbits with linear center x y y x the perturbation of these systems is made inside the class of quartic polynomial differential systems without quadratic and cubic terms

kukles revisited advances in computing techniques we revisit the kukles system to show how advances in computer hardware and software have improved symbolic calculations we confirm directly that there are no non persistent centres for the kukles system we also prove an earlier conjecture regarding the isochronous centres for the extended kukles system we introduce a technique whereby modular resultants can be used to investigate the properties of resultants that cannot be readily calculated using the existing software

well posedness for a class of 2 2 conservation laws withl data abstract the cauchy problem for a special class of 2 2 systems of conservation laws with data in l 1 l is considered in the strictly hyperbolic case we prove the existence of a weak solution which depends continuously on the initial data with respect to the l 1 norm this solution can be characterized in terms of a kru kov type entropy condition which is introduced here

delta shock waves in chromatography equations abstract the previous investigations on delta shock waves were mostly focused on those with dirac delta function in only one state variable in this paper we obtain another kind from the nonlinear chromatography equations in which the dirac delta functions develop simultaneously in both state variables it is strictly proved to satisfy the system in the sense of distributions the generalized rankine hugoniot relation and entropy condition are clarified the numerical results completely coinciding with the theoretical analysis are presented

solution of lambda omega systems theta schemes and multigrid methods the numerical solution of a time dependent reaction diffusion lambda omega system discretized by theta schemes and finite differences is considered stability and accuracy of finite difference theta schemes for this system are established to solve the time implicit evolution equations a nonlinear multigrid method is applied the convergence properties of this solver are investigated considering a linearized lambda omega model

periodic orbits in analytically perturbed poisson systems abstract analytical perturbations of a family of finite dimensional poisson systems are considered it is shown that the family is analytically orbitally conjugate in u r n to a planar harmonic oscillator defined on the symplectic leaves as a consequence the perturbed vector field can be transformed in the domain u to the lagrange standard form on the latter use can be made of averaging theory up to second order to study the existence number and bifurcation phenomena of periodic orbits examples are given ranging from harmonic oscillators with a potential and duffing oscillators to a kind of zero hopf singularity analytic normal form