non existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities abstract in this paper we study the limit cycles of the planar polynomial differential systems x a x y p n x y y x a y q n x y where p n and q n are homogeneous polynomials of degree n 2 and a r consider the functions p n cos sin cos q n cos sin sin q n cos sin cos p n cos sin sin 1 a 2 n 1 2 a first we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of 1 and 2 with definite sign this result improves previous known results furthermore if 1 1 a 2 0 for some 1 2 0 we provide necessary and sufficient conditions for the non existence and the existence and uniqueness of the limit cycles of these differential systems when one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin

upper bounds of limit cycles in abel differential equations with invariant curves abstract new criteria are established for upper bounds on the number of limit cycles of periodic abel differential equations having two periodic invariant curves one of them bounded the criteria are applied to obtain upper bounds of either zero or one limit cycle for planar differential systems

high order exactly well balanced numerical methods for shallow water systems abstract this paper is concerned with the development of high order methods for the shallow water system that preserve every stationary solution of the system and not only the water at rest ones the pursued strategy is based on the generalized hydrostatic reconstruction introduced in 10 and on the technique proposed in 8 to obtain high order well balanced reconstruction operators the more difficult step in the reconstruction procedure is the first one given a cell value a stationary solution of the system has to be found whose average at the cell equals the given value the implementation of this step is discussed in detail as well as the similarities and differences with some previous approaches to obtain well balanced methods for the shallow water system finally the well balanced properties of the developed numerical methods are checked in some test cases

a faster numerical scheme for a coupled system modeling soil erosion and sediment transport overland flow and soil erosion play an essential role in water quality and soil degradation such processes involving the interactions between water flow and the bed sediment are classically described by a well established system coupling the shallow water equations and the hairsine rose model numerical approximation of this coupled system requires advanced methods to preserve some important physical and mathematical properties in particular the steady states and the positivity of both water depth and sediment concentration recently finite volume schemes based on roe s solver have been proposed by heng et al 2009 and kim et al 2013 for one and twodimensional problems in their approach an additional and artificial restriction on the time step is required to guarantee the positivity of sediment concentration this artificial condition can lead the computation to be costly when dealing with very shallow flow and wet dry fronts the main result of this paper is to propose a new and faster scheme for which only the cfl condition of the shallow water equations is sufficient to preserve the positivity of sediment concentration in addition the numerical procedure of the erosion part can be used with any well balanced and positivity preserving scheme of the shallow water equations the proposed method is tested on classical benchmarks and also on a realistic configuration

a fast and stable well balanced scheme with hydrostatic reconstruction for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom it is a hyperbolic system of conservation laws that approximately describes various geophysical flows such as rivers coastal areas and oceans when completed with a coriolis term or granular flows when completed with friction numerical approximate solutions to this system may be generated using conservative finite volume methods which are known to properly handle shocks and contact discontinuities however in general these schemes are known to be quite inaccurate for near steady states as the structure of their numerical truncation errors is generally not compatible with exact physical steady state conditions this difficulty can be overcome by using the so called well balanced schemes we describe a general strategy based on a local hydrostatic reconstruction that allows us to derive a well balanced scheme from any given numerical flux for the homogeneous problem whenever the initial solver satisfies some classical stability properties it yields a simple and fast well balanced scheme that preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality

a well balanced flux vector splitting scheme designed for hyperbolic systems of conservation laws with source terms abstract we propose a way to construct robust numerical schemes for the computations of numerical solutions of one and two dimensional hyperbolic systems of balance laws in order to reduce the computational cost we selected the family of flux vector splitting schemes we reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations this is applied to a 1d shallow water system with topography and to a 2d simplified model of two phase flows with damping effects numerical results and comparisons with a classical centered discretizations scheme are supplied

periodic solutions for equation x a t x m b t x n c t x l with a t and b t changing signs in this paper we investigate the differential equation x s x t a t xm b t xn c t xl where a b c c 0 1 m n l and m n l z a solution x t with x 1 x 0 is called a periodic solution under some hypotheses which admit a t and b t without fixed sign we obtain the upper bound sometimes sharp for the number of isolated periodic solutions of the equation applying these results for the abel equation i e m 3 n 2 l 1 we get that if there exists 0 such that s t c t 0 resp s t a t b t 0 then the equation has at most 2 resp 4 non zero isolated periodic solutions furthermore suppose that a t t is a smooth curve which lies in r 0 0 1 with a 0 a 1 we obtain that if vector fields s x t 1 resp and c t x 1 are transverse to resp s x t 1 on in opposite directions then the number of non zero isolated periodic solutions of this abel equation is still no more than 2 resp 4 these conclusions generalize the known criteria about the abel equation which only refer to the cases with either a t or b t keeping sign finally as an application we study a kind of trigonometrical abel equation

well balanced adaptive mesh refinement for shallow water flows well balanced shock capturing wbsc schemes constitute nowadays the state of the art in the numerical simulation of shallow water flows they allow to accurately represent discontinuous behavior known to occur due to the non linear hyperbolic nature of the shallow water system and at the same time numerically maintain stationary solutions in situations of practical interest these schemes often need to be combined with some kind of adaptivity in order to speed up computing times in this paper we discuss what ingredients need to be modified in a block structured amr technique in order to ensure that when combined with a wbsc scheme the so called water at rest stationary solutions are exactly preserved

equilibrium schemes for scalar conservation laws with stiff sources we consider a simple model case of stiff source terms in hyperbolic conservation laws namely the case of scalar conservation laws with a zeroth order source with low regularity it is well known that a direct treatment of the source term by finite volume schemes gives unsatisfactory results for both the reduced cfl condition and refined meshes required because of the lack of accuracy on equilibrium states the source term should be taken into account in the upwinding and discretized at the nodes of the grid in order to solve numerically the problem we introduce a so called equilibrium schemes with the properties that i the maximum principle holds true ii discrete entropy inequalities are satisfied iii steady state solutions of the problem are maintained one of the difficulties in studying the convergence is that there are no bv estimates for this problem we therefore introduce a kinetic interpretation of upwinding taking into account the source terms based on the kinetic formulation we give a new convergence proof that only uses property ii in order to ensure desired compactness framework for a family of approximate solutions and that relies on minimal assumptions the computational efficiency of our equilibrium schemes is demonstrated by numerical tests that show that in comparison with an usual upwind scheme the corresponding equilibrium version is far more accurate furthermore numerical computations show that equilibrium schemes enable us to treat efficiently the sources with singularities and oscillating coefficients

a fully well balanced positive and entropy satisfying godunov type method for the shallow water equations this work is devoted to the derivation of a fully well balanced numerical scheme for the well known shallow water model during the last two decades several well balanced strategies have been introduced with a special attention to the exact capture of the stationary states associated with the so called lake at rest by fully well balanced we mean here that the proposed godunov type method is also able to preserve stationary states with non zero velocity the numerical procedure is shown to preserve the positiveness of the water height and satisfies a discrete entropy inequality

a multi well balanced scheme for the shallow water mhd system with topography the shallow water magnetohydrodynamic system involves several families of physically relevant steady states in this paper we design a well balanced numerical scheme for the one dimensional shallow water magnetohydrodynamic system with topography that resolves exactly a large range of steady states two variants are proposed with slightly different families of preserved steady states they are obtained by a generalized hydrostatic reconstruction algorithm involving the magnetic field and with a cutoff parameter to remove singularities the solver is positive in height and semi discrete entropy satisfying which ensures the robustness of the method

a subsonic well balanced reconstruction scheme for shallow water flows we consider the saint venant system for shallow water flows with nonflat bottom in past years efficient well balanced methods have been proposed in order to well resolve solutions close to steady states at rest here we describe a strategy based on a local subsonic steady state reconstruction that allows one to derive a subsonic well balanced scheme preserving exactly all the subsonic steady states it generalizes the now well known hydrostatic solver and like the latter it preserves the nonnegativity of the water height and satisfies a semidiscrete entropy inequality an application to the euler poisson system is proposed

reliability of first order numerical schemes for solving shallow water system over abrupt topography we compare some first order well balanced numerical schemes for shallow water system with special interest in applications where there are abrupt variations of the topography we show that the space step required to obtain a prescribed error depends on the method moreover the solutions given by the numerical scheme can be significantly different if not enough space resolution is used we shall pay special attention to the well known hydrostatic reconstruction technique where it is shown that the effect of large bottom discontinuities might be missed and a modification is proposed to avoid this problem

cubic systems and abel equations abstract we investigate cubic systems which can be transformed to an equation of abel form the conditions for the origin to be a centre and in particular an isochronous centre are obtained the maximum number of limit cycles which can bifurcate from a fine focus is determined and some information is obtained about the global phase portrait

a well balanced scheme for the numerical processing of source terms in hyperbolic equations in a variety of physical problems one encounters source terms that are balanced by internal forces and this balance supports multiple steady state solutions that are stable typical of these are gravity driven flows such as those described by the shallow water equations over a nonuniform ocean bottom 1 10 qquad h t hu x 0 text and hu t left hu 2 frac gh 2 2 right x ga x x h 0 many classic numerical schemes cannot maintain these steady solutions or achieve them in the long time limit with an acceptable level of accuracy because they do not preserve the proper balance between the source terms and internal forces we propose here a numerical scheme adapted to a scalar conservation law that preserves this balance and that can it is hoped be extended to more general hyperbolic systems the proof of convergence of this scheme toward the entropy solution is given and some numerical tests are reported

exactly well balanced discontinuous galerkin methods for the shallow water equations with moving water equilibrium hyperbolic conservation laws with source terms often admit steady state solutions where the fluxes and source terms balance each other to capture this balance and near equilibrium solutions well balanced methods have been introduced and performed well in many numerical tests shallow water equations have been extensively investigated as a prototype example in this paper we develop well balanced discontinuous galerkin methods for the shallow water system which preserve not only the still water at rest steady state but also the more general moving water equilibrium the key idea is the recovery of well balanced states a special source term approximation and the approximation of the numerical fluxes based on a generalized hydrostatic reconstruction we also study the extension of the positivity preserving limiter presented in 40 in this framework numerical examples are provided at the end to verify the well balanced property and good resolution for smooth and discontinuous solutions

anticommuting selfadjoint operators abstract several equivalent definitions of anticommutativity for selfadjoint operators are presented it is shown that the anticommutativity of two selfadjoint operators a and b satisfying af bg bf ag 0 for all and g in the intersection of the domains of a and b is equivalent to the selfadjointness of their sum and difference in studying the structure of anticommuting selfadjoint operators a and b attention may be restricted to the case where both a and b are injective in that case a is unitarily equivalent to a moreover a and b are uniquely determined by a triplet b p q where b is unitary and p and q are commuting injective positive and selfadjoint operators this is a natural correspondence

a new uniqueness criterion for the number of periodic orbits of abel equations abstract a solution of the abel equation x a t x 3 b t x 2 such that x 0 x 1 is called a periodic orbit of the equation our main result proves that if there exist two real numbers a and b such that the function a a t b b t is not identically zero and does not change sign in 0 1 then the abel differential equation has at most one non zero periodic orbit furthermore when this periodic orbit exists it is hyperbolic this result extends the known criteria about the abel equation that only refer to the cases where either a t 0 or b t 0 does not change sign we apply this new criterion to study the number of periodic solutions of two simple cases of abel equations the one where the functions a t and b t are 1 periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials finally we give an upper bound for the number of isolated periodic orbits of the general abel equation x a t x 3 b t x 2 c t x when a t b t and c t satisfy adequate conditions

a uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities abstract this paper is devoted to study the planar polynomial system x a x y p n x y y x a y q n x y where a r and p n q n are homogeneous polynomials of degree n 2 denote cos q n cos sin sin p n cos sin we prove that the system has at most 1 limit cycle surrounding the origin provided n 1 a 0 furthermore this upper bound is sharp this is maybe the first uniqueness criterion which only depends on a linear condition of for the limit cycles of this kind of systems we show by examples that in many cases the criterion is applicable while the classical ones are invalid the tool that we mainly use is a new estimate for the number of limit cycles of abel equation with coefficients of indefinite signs employing this tool we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin

existence of non trivial limit cycles in abel equations with symmetries abstract we study the periodic solutions of the generalized abel equation x a 1 a 1 t x n 1 a 2 a 2 t x n 2 a 3 a 3 t x n 3 where n 1 n 2 n 3 1 are distinct integers a 1 a 2 a 3 r and a 1 a 2 a 3 are 2 periodic analytic functions such that a 1 t sin t a 2 t cos t a 3 t sin t cos t are periodic positive even functions when n 3 n 1 n 3 n 2 0 we prove that the equation has no non trivial different from zero limit cycle for any value of the parameters a 1 a 2 a 3 when n 3 n 1 n 3 n 2 0 we obtain under additional conditions the existence of non trivial limit cycles in particular we obtain limit cycles not detected by abelian integrals

periodic orbits in complex abel equations abstract this paper is devoted to prove two unexpected properties of the abel equation d z d t z 3 b t z 2 c t z where b and c are smooth 2 periodic complex valuated functions t r and z c the first one is that there is no upper bound for its number of isolated 2 periodic solutions in contrast recall that if the functions b and c are real valuated then the number of complex 2 periodic solutions is at most three the second property is that there are examples of the above equation with b and c being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of b and c this result is also in contrast with the characterization of the center variety for the examples of abel equations d z d t a t z 3 b t z 2 studied in the literature where the center variety is located in a finite number of connected components

limit cycles of abel equations of the first kind abstract consider the scalar differential equation x i 0 m a i t x n i where a i t are t periodic analytic functions and 1 n i n for any polynomial q x x n 0 i 1 m i x n i the equation can be written as x a 0 q x r t x let w be the wronskian of q and r with respect to x and q w the previous polynomials after removing multiplicity of roots and solutions of the differential equation we prove that if the vector field defined by the differential equation is transversal at every point of q x 0 or w t x 0 then the number of limit cycles isolated periodic solutions in the set of periodic solutions of the differential equation is at most 3 n 1

moving water equilibria preserving partial relaxation scheme for the saint venant system we develop a new moving water equilibria preserving numerical scheme for the saint venant system the new scheme is designed in two major steps first the geometric source term is incorporated int

a geometric criterion for equation x i 0mai t xi having at most m isolated periodic solutions abstract this paper is devoted to the investigation of generalized abel equation x s x t i 0 m a i t x i where a i c 0 1 a solution x t is called a periodic solution if x 0 x 1 in order to estimate the number of isolated periodic solutions of the equation we propose a hypothesis h which is only concerned with s x t on m straight lines there exist m real numbers 1 m such that either 1 i s i t 0 for i 1 m or 1 i s i t 0 for i 1 m by means of lagrange interpolation formula we prove that the equation has at most m isolated periodic solutions counted with multiplicities if hypothesis h holds and the upper bound is sharp furthermore this conclusion is also valid under some weaker geometric hypotheses applying our main result for the trigonometrical generalized abel equation with coefficients of degree one we give a criterion to obtain the upper bound for the number of isolated periodic solutions this criterion is almost equivalent to hypothesis h and can be much more effectively checked