degenerate hopf bifurcations in discontinuous planar systems abstract we study the stability of a singular point for planar discontinuous differential equations with a line of discontinuities this is done for the most generic cases by computing some kind of lyapunov constants our computations are based on the so called r p q generalized polar coordinates introduced by lyapunov and they are essentially different from the ones used in the smooth case these lyapunov constants are also used to generate limit cycles for some concrete examples

melnikov method for discontinuous planar systems abstract this paper treats the occurrence of homoclinic solutions in planar systems with discontinuous right hand side more precisely we deal with a t periodic perturbed system such that the unperturbed system is an autonomous possessing homoclinic orbit by means of the so called non smooth melnikov function there is shown the existence of a homoclinic solution for a perturbed system the non smooth melnikov function is derived and the method of how to find it in concrete problems is also introduced

bifurcation analysis on a class of z2 equivariant cubic switching systems showing eighteen limit cycles abstract in this paper the method developed for computing the lyapunov constants of planar switching systems associated with an elementary singular point is applied to study bifurcation of limit cycles in a cubic switching system a complete classification on the center conditions and 16 limit cycles of this system are obtained around the two foci 1 0 and 1 0 further with the method an example of cubic switching systems is constructed to show the existence of 18 small amplitude limit cycles bifurcating from centers this is a new lower bound on the maximal number of small amplitude limit cycles obtained in such cubic switching systems finally a method is present to show the realization of the 18 limit cycles

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

bifurcations of equilibria in non smooth continuous systems abstract the aim of this paper is to show a variety of bifurcation phenomena of equilibria that can be observed in non smooth continuous systems in non smooth systems so called multiple crossing bifurcations can occur for which the eigenvalues jump more than once over the imaginary axis and which do not have a classical bifurcation as counterpart novel theoretical results are given for a class of planar systems but no general theory is available for the multi dimensional case a number of well chosen examples of multiple crossing bifurcations are discussed in detail

bifurcation of a non smooth planar limit cycle from a vertex abstract we investigate non smooth planar systems of differential equations with discontinuous right hand sides discontinuity sets intersect at a vertex and are of quasilinear nature by means of the b equivalence method which was introduced in m akhmetov asymptotic representation of solutions of regularly perturbed systems of differential equations with a nonclassical right hand side ukrainian math j 43 1991 1209 1214 m akhmetov on the expansion of solutions to differential equations with discontinuous right hand side in a series in initial data and parameters ukrainian math j 45 1993 786 789 m akhmetov n a perestyuk differential properties of solutions and integral surfaces of nonlinear impulse systems differential equations 28 1992 445 453 see also e akalin m u akhmet the principles of b smooth discontinuous flows math comput simul 49 2005 981 995 m u akhmet perturbations and hopf bifurcation of the planar discontinuous dynamical system nonlinear anal 60 2005 163 178 these systems are reduced to impulsive differential equations sufficient conditions are established for the existence of foci and centers both in the noncritical and critical cases hopf bifurcation is considered from a vertex which unites several curves in the critical case an appropriate example is provided to illustrate the results

mixed finite elements for elliptic problems with tensor coefficients as cell centered finite differences we present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient the mixed method is expanded in the sense that three variables are explicitly approximated namely the scalar unknown the negative of its gradient and its flux the tensor coefficient times the negative gradient the resulting linear system is a saddle point problem in the case of the lowest order raviart thomas elements on rectangular parallelepipeds we approximate this expanded mixed method by incorporating certain quadrature rules this enables us to write the system as a simple cell centered finite difference method requiring the solution of a sparse positive semidefinite linear system for the scalar unknown for a general tensor coefficient the sparsity pattern for the scalar unknown is a 9 point stencil in two dimensions and 19 points in three dimensions existing theory shows that the expanded mixed method gives optimal order approximations in the l 2 and h s norms and superconvergence is obtained between the l 2 projection of the scalar variable and its approximation we show that these rates of convergence are retained for the finite difference method if h denotes the maximal mesh spacing then the optimal rate is o h the superconvergence rate o h 2 is obtained for the scalar unknown and rate o h 3 2 for its gradient and flux in certain discrete norms moreover the full o h 2 is obtained in the strict interior of the domain computational results illustrate these theoretical results

numerical methods for flows through porous media i the degenerate parabolic equation aut v oulvvu v 1 has been used to model the flow of gas through a porous medium error estimates for continuous and discrete time finite element procedures to approximate the solution of this equation are proved and several new regularity results are given 1 a porous medium equation introduction we shall study the porous medium equation 1 1 au at v i u i vu on 2 x 0 t 1 2 au an 0 on aa x o t 1 3 u x 0 uo x on 2 where v 1 is a parameter and q is a bounded domain in rn n 3 with a smooth boundary the initial function u0 is assumed to be nonnegative and four times continuously differentiable on u notice that the compatibility condition auo an 0 holds on asa our main result is the derivation of error estimates for numerical approximations to the problem 1 1 1 3 which we shall refer to as the porous medium equation or ipme the pme does not in general admit classical solutions existence and uniqueness of weak solutions was proved in one space dimension by oleinik kalashnikov and czou 15 16 and in several space dimensions by lions 12 these proofs concern the pme with different boundary conditions but the arguments carry over to the pme 1 1 1 3 the maximum principle implies that since u0 is nonnegative on 92 u x t is nonnegative for all x t e q x 0 t see 15 16 if u0 is nonzero the neumann boundary condition implies that u will eventually become strictly positive and 1 1 will become nondegenerate for all time t to to sufficiently large we can rewrite 1 1 in the form 1 4 au at ak u on qx 0 t received may 11 1979 revised july 24 1981 1980 mathematics subject classification primary 35k65 65n30 65n15 current address reservoir engineering branch petroleum engineering division bp exploration dos britannic house moore lane london ec2y 9bu united kingdom d1983 american mathematical society 0025 5718 82 0000 1 063 08 25 435 this content downloaded from 157 55 39 55 on tue 23 aug 2016 04 12 54 utc all use subject to http about jstor org terms

the qualitative analysis of a class of planar filippov systems abstract we use the theory of differential inclusions filippov transformations and some appropriate poincare maps to discuss the special case of two dimensional discontinuous piecewise linear differential systems with two zones this analysis applies to uniqueness and non uniqueness for the initial value problem stability of stationary points sliding motion solutions number of closed trajectories existence of heteroclinic trajectories connecting two saddle points forming a heteroclinic cycle and existence of the homoclinic trajectory

concurrent homoclinic bifurcation and hopf bifurcation for a class of planar filippov systems abstract this paper investigates both homoclinic bifurcation and hopf bifurcation which occur concurrently in a class of planar perturbed discontinuous systems of filippov type firstly based on a geometrical interpretation and a new analysis of the so called successive function sufficient conditions are proposed for the existence and stability of homoclinic orbit of unperturbed systems then with the discussion about poincare map bifurcation analyses of homoclinic orbit and parabolic parabolic pp type pseudo focus are presented it is shown that two limit cycles can appear from the two different kinds of bifurcation in planar filippov systems

on the origin and bifurcations of stick slip oscillations a recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis particular emphasis is put on the bifurcations associated with the appearance of stick

a monotone version of the sokolov property and monotone retractability in function spaces abstract we introduce the monotone sokolov property and show that it is dual to monotone retractability in the sense that x is monotonically retractable if and only if c p x is monotonically sokolov besides a space x is monotonically sokolov if and only if c p x is monotonically retractable monotone retractability and monotone sokolov property are shown to be preserved by r quotient images and f subspaces furthermore every monotonically retractable space is sokolov so it is collectionwise normal and has countable extent we also establish that if x and c p x are lindelof spaces then they are both monotonically retractable and have the monotone sokolov property an example is given of a space x such that c p x has the lindelof property but neither x nor c p x is monotonically retractable we also establish that every lindelof space with a unique non isolated point is monotonically retractable on the other hand each lindelof space with a unique non isolated point is monotonically sokolov

hopf and homoclinic bifurcations on the sliding vector field of switching systems in r3 a case study in power electronics abstract in this paper hopf and homoclinic bifurcations that occur in the sliding vector field of switching systems in r 3 are studied in particular a dc dc boost converter with sliding mode control and washout filter is analyzed this device is modeled as a three dimensional filippov system characterized by the existence of sliding movement and restricted to the switching manifold the operating point of the converter is a stable pseudo equilibrium and it undergoes a subcritical hopf bifurcation such a bifurcation occurs in the sliding vector field and creates in this field an unstable limit cycle the limit cycle is connected to the switching manifold and disappears when it touches the visible invisible two fold point resulting in a homoclinic loop which itself closes in this two fold point the study of these dynamic phenomena that can be found in different power electronic circuits controlled by sliding mode control strategies are relevant from the viewpoint of the global stability and robustness of the control design

families of continuous retractions and function spaces in this paper we mainly study certain families of continuous retractions r skeletons having some rich properties it is shown that if the space x has a full r skeleton then its alexandroff duplicate also has a full r skeleton in a very similar way it is proved that the alexandroff duplicate of a monotonically retractable space is monotonically retractable by using monotonically retractable spaces we solve a question posed by r z buzyakova in 7 concerning the alexandroff duplicate of a space the notion of q skeleton is introduced and it is shown that every compact subspace of cp x is corson when x has a full q skeleton the notion of strong r skeleton is also introduced to answer a question suggested by f casarrubias segura and the second author in 8 certainly it is established that a space x is monotonically sokolov if and only if it is monotonically monolithic and has a strong r skeleton the techniques used here allow us to give a topological proof of a result of i bandlow 5 who used elementary submodels and uniform spaces

generic bifurcations of low codimension of planar filippov systems abstract in this article some qualitative and geometric aspects of non smooth dynamical systems theory are discussed the main aim of this article is to develop a systematic method for studying local and global bifurcations in non smooth dynamical systems our results deal with the classification and characterization of generic codimension 2 singularities of planar filippov systems as well as the presentation of the bifurcation diagrams and some dynamical consequences

on hopf bifurcation in non smooth planar systems abstract as we know for non smooth planar systems there are foci of three different types called focus focus ff focus parabolic fp and parabolic parabolic pp type respectively the poincare map with its analytical property and the problem of hopf bifurcation have been studied in coll et al 2001 3 and filippov 1988 6 for general systems and in zou et al 2006 13 for piecewise linear systems in this paper we also study the problem of hopf bifurcation for non smooth planar systems obtaining new results more precisely we prove that one or two limit cycles can be produced from an elementary focus of the least order order 1 for foci of ff or fp type and order 2 for foci of pp type theorem 2 3 different from the case of smooth systems for piecewise linear systems we prove that 2 limit cycles can appear near a focus of either ff fp or pp type theorem 3 3

perturbations and hopf bifurcation of the planar discontinuous dynamical system abstract the objective of the paper is to obtain results on the behavior of a specific plane discontinuous dynamical system in the neighbourhood of the singular point a new technique of investigation is presented conditions for existence of the foci and centres are proposed the focus centre problem and hopf bifurcation are considered appropriate examples are given to ilustrate the bifurcation theorem

characterizing corson and valdivia compact spaces abstract we give a new characterization of valdivia compact spaces a compact space is valdivia if and only if it has a dense commutatively monotonically retractable subspace this result solves problem 5 12 from 6 besides we introduce the notion of full c skeleton and prove that a compact space is corson if and only if it has a full c skeleton

a local discontinuous galerkin method for the burgers poisson equation in this work we design analyze and test a local discontinuous galerkin method for solving the burgers poisson equation this model proposed by whitham linear and nonlinear waves 1974 as a simplified model for shallow water waves admits conservation of both momentum and energy as two invariants the proposed numerical method is high order accurate and preserves two invariants hence producing solutions with satisfying long time behavior the l 2 l 2 stability of the scheme for general solutions is a consequence of the energy preserving property the optimal order of accuracy for polynomial elements of even degree is proven a series of numerical tests is provided to illustrate both accuracy and capability of the method

banach spaces with projectional skeletons a projectional skeleton in a banach space is a directed family of projections onto separable subspaces covering the entire space the class of banach spaces with projectional skeletons is strictly larger than the class of plichko spaces i e banach spaces with a countably norming markushevich basis we show that every space with a projectional skeleton has a projectional resolution of the identity and has a norming space with similar properties to spaces we characterize the existence of a projectional skeleton in terms of elementary substructures providing simple proofs of known results concerning weakly compactly generated spaces and plichko spaces we prove a preservation result for plichko banach spaces involving transfinite sequences of projections as a corollary we show that a banach space is plichko if and only if it has a commutative projectional skeleton

analysis of expanded mixed finite element methods for a nonlinear parabolic equation modeling flow into variably saturated porous media we present an analysis of expanded mixed finite element methods applied to richards equation a nonlinear parabolic partial differential equation modeling the flow of water into a variably saturated porous medium we consider the full range of saturated to completely unsaturated media in the case of the lowest order raviart thomas spaces and the range of all possible saturations we bound the h 1 norm of the error in capacity in terms of approximation error this estimate uses a time integrated scheme and the kirchhoff transformation to handle a degeneracy in the case of completely unsaturated flow optimal convergence is then shown for a nonlinear form related to the error in the capacity for the case of saturated to partially saturated flow convergence rates depending on the holder continuity of the capacity term are derived last optimal convergence of pressures and fluxes is stated for the case of strictly partially saturated flow