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

posynomial geometric programming with parametric uncertainty geometric programming provides a powerful tool for solving nonlinear problems where nonlinear relations can be well presented by exponential or power function this paper develops a procedure to derive the lower and upper bounds of the objective of the posynomial geometric programming problem when the cost and constraint parameters are uncertain the imprecise parameters are represented by ranges instead of single values an imprecise geometric program is transformed to a family of conventional geometric programs to calculate the objective value the derived result is also in a range where the objective value would appear the ability of calculating the bounds of the objective value developed in this paper might help lead to more realistic modeling efforts in engineering design areas

degenerate hopf bifurcations in a family of ff type switching systems abstract the theory of hopf bifurcation for switching systems is different from that for smooth systems because the nonsmoothness may produce more limit cycles although some results were obtained and several versions of order were given for switching weak foci few works were found for switching centers or the general relationship between order and cyclicity in this paper we study degenerate hopf bifurcations in a family of ff type switching systems whose unperturbed upper system and lower one both have the same equilibrium of linear center type on the sole switching line we use fractional orders to uniform the concept of order of weak foci for both the smooth case and the switching one and generally discuss the relationship between the order of weak foci and the cyclicity and give the uniform upper bound for the cyclicity in terms of the bautin depth of the ideal generated by the switching lyapunov quantities moreover we investigate the independence of those switching lyapunov quantities we finally apply our results to a family of switching quadratic bautin systems and prove that the cyclicity is at least 5 and 8 in the cases of weak foci and centers respectively

weighted pseudo antiperiodic solutions for fractional integro differential equations in banach spaces in this paper we prove the existence of weighted pseudo antiperiodic mild solutions for fractional integro differential equations in the form d u t au t t a t s au s ds f t u t t r where f u is stepanov like weighted pseudo antiperiodic and a generates a resolvent family of bounded and linear operators on a banach space x a l loc 1 r and 0 here the fractional derivative is considered in the sense of weyl also we give a short proof to show that the vector valued space of stepanov like weighted pseudo antiperiodic functions is a banach space

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

limit cycles bifurcate from centers of discontinuous quadratic systems like for smooth quadratic systems it is important to determine the maximum order of a fine focus and the cyclicity of discontinuous quadratic systems previously examples of discontinuous quadratic systems with five limit cycles bifurcated from a fine focus of order 5 have been constructed in this paper we construct a class of discontinuous quadratic systems with a fine focus of order 9 in addition by using a method similar to that developed by c christopher for smooth systems which allows one to estimate the cyclicity just from the lower order terms of lyapunov constants we show that the cyclicity of discontinuous quadratic systems is at least 9 thus improving on previous results

composition of pseudo almost automorphic and asymptotically almost automorphic functions this paper is concerned with pseudo almost automorphic functions which are more general and complicated than pseudo almost periodic functions and asymptotically almost automorphic functions new results concerning the composition of pseudo almost automorphic functions are established

weighted pseudo almost periodic solutions to some differential equations abstract the paper considers some new classes of functions called weighted pseudo almost periodic functions which implement in a natural fashion the classical pseudo almost periodic functions due to zhang properties of these weighted pseudo almost periodic functions are discussed including a composition result for weighted pseudo almost periodic functions the results obtained are subsequently utilized to study the existence and uniqueness of a weighted pseudo almost periodic solution to the heat equation with dirichlet conditions

a linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system abstract in this paper we study a class of planar piecewise smooth quadratic integrable non hamiltonian systems which have a center by using the averaging method we give an estimation of the number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n our estimation is linear depending on n and it is at least twice the associated estimation of smooth systems

stepanov like almost automorphic functions and monotone evolution equations abstract in this paper we are concerned with a new class of stepanov like almost automorphic s p a a functions with values in a banach space x this class contains the space aa x of all bochner almost automorphic functions we use the results obtained to prove the existence and uniqueness of a weak s p a a solution to the parabolic equation u t a t u f t in a reflexive banach space assuming some appropriate conditions of monotonicity coercitivity of the operators a t and s p almost automorphy of the forced term f t this result extends a known result in the case of almost periodicity an application is also given

the focus center limit cycle bifurcation in symmetric 3d piecewise linear systems the birth of limit cycles in 3d three dimensional piecewise linear systems for the relevant case of symmetrical oscillators is considered a technique already used by the authors in planar systems is extended to cope with 3d systems where a greater complexity is involved under some given nondegeneracy conditions the corresponding theorem characterizing the bifurcation is stated in terms of the deviation from the critical value of the bifurcation parameter expressions in the form of power series for the period amplitude and the characteristic multipliers of the bifurcating limit cycle are also obtained the results are applied to accurately predict the birth of symmetrical periodic oscillations in a 3d electronic circuit genealogically related to the classical van der pol oscillator

weighted pseudo almost automorphic mild solutions for two term fractional order differential equations using the notion of weyl fractional derivative we study the existence and uniqueness of weighted pseudo almost automorphic mild solutions for a class of two term fractional order abstract differential equation in the form d t u t d t u t a u t d t f t u t t 0 0 1 0 where a is a closed linear operator defined in a banach space x and the forcing term f is sp weighted pseudo almost automorphic

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

jumps of the fundamental solution matrix in discontinuous systems and applications abstract this paper deals with differential equations with discontinuous right hand side the concept of a solution for a discontinuous system is defined on the basis of differential inclusions using filippov s method we study in particular the behaviour of solutions crossing a discontinuity surface transversally a formula characterizing jumps of the fundamental solution matrix is derived as an application of it the concept of poincare mapping is defined for such systems

weighted pseudo almost automorphic functions and applications to abstract differential equations abstract this paper is concerned with weighted pseudo almost automorphic functions which are more general and complicated than weighted pseudo almost periodic functions using these properties we establish an existence and uniqueness theorem for pseudo almost automorphic mild solutions to semilinear differential equations in a banach space which extends known results

bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems abstract the generalized homoclinic loop appears in the study of dynamics on piecewise smooth differential systems during the past two decades for planar piecewise smooth differential systems there are concrete examples showing that under suitable perturbations of a generalized homoclinic loop one or two limit cycles can appear but up to now there is no a general theory to study the cyclicity of a generalized homoclinic loop that is the maximal number of limit cycles which are bifurcated from it in this paper we provide some sufficient conditions on the cyclicity of some homoclinic loops especially we prove the existence of one or two limit cycles which are bifurcated from a generalized homoclinic loop of an unperturbed piecewise smooth differential systems also we obtain a class of piecewise differential system in which two limit cycles can be bifurcated from a generalized homoclinic loop with multiplicity one the phenomena cannot happen in the smooth differential systems finally we provide five concrete piecewise smooth differential systems showing the applications of our theories on this phenomenon

new global bifurcation diagrams for piecewise smooth systems transversality of homoclinic points does not imply chaos abstract in this paper we consider some piecewise smooth 2 dimensional systems having a possibly non smooth homoclinic t we assume that the critical point 0 lies on the discontinuity surface 0 we consider 4 scenarios which differ for the presence or not of sliding close to 0 and for the possible presence of a transversal crossing between t and 0 we assume that the systems are subject to a small non autonomous perturbation and we obtain 4 new bifurcation diagrams in particular we show that in one of these scenarios the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinics

measure theory and pseudo almost automorphic functions new developments and applications abstract in this work we establish a new concept of weighted pseudo almost automorphic functions using the measure theory we present new results on weighted ergodic functions like completeness and composition theorems the theory of this work generalizes the classical results on weighted pseudo almost periodic and automorphic functions for illustration we provide some applications for evolution equations which include reaction diffusion systems and partial functional differential equations

decomposition of weighted pseudo almost periodic functions abstract first we show by constructing two counterexamples that the decomposition of weighted pseudo almost periodic functions is not unique in general then we prove that the decomposition of such functions is unique if p a p 0 x is translation invariant but not necessarily unique without the assumption moreover we give an example to show that the mean value under a certain weight may not exist for all almost periodic functions with these results we answer some fundamental questions on weighted pseudo almost periodic functions

low dimensional maps for piecewise smooth oscillators abstract dynamics of the piecewise smooth nonlinear oscillators is considered for which general methodology of reducing multidimensional flows to low dimensional maps is proposed this includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution the global map is comprised of local maps which are constructed in the smooth sub regions of phase space to construct this low dimensional map it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory the dimension reduction is directly related to the dimension of the chosen boundary and the lower its dimension is the larger dimension reduction can be achieved full details are given for a drifting impact oscillator where the five dimensional flow is reduced to one dimensional 1d approximate analytical map first an exact two dimensional map has been formulated and analysed a further reduction to 1d approximate map is introduced and discussed standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos and co existence of multiple attractors accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters

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

weighted pseudo almost periodic functions and applications to evolution equations with delay in this paper we introduce a new class of vector valued weighted pseudo almost periodic functions and present some basic properties about these new functions moreover we obtain an new existence theorem of vector valued weighted pseudo almost periodic solution to the evolution equations x t ax t f t x t with delay in a banach space x

on hopf bifurcations of piecewise planar hamiltonian systems abstract in this paper we study the number of limit cycles appearing in hopf bifurcations of piecewise planar hamiltonian systems for the case that the hamiltonian function is a piecewise polynomials of a general form we obtain lower and upper bounds of the number of limit cycles near the origin respectively for some systems of special form we obtain the hopf cyclicity

melnikov method for homoclinic bifurcation in nonlinear impact oscillators based on an inverted pendulum impacting on rigid walls under external periodic excitation a class of nonlinear impact oscillators is discussed for its homoclinic bifurcation the melnikov method established for smooth dynamical systems is extended to be applicable to the nonsmooth one for nonlinear impact systems closed form solutions between impacts are generally unavailable the absence of closed form solutions makes difficulties in estimation of the gap between the stable manifold and unstable manifold in this paper we give a method to compute the melnikov functions up to the n t h order so as to obtain conditions of parameters for the persistence of homoclinic cycles which are formed via the identification given by the impact rule

melnikov s method for a general nonlinear vibro impact oscillator abstract melnikov s method is presented for a general nonlinear vibro impact oscillator and melnikov function for homoclinic orbits is obtained analytically a duffing vibro impact oscillator is given to illustrate the application of the procedures the obtained results are verified by the phase portrait poincare surface of section and bifurcation diagrams

existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations this paper is concerned with the study of weighted pseudo almost periodic solutions to some classes of hyperbolic differential equations upon making some suitable assumptions the existence and uniqueness of a weighted pseudo almost periodic solution is obtained to illustrate our main result we study the existence of a weighted pseudo almost periodic solution to a system of partial differential equations arising in control systems described by abstract retarded functional differential equations with feedback control

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

isochronicity of centers in a switching bautin system abstract in this paper isochronicity of centers is discussed for a class of discontinuous differential system simply called switching system we give some sufficient conditions for the system to have a regular isochronous center at the origin and on the other hand construct a switching system with an irregular isochronous center at the origin we give a computation method for periods of periodic orbits near the center and use the method to discuss a switching bautin system for center conditions and isochronous center conditions we further find all of those systems which have an irregular isochronous center

almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional applications to heat and wave equations with nonlinearities in this work we study the existence of bounded and almost automorphic solutions for evolution equations in banach spaces we suppose that the linear part is the infinitesimal generator of a compact c0 semigroup of bounded linear operators and the nonlinear part is an almost automorphic function with respect to the second argument we give sufficient conditions ensuring the existence of an almost automorphic solution when there is at least one bounded solution on r we use the subvariant functional method to show that every k minimizing mild solution is compact almost automorphic applications are provided for both heat and wave equations with nonlinearities in several functional spaces

limit cycles bifurcating from piecewise quadratic systems separated by a straight line abstract in this paper we are concerned with limit cycles bifurcating from piecewise quadratic systems separated by a straight line by means of the first order averaging theory of discontinuous differential systems and the argument principle we present an estimate of the maximum number of limit cycles bifurcating from the period annulus around the center of piecewise smooth systems

weighted pseudo almost automorphic functions and wpaa solutions to semilinear evolution equations abstract in this paper we reveal several basic properties about nonlinear vector valued weighted pseudo almost automorphic functions including equivalence completeness translation invariance composition theorem and convolution theorem of these functions we also give some concrete examples to illustrate our results finally we obtain a new existence theorem of nonlinear weighted pseudo almost automorphic solutions for semilinear evolution equations in banach spaces

border collision bifurcations including period two to period three for piecewise smooth systems abstract we examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter in the simplest case there is a surface in phase space along which the map has no derivative or has two one sided derivatives is the border of two regions in which the map is smooth as the parameter is varied a fixed point e may collide with the border and we may assume that this collision occurs at 0 a variety of bifurcations occur frequently in such situations but never or almost never occur in smooth systems in particular e may cross the border and so will exist for 0 but it may be a saddle in one case say 0 for 0 there may be a stable period 3 orbit which similarly shrinks to e0 as 0 hence one observes the following stable periodic orbits a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit we also see analogously stable period 2 to stable period p orbit bifurcations with p 5 11 52 or period 2 to quasi periodic or even to a chaotic attractor we believe this phenomenon will be seen in many applications

on generalized geometric programming problems with non positive variables generalized geometric programming ggp problems occur frequently in engineering design and management some exponential based decomposition methods have been developed for solving global optimization of ggp problems however the use of logarithmic exponential transformations restricts these methods to handle the problems with strictly positive variables this paper proposes a technique for treating non positive variables with integer powers in ggp problems by means of variable transformation the ggp problem with non positive variables can be equivalently solved with another one having positive variables in addition we present some computationally efficient convexification rules for signomial terms to enhance the efficiency of the optimization approach numerical examples are presented to demonstrate the usefulness of the proposed method in ggp problems with non positive variables

a cubic system with thirteen limit cycles abstract we construct a planar cubic system and demonstrate that it has at least 13 limit cycles the construction is essentially based on counting the number of zeros of some abelian integrals

non smooth quadratic centers defined in two arbitrary sectors in this paper we analyze the center focus problem of some families of piecewise planar quadratic vector fields on two zones of r 2 the zones we consider are two unbounded sectors defined by an arbitrary angle and a fixed vertex we also assume that each vector field share a common weak focus singularity at the vertex of the boundary we observe how the center variety depends on the angle

limit cycles for discontinuous quadratic differential systems with two zones abstract in this paper we study the maximum number of limit cycles given by the averaging theory of first order for discontinuous differential systems which can bifurcate from the periodic orbits of the quadratic isochronous centers x y x 2 y x x y and x y x 2 y 2 y x 2 x y when they are perturbed inside the class of all discontinuous quadratic polynomial differential systems with the straight line of discontinuity y 0 comparing the obtained results for the discontinuous with the results for the continuous quadratic polynomial differential systems this work shows that the discontinuous systems have at least 3 more limit cycles surrounding the origin than the continuous ones

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

bifurcation and chaos near sliding homoclinics we study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface we assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations following a functional analytic approach we construct a melnikov like function m in such a way that if m has a simple zero at some point then the system has solutions that behave chaotically applications of this result to quasi periodic systems are also given

impact oscillators with homoclinic orbit tangent to the wall abstract homoclinic bifurcation for a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation is analyzed under the hypothesis that the unperturbed system has a homoclinc orbit tangent to the wall consequently the impact surface cannot be chosen as the poincare section to measure the distance between the perturbed stable and unstable manifolds furthermore compared to the case that the unperturbed homoclinic orbit intersects the wall transversally more cases are involved as the parameters vary thus the analysis of the homoclinic orbit tangent to the wall is much more difficult in this paper by using a method of melnikov type we derive sufficient conditions under which the perturbed stable and unstable manifolds intersect transversally as an application an impact oscillator of duffing type is studied in detail

bifurcation of critical periods for plane vector fields on etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l origine pour toutes les valeurs d un parametre r n on definit la fonction periode p on determine les points critiques de cette fonction pres du centre avec comme parametre de bifurcation

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

bifurcation of limit cycles by perturbing a piecewise linear hamiltonian system with a homoclinic loop abstract in this paper we study limit cycle bifurcations for a kind of non smooth polynomial differential systems by perturbing a piecewise linear hamiltonian system with the center at the origin and a homoclinic loop around the origin by using the first melnikov function of piecewise near hamiltonian systems we give lower bounds of the maximal number of limit cycles in hopf and homoclinic bifurcations and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in e in the case when the degree of perturbing terms is low we obtain a precise result on the number of zeros of the first melnikov function

investigation of path following algorithms for signomial geometric programming problems this paper considers signomial geometric programming gp dual problems a class of nonconvex nonlinear programming problems possessing multiple locally optimal solutions the primary purpose of this paper is to investigate the quality of solutions found by use of a path following algorithm the path following method may be applied to either the original nonconvex problem or to each of a sequence of convex posynomial gp problems approximating the original problem for each test problem the algorithms were initiated with thousands of different starting points it was determined that when the stopping criterion was relaxed for early posynomial gp problems in the sequence the ultimate solution tended to be of better quality and more frequently globally optimal

on normal form calculations in impact oscillators normal form calculations are useful for analysing the dynamics close to bifurcations however the application to non smooth systems is a topic for current research here we consider a class of imp

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

nonsmooth homoclinic orbits melnikov functions and chaos in discontinuous systems abstract we first study the problem of chaotic behaviour in time perturbed discontinuous systems whose unperturbed part has a piecewise c 1 homoclinic solution transversally crossing the discontinuity manifolds we show that if a certain melnikov function has a simple zero at some point then the system has solutions that behave chaotically the melnikov function is explicitly constructed for two dimensional systems and extends the usual melnikov function for the smooth case in the second part we extend these results to sliding homoclinic bifurcations we also mention some possibilities for further research

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

new lower bound for the hilbert number in piecewise quadratic differential systems abstract we study the number of limit cycles bifurcating from a piecewise quadratic system all the differential systems considered are piecewise in two zones separated by a straight line we prove the existence of 16 crossing limit cycles in this class of systems if we denote by h p n the extension of the hilbert number to degree n piecewise polynomial differential systems then h p 2 16 as fas as we are concerned this is the best lower bound for the quadratic class moreover in the studied cases all limit cycles appear nested bifurcating from a period annulus of a isochronous quadratic center

bifurcation of periodic orbits in a class of planar filippov systems abstract in this paper we discuss the perturbations of a general planar filippov system with exactly one switching line when the system has a limit cycle we give a condition for its persistence when the system has an annulus of periodic orbits we give a condition under which limit cycles are bifurcated from the annulus we also further discuss the stability and bifurcations of a nonhyperbolic limit cycle when the system has an annulus of periodic orbits we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching

an interior point algorithm to solve computationally difficult set covering problems we present an interior point approach to the zero one integer programming feasibility problem based on the minimization of a nonconvex potential function given a polytope defined by a set of linear inequalities this procedure generates a sequence of strict interior points of this polytope such that each consecutive point reduces the value of the potential function an integer solution not necessarily feasible is generated at each iteration by a rounding scheme the direction used to determine the new iterate is computed by solving a nonconvex quadratic program on an ellipsoid we illustrate the approach by considering a class of difficult set covering problems that arise from computing the 1 width of the incidence matrix of steiner triple systems

a class of reversible quadratic systems with piecewise polynomial perturbations abstract this paper investigates a class of reversible quadratic systems perturbed inside piecewise polynomial differential systems of arbitrary degree n all possible phase portraits of the reversible quadratic systems on the plane are obtained by analytic techniques then we present the algebraic structure of its corresponding first order melnikov function which can be used to study the limit cycle bifurcation problem in this direction we use it to investigate the hopf bifurcation of the perturbed systems and obtain an upper bound for the hopf cyclicity meanwhile we employ it to discuss the maximal number of limit cycles emerging from the periodic orbits around the origin for a special case and also provide its upper bound

on completeness of the space of weighted pseudo almost automorphic functions abstract in this paper we prove that for every u the space of weighted pseudo almost automorphic functions is complete under the supremum norm this gives an affirmative answer to a key and fundamental problem for weighted pseudo almost automorphic functions and fills a gap in the proof of liu and song 2010 9

center conditions in a switching bautin system a new method with an efficient algorithm is developed for computing the lyapunov constants of planar switching systems and then applied to study bifurcation of limit cycles in a switching bautin system a complete classification on the conditions of a singular point being a center in this bautin system is obtained further an example of switching systems is constructed to show the existence of 10 small amplitude limit cycles bifurcating from a center this is a new lower bound of the maximal number of small amplitude limit cycles obtained in quadratic switching systems near a singular point

a polynomial barrier algorithm for linearly constrained convex programming problems we consider the standard linearly constrained convex program min fx x n ax b x 0 where fx satisfies a certain scaled lipschitz condition using the classical logarithmic barrier method our algorithm obtains an e solution within o n ln e or on ln e total iterations depending on the updating of the barrier parameter

control of near grazing dynamics in impact oscillators a method is presented for controlling the persistence of a local attractor near a grazing periodic trajectory in a piecewise smooth dynamical system in the presence of discontinuous jumps in the state associated with intersections with system discontinuities in particular it is shown that a discrete linear feedback strategy may be employed to retain the existence of an attractor near the grazing trajectory such that the deviation of the attractor from the grazing trajectory goes to zero as the system parameters approach those corresponding to grazing contact the implementation relies on a local analysis of the near grazing dynamics using the concept of discontinuity mappings numerical results are presented for a linear and a nonlinear oscillator