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

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

logarithmic lower bounds in the cell probe model we develop a new technique for proving cell probe lower bounds on dynamic data structures this technique enables us to prove an amortized randomized omega lg n lower bound per operation for several data structural problems on n elements including partial sums dynamic connectivity among disjoint paths or a forest or a graph and several other dynamic graph problems by simple reductions such a lower bound breaks a long standing barrier of omega lg n lg lg n for any dynamic language membership problem it also establishes the optimality of several existing data structures such as sleator and tarjan s dynamic trees we also prove the first omega log b n lower bound in the external memory model without assumptions on the data structure such as the comparison model our lower bounds also give a query update trade off curve matched e g by several data structures for dynamic connectivity in graphs we also prove matching upper and lower bounds for partial sums when parameterized by the word size and the maximum additive change in an update

two parameter degenerate sliding bifurcations in filippov systems this paper is concerned with the extension to the case of codimension 2 degenerate sliding bifurcations of the theory of sliding bifurcations in filippov systems presented in m di bernardo p kowalczyk a nordmark bifurcations of dynamical systems with sliding derivation of normal form mappings physica d 170 2002 175 205 these bifurcations were detected in experimental systems such as the dry friction oscillator and turn out to be organising centres for branches of codimension 1 sliding bifurcations the analysis is carried out for generic n dimensional piecewise smooth systems the possible degenerate scenarios are classified it is shown that several branches of codimension 1 sliding bifurcations originate from the degenerate codimension 2 points such branches are appropriately classified in the degenerate crossing sliding case a friction oscillator is used as a representative example to illustrate and confirm the theoretical derivations the importance is discussed of the unfolding of the degenerate sliding bifurcations for the development of continuation techniques

limit cycles of a lienard system with symmetry allowing for discontinuity abstract this paper presents new results on the limit cycles of a lienard system with symmetry allowing for discontinuity our results generalize and improve the results in 32 theorem 1 and 2 or the monograph 33 chapter 4 theorem 5 2 the results in 33 are only valid for the smooth system we emphasize that our main results are valid for discontinuous systems moreover we show the presence and an explicit upper bound for the amplitude of the two limit cycles and we estimate the position of the double limit cycle bifurcation surface in the parameter space until now there is no result to determine the amplitude of the two limit cycles the existing results on the amplitude of limit cycles guarantee that the lienard system has a unique limit cycle finally some applications and examples are provided to show the effectiveness of our results we revisit a co dimension 3 lienard oscillator see 22 34 in application 1 li and rousseau 22 studied the limit cycles of such a system when the parameters are small however for the general case of the parameters in particular the parameters are large the upper bound of the limit cycles remains open we completely provide the bifurcation diagram for the one equilibrium case moreover we determine the amplitude of the two limit cycles and estimate the position of the double limit cycle bifurcation surface for the one equilibrium case application 2 is presented to study the limit cycles of a class of the filippov system

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

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

bifurcations of dynamical systems with sliding derivation of normal form mappings this paper is concerned with the analysis of so called sliding bifurcations in n dimensional piecewise smooth dynamical systems with discontinuous vector field these novel bifurcations occur when the system trajectory interacts with regions on the discontinuity set where sliding is possible the derivation of appropriate normal form maps is detailed it is shown that the leading order term in the map depends on the particular bifurcation scenario considered this is in turn related to the possible bifurcation scenarios exhibited by a periodic orbit undergoing one of the sliding bifurcations discussed in the paper a third order relay system serves as a numerical example

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

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

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

normal form and limit cycle bifurcation of piecewise smooth differential systems with a center abstract in this paper we prove that any center either nondegenerate or degenerate of a planar piecewise c r smooth vector field z is topologically equivalent to that of z 0 x y 1 2 x for y 0 x y 1 2 x for y 0 and that the homeomorphism between z and z 0 is c r smoothness when restricted to each side of the switching line except at the center p we illustrate by examples that there are degenerate centers whose flows are conjugate to that of z 0 and also there exist nondegenerate centers whose flows cannot be conjugate to that of z 0 finally applying the normal form z 0 together with the piecewise smooth equivalence we study the number of limit cycles which can be bifurcated from the center of z

on the difference between real and complex arrangements if b is an arrangement of linear complex hyperplanes in c d then the following can be constructed from knowledge of its intersection lattice a the cohomology groups of the complement br b the cohomology algebra of the complement os c the fundamental group of the complement if d 2 d the singularity link up to homeomorphism if d 3 e the singularity link up to homotopy type zz if bis more generally a 2 arrangement in ir 2d an arrangement of real subspaces of codimension 2 with even dimensional intersections then the intersection lattice still determines a the cohomology groups of the complement gm and e the homotopy type of the singularity link zz we show however that for 2 arrangements the data b c and d are not deter mined by the intersection lattice they require the knowledge of extra information on sign patterns which can be computed as determinants of linear relations or equiva lently as linking coefficients in the sense of knot theory

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

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

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

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

degenerate grazing sliding bifurcations in planar filippov systems abstract in this paper we contribute to a class of degenerate grazing sliding bifurcations in planar filippov systems assuming that the unperturbed system has a generalized homoclinic loop to a fold fold point depending on the type of this fold fold point we exhibit some codimension 2 bifurcation diagrams as well as the asymptotics of all bifurcation curves our work not only reports several complete unfoldings in the investigation of degenerate grazing sliding bifurcations but also answers the open problem about the maximum number of limit cycles bifurcating from the generalized homoclinic loop for the case of fold fold presented in liang et al 2013 30

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

a note on finding minimum cost edge disjoint spanning trees let g be an undirected graph with n vertices and m edges such that each edge has a real valued cost we consider the problem of finding a set of k edge disjoint spanning trees in g of minimum total edge cost this problem can be solved in polynomial time by the matroid greedy algorithm we present an implementation of this algorithm that runs in o m log m k2n2 time if all edge costs are the same the algorithm runs in o k2n2 time the algorithm can also be extended to find the largest k such that k edge disjoint spanning trees exist in o m2 time we mention several applications of the algorithm

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

bifurcations in piecewise smooth feedback systems this paper is concerned with the analysis and classification of non standard bifurcations in piecewise smooth feedback systems bifurcations involving fixed points in maps equilibria and limit cycles in flows are considered with particular attention to border collisions of fixed points in maps non smooth bifurcations of equilibria grazing bifurcations and sliding bifurcations of limit cycles in flows the aim is to describe existing and novel results to form the basis of a consistent theory of bifurcations in such systems in so doing a novel approach to classify non smooth bifurcations of equilibria in flows is presented experimental results on pulse width modulated voltage mode controlled dc dc power converters motivates the relevance to applications of the analysis proposed

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

homotopy types of subspace arrangements via diagrams of spaces we prove combinatorial formulas for the homotopy type of the union of the subspaces in an affine compactified affine spherical or projective subspace arrangement from these formulas we derive results of goresky macpherson on the homology of the arrangement and the cohomology of its complement the union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset a closely related space is obtained by taking the homotopy direct limit of this diagram our method consists in constructing a combinatorial model diagram over the same poset whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces

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

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

fully dynamic arboricity maintenance abstract given an undirected graph its arboricity is the minimum number of edge disjoint forests that its edge set can be partitioned into we develop the first fully dynamic algorithms to determine the arboricity of a graph under edge insertions and deletions while our insertion algorithm is based on known static algorithms to determine the arboricity our deletion algorithm is to the best of our knowledge new our algorithms take o m log n time to insert or delete an edge where m is the number of edges in the graph while the best static algorithm to compute arboricity takes o m 3 2 log n 2 m time 9 we complement our upper bound with a lower bound of amortized log n time for an update for an algorithm that maintains a forest decomposition of size arboricity of the graph under edge insertions and deletions

grazing and border collision in piecewise smooth systems a unified analytical framework a comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes this links grazings with border collisions in nonsmooth maps contrary to previous literature piecewise linear maps correspond only to nonsmooth discontinuity boundaries all other maps have either square root or 3 2 type singularities

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

on representations and k theory of the braid groups let be the fundamental group of the complement of a k 1 hyperplane arrangement such as artin s pure braid group or more generally a homologically toroidal group as defined below the triviality of bundles arising from orthogonal representations of is characterized completely as follows an orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups furthermore the subgroup of elements in the complex k theory of b which arises from complex unitary representations of is shown to be trivial in the case of real k theory the subgroup of elements which arises from real orthogonal representations of is an elementary abelian 2 group which is characterized completely in terms of the first two stiefel whitney classes of the representation in addition quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the jacobi identity for certain choices of poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized heisenberg groups these cohomology relations correspond to non trivial spin representations of the pure braid groups which give rise to trivial bundles

non periodic motion caused by grazing incidence in an impact oscillator the motion of a single degree of freedom periodically forced oscillator subjected to a rigid amplitude constraint is considered using analytical methods the singularities caused by grazing impact are studied it is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter a special type of bifurcation occurs the motion after the bifurcation may be non periodic and a criterion for this based on orientation and eigenvalues is given

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

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

does distribution theory contain means for extending poincare bendixson theory we use the theory of distributions to extend the poincare bendixson theorem and the bendixson criterion to piecewise lipschitz continuous system possessing unique and continuous solutions we demonstrate the use of these extensions by several examples that have recently appeared in the literature

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