uniform bounds for strongly competing systems the optimal lipschitz case for a class of systems of semi linear elliptic equations including delta u i f i x u i beta u i sum j neq i a ij u j p quad i 1 dots k for p 2 variational type interaction or p 1 symmetric type interaction we prove that uniform l infty boundedness of the solutions implies uniform boundedness of their lipschitz norm as beta to infty that is in the limit of strong competition this extends known quasi optimal regularity results and covers the optimal case for this class of problems the proofs rest on monotonicity formulae of alt caffarelli friedman and almgren type in the variational setting and on the caffarelli jerison kenig almost monotonicity formula in the symmetric one

analysis of grazing bifurcations of quasiperiodic system attractors this paper presents the first application of the discontinuity mapping approach to the study of near grazing bifurcations of originally quasiperiodic co dimension two system attractors the paper establishes an exact formulation for the discontinuity mapping methodology under the assumption that a poincare section can be found that is everywhere transversal to the grazing attractor in particular it is shown that while a reduced formulation may be employed successfully in the case of co dimension one attractors it fails to capture dynamics in directions transversal to the original quasiperiodic attractor this shortcoming necessitates the full machinery presented here the generality of the proposed approach is illustrated through numerical analysis of two nonlinear dynamical systems of dimension three and four

variational problems with two phases and their free boundaries the problem of minimizing vu 2 q2 x 2 v dx in an appropriate class of functions v is considered here q x 0 and a2 t x2 if v 0 any minimizer u is harmonic in u 0 and vu 2 has a jump a2 x 22 across the free boundary u 0 regularity and various properties are established for the minimizer u and for the free boundary introduction in this paper we consider the problem of minimizing j v f vv 2 q2 x x2 v dx vgk where q2 x 0 a22 ifu 0 and x2 u is lower semicontinuous at v 0 it is assumed that a2 0 and a a2 a22 0 the class k consists of all functions v in l loc s2 with vu g l2 i2 such that v u on a given open subset s of 3i2 and s is a domain in r the analogous problem for functions in k v k v 0 a e was studied in 1 in that paper it was proved that any local minimizer of j v in k is lipschitz continuous and if n 2 the free boundary 8 w 0 is analytic if q x is analytic the present variational problem is motivated by applications to the now of two liquids in models of jets and cavities these applications will be studied in other forthcoming papers 5 6 the present work is aimed at extending results of 1 in particular we shall establish nondegeneracy theorems the lipschitz continuity of the solution and some properties of the free boundary for n 2 the free boundary is proved to be continuously differentiable a new and rather powerful tool introduced in this paper is the monotonicity formula lemma 5 1 asserting that for a minimizer u if u x0 0 then

degenerate discontinuity induced bifurcations in tapping mode atomic force microscopy abstract this paper documents the existence of degenerate bifurcation scenarios in the low contact velocity dynamics during tapping mode atomic force microscopy specifically numerical analysis of a model of the microscope dynamics shows branch point and isola bifurcations associated with the emergence of two families of saddle node bifurcation points along a branch of low amplitude oscillations the paper argues for the origin of the degenerate bifurcations in the existence of a periodic steady state trajectory that i achieves tangential contact with a discontinuity surface in a piecewise smooth model of the cantilever response and ii retracts from the surface under variations in either direction along a line segment in parameter space specifically the discontinuity mapping technique is here rigorously applied to a general situation of such degenerate contact showing the codimension two nature of these bifurcations for appropriately chosen parameter values the discontinuity mapping based normal form derived here is a novel extension of that derived in dankowicz and nordmark 2000 28 in the case that ii does not hold in addition the paper includes a quantitative reflection on the relative importance of discontinuities in the attractive and repulsive force components in producing the predicted bifurcations

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

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

uniform holder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames the main result of this paper is a general holder estimate in a class of singularly perturbed elliptic systems this estimate is applied to the study of the well known burke schuman approximation in flame theory after reviewing some classical cases equidiffusional case high activation energy approximation we turn to the non equidiffusional case and to nonlinear diffusion models the limiting problems are nonlinear elliptic equations they have holder or lipschitz maximal global regularity a natural question is then whether this regularity is kept uniformly throughout the approximation process and we show that this is true in general

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

corner collision implies border collision bifurcation abstract this paper analyses a so called corner collision bifurcation in piecewise smooth systems of ordinary differential equations odes for which a periodic solution grazes with a corner of the discontinuity set it is shown under quite general circumstances that this leads to a normal form that is to lowest order a piecewise linear map this is the first generic derivation from ode theory of the so called c bifurcation or border collision for piecewise linear maps the result contrasts with the equivalent results when a periodic orbit grazes with a smooth discontinuity set which has recently been shown to lead to maps that have continuous first derivatives but not second moreover it is shown how to calculate the piecewise linear map for arbitrary dimensional systems using only properties of the single periodic trajectory undergoing corner collision the calculation is worked out for two examples including a model for a commonly used power electronic converter where complex dynamics associated with corner collision was previously found numerically but is explained analytically here for the first time

existence of multiple solutions to a class of nonlinear schrodinger system with external sources terms abstract we study a class of nonlinear schrodinger system with external sources terms as perturbations in order to obtain existence of multiple solutions this system arises from bose einstein condensates etc as these external sources terms are positive functions and small in some sense we use nehari manifold to get the existence of a positive ground state solution and a positive bound state solution

a periodically forced piecewise linear oscillator a single degree of freedom non linear oscillator is considered the non linearity is in the restoring force and is piecewise linear with a single change in slope such oscillators provide models for mechanical systems in which components make intermittent contact a limiting case in which one slope approaches infinity an impact oscillator is also considered harmonic subharmonic and chaotic motions are found to exist and the bifurcations leading to them are analyzed

angle theorems for the lagrangian mean curvature flow we prove that symplectic maps between riemann surfaces l m of constant nonpositive and equal curvature converge to minimal symplectic maps if the lagrangian angle alpha for the corresponding lagrangian submanifold in the cross product space l times m satisfies text osc alpha le pi if one considers a 4 dimensional kahler einstein manifold overline m of nonpositive scalar curvature that admits two complex structures j k which commute and assumes that l subset overline m is a compact oriented lagrangian submanifold w r t j such that the kahler form overline kappa w r t k restricted to l is positive and text osc alpha le pi then l converges under the mean curvature flow to a minimal lagrangian submanifold which is calibrated w r t overline kappa

local analysis of co dimension one and co dimension two grazing bifurcations in impact microactuators impact microactuators rely on repeated collisions to generate gross displacements of a microelectromechanical machine element without the need for large applied forces their design and control rely on an understanding of the critical transition between non impacting and impacting long term system dynamics and the associated changes in system behavior in this paper we present three co dimension one characteristically distinct transition scenarios associated with grazing conditions for a periodic response of an impact microactuator a discontinuous jump to an impacting periodic response associated with parameter hysteresis a continuous transition to an impacting chaotic attractor and a discontinuous jump to an impacting chaotic attractor using the concept of discontinuity mappings a theoretical analysis is presented that predicts the character of each transition from a set of quantities that are computable in terms of system properties at grazing specifically we show how this analysis can be applied to predict the bifurcation behavior on neighborhoods of two co dimension two bifurcation points that separate the co dimension one bifurcation scenarios the predictions are validated against results from numerical simulations of a model impact microactuator

numerical and experimental investigations of the nonlinear dynamics and chaos in non smooth systems nature is full of non smooth nonlinearities that are usually related to either friction phenomena or the discontinuous characteristics of intermittent contacts in general non smooth characteristics cause difficulties in the modeling and simulation of natural systems a smoothed switch model is used in this contribution in order to analyze non smooth systems the procedure is shown to be an effective method of dealing with this kind of system and has advantages in numerical implementation as an application of the general formulation a single degree of freedom oscillator with discontinuous support is analyzed an experimental apparatus is developed in order to verify the capability of the mathematical formulation and the numerical procedure used to describe the system general behavior this apparatus is instrumented to obtain all system state variables making it possible to compare experimental results with those obtained through numerical simulations the system s dynamical behavior shows a rich response that includes dynamical jumps bifurcations and chaos different aspects related to nonlinear dynamics are carried out showing that numerical simulations are in close agreement with experimental data

discontinuity induced bifurcations in models of mechanical contact capillary adhesion and cell division a common framework abstract this paper collects four distinct instances of grazing contact of a periodic trajectory in a hybrid dynamical system under a common abstract framework and establishes selected general properties of the associated near grazing dynamics in particular it is shown that for critical choices of parameter values commonly used physical models of rigid or compliant mechanical contact capillary adhesion and cell division satisfy the conditions required by the general framework the paper relies on the well known discontinuity mapping formalism in contrast to previous treatments the proposed abstract framework more clearly establishes the origin of the large state space stretching in the initial and possibly only step of the construction of a discontinuity mapping it further highlights the nonuniqueness in the formulation of the discontinuity mapping and its connection to the choice of a locally smooth map with which the discontinuity mapping is composed to describe the near grazing dynamics the analysis is illustrated with examples from tapping mode atomic force microscopy in the presence of thin fluid layers on the sample and the probe tip and from the study of protein activity during a eukaryotic cell cycle

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

periodic solution finder for an impact oscillator with a drift in this paper an efficient semi analytical method is developed to compute periodic solutions for a new model of an impact oscillator with a drift which explains the progression mechanism in vibro impact systems and can be used to optimize their performance the method constructs a periodic response assuming that each period is comprised of a sequence of distinct phases for which analytical solutions are known for example a period may consist of the following sequential phases i contact with progression ii contact without progression iii no contact and iv contact without progression using this information a system of four piecewise linear first order differential equations is transformed to a system of non linear algebraic equations the method allows one to accurately predict a range of control parameters for which the best progression rates are obtained

addendum to the limit equation for the gross pitaevskii equations and s terraciniʼs conjecture j funct anal 262 3 2012 1087 1131 abstract we study the differential systems arising from bose einstein condensates under the assumption that all the diffusion constants d i are not equal we give interesting addenda to our previous papers j funct anal 262 3 2012 1087 1131 and j differential equations 251 2011 2737 2769

existence and analytical predictions of periodic motions in a periodically forced nonlinear friction oscillator the grazing bifurcation stick phenomena and periodic motions in a periodically forced nonlinear friction oscillator are investigated the nonlinear friction force is approximated by a piecewise linear kinetic friction model with the static force the total forces for the input and output flows to the separation boundary are introduced and the force criteria for the onset and vanishing of stick motions are developed through such input and output flow forces the periodic motions of such an oscillator are predicted analytically through the corresponding mapping structure illustrations of the periodic motions in such a piecewise friction model are given for a better understanding of the stick motion with the static friction the force responses are presented which agreed very well with the force criteria if the fully nonlinear friction force is modeled by several portions of piecewise linear functions the periodically forced nonlinear friction oscillator can be predicted more accurately however for the fully nonlinear friction force model only the numerical investigation can be carried out

hypersurfaces of prescribed mean curvature in lorentzian manifolds we give a new existence proof for closed hypersurfaces of prescribed mean curvature in lorentzian manifolds

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

grazing bifurcations in an elastic structure excited by harmonic impactor motions abstract in this article non smooth dynamics of an elastic structure excited by a harmonic impactor motion is studied through a combination of experimental numerical and analytical efforts the test apparatus consists of a stainless steel cantilever structure with a tip mass that is impacted by a shaker soft impact between the impactor and the structure is considered and bifurcations with respect to quasi static variation of the shaker excitation frequency are examined in the experiments qualitative changes that can be associated with grazing and corner collision bifurcations are observed aperiodic motions are also observed in the vicinity of the non smooth bifurcation points assuming the system response to be dominated by the structure s fundamental mode a non autonomous single degree of freedom model is developed and used for local analysis and numerical simulations the predicted grazing and corner collision bifurcations are in agreement with the experimental results to study the local bifurcation behavior at the corner collision point and explore the mechanism responsible for the aperiodic motions a derivation is carried out to construct local poincare maps of periodic orbits at a corner collision point such as the one observed in the soft impact oscillator

mean curvature flows and isotopy of maps between spheres let f be a smooth map between unit spheres of possibly different dimensions we prove the global existence and convergence of the mean curvature flow of the graph of f under various conditions a corollary is that any area decreasing map between unit spheres of possibly different dimensions is isotopic to a constant map c 2004 wiley periodicals inc

border collision bifurcations an explanation for observed bifurcation phenomena recently physical and computer experiments involving systems describable by continuous maps that are nondifferentiable on some surface in phase space have revealed novel bifurcation phenomena these phenomena are part of a rich new class of bifurcations which we call border collision bifurcations a general criterion for the occurrence of border collision bifurcations is given illustrative numerical results including transitions to chaotic attractors are presented these border collision bifurcations are found in a variety of physical experiments

normal form maps for grazing bifurcations in n dimensional piecewise smooth dynamical systems abstract this paper presents a unified framework for performing local analysis of grazing bifurcations in n dimensional piecewise smooth systems of odes these occur when a periodic orbit has a point of tangency with a smooth n 1 dimensional boundary dividing distinct regions in phase space where the vector field is smooth it is shown under quite general circumstances that this leads to a normal form map that contains to lowest order either a square root or a 3 2 type singularity according to whether the vector field is discontinuous or not at the grazing point in particular contrary to what has been reported in the literature piecewise linear local maps do not occur generically first the concept of a grazing bifurcation is carefully defined using appropriate non degeneracy conditions next complete expressions are derived for calculating the leading order term in the normal form poincare map at a grazing bifurcation point in arbitrary systems using the concept of a discontinuity mapping finally the theory is compared with numerical examples including bilinear oscillators a relay feedback controller and general third order systems

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

dynamics of strongly competing systems with many species in this paper we prove that the solution of the lotka volterra competing species system with strong competition converges to a stationary point under some natural conditions we also study the moving boundary problem of the singular limit equation which plays an important role in our proof

uniform holder estimate for singularly perturbed parabolic systems of bose einstein condensates and competing species abstract we prove the uniform holder continuity of solutions for two classes of singularly perturbed parabolic systems these systems arise in bose einstein condensates and in competing models in population dynamics the proof relies upon the blow up technique and the monotonicity formulas by almgren and alt caffarelli and friedman

the mapping dynamics of periodic motions for a three piecewise linear system under a periodic excitation in this paper the mapping dynamics of periodic motions for a three piecewise linear system under a periodic excitation is developed and the mapping structures for specified periodic motions are constructed based on the mapping structures an analytical prediction of all possible stable and unstable periodic motions is given the symmetry for the stable asymmetrical periodic motions of such a system is observed however the unstable periodic motions do not have such symmetry the methodology presented in this paper is applicable to other non smooth systems such as friction induced vibration impact oscillator and power control systems in addition the mapping dynamics provides a useful and efficient tool for the co existence of periodic motions and or chaotic motions in nonlinear dynamical systems

asymptotic estimates for the spatial segregation of competitive systems abstract for a class of population models of competitive type we study the asymptotic behavior of the positive solutions as the competition rate tends to infinity we show that the limiting problem is a remarkable system of differential inequalities which defines the functional class s in 2 by exploiting the regularity theory recently developed in conti et al indiana univ math j to appear for the elements of functional classes of the form s we provide some qualitative and regularity property of the limiting configurations besides for the case of two competing species we obtain a full description of the limiting states and we prove some quantitative estimates for the rate of convergence finally we prove some new liouville type results which allow to have uniform regularity estimates of the solutions

a strong unique continuation theorem for parabolic equations the unique continuation is best understood for second order elliptic operators the classical paper by carleman 7 established the strong unique continuation theorem for second order elliptic operators which need not have analytic coefficients this was remarkable since it indicates that even a non analytic solution of an elliptic equation can behave somewhat in an analytic manner the powerful technique he used the so called carleman weighted inequality has played a central role in later developments in 1950 s aronszajn 4 and cordes 11 generalized his result to higher dimensions in recent years this classical subject caught new attention from a great number of people much efforts have been made to relax the smoothness hypotheses on the coefficients see 6 17 43 1 22 a seminal paper by jerison and kenig 26 has shown the unique continuation property for operators of the form c x with c l 2 loc where n 3 is the space dimension further improvements have been carried out in considering other classes of coefficients fefferman phong class and kato class 42 8 12 29 in extending the result to operators with first derivative terms and variable leading coefficients 46 47 51 52 and etc see 25 15 16 28 5 27 and the references therein for second order linear parabolic operators with time independent coefficients the strong unique continuation property along with detailed estimates on the hausdorff measures of nodal sets was reduced in 32 33 to the previously established elliptic counterparts the reduction from time independent parabolic

parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes spacelike hypersurfaces of prescribed mean curvature in cosmological spacetimes are constructed as asymptotic limits of a geometric evolution equation in particular an alternative constructive proof is given for the existence of maximal and constant mean curvature slices

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

mean curvature flow of surfaces in einstein four manifolds let be a compact oriented surface immersed in a four dimensional kahler einstein manifold m w we consider the evolution of in the direction of its mean curvature vector it is proved that being symplectic is preserved along the flow and the flow does not develop type i singularity when m has two parallel kahler forms w and w that determine different orientations and is symplectic with respect to both w and w we prove the mean curvature flow of exists smoothly for all time in the positive curvature case the flow indeed converges at infinity

a nonsmooth coulomb friction oscillator abstract a forced coulomb friction oscillator whose frictional force is allowed to vary with displacement is analyzed geometrically the equation of motion for the oscillator is piecewise linear we geometrically observe the nature of the flow in each region of solvability and then see how these solutions interact at the boundary of the regions the dynamics of the flow is viewed in terms of a map on the boundary between the regions for chaotic motion we geometrically construct the strange attractor and show that its exact behavior is that of a one dimensional map the following dynamical properties arise from the nonsmooth nature of the coulomb friction law the flow may not be invertible the flow may reach its attractor in finite time the dimension of the attractor may be less than or equal to two embeddings of an observable may not be diffeomorphic to the full phase flow

forced vibrations of a beam with one sided amplitude constraint theory and experiment abstract an elastic beam with one sided amplitude constraint subjected to periodic excitation is considered experimental results are obtained and compared with results given by a theoretical model based on a single mode analysis of the beam following the work of moon and shaw 1 this model has been studied in some detail by the author in previous works 2 4 it is a single degree of freedom oscillator with periodic excitation and a piecewise linear restoring force this single mode model is shown to provide good overall qualitative information about the actual physical system it predicts the multiple subharmonic resonances period doublings and some chaotic regimes found experimentally

stochastic responses of vibro impact duffing oscillator excited by additive gaussian noise the stationary responses of vibro impact duffing oscillator excited by additive gaussian white noise are studied by using the quasi conservative averaging method with the help of a non smooth variable transformation and the dirac delta function the response probability density functions pdfs are formulated analytically meanwhile the results are validated numerically using monte carlo simulations

construction of a solution for the two component radial gross pitaevskii system with a large coupling parameter abstract we consider strongly coupled competitive elliptic systems that arise in the study of two component bose einstein condensates as the coupling parameter tends to infinity solutions that remain uniformly bounded are known to converge to a segregated limiting profile with the difference of its components satisfying a limit scalar pde in the case of radial symmetry under natural non degeneracy assumptions on a solution of the limit problem we establish by a perturbation argument its persistence as a solution to the elliptic system

singularity of mean curvature flow of lagrangian submanifolds in this article we study the tangent cones at first time singularity of a lagrangian mean curvature flow if the initial compact submanifold 0 is lagrangian and almost calibrated by re in a calabi yau n fold m and t 0 is the first blow up time of the mean curvature flow then the tangent cone of the mean curvature flow at a singular point x 0 t is a stationary lagrangian integer multiplicity current in r 2n with volume density greater than one at x 0 when n 2 the tangent cone is a finite union of at least two 2 planes in r 4 which are complex in a complex structure on r 4

on a special type of border collision bifurcations occurring at infinity abstract in piecewise smooth dynamical systems the regions of existence of a periodic orbit are typically parameter sub spaces confined by border collision bifurcations of this orbit we demonstrate that additionally to the usual border collision bifurcations occurring at finite points in the state space there exist also border collision bifurcations occurring at infinity

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

flow by mean curvature of convex surfaces into spheres the motion of surfaces by their mean curvature has been studied by brakke 1 from the viewpoint of geometric measure theory other authors investigated the corresponding nonparametric problem 2 5 9 a reason for this interest is that evolutionary surfaces of prescribed mean curvature model the behavior of grain boundaries in annealing pure metal in this paper we take a more classical point of view consider a compact uniformly convex w dimensional surface m mo without boundary which is smoothly imbedded in r let mo be represented locally by a diffeomorphism

long time existence and convergence of graphic mean curvature flow in arbitrary codimension let f 1 2 be a map between compact riemannian manifolds of constant curvature this article considers the evolution of the graph of f in 1 2 by the mean curvature flow under suitable conditions on the curvature of 1 and 2 and the differential of the initial map we show that the flow exists smoothly for all time at each instant t the flow remains the graph of a map ft and ft converges to a constant map as t approaches infinity this also provides a regularity estimate for lipschitz initial data

global dynamics of a vibro impacting linear oscillator abstract the steady state vibro impacting responses of one dimensional harmonically excited linear oscillators are studied by using a modern dynamical systems approach allied with numerical simulation the steady state motions are attracting sets in the system phase space and capture initial conditions in their domains of attraction unlike the free harmonically excited oscillator the phase space of a vibro impacting system may be inhabited by many attracting sets for example there are sub harmonic multi impact periodic orbits and chaotic steady state responses in order to build a qualitative understanding of vibro impact response an attempt is made to build generic topological models of their phase spaces for physically significant parameter ranges use is made of the poincare section or stroboscopic mapping technique essentially following an initial impact forwards or backwards in time to subsequent or previous impacts using a computer the qualitative understanding gained from the analysis and simulations is discussed in an engineering context

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

two dimensional map for impact oscillator with drift an impact oscillator with a drift is considered using a simple co ordinates transformation the bounded oscillations are separated from the drift in general the dynamic state of the system is fully described by four variables time relative displacement p and velocity y of the mass and relative displacement of the slider top q however this number can be reduced by two if the beginning of the progression phase is monitored in this paper a new two dimensional numerical map is developed and its dynamics is discussed

singularities in soft impacting systems abstract in this paper the character of the normal form map in the neighbourhood of a grazing orbit is investigated for four possible configurations of soft impacting systems it is shown that if the spring in the impacting surface is relaxed the impacting side of the map has a power of 3 2 but if the spring is pre stressed the map has a square root singularity the singularity appears only in the trace of the jacobian matrix and not in the determinant under all conditions the determinant of the jacobian matrix varies continuously across the grazing condition however if the impacting surface has a damper the determinant decreases exponentially with increasing penetration it is found that the system behaviour is greatly dependent upon a parameter m given by 2 0 forcing and that the singularity disappears for integer values of m thus if the parameters are chosen to obtain an integer value of m one can expect no abrupt change in behaviour as the system passes through the grazing condition from a non impacting mode to an impacting mode with increasing excitation amplitude the above result has been tested on an experimental rig which showed a persistence of a period 1 orbit across the grazing condition for integer values of m but an abrupt transition to a chaotic orbit or a high period orbit for non integer values of m finally through simulation it is shown that the condition for vanishing singularity is not a discrete point in the parameter space this property is valid over a neighbourhood in the parameter space which shrinks for larger values of the stiffness ratio k 2 k 1

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

the limit equation for the gross pitaevskii equations and s terraciniʼs conjecture abstract we establish the limit system for the gross pitaevskii equations when the segregation phenomenon appears and shows this limit is the one arising from the competing systems in population dynamics this covers and verifies a conjecture of s terracini et al both in the parabolic case and the elliptic case

a local regularity theorem for mean curvature flow this paper proves curvature bounds for mean curvature flows and other related flows in regions of spacetime where the gaussian densities are close to 1

bifurcations in nonsmooth dynamical systems a review is presented of the one parameter nonsmooth bifurcations that occur in a variety of continuous time piecewise smooth dynamical systems motivated by applications a pragmatic approach is taken to defining a discontinuity induced bifurcation dib as a nontrivial interaction of a limit set with respect to a codimension one discontinuity boundary in phase space only dibs that are local are considered that is bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for flows three classes of systems are considered involving either state jumps jumps in the vector field or jumps in some derivative of the vector field a rich array of dynamics are revealed involving the sudden creation or disappearance of attractors jumps to chaos bifurcation diagrams with sharp corners and cascades of period adding for each kind of bifurcation identified where possible a kind of normal form or discontinuity mapping dm is given together with a canonical example and an application the goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillators impact oscillators dc dc converters and problems in control theory

uniform holder bounds for nonlinear schrodinger systems with strong competition for the positive solutions of the gross pitaevskii system u u 1 u 3 u v 2 v v 2 v 3 u 2 v we prove that l boundedness implies c 0 boundedness for every 0 1 uniformly as moreover we prove that the limiting profile as is lipschitz continuous the proof relies upon the blowup technique and the monotonicity formulae by almgren and alt caffarelli and friedman this system arises in the hartree fock approximation theory for binary mixtures of bose einstein condensates in different hyperfine states extensions to systems with k 2 densities are given

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

contracting convex hypersurfaces in riemannian manifolds by their mean curvature which shrink towards the center of the initial sphere in finite time it was shown in 3 that this behaviour is very typical if the initial hypersurface m o o r 1 is uniformly convex then the surfaces m t contract smoothly to a single point in finite time and the shape of the surfaces becomes spherical at the end of the contraction if the ambient space n is a general riemannian manifold the curvature of n will interfere with the mot ion of the surfaces m r we want to show here that the contract ion first to a small sphere and then to a single point is still

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

singular limit of a competition diffusion system with large interspecific interaction abstract we consider a competition diffusion system for two competing species the density of the first species satisfies a parabolic equation together with an inhomogeneous dirichlet boundary condition whereas the second one either satisfies a parabolic equation with a homogeneous neumann boundary condition or an ordinary differential equation under the situation where the two species spatially segregate as the interspecific competition rate becomes large we show that the resulting limit problem turns out to be a free boundary problem we focus on the singular limit of the interspecific reaction term which involves a measure located on the free boundary

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

topology of vibro impact systems in the neighborhood of grazing abstract the grazing bifurcation is considered for the newtonian model of vibro impact systems a brief review on the conditions sufficient for the existence of a grazing family of periodic solutions is given the properties of these periodic solutions are discussed a plenty of results on the topological structure of attractors of vibro impact systems is known however since the considered system is strongly nonlinear these attractors must be very sensitive to changes of parameters of the system on the other hand they are observed in experiments and numerical simulations we offer theorem 2 an approach which allows to explain this contradiction and give a new robust mathematical model of the non hyperbolic dynamics in a neighborhood of grazing

on phase separation models asymptotics and qualitative properties in this paper we study bound state solutions of a class of two component nonlinear elliptic systems with a large parameter tending to infinity the large parameter giving strong intercomponent repulsion induces phase separation and forms segregated nodal domains divided by an interface to obtain the profile of bound state solutions near the interface we prove the uniform lipschitz continuity of bound state solutions when the spatial dimension is n 1 furthermore we show that the limiting nonlinear elliptic system that arises has unbounded solutions with symmetry and monotonicity these unbounded solutions are useful for rigorously deriving the asymptotic expansion of the minimizing energy which is consistent with the hypothesis of du and zhang discontin dynam sys 2012 when the spatial dimension is n 2 we establish the de giorgi type conjecture for the blow up nonlinear elliptic system under suitable conditions at infinity on bound state solutions these results naturally lead us to formulate de giorgi type conjectures for these types of systems in higher dimensions

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

bifurcations of piecewise smooth flows perspectives methodologies and open problems abstract in this paper the theory of bifurcations in piecewise smooth flows is critically surveyed the focus is on results that hold in arbitrarily but finitely many dimensions highlighting significant areas where a detailed understanding is presently lacking the clearest results to date concern equilibria undergoing bifurcations at switching boundaries and limit cycles undergoing grazing and sliding bifurcations after discussing fundamental concepts such as topological equivalence of two piecewise smooth systems discontinuity induced bifurcations are defined for equilibria and limit cycles conditions for equilibria to exist in n dimensions are given followed by the conditions under which they generically undergo codimension one bifurcations the extent of knowledge of their unfoldings is also summarized codimension one bifurcations of limit cycles and boundary intersection crossing are described together with techniques for their classification codimension two bifurcations are discussed with suggestions for further study

mean curvature flow of spacelike graphs we prove the mean curvature flow of a spacelike graph in 1 2 g1 g2 of a map f 1 2 from a closed riemannian manifold 1 g1 with ricci1 0 to a complete riemannian manifold 2 g2 with bounded curvature tensor and derivatives and with sectional curvatures satisfying k2 k1 remains a spacelike graph exists for all time and converges to a slice at infinity we also show with no need of the assumption k2 k1 that if k1 0 or if ricci1 0 and k2 c c 0 constant any map f 1 2 is trivially homotopic provided f g2 0 and in case k2 0 this largely extends some known results for ki constant and 2 compact obtained using the riemannian structure of 1 2 and also shows how regularity theory on the mean curvature flow is simpler and more natural in pseudo riemannian setting then in the riemannian one

lorentzian legendrian mean curvature flow in this paper we generalize the legendrian mean curvature flow to lorentzian geometry more precisely we study the case where the ambient manifold is a lorentzian sasaki eta einstein manifold for legendrian curves we establish convergence results in theorems 1 1 and 1 2 and we derive estimates for the legendrian angle for arbitrary dimensions in theorem 1 3

universal limit mapping in grazing bifurcations grazing bifurcations are nonsmooth bifurcations that occur in impacting mechanical oscillators a sign of the lack of smoothness is a square root expression that arises in mappings describing the dynamics certain types of grazing bifurcations involve chaotic bands or period adding cascades with or without chaotic bands there is also a characteristic scaling behavior present here this type of bifurcation is investigated and the self similarity under scaling is used to derive a renormalized limit mapping a study of the dynamics of the limit mapping identifying all attractors for all parameter values is also presented

mean curvature flow of compact spacelike submanifolds in higher codimension we prove the longtime existence for mean curvature flow of a smooth n dimensional spacelike submanifold of an n m dimensional manifold whose metric satisfies the timelike curvature condition

continuous and discontinuous grazing bifurcations in impacting oscillators this paper seeks to formulate conditions for the persistence of a local attractor in the immediate vicinity of periodic and quasiperiodic grazing trajectories in an impacting mechanical system a local analysis based on the discontinuity mapping approach is employed to derive a normal form description of the dynamics near the grazing trajectory in agreement with previous studies of grazing periodic trajectories it is found that the catastrophic loss of a local attractor and strong instability characteristic of grazing bifurcations is directly associated with the repeated application of a square root term that appears to lowest order in the normal form expansion specifically it is found that the square root term is absent in the description of the dynamics normal to a quasiperiodic trajectory covering a co dimension one invariant torus resulting in a piecewise linear description of the normal dynamics and at most a weak instability in contrast for co dimension two or higher the square root term is generically present in the normal dynamics here however the quasiperiodicity of the grazing motion implies that there is no upper limit to the time between impacts on nearby trajectories suggesting the persistence of a local attractor for some interval about the parameter value corresponding to grazing the results of the analysis are illustrated through a series of model examples

generalized hopf bifurcation emanated from a corner for piecewise smooth planar systems abstract in this paper we study the existence of periodic orbits bifurcating from a corner in a piecewise smooth planar dynamical system this phenomenon is interpreted as generalized hopf bifurcation philos trans r soc london ser a 359 1789 2001 2483 2496 hopf bifurcation for nonsmooth planar systems universitat zu koln 2000 northeast math j 17 4 2001 383 386 northeast math j 17 3 2001 261 264 generalized hopf bifurcation for non smooth planar dynamical systems university cologne 2003 in the case of smoothness hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis this characterization does not apply to a piecewise smooth system due to the lack of linearization in fact the generalized hopf bifurcation is determined by interactions between the geometrical structure of the corner and the eigenstructure of each smooth subsystem we combine a geometrical method and an analytical method to investigate the generalized hopf bifurcation the bifurcating periodic orbits are obtained by studying the fixed points of a return map

bifurcations caused by grazing incidence in many degrees of freedom impact oscillators the transition from stable periodic non impacting motion to impacting motion is analysed for a mechanical oscillator by using local methods it is shown that a grazing impact leads to an almost one dimensional stretching in state space a condition can then be formulated such that a grazing trajectory will be stable if the condition is fulfilled if this is the case the bifurcation will be continuous and the motion after the bifurcation can be understood by a one dimensional mapping this mapping is known to exhibit chaotic solutions as well as arbitrary long stable cycles depending on parameters

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

impact oscillations linear theory of stability and bifurcations abstract a unified approach to analysis of impact oscillations is proposed its mathematical essence is the continuous representation of the impulsive motion in some auxiliary variables as a result an explicit formula for the fundamental matrix is obtained which allows one to construct the characteristic equation then the stability and bifurcations can be studied by the usual techniques the exceptional case of bifurcations related to grazing impacts is investigated by taking into account a non zero impact duration due to this approach linearization is also possible and the complex bifurcation becomes a sequence of ordinary ones some algebraic conditions are obtained which allow one to determine the type of the resulting motion a mechanical example is considered a disc with an offset centre of gravity bouncing on an oscillating surface