comparison of several numerical methods for mechanical systems with impacts the modellization of mechanical systems with non linearities of impact type leads to discontinuities in the velocity in this paper we study some numerical methods adapted to the occurrence of such discontinuities from a single degree of freedom vibro impact system theoretical results of consistency are given for numerical methods valid for smooth ordinary differential equations with several kinds of procedures for approximating impact times the algorithms are applied to classical newmark and runge kutta schemes and the numerical behaviour is investigated for two categories of periodic response either with finite or infinite number of impacts per cycle these numerical methods are compared to a scheme without explicit computation of impact times we show that high orders of convergence can be obtained with appropriate schemes and we discuss the time of computation needed in each case copyright 2001 john wiley sons ltd

codimension 2 hopf bifurcation of a two degree of freedom vibro impact system abstract codimension 2 hopf bifurcation problem of a two degree of freedom system vibrating against a rigid surface is investigated in this paper the four dimensional poincare map of the vibro impact system is reduced to a two dimensional normal form by virtue of a center manifold reduction and a normal form technique then the theory of hopf bifurcation of maps in r 2 is applied to conclude the existence of codimension 2 hopf bifurcation of the vibro impact system the quasi periodic response of the system by theoretical analysis is well supported by numerical simulations it is shown that there exists codimension 2 hopf bifurcation in multi degree of freedom vibro impact systems the codimension 2 tori doubling phenomenon and the routes of quasi periodic impacts to chaos are reported briefly

symmetry and bifurcations of a two degree of freedom vibro impact system abstract a two degree of freedom system with impact is considered the symmetry of the system and its poincare map is described the symmetric period n 2 motion corresponding to the symmetric fixed point of the poincare map is obtained if the jacobian matrix of the poincare map at the fixed point has a real eigenvalue crossing the unit circle at 1 the symmetric fixed point will bifurcate into two antisymmetric fixed points which have the same stability via pitchfork bifurcation the numerical simulation shows that the symmetric fixed points may have pitchfork bifurcations and hopf bifurcations while the control parameter changes continuously the two antisymmetric fixed points will give birth to two synchronous bifurcation sequences

an existence result for non smooth vibro impact problems we are interested in mechanical systems with a finite number of degrees of freedom submitted to frictionless unilateral constraints we consider the case of a convex non smooth set of admissible positions given by k q mathbb r d phi alpha q geq 0 1 leq alpha leq v v geq 1 and we assume inelastic shocks at impacts we propose a time discretization of the measure differential inclusion which describes the dynamics and we prove the convergence of the approximate solutions to a limit motion which satisfies the constraints moreover if the geometric properties ensuring continuity on data hold at the limit we show that the transmission of velocities at impacts follows the inelastic shocks rule

a numerical scheme for impact problems i the one dimensional case we consider a mechanical system with impact and one degree of freedom the system is not necessarily lagrangian the representative point is subject to the constraint u t in er for all t we assume that at impact the velocity is reversed and multiplied by a given coefficient of restitution e in 0 1 we define a numerical scheme which enables us to approximate the solutions of the cauchy problem this is an ad hoc scheme which does not require a systematic search for the times of impact we prove the convergence of this numerical scheme to a solution many of the features of this proof will be reused in the nonconvex multidimensional case written in generalized coordinates given in the companion paper l paoli and m schatzman siam j numer anal 40 2002 pp 734 768 we present some numerical results obtained with the scheme for a spring dashpot system and we compare them to the results obtained by impact detection and penalization

time integration schemes for the finite element dynamic signorini problem a large variety of discretizations have been proposed in the literature for the numerical solution of the dynamic signorini problem we classify the different discretizations into four groups the first three groups correspond to different ways of enforcing the contact condition exact enforcement enforcement with penalty and enforcement with contact condition in velocity the fourth approach is based on a modification of the mass matrix numerical simulations on two one dimensional benchmark problems with analytical solutions illustrate the properties of representative methods within each class focusing first on spurious oscillations triggered by contact and then on energy behavior after multiple impacts selected schemes are also tested on a two dimensional benchmark

time stepping approximation of rigid body dynamics with perfect unilateral constraints i the inelastic impact case we consider a discrete mechanical system with a non trivial mass matrix subjected to perfect unilateral constraints described by the geometrical inequalities f alpha q geqq 0 alpha in 1 dots nu nu geqq 1 we assume that the transmission of the velocities at impact is governed by newton s law with a coefficient of restitution e 0 so that the impact is inelastic we propose a time discretization of the second order differential inclusion describing the dynamics which generalizes the scheme proposed in paoli j differ equ 211 247 281 2005 and for any admissible data we prove the convergence of approximate motions to a solution of the initial value problem

dynamical behavior of vibro impact machinery near a point of codimension two bifurcation a dual component system with vibro impact is considered local codimension two bifurcation of maps involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously is analyzed by using the center manifold and normal form method for maps the period n single impact motion and poincare map of the vibro impact system are derived analytically stability and local bifurcation of single impact periodic motion are analyzed by using the poincare map a center manifold theorem technique is applied to reduce the poincare map to a three dimensional one and the normal form map associated with the codimension two bifurcation is obtained local behavior of the dual component system with vibro impact near the point of codimension two bifurcation is analyzed it is found that near the point of codimension two bifurcation there exists not only hopf bifurcation of period one single impact motion but also hopf bifurcation of period two double impact motion period doubling bifurcation of period one single impact motion commonly exists near the point of codimension two bifurcation however no period doubling cascade emerges due to change of the type of period two fixed points and occurrence of hopf bifurcation associated with period two fixed points the period two fixed points are symmetrical about the corresponding period one point the results from simulation show that the attracting invariant circles associated with period two points are symmetrical about the corresponding period one point near the value of hopf bifurcation of period two points two actual examples the impact forming machinery and inertial shaker are chosen to analyze further the phenomena of codimension two bifurcation of maps and local bifurcation analysis and numerical simulation are carried out to unfold dynamical behavior of these vibro impact systems near the points of codimension two bifurcations

on non standard finite difference models of reaction diffusion equations reaction diffusion equations arise in many fields of science and engineering often their solutions enjoy a number of physical properties we design in a systematic way new non standard finite difference schemes which replicate three of these properties the first property is the stability instability of the fixed points of the associated space independent equation this property is preserved by non standard one and two stage theta methods presented in the general setting of stiff or non stiff systems of differential equations schemes which preserve the principle of conservation of energy for the corresponding stationary equation second property are constructed by nonlocal approximation of nonlinear reactions assembling of theta methods in the time variable with energy preserving schemes in the space variable yields non standard schemes which under suitable functional relation between step sizes display the boundedness and positivity of the solution third property a spectral method in the space variable coupled with a suitable non standard scheme in the time variable is also presented numerical experiments are provided

the corresponding phenomena of mechanical and electronic impact oscillator a mechanical impact oscillator and a diode rectifier circuit of an impact oscillator are presented time series and phase portrait analysis are performed for both oscillators the results show that the proposed electronic impact oscillator is in agreement with the mechanical impact oscillator for varying the driving frequency a fixed harmonic input force can generate a periodic or non periodic impact oscillation the phenomena of bifurcation are also observed in both impact oscillators

double impact orbits of periodically forced impact oscillators we consider a periodically forced oscillator which impacts against a rigid constraint the forcing is both sinusoidal and non sinusoidal and is made up of a combination of the first and second harmonics of a fourier series a simple restitution law is used to describe the impact this paper centres around the analysis of double impact periodic solutions that is motions which repeat after every second impact solutions for these double impact orbits are found including non physical orbits and unstable orbits behaviour typical of nonlinear dynamical systems such as period doubling bifurcations are detected and grazing bifurcations which are unique to discontinuous dynamical systems are also observed a combination of analytical and numerical techniques are used to uncover the behaviour of the system at odd multiples of the natural frequency where numerical experiments tell us that stable double impact periodic solutions exist the persistence of these stable double impact orbits are also examined as parameters are varied we conclude by presenting some numerical calculations of two more general cases of impacting systems

the isomorphism problem for some universal operator algebras abstract this paper addresses the isomorphism problem for the universal non self adjoint operator algebras generated by a row contraction subject to homogeneous polynomial relations we find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables and that this happens if and only if the associated subproduct systems are isomorphic the proof makes use of the complex analytic structure of the character space together with some recent results on subproduct systems restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry these commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties we prove a projective nullstellensatz connecting closed ideals and their zero sets under some technical assumptions we find that two such algebras are isomorphic as algebras if and only if they are similar and we obtain a clear geometrical picture of when this happens this result is obtained with tools from algebraic geometry reproducing kernel hilbert spaces and some new complex geometric rigidity results of independent interest the c envelopes of these algebras are also determined the banach algebraic and the algebraic classification results are shown to hold for the wot closures of these algebras as well

controlling chaos of a dynamical system with discontinuous vector field abstract the paper presents how to control the chaos of dynamical systems with discontinuous vector field through the paradigm of a harmonically forced oscillator having a set up elastic stop it is first shown that the poincare mapping of this oscillator is not smooth near the fixed point corresponding to a periodic motion that grazes the stop thus the current control strategies based on the smooth mapping cannot be directly used to stabilize the chaotic motion near a periodic grazing motion then proposed is a piecewise linear control strategy based on the piecewise linearized poincare mapping reconstructed from sampled data and on the pole assignment in two regions near the fixed point the efficacy of the strategy is finally demonstrated via numerical simulations

non standard finite difference methods for vibro impact problems impact oscillators are non smooth systems with such complex behaviours that their numerical treatment by traditional methods is not always successful we design non standard finite difference schemes in which the intrinsic qualitative parameters of the system the restitution coefficient the oscillation frequency and the structure of the nonlinear terms are suitably incorporated the schemes obtained are unconditionally stable and replicate a number of important physical properties of the involved oscillator system such as the conservation of energy between two consecutive impact times numerical examples including the duffing oscillator that develops a chaotic behaviour for some positions of the obstacle are presented it is observed that the cpu times of computation are of the same order for both the standard and the non standard schemes

dynamical systems with active singularities of elastic type a modeling and controller synthesis framework a new class of systems characterized by admitting impulsive control action within their singular motion phases such as dimension changes state discontinuities and other irregularities is introduced this class termed dynamical systems with active or controlled singularities encompasses various applications with sensing and or actuation ultra fast in comparison with the natural system time scale including mechanisms with impact induced motion power and sensor networks under faults fast positioning devices smart skins and switching electronic circuits the present work focuses on systems with impact type singularities induced through system interaction with controlled or active state constraints the latter are assumed to be characterized by parameter dependent elastic type constraint violation that vanishes in the limit as the parameter value tends to infinity a physically well justified representation of this class of systems is proposed but found to be not well suited for controller synthesis in the singular phase to address this problem two equations are derived one describing controlled rigid impacts in auxiliary stretched time and compatible with the regular control design techniques termed controlled infinitesimal dynamics equation and another incorporating controlled impact dynamics in the infinitesimal form in terms of the shift operator along the trajectories of the first equation termed the limit model the latter is demonstrated to generate a unique isolated discontinuous system motion i e to provide a tight and well behaved description of the collision with rigid constraint it is then shown that the corresponding paths generated by the original physically based and the limit representations can be made arbitrarily close to each other uniformly except possibly in the vicinities of the jump points by the appropriate choice of the value of the constraint violation parameter this is shown to permit enforcing for sufficiently large values of the parameter the desired limit system behavior onto the original system by simply taking the control signals found through the limit representation time rescaling them and inserting the resulting signals directly into the original system these features show that the procedure developed in fact provides a well posed controller synthesis framework for the class of systems considered using this framework a ball racket rotationally controlled impact representation is developed and on its basis a singular phase control signal in the form of racket rotation velocity is designed demonstrating that the soft racket provides the specified bounce off angle increment under a noticeably lower racket rotation velocity and a longer rotation phase than the hard racket

essential normality essential norms and hyperrigidity abstract let s s 1 s d denote the compression of the d shift to the complement of a homogeneous ideal i of c z 1 z d arveson conjectured that s is essentially normal in this paper we establish new results supporting this conjecture and connect the notion of essential normality to the theory of the c envelope and the noncommutative choquet boundary the unital norm closed algebra b i generated by s 1 s d modulo the compact operators is shown to be completely isometrically isomorphic to the uniform algebra generated by polynomials on v z i b d where z i is the variety corresponding to i consequently the essential norm of an element in b i is equal to the sup norm of its gelfand transform and the c envelope of b i is identified as the algebra of continuous functions on v b d which means it is a complete invariant of the topology of the variety determined by i in the ball motivated by this determination of the c envelope of b i we suggest a new more qualitative approach to the problem of essential normality we prove the tuple s is essentially normal if and only if it is hyperrigid as the generating set of a c algebra which is a property closely connected to arveson s notion of a boundary representation we show that most of our results hold in a much more general setting in particular for most of our results the ideal i can be replaced by an arbitrary not necessarily homogeneous invariant subspace of the d shift

projected event capturing time stepping schemes for nonsmooth mechanical systems with unilateral contact and coulomb s friction this work addresses the problem of the numerical time integration of nonsmooth mechanical systems subjected to unilateral contacts impacts and coulomb s friction the considered systems are the space discretized continuous systems obtained by using a finite element method fem approach or the multi body systems or a mix of them as in flexible multibody dynamics up to now two main numerical schemes are available for this purpose the moreau jean scheme which solves the constraints at the velocity level together with a newton impact law and the schatzman paoli scheme which directly considers the constraints at the position level in both schemes the position and velocity constraints are not both satisfied in discrete time a first attempt is made by directly using the gear gupta leimkuhler ggl approach for differential algebraic equations dae that solves in discrete time the constraints on both position and velocity levels the obtained direct projection scheme succeeds in solving in discrete time both position and velocity constraints but introduces some chattering at contact after a finite accumulation of impacts the proposed scheme improves the direct projected scheme by combining several steps of activation and projection to avoid the chattering effect the stability and the local order of the scheme will be discussed the usefulness of the scheme is demonstrated on several academic examples and is illustrated on an industrial application the modeling and simulation of an electrical circuit breaker

on the equidistribution of some hodge loci we prove the equidistribution of the hodge locus for certain non isotrivial polarized variations of hodge structure of weight 2 with h 2 0 1 over complex quasi projective curves given some norm condition we also give an asymptotic on the growth of the hodge locus in particular this implies the equidistribution of elliptic fibrations in quasi polarized non isotrivial families of k3 surfaces

qualitatively stable finite difference schemes for advection reaction equations a systematic procedure is proposed and implemented for the design of nonstandard finite difference methods as reliable numerical simulations that preserve significant properties inherent to the solutions of advection reaction equations in the case of hyperbolic fixed points a renormalization of the denominators of the discrete derivatives is performed for the numerical solutions to display the linear stability properties of the exact solutions non hyperbolic fixed points are described with the help of two new monotonic properties the construction of schemes which preserve these properties being done by nonlocal approximation of nonlinear terms in the reaction terms

a proximal like method for a class of second order measure differential inclusions describing vibro impact problems abstract we are interested in the study of discrete mechanical systems subjected to frictionless unilateral constraints the dynamics is described by a second order measure differential inclusion for the unknown positions completed by a newton s impact law describing the transmission of the velocities when the constraints are saturated by using another formulation of the problem at the velocity level we introduce a time stepping algorithm inspired by the proximal methods for differential inclusions and we prove the convergence of the approximate solutions to a solution of the cauchy problem

a new form of the circle method and its application to quadratic forms if the coefficients r n satisfy suitable arithmetic conditions the behaviour of f will be determined by an appropriate rational approximation a q to with small values of q usually producing large values of f when lies in an interval a q a q with q small a major arc one hopes to estimate f asymptotically while if the corresponding q is large for the minor arcs one hopes that f will be small at least on average one can distinguish two different types of application of the circle method the first type following the above pattern uses one of a variety of mean value techniques hua s inequality for example to give an upper bound for the minor arc contribution results on waring s problem are generally proved this way for example the second type of application on which we shall be focusing our attention avoids the minor arcs completely and treats f asymptotically for all here one cannot take advantage of the mean value ideas used in the first type of problem the savings come instead from the availability of good estimates for complete exponential sums kloosterman s work 12 on quaternary quadratic forms and the papers of the author 5 and hooley 8 and 9 on cubic forms are of this latter type we should point out that such applications do not include any problems in which f is the generating function for a polynomial of degree 4 or more there are

quadratic forms representing all odd positive integers we consider the problem of classifying all positive definite integer valued quadratic forms that represent all positive odd integers kaplansky considered this problem for ternary forms giving a list of 23 candidates and proving that 19 of those represent all positive odds jagy later dealt with a 20th candidate assuming that the remaining three forms represent all positive odds we prove that an arbitrary positive definite quadratic form represents all positive odds if and only if it represents the odd numbers from 1 up to 451 this result is analogous to bhargava and hanke s celebrated 290 theorem in addition we prove that these three remaining ternaries represent all positive odd integers assuming the generalized riemann hypothesis this result is made possible by a new analytic method for bounding the cusp constants of integer valued quaternary quadratic forms q with fundamental discriminant this method is based on the analytic properties of rankin selberg l functions and we use it to prove that if q is a quaternary form with fundamental discriminant the largest locally represented integer n for which q vec x n has no integer solutions is o d 2 epsilon

operator algebras and subproduct systems arising from stochastic matrices we study subproduct systems in the sense of shalit and solel arising from stochastic matrices on countable state spaces and their associated operator algebras we focus on the non self adjoint tensor algebra and viselter s generalization of the cuntz pimsner c algebra to the context of subproduct systems suppose that x and y are arveson stinespring subproduct systems associated to two stochastic matrices over a countable set and let t x and t y be their tensor algebras we show that every algebraic isomorphism from t x onto t y is automatically bounded furthermore t x and t y are isometrically isomorphic if and only if x and y are unitarily isomorphic up to a automorphism of l when is finite we prove that t x and t y are algebraically isomorphic if and only if there exists a similarity between x and y up to a automorphism of l moreover we provide an explicit description of the cuntz pimsner algebra o x in the case where is finite and the stochastic matrix is essential

a variational scheme for hyperbolic obstacle problems abstract we consider an obstacle problem for possibly non local wave equations and we prove existence of weak solutions through a convex minimization approach based on a time discrete approximation scheme we provide the corresponding numerical implementation and raise some open questions

the effect of frequency and clearance variations on single degree of freedom impact oscillators abstract a simple piecewise linear forced oscillator with impacts is taken as a model for more complicated vibro impacting systems the presence of impacts makes the system discontinuous and non linear and to study it effectively the theory of dynamical systems is used by a combination of analytical and numerical techniques many phenomena are identified which are characteristic of non linear dynamical systems as the system parameters are varied in particular these include chaotic regimes and multiply coexisting stable states the domains of attraction of the periodic states are plotted and compared with the strange attractors of the chaotic states in addition phenomena are observed related to the discontinuity which have no counterpart in smooth dynamical systems chatter when infinitely many impacts occur in a finite time and grazing when a stable periodic orbit encounters a discontinuity and disappears catastrophically under a smooth change in the parameters

a nonsmooth generalized α scheme for flexible multibody systems with unilateral constraints summary mechanical systems are usually subjected not only to bilateral constraints but also to unilateral constraints inspired by the generalized time integration method for smooth flexible multibody dynamics this paper presents a nonsmooth generalized method which allows a consistent treatment of the nonsmooth phenomena induced by unilateral constraints and an accurate description of the structural vibrations during free motions both the algorithm and the implementation are illustrated in detail numerical example tests are given in the scope of both rigid and flexible body models taking account for both linear and nonlinear systems and comprising both unilateral and bilateral constraints the extended nonsmooth generalized method is verified through comparison to the traditional moreau jean method and the fully implicit newmark method results show that the nonsmooth generalized method benefits from the accuracy and stability property of the classical generalized method with controllable numerical damping in particular when it comes to the analysis of flexible systems the nonsmooth generalized method shows much better accuracy property than the other two methods copyright 2013 john wiley sons ltd

counting rational points on biquadratic hypersurfaces abstract an asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables the proof uses the hardy littlewood circle method

step size adjustment and extrapolation for time stepping schemes in non smooth dynamics in this paper we use step size adjustment and extrapolation methods to improve moreau s time stepping scheme for the numerical integration of non smooth mechanical systems i e systems with impact and friction the scheme yields a system of inclusions which is transformed into a system of projective equations these equations are solved iteratively switching points are time instants for which the structure of the mechanical system changes for example time instants for which a sticking friction element begins to slide we show how switching points can be localized and how these points can be resolved by choosing a minimal step size in order to improve the integration of non smooth systems in the smooth parts we show how the time stepping method can be used as a base integration scheme for extrapolation methods which allow for an increase in the integration order switching points are processed by a small time step while time intervals during which the structure of the system does not change are computed with a larger step size and improved integration order the overall algorithm which consists of a time stepping module an extrapolation module and a step size adjustment module is discussed in detail and some examples are given

on the representation of integers by quadratic forms let n 4 and let q 2 z x1 xn be a non singular quadratic form when q is indefinite we provide new upper bounds for the least non trivial integral solution to the equation q 0 and when q is positive definite we provide improved upper bounds for the greatest positive integer k for which the equation q k is insoluble in integers despite being soluble modulo every prime power

operator algebras of monomial ideals in noncommuting variables abstract we study operator algebras arising from monomial ideals in the ring of polynomials in noncommuting variables through the apparatus of subproduct systems and c correspondences we provide a full comparison amongst the related operator algebras for our analysis we isolate a partially defined dynamical system to which we refer as the quantised dynamics of the monomial ideal in addition we revisit several previously considered constructions these include matsumoto s subshift c algebras as well as the tensor and the pimsner algebras associated with dynamical systems or graphs we sort out the various relations by giving concrete conditions and counterexamples that orientate the operator algebras of our context it appears that the boundary c algebras do not arise as the quotient with the compact operators unconditionally we establish a dichotomy to this effect by examining the resulting tensor algebras we identify their boundary representations we analyse their c envelopes and we give criteria for hyperrigidity moreover we completely classify them in terms of the data provided by the monomial ideals for tensor algebras of c correspondences and bounded isomorphisms this is achieved up to the level of local conjugacy in the sense of davidson and roydor for the quantised dynamics for tensor algebras of subproduct systems and algebraic isomorphisms this is achieved up to the level of equality of monomial ideals modulo permutations of the variables in the process we accomplish more in different directions most notably we show that tensor algebras form a complete invariant for isomorphic resp similar subproduct systems of homogeneous ideals up to isometric resp bounded isomorphisms the results on local conjugacy are obtained via an alternative proof of the breakthrough result of davidson and katsoulis on piecewise conjugate systems for our purposes we use appropriate compressions of the fock representation we then apply this alternative proof locally for the partially defined quantised dynamics in this way we avoid the topological graphs machinery and pave the way for further applications these include operator algebras of dynamical systems over commuting contractions or over row commuting contractions

a novel route to a hopf bifurcation scenario in switched systems with dead zone abstract planar switched systems with dead zone are analyzed in particular we consider the effects of a perturbation which is applied to a linear control law and due to the perturbation the control changes from purely positional to position velocity control this type of a perturbation leads to a novel hopf like discontinuity induced bifurcation we show that this bifurcation leads to the creation of a small scale limit cycle attractor which scales as the square root of the bifurcation parameter we then investigate numerically a planar switched system with a positional feedback law dead zone and time delay in the switching function using the same parameter values as for the switched system without time delay in the switching function we show a hopf like bifurcation scenario which exhibits a qualitative and a quantitative agreement with the scenario analyzed for the non delayed system

small solutions of quadratic diophantine equations we consider quadratic diophantine equations of the shape q x 1 ldots x s 0 qquad 1 for a polynomial q x 1 ldots x s in mathbf z x 1 ldots x s of degree 2 let h be an upper bound for the absolute values of the coefficients of q and assume that the homogeneous quadratic part of q is non singular we prove for all s ge 3 the existence of a polynomial bound lambda s h with the following property if equation 1 has a solution mathbf x in mathbf z s at all then it has one satisfying x i le lambda s h quad 1 le i le s for s 3 and s 4 no polynomial bounds lambda s h were previously known and for s ge 5 we have been able to improve existing bounds quite significantly

least zero of a cubic form an effective search bound is established for the least non trivial integer zero of an arbitrary cubic form c in mathbb z x 1 ldots x n provided that n 17

half explicit timestepping schemes on velocity level based on time discontinuous galerkin methods this paper presents a time discretization scheme for the simulation of nonsmooth mechanical systems these consist of rigid and flexible bodies joints as well as contacts and impacts with dry friction the benefit of the proposed formalism is both the consistent treatment of velocity jumps e g due to impacts and the automatic local order elevation in non impulsive intervals at the same time for an appropriate treatment of constraints in impulsive and non impulsive intervals constraints are implicitly formulated on velocity level in terms of an augmented lagrangian technique 5 they are satisfied exactly without any penetration for efficiency reasons all other evaluations are explicit which yields a half explicit method 9 10 43 44 7 29 30 the numerical scheme is an extended timestepping scheme for nonsmooth dynamics according to moreau 42 it is based on time discontinuous galerkin methods to carry over higher order trial functions of event driven integration schemes to consistent timestepping schemes for nonsmooth dynamical systems with friction and impacts splitting separates the portion of impulsive contact forces from the portion of non impulsive contact forces impacts are included within the discontinuity of the piecewise continuous trial functions i e with first order accuracy non impulsive contact forces are integrated with respect to the local order of the trial functions in order to satisfy the constraints a set of nonsmooth equations has to be solved in each time step depending on the number of stages the solution of the velocity jump together with the corresponding impulse yields another nonsmooth equation all nonsmooth equations are treated separately by semi smooth newton methods the integration scheme on acceleration level was first introduced in 51 labeled forecasting trapezoidal rule it was analyzed and applied to a decoupled bouncing ball example concerning principal suitability without taking friction into account in this work the approach is algorithmically specified improved and applied to nonlinear multi contact examples with friction it is compared to other numerical schemes and it is shown that the newly proposed integration scheme yields a unified behavior for the description of contact mechanical problems

quadratic forms and semiclassical eigenfunction hypothesis for flat tori let q x be any integral primitive positive definite quadratic form in k variables where k geq4 and discriminant d for any integer n we give an upper bound on the number of integral solutions of q x n in terms of n k and d as a corollary we prove a conjecture of lester and rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the laplacian on the flat torus mathbb t d for d geq 5 this conjecture is motivated by the work of berry 2 3 on the semiclassical eigenfunction hypothesis

the effect of multiple impacts on the dynamics of an impact system the dynamics of an impact system of two concentric hollow cylinders with a clearance is investigated by using a continuous model and the theory of wave propagation exact solutions are obtained by the expansion of the transient wave function in a series of eigenfunctions and their numerical results are presented by selecting a suitable number of truncation terms of eigenfunctions the effect of multiple impact is considered the impact clusters sensitive to the time step length truncation terms of eigenfunctions and system parameters are studied in comparison to single impacts two evolving ways of impact clusters are observed the impact responses frequency spectra and reconstructed phase portraits demonstrate complicated motions including quasi periodic and chaotic motions two routes to chaos through two frequency quasi periodicity and intermittency phase locking and the low dimensional behavior of the system this investigation also verifies the validity of the simplified models for describing impact systems

neimark sacker bifurcations in a non standard numerical scheme for a class of positivity preserving odes we discuss the nature of neimark sacker bifurcations occurring in a non standard numerical scheme for a class of positivity preserving systems of ordinary differential equations odes which undergoes a corresponding hopf bifurcation extending previous work alexander moghadas 2005a electron j diff eqn conf 12 9 19 it is shown that the type of neimark sacker bifurcation supercritical or subcritical may be affected by the integration time step the general form of the scheme in the vicinity of a fixed point is given from which the expression for the first lyapunov coefficient for two dimensional systems valid for arbitrary time step is explicitly derived the analysis shows that this coefficient undergoes an o shift with respect to the corresponding coefficient of the original ode this could lead to a type of bifurcation which differs from the corresponding hopf bifurcation in the ode due to changes in the sign of the first lyapunov coefficient as varies this is especially problematic in the vicinity of certain types of degenerate hopf bifurcation at which this coefficient may vanish we also present a general method to eliminate the possible o shift in the bifurcation parameter of the scheme however the first lyapunov coefficient may still be subjected to an o shift leading to a possibly erroneous type of bifurcation examples are given to illustrate the theoretical results of the paper with applications to mathematical biology

dynamically consistent nonstandard finite difference schemes for epidemiological models this work is the numerical analysis and computational companion of the paper by kamgang and sallet j c kamgang g sallet computation of threshold conditions for epidemiological models and global stability of the disease free equilibrium dfe mathematical biosciences 213 2008 1 12 where threshold conditions for epidemiological models and the global stability of the disease free equilibrium dfe are studied we establish a discrete counterpart of the main continuous result that guarantees the global asymptotic stability gas of the dfe for general epidemiological models then we design nonstandard finite difference nsfd schemes in which the metzler matrix structure of the continuous model is carefully incorporated and both mickens rules world scientific singapore 1994 on the denominator of the discrete derivative and the nonlocal approximation of nonlinear terms are used in an innovative way as a result of these strategies our nsfd schemes are proved to be dynamically consistent with the continuous model i e they replicate their basic features including the gas of the dfe the linear stability of the endemic equilibrium ee the positivity of the solutions the dissipativity of the system and its inherent conservation law the general analysis is made detailed for the mseir model for which the nsfd theta method is implemented with emphasis on the computational aspects such as its convergence or local truncation error numerical simulations that illustrate the theory are provided

uniqueness in the elastic bounce problem ii in a recent paper uniqueness for the elastic bounce problem has been studied in a very general framework the aim of this paper is to show that if u is assumed to be real analytic and u equiv 0 then these phenomena disappear and the solution to the elastic bounce problem is unique for every choice of the cauchy data

operator algebras associated with unitary commutation relations abstract we define nonselfadjoint operator algebras with generators l e 1 l e n l f 1 l f m subject to the unitary commutation relations of the form l e i l f j k l u i j k l l f l l e k where u u i j k l is an n m n m unitary matrix these algebras which generalise the analytic toeplitz algebras of rank 2 graphs with a single vertex are classified up to isometric isomorphism in terms of the matrix u

subproduct systems over n n abstract we develop the theory of subproduct systems over the monoid n n and the non self adjoint operator algebras associated with them these are double sequences of hilbert spaces x m n m n 0 equipped with a multiplication given by coisometries from x i j x k l to x i k j l we find that the character space of the norm closed algebra generated by left multiplication operators the tensor algebra is homeomorphic to a complex homogeneous affine algebraic variety intersected with a unit ball certain conditions are isolated under which subproduct systems whose tensor algebras are isomorphic must be isomorphic themselves in the absence of these conditions we show that two numerical invariants must agree on such subproduct systems additionally we classify the subproduct systems over n n by means of ideals in algebras of non commutative polynomials

dynamical behavior of a class of vibratory systems with symmetrical rigid stops near the point of codimension two bifurcation a multi degree of freedom vibratory system having symmetrically placed rigid stops is considered the system consists of linear components but the maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops repeated impacts usually occur in the vibratory system due to the rigid amplitude constraints such models play an important role in the studies of mechanical systems with clearances or gaps local codimension two bifurcation of maps involving a real eigenvalue and a complex conjugate pair escaping the unit circle simultaneously is analyzed by using the center manifold theorem technique and normal form method for maps symmetrical double impact periodic motion and poincare map of the system are derived analytically a center manifold theorem technique is applied to reduce the poincare map to a three dimensional one and the normal form map associated with the codimension two bifurcation is obtained local behaviors of the vibratory systems with symmetrical rigid stops near the points of codimension two bifurcations are reported by the presentation of results for a two degree of freedom vibratory system with symmetrical stops the existence and stability of symmetrical double impact periodic motion are analyzed explicitly also local bifurcations at the points of change in stability are analyzed near the point of codimension two bifurcation there exists not only hopf bifurcation of period one double impact motion but also pitchfork bifurcation of the motion pitchfork bifurcation of period one double impact symmetrical motion results in the period one double impact unsymmetrical motion the unsymmetrical double impact motion is of two antisymmetrical forms due to different initial conditions and symmetrical stops with change of the forcing frequency the unsymmetrical double impact periodic motion will undergo hopf bifurcation moreover the period one double impact symmetrical motion will undergo hopf bifurcation directly as the forcing frequency is changed in the contrary direction the routes of quasi periodic impact motions to chaos are observed by results from simulation

numerical simulation of the dynamics of an impacting bar we calculate numerically the motion of a slender bar dropped on a rigid foundation for the computation the bar is discretized by a system of rigid bodies linked by spiral springs or by a pair of linear springs we assume that the impact is frictionless and we model it by newton s law we compute the motion by using either an event driven method based on the detection of impacts or a time stepping scheme avoiding the detection of impacts we calculate also the apparent restitution coefficient and we compare our results with the experimental and numerical results of stoianovici and hurmuzlu

convergence of a time stepping scheme for rigid body dynamics and resolution of painleve s problem this paper gives convergence theory for a new implicit time stepping scheme for general rigid body dynamics with coulomb friction and purely inelastic collisions and shocks an important consequence of this work is the proof of existence of solutions of rigid body problems which include the famous counterexamples of painleve the mathematical basis for this work is the formulation of the rigid body problem in terms of measure differential inclusions of moreau and monteiro marques the implicit time stepping method is based on complementarity problems and is essentially a particular case of the algorithm described in anitescu potra 2 which in turn is based on the formulation in stewart trinkle 47

on the approximate solution of a piecewise nonlinear oscillator under super harmonic resonance abstract an approximate solution for the super harmonic resonance response of a periodically excited nonlinear oscillator with a piecewise nonlinear linear characteristic is constructed using both a matching method and a modified averaging method the validity of the developed analysis is confirmed by comparing the approximate solutions with the results of direct numerical integration of the original equation

bifurcations and chaos in a system with impacts abstract a vibro impact system is considered a body bounces on a flat horizontal surface of the vibro bench hopf bifurcations of the vibro impact system in two kinds of strong resonance cases 03 1 and 04 1 are investigated a poincare map of the system is established the period 1 single impact motion of the system and its stability are studied by analytical methods and hopf bifurcation values and intersecting conditions of the system in strong resonance cases are determined a center manifold theorem technique is applied to reduce the poincare map to a two dimensional one which is put into normal form by theory of normal forms by the theory of hopf and subharmonic bifurcations of fixed points in r 2 strong resonance dynamic behavior of the vibro impact system near points of resonance is analyzed in the resonance case of 03 1 the system generally exhibits unstable period 3 three impact motion in the resonance case of 04 1 the system can exhibit stable period 4 four impact motion and quasi periodic motion the theoretical analyses are verified by numerical solutions routes to chaos in two kinds of strong resonance cases considered are obtained by numerical simulations quasi periodic impacts in non resonance and weak resonance cases doubling periodic bifurcations and chaotic motions are also stated briefly by numerical analyses

from enzyme kinetics to epidemiological models with michaelis menten contact rate design of nonstandard finite difference schemes we consider the basic sir epidemiological model with the michaelis menten formulation of the contact rate from the study of the michaelis menten basic enzymatic reaction we design two types of nonstandard finite difference nsfd schemes for the sir model exact related schemes based on the lambert w function and schemes obtained by using mickens s rules of more complex denominator functions for discrete derivatives and nonlocal approximations of nonlinear terms we compare and investigate the performance of the two types of schemes by showing that they are dynamically consistent with the continuous model numerical simulations that support the theory and demonstrate computationally the power of nsfd schemes are presented

application of non smooth modelling techniques to the dynamics of a flexible impacting beam non smooth modelling techniques have been successfully applied to lumped mass type structures for modelling phenomena such as vibro impact and friction oscillators in this paper the application of these techniques to continuous elements using the example of a cantilever beam is considered employing a galerkin reduction to form an n degree of freedom modal model a technique for modelling impact phenomena using a non smooth dynamics approach is demonstrated numerical simulations computed using the non smooth model are compared with experimentally recorded data for a flexible beam constrained to impact on one side a method for dealing with sticking motions when numerically simulating the beam motion is presented in addition choosing the dimension of the model based on power spectra of experimentally recorded time series is discussed

optimal control problems in hybrid systems with active singularities abstract the optimal control problem for systems with controlled unilateral phase constraints is considered the definition of the generalized solutions is introduced the transformation method for the original optimal control problem within the class of generalized solution to a standard optimal control problem is proposed and the necessary optimality conditions are found

observer design for unilaterally constrained lagrangian systems a passivity based approach this paper addresses the problem of state estimation in nonlinear lagrangian dynamical systems with frictionless unilateral constraints using the position measurement as output the discontinuous velocity variable in such systems is modeled as a function of bounded variation so that zeno phenomenon is not ruled out since the derivative of such functions is represented with the lebesgue stieltjes measure the framework of measure differential inclusions mdis is used to describe the dynamics a class of estimators is proposed which also uses the framework of mdis and is shown to generate asymptotically converging state estimates the existence and uniqueness of solutions for the proposed estimators is rigorously proven the global stability of error dynamics is analyzed using the generalized lyapunov methods for functions of bounded variation as particular cases of our estimators we provide an explicit construction of a full order observer and a reduced order observer

simultaneous enforcement of constraints at position and velocity levels in the nonsmooth generalized α scheme abstract this paper presents a formalism for the transient simulation of nonsmooth dynamic mechanical systems composed of rigid and flexible bodies kinematic joints and frictionless contact conditions the proposed algorithm guarantees the exact satisfaction of the bilateral and unilateral constraints both at position and velocity levels thus it significantly differs from penalty techniques since no penetration is allowed the numerical scheme is obtained in two main steps firstly a splitting method is used to isolate the contributions of impacts which shall be integrated with only first order accuracy from smooth contributions which can be integrated using a higher order scheme secondly following the idea of gear gupta and leimkuhler the equation of motion is reformulated so that the bilateral and unilateral constraints appear both at position and velocity levels after time discretization the equation of motion involves two complementarity conditions and it can be solved at each time step using a monolithic semi smooth newton method the numerical behavior of the proposed method is studied and compared to other approaches for a number of numerical examples it is shown that the formulation offers a unified and valid approach for the description of contact conditions between rigid bodies as well as between flexible bodies