stability of periodic motion bifurcations and chaos of a two degree of freedom vibratory system with symmertical rigid stops a two degree of freedom system having symmetrically placed rigid stops and subjected to periodic excitation is considered such models play an important role in the studies of mechanical systems with clearances or gaps the period one double impact symmetrical motion and its poincare map are derived analytically stability and local bifurcations of the period one double impact symmetrical motion are analyzed by the equation of poincare map the routes from period one double impact symmetrical motion to chaos via pitchfork bifurcations and period doubling bifurcation are studied by numerical simulation some non typical routes to chaos caused by grazing the stops and hopf bifurcation of period two four impact motion are analyzed hopf bifurcations of period one double impact symmetrical and antisymmetrical motions are shown to exist in the two degree of freedom vibratory system with two sided stops interesting feature like the period one four impact symmetrical motion is also found and its route to chaos is analyzed it is of special interest to acquire an overall picture of the system dynamics for some extreme values of parameters especially those which relate to the degenerated case of a single degree of freedom system and these analyses are presented here

the dynamics of repeated impacts with a sinusoidally vibrating table a deceptively simple difference equation is derived which approximately describes the motion of a small ball bouncing vertically on a massive sinusoidally vibrating plate in the case of perfect elastic impacts the equation reduces to the standard mapping which has been extensively studied by physicists in connection with the motions of particles constrained in potential wells it is shown that for sufficiently large excitation velocities and a coefficient of restitution close to one this deterministic dynamical system exhibits large families of irregular non periodic solutions in addition to the expected harmonic and subharmonic motions the physical significance of these and other chaotic motions which appear to occur frequently in non linear oscillations is discussed

comparison results between linear and quasilinear elliptic boundary value problems on considere des resultats de comparaison du type suivant on suppose que u c 2 c satisfait un probleme aux valeurs limites elliptique quasi lineaire div a x u u u b x u u dans u sur et on suppose que v est la solution classique du probleme lineaire v const dans v sur on etudie l existence de situations ou u x v x x

symmetry of the poincare map and its influence on bifurcations in a vibro impact system abstract symmetric period n 2 motion of a three degree of freedom 3 dof vibro impact system with symmetric rigid constraints is considered the poincare map of the system is established and the symmetric fixed point of the poincare map corresponds to the associated symmetric period n 2 motion it is shown that the poincare map exhibits some symmetry property and can be expressed as the second iteration of another unsymmetric implicit map the symmetry of the poincare map influence bifurcation behaviors in vibro impact system significantly and suppresses not only period doubling bifurcation but also hopf flip bifurcation and pitchfork flip bifurcation based on the second iteration of another unsymmetric implicit map the normal forms in the case of hopf hopf bifurcation and hopf bifurcation satisfying 1 2 resonant conditions are obtained by numerical simulation general hopf bifurcation hopf hopf bifurcation and hopf bifurcation satisfying 1 2 resonant conditions of the symmetric period n 2 motion are represented however period doubling bifurcation hopf flip and pitchfork flip bifurcation have not been obtained which reflects upon the effect of the symmetry property on possible bifurcations it is interesting that the system can exhibit both the characteristic of 1 2 resonance and that of torus t2 under some parameter combination and 2 t1 torus is also obtained

on saint venant s principle in finite anti plane shear an energy approach some time ago a version of saint venant s principle was formulated and established for finite elastostatics 1 as was discussed in 1 the issues of concern in connection with saint venant s principle in the nonlinear theory of elasticity are considerably more involved then those arising in the linear theory for a survey of results on saint venant s principle primarily for linear theories see e g 2 5 one difficulty is that the appropriate saint venant solutions need to be carefully characterized see e g 6 13 and the references cited therein secondly in the absence of superposition consideration of self equilibrated end loads is no longer sufficient furthermore instabilities may have to be taken into account also the decay rate for end effects even if exponential might depend on the overall loading as well as on geometry and material characteristics several of these issues have been considered in recent studies in the nonlinear elasticity context 1 14 23 as well as in investigations of spatial decay of solutions of nonlinear elliptic partial differential equations 24 32

principal minors and diagonal similarity of matrices abstract let a b be n n matrices with entries in a field f our purpose is to show the following theorem suppose n 4 a is irreducible and for every partition of 1 2 n into subsets with 2 2 either rank a 2 or rank a 2 if a and b have equal corresponding principal minors of all orders then b or b t is diagonally similar to a

singularities in vibro impact dynamics abstract the non differentiable nature of vibro impact dynamics can lead to breakdown of the global stable manifold theorem applicable to smooth dynamical systems in one dimensional periodically excited system the breakdown leads to the shredding of the stable manifolds and this process is analyzed in detail bodily translation of filaments of stable manifolds can lead to homoclinic intersections and this topic is discussed in the context of a new bifurcation due to re entrance shredding leads to a strong similarity between the geometries of the singular subspaces and the geometries of homoclinic tangles and strange attracting sets

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

principal minor assignment isometries of hilbert spaces volumes of parallelepipeds and rescaling of sesqui holomorphic functions abstract in this article we consider the following equivalence relation on the class of all functions of two variables on a set x we will say that l m x x c are rescalings if there are non vanishing functions f g on x such that m x y f x g y l x y for any x y x we give criteria for being rescalings when x is a topological space and l and m are separately continuous or when x is a domain in c n and l and m are sesqui holomorphic a special case of interest is when l and m are symmetric and f g only has values 1 this relation between m and l in the case when x is finite and so l and m are square matrices is known to be characterized by the equality of the principal minors of these matrices we extend this result to the case when x is infinite as an application we characterize restrictions of isometries of hilbert spaces on weakly connected sets as the maps that preserve the volumes of parallelepipeds spanned by finite collections of vectors

on the spatial decay of the energy for some quasilinear boundary value problems in semi infinite cylinders dans un cylindre semi infini a 3 dimensions r on considere le probleme p x u u u j j 0 avec des donnees aux limites de type dirichlet ou neumann on deduit une borne d energie

the hilbert phenomenon in chaotic motions abstract it is proved that chaotic dynamical systems have d 1 singularity due to such singularity and its periodicity in hausdorff metric spaces a chaotic dynamical system induces a hilbert iterated system which was discovered at the end of the 19th century this implies that the most puzzling discovery in nature in the 20th century can be put into deep correspondence with that in thoughts in the 19th century all this is verified by the numerical experiments of chaotic vibrations of a beam and a tension slack oscillator

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

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

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

type i periodic motions for nonlinear impact oscillators abstract in this paper a general class of nonlinear impact oscillators is considered for subharmonic bifurcation this system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation and its unperturbed system possesses a pair of homoclinic cycles via the identification given by the impact law and three separate families of periodic orbits inside and outside the homoclinic cycles by discussing the subharmonic orbits inside the homoclinic cycles the subharmonic melnikov method established for smooth dynamical systems is extended to be applicable to the non smooth system

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

periodic motions and global bifurcations of a two degree of freedom system with plastic vibro impact a two degree of freedom d o f impact system with proportional damping is considered the maximum displacement of one of the masses is limited to a threshold value by a rigid wall which gives rise to a non linearity in the system a limiting case of a dynamical problem arising in the mechanical systems with amplitude constraints is investigated for the perfectly plastic vibro impact case dynamics of a two degree of freedom vibratory system contacting a single stop is represented by a three dimensional map existence and stability of period n single impact motions are analyzed by analytical and numerical methods it is shown that the vibro impact system may exhibit two different types of periodic motions due to the piecewise property of the map transitions of two types of period n single impact motions are demonstrated the singularities of the poincare map caused by grazing boundary motion of the impact oscillator are considered due to the piecewise discontinuities and singularities of the map the vibro impact system is shown to undergo period doubling bifurcations followed by complex sequence of transitions in which the period doubling cascades do not occur and extremely long periodic and chaotic motions are generated directly with the motions with grazing boundary occurring finally the influence of system parameters on periodic impacts and global bifurcations is discussed

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

phragmen lindelof principles for solutions of elliptic differential inequalities on considere des operateurs differentiels elliptiques lineaires d ordre 2 lu i j 1 n a i j u 2u x i x j i 1 n bi u x i cu ou les coefficients a i j b i c sont definis dans un domaine de r n on etudie les fonctions barrieres pour de tels operateurs on etablit des principes de phragmen lindelof pour des points limites finis et le point limite a l infini

dynamics of a two degree of freedom periodically forced system with a rigid stop diversity and evolution of periodic impact motions abstract a two degree of freedom periodically forced system with a clearance is considered the correlation between dynamic performance and system parameters is studied to find their feasible matching law the fundamental group of impact motions is defined which have the excitation period and differ by the number p of impacts the generation mechanism of complete and incomplete chattering impact vibration of the system is studied a series of singular points between any two adjacent fundamental impact motions are found at which real grazing and bare grazing bifurcation boundaries of one of them saddle node and period doubling bifurcation boundaries of the other alternately intersect and create two types of transition regions hysteresis and tongue shaped regions subharmonic impact motions are found to regularly appear in the tongue shaped regions based on the sampling ranges of parameters dynamics of the system is studied with emphasis on the influence of system parameters on impact velocities existence regions and correlative distribution of single impact periodic motions by multi parameter and multi performance co simulation analysis

frequency response characteristics of a multi degree of freedom system with clearances abstract in an earlier study the frequency response characteristics of a single degree of freedom system with a clearance non linearity were studied the current study an extension of the earlier work is concerned with the frequency response characteristics of a multi degree of freedom system with clearances the method of harmonic balance is used to develop approximate analytical solutions of the undamped equations of motion of a multi degree of freedom system composed of three coupled non linear oscillators for primary resonances and a harmonic excitation general formulations are presented which can be used to study both the existence and the stability of the solutions these general formulations are used to discuss a number of modal spacing and modal coupling issues an analysis methodology for multi degree of freedom systems is introduced and illustrated by two special but practical cases a strongly non linear system and a weakly non linear system the results of the analysis of the special cases are validated by using analog simulation it is shown that an analysis of the special cases of the general multi degree of freedom non linear formulation can provide not only an improved understanding of the dynamic behavior of non linear systems but also can be used as the basis for the development of simplified approximate solutions the limitations of the current study and areas for further research are discussed

an efficient algorithm for the symmetric principal minor assignment problem abstract we consider the problem of reconstructing a symmetric matrix from its principal minors which has several applications in information theory and statistical modeling we develop a theory of symmetric matrices with equal corresponding principal minors based on a simple equivalent property due to oeding 2011 10 we then use this theory to provide a method for choosing a canonical representative from the class of all symmetric matrices with specified principal minors finally we provide an efficient algorithm for computing this canonical representative given its principal minors as input

skew symmetric matrices and their principal minors abstract in this article we study the following problem given a positive integer k what is the relationship between two matrices with entries in a field k and having equal corresponding principal minors of order at most k our main theorem improves a result of loewy 13 for skew symmetric matrices all of whose off diagonal entries are nonzero

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

a discontinuity geometry view of the relationship between saddle node and grazing bifurcations abstract in this study we will introduce a topological approach termed as discontinuity geometry for both analysis and visualization of the dynamics of periodically forced impact oscillators after a brief outline of the methodology and the general representation of trajectories with impacts the framework will be used to analyse grazing bifurcations in particular it will be used to geometrically explain why there is sometimes a saddle node bifurcation present in the vicinity of grazing bifurcations and why it is sometimes absent

stability and dynamics of a harmonically excited elastic perfectly plastic oscillator this paper deals with the stability and the dynamics of a harmonically excited elastic perfectly plastic oscillator the hysteretical system is written as a non smooth forced autonomous system it is shown that the dimension of the phase space can be reduced using adapted variables free vibrations of such a system are then considered for the damped system the extended direct method of liapounov is applied to this non smooth mechanical system and the asymptotic stability of the origin is proven in the new phase space the forced vibration of such an oscillator is treated by numerical approach by using the time locating techniques the stability of the limit cycles is analytically investigated with a perturbation approach the boundary between elastoplastic shakedown and alternating plasticity is given in closed form it is shown that this boundary corresponds to a bifurcation boundary for the undamped system period doubling bifurcation finally the equivalent damping of this hysteretic system is characterized from dynamical properties

double neimark sacker bifurcation and torus bifurcation of a class of vibratory systems with symmetrical rigid stops abstract a multidegree of freedom system having symmetrically placed rigid stops and subjected to periodic excitation 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 double neimark sacker bifurcation of the system is analyzed by using the center manifold and normal form method of maps the period one double impact symmetrical motion and homologous disturbed map of the system are derived analytically a center manifold theorem technique is applied to reduce the poincare map to a four dimensional one and the normal form map associated with double neimark sacker bifurcation is obtained the bifurcation sets for the normal form map are illustrated in detail local behavior of the vibratory systems with symmetrical rigid stops near the points of double neimark sacker bifurcations is reported by the presentation of results for a three degree of freedom vibratory system with symmetrical stops the existence and stability of period one double impact symmetrical motion are analyzed explicitly also local bifurcations at the points of change in stability are analyzed thus giving some information on dynamical behavior near the points of double neimark sacker bifurcations near the value of double neimark sacker bifurcation there exist period one double impact symmetrical motion and quasi periodic impact motions the quasi periodic impact motions are represented by the closed circle and tire like attractor in projected poincare sections with change of system parameters the quasi periodic impact motions usually lead to chaos via tire like torus doubling

stability and bifurcation analysis for oscillators with motion limiting constraints abstract the stability and bifurcation analysis is presented for a class of non linear single degree of freedom vibrating systems excited by harmonic forcing the common characteristic of these systems is that their damping and stiffness parameters can change abruptly at certain displacement values the most general n periodic steady state solutions with an arbitrary number of contacts per response cycle are analyzed it is shown that when the damping is positive the change of stability of the steady state response cannot happen through a hopf bifurcation

dynamical analysis of a two parameter family for a vibro impact system in resonance cases abstract dynamics of a two degree of freedom vibro impact system in resonance is considered the dynamical model and poincare maps are established when a pair of complex conjugate eigenvalues of the jacobian matrix cross the unit circle and satisfy the resonant condition 0 4 1 or 0 3 1 the four dimensional map is reduced to a two dimensional one by the center manifold theorem and the reduced map is put into its normal form by the method of normal form the two parameter unfoldings of local dynamical behavior studied in this paper develop the results of one parameter family analysis the hopf and subharmonic bifurcation conditions of period n 1 motion are given the numerical simulation method confirms the theoretical analysis it is shown that there exists an invariant torus via hopf bifurcation and period 4 4 motions via subharmonic bifurcation as two controlling parameters varying near the critical point for the resonance 0 4 1 and that there exists an invariant circle and unstable fixed points of order 3 bifurcating from the fixed point and the system leads eventually to chaos in the resonance 0 3 1

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

decay estimates for a class of second order quasilinear equations in three dimensions in a recent paper 1 the authors made use of differential inequality techniques to derive exponential decay estimates for a class of second order quasilinear equations not necessarily elliptic in the plane specifically they were concerned with solutions of equations in divergence form defined in a semi infinite strip with non zero boundary data on one end only decay estimates in various norms were derived

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

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

torus doubling bifurcations and strange nonchaotic attractors in a vibro impact system whether strange nonchaotic attractors snas can occur typically in dynamical systems other than quasiperiodically driven systems has been an open problem here we show that the snas can be induced by the interruption of torus doubling bifurcation in a periodically driven vibro impact system it is found that the creation of snas occurs due to the collision of doubled torus with some unstable periodic orbits based on the poincare map technique two types of codimension 3 bifurcations are represented and typical torus doubling bifurcations can occur in the neighborhood of these bifurcation points we use the singular continuous fourier spectrum and its scaling to verify the strangeness nondifferentiability of the attractors the implicitly transcendental map is used to calculate the largest lyapunov exponent by qr based algorithm which ensures that the underlying dynamics with nonpositive lyapunov exponents is nonchaotic the associated mechanism is described for the creation of snas through detecting unstable periodic orbits from poincare map series

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

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

existence and stability of the grazing periodic trajectory in a two degree of freedom vibro impact system this paper investigates the existence and stability of the grazing periodic trajectory in a two degree of freedom vibro impact system the criterion for existence of grazing period n motion is presented a local analysis based on the discontinuity mapping approach is employed to derive a normal form poincare mapping near the grazing trajectory based on the above approach a condition of stability can be formulated such that a grazing trajectory will be discontinuous if the condition is unfulfilled the predicted grazing bifurcations are in agreement with the numerical results in particular comparison of the grazing bifurcation diagrams of the normal form poincare mapping and the simulation diagrams of the original differential equation illustrates the validity of the discontinuity mapping approach

modal analysis of mechanical systems with impact non linearities limitations to a modal superposition this paper presents an attempt to generalize the modal superposition formula to mechanical systems with impact type non linearities following the procedure introduced for smooth non linearities the study is restricted to simple one and two degree of freedom systems with a unilateral constraint on one of the degrees of freedom for which the response can be analytically determined generalized frequencies modes and masses are built in the procedure the results obtained for various sets of parameters point out some limitations to the validity of a general modal superposition formula

direct and inverse problems encountered in vibro impact oscillations of a discrete system abstract we study direct and inverse problems that arise in the vibro impact oscillations of a discrete system specifically we examine a class of systems with two coordinates undergoing single or double sided impacts however the presented techniques are sufficiently general to apply to systems with multiple impacts the analytical methods employed are a nonlinear normal mode nnm type analysis and a boundary value problem bvp formulation and enable the computation of various branches of bifurcating periodic solutions with different impacting characteristics additional insight on the dynamics of these systems is obtained by direct integrations of the equations of motion and by numerical poincare maps it is found that the vibro impact systems considered possess rich nonlinear dynamics including vibro impact localized and nonlocalized time periodic motions complicated bifurcation structures giving rise to new types of single and double sided impacting motions mode instabilities and chaotic responses we also formulate inverse vibro impact problems whereby we seek the class of dynamical systems that produce specified orbits in the configuration plane the solutions of the inverse problems are generally non unique since they can be reduced to underdetermined sets of algebraic equations with multiple infinities of unknowns numerical applications are provided to demonstrate the techniques and validate the analytical results