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

brief minimal and non minimal optimal fixed order compensators for time varying discrete time systems using the minimality property of finite horizon time varying compensators established in this paper and the moore penrose pseudo inverse instead of the standard inverse strengthened discrete time optimal projection equations sdope and associated boundary conditions are derived for finite horizon fixed order lqg compensation they constitute a two point boundary value problem explicit in the lqg problem parameters which is equivalent to first order necessary optimality conditions and which is suitable for numerical solution the minimality property implies that minimal compensators have time varying dimensions and that the finite horizon optimal full order compensator is not minimal the use of the moore penrose pseudo inverse is further exploited to reveal that the optimal projection approach can be generalised but only to partially include non minimal compensators furthermore the structure of the space of optimal compensators with arbitrary dimensions is revealed to a large extent max min compensator dimensions are introduced and their significance in solving numerically the two point boundary value problem is explained the numerical solution is presented in a recently published companion paper which relies on the results of this paper

harmonic analysis on nilpotent groups and singular integrals ii singular kernels supported on submanifolds on etudie des operateurs de convolution singuliers sur des groupes de lie nilpotents generaux et plus specifiquement des operateurs dont les noyaux ont pour supports des varietes de faible dimension et qui peuvent aussi contenir des facteurs polynomiaux exponentiels oscillatoires

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

sub hermitian geometry and the quantitative newlander nirenberg theorem abstract given a finite collection of c 1 complex vector fields on a c 2 manifold m such that they and their complex conjugates span the complexified tangent space at every point the classical newlander nirenberg theorem gives conditions on the vector fields so that there is a complex structure on m with respect to which the vector fields are t 0 1 in this paper we give intrinsic diffeomorphic invariant necessary and sufficient conditions on the vector fields so that they have a desired level of regularity with respect to this complex structure i e smooth real analytic or have zygmund regularity of some finite order by addressing this in a quantitative way we obtain a holomorphic analog of the quantitative theory of sub riemannian geometry initiated by nagel stein and wainger we call this sub hermitian geometry moreover we proceed more generally and obtain similar results for manifolds which have an associated formally integrable elliptic structure this allows us to introduce a setting which generalizes both the real and complex theories

models of degenerate fourier integral operators and radon transforms fourier integral operators whose lagrangians project with singularities arise frequently in many areas of analysis and geometry 3 5 9 10 18 20 however there are as yet few analytic tools available for their study and even their simplest regularity properties with respect to the singularities of the lagrangian are still obscure 4 5 13 16 in this paper we study a class of oscillatory integrals ta and fourier integral operators r which can be expected to model the higher order singularities of the lagrangian they have homogeneous polynomial phases in two variables of order n and indeed the case n 3 is the model for lagrangians which project with whitney folds 3 10 11 the main difficulty which sets the higher order degeneracy cases n 4 apart from the the lower ones n 2 3 is that the critical varieties which arise there are usually not smooth manifolds a systematic study of these oscillatory integrals was begun in 14 the main idea in that work was to treat the critical varieties as smooth manifolds away from a lower dimensional subvariety and to keep track of the distance to this lower dimensional subvariety to achieve this we introduced a method of stationary phase which exhibited clearly the dependence on the distance between critical points the method gave sharp bounds on the size of the kernel k x y of tat but no information on its phase and the resulting cancellations it does not seem possible to refine this approach to the phase level and this suggests looking instead for decompositions of the operators ta which can incorporate indirectly the required cancellations the main goal of this paper is to introduce such decompositions these decompositions which we will try to describe momentarily reflect the singular nature of the critical points and are powerful enough to yield sharp bounds for the above oscillatory integrals and radon transforms although the models under consideration are very

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

an optimal controller reduction by covariance equivalent realizations based upon the theory of covariance equivalent realizations 1 this note presents a technique to obtain reduced order controllers via an optimal projection of full order lqg controllers this projection is optimal in the sense that it is a weighted least square approximation to a covariance equivalent projection of the full order optimal closed loop system the technique which is capable of extracting optimal minimal lqg controllers whenever they exist is also shown to yield controllers that are equivalent to those obtained by component cost analysis 12

coprime factorization controller reduction with bezout identity induced frequency weighting abstract this paper introduces a frequency weighted controller reduction scheme which offers good closed loop stability under use of the reduced order controller by combining the use of a stable fractional representation of the controller with a frequency weighting in the reduction procedure derived by consideration of the closed loop stability margin we formulate a frequency weighted controller reduction procedure with the weight simplified by the bezout identity some examples are used to illustrate the reduction procedure

on direct versus indirect methods for reduced order controller design comparisons are obtained between direct parameter optimization and indirect controller reduction methods in reduced order controller design in particular the authors compare the controller canonical correlation coefficients method of c de villemagne and r e skelton 1988 to the optimal projection theory of d c hyland and d s bernstein 1984 it is noted that suboptimal methods may in certain cases be simpler to implement than optimal methods thus the choice of direct versus indirect methods can be viewed as a tradeoff between design simplicity and actual control system performance

frequency weighted controller reduction methods and loop transfer recovery abstract this paper shows that if one designs an lqg controller using the conventional technique of loop transfer recovery ltr then two frequency weighted controller reduction methods the enns frequency weighted balanced truncation enns 1984a ph d thesis stanford university ca 1984b proc 23rd cdc las vegas nv 127 132 and the bezout identity induced frequency weighted reduction method anderson and liu 1987 proc amer control conf mn will be equivalent under the condition that the plant transfer function is square nonsingular and minimum phase we also show that enns method is equivalent to the bezout identity induced frequency weighted reduction method if the controller itself is stable and a particular representation for the controller is assumed

balanced forms canonicity and parameterization we show that balanced forms are canonical and continuously parameterize the set of systems of a given order which are asymptotically stable minimal and have distinct singular values the advantage of the balanced canonical form over other classical canonical forms lies precisely in the guaranteed stability and minimality of the parameterization we also derive formulas allowing the synthesis of systems using gradient methods in balanced form

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

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

multi parameter singular radon transforms ii the lp theory the purpose of this paper is to study the l2 boundedness of operators of the form f x f t x k t dt where t x is a c function defined on a neighborhood of the origin in t x n n satisfying 0 x x is a c cut off function supported on a small neighborhood of 0 n and k is a multi parameter singular kernel supported on a small neighborhood of 0 n the goal is given an appropriate class of kernels k to give conditions on such that every operator of the above form is bounded on l2 the case when k is a calderon zygmund kernel was studied by christ nagel stein and wainger we generalize their conditions to the case when k has a multi parameter structure for example when k is given by a product kernel even when k is a calderon zygmund kernel our methods yield some new results this is the first paper in a three part series the later two of which are joint with e m stein the second paper deals with the related question of lp boundedness while the third paper deals with the special case when is real analytic

sharp regularity for the integrability of elliptic structures abstract as part of his celebrated complex frobenius theorem nirenberg showed that given a smooth elliptic structure on a smooth manifold the manifold is locally diffeomorphic to an open subset of r r c n for some r and n in such a way that the structure is locally the span of t 1 t r z 1 z n where r r c n has coordinates t 1 t r z 1 z n in this paper we give optimal regularity for the coordinate charts which achieve this realization namely if the manifold has zygmund regularity of order s 2 and the structure has zygmund regularity of order s 1 for some s 0 then the coordinate charts may be taken to have zygmund regularity of order s 2 we do this by generalizing malgrange s proof of the newlander nirenberg theorem to this setting

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

on the nonlinear brascamp lieb inequality we prove a nonlinear variant of the general brascamp lieb inequality our proof consists of running an efficient or tight induction on scales argument which uses the existence of gaussian near extremizers to the underlying linear brascamp lieb inequality lieb s theorem in a fundamental way a key ingredient is an effective version of lieb s theorem which we establish via a careful analysis of near minimizers of weighted sums of exponential functions instances of this inequality are quite prevalent in mathematics and we illustrate this with some applications in harmonic analysis

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

a note on balanced controller reduction the standard linear quadratic gaussian lqg optimal controller is not minimal in general even if the plant is minimal hence the order of the optimal controller can be less than that of the plant this note shows that the original balancing method of moore 1 can be used to generate these reduced order optimal controllers but the newer balancing methods recently proposed for controller reduction might not generate the reduced order optimal controllers even when they exist

a kakeya type problem for circles we prove full hausdorff dimension in a variant of the kakeya problem involving circles in the plane and also sharp estimates for the relevant maximal function these results can also be formulated in terms of the wave equation in two space variables a novelty in our approach is the use of ideas from computational geometry

corner bifurcations in non smoothly forced impact oscillators abstract we consider an impacting mechanical system in which a particle at position u t impacts with a periodically moving obstacle at position z t the motion of which is non smooth in particular we look at corner events when u impacts with z very close to a point where z loses smoothness we show that this leads through a corner bifurcation to complex dynamics in u which can include periodic orbits of arbitrary period and period adding cascades by analysing associated maps close to the corner event we show that this dynamics can be understood in terms of the iterations of a two dimensional piecewise linear discontinuous map we also show some links between this analysis and the difficult problem of understanding the motion of three objects which may have simultaneous impacts

singular and maximal radon transforms analysis and geometry part 2 geometric theory 8 curvature introduction 8 1 three notions of curvature 8 2 theorems 8 3 examples 9 curvature some details 9 1 the exponential representation 9 2 diffeomorphism invariance 9 3 curvature condition cy 9 4 two lemmas 9 5 double fibration formulation 10 equivalence of curvature conditions 10 1 invariant submanifolds and deficient lie algebras 10 2 vanishing jacobians 10 3 construction of invariant submanifolds

controller reduction via stable factorization and balancing abstract a controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper for a certain right coprime factorization of an lqg designed controller k s n s d 1 s we approximate using a balancing technique the pair d s n s t by a low order pair d1 s n1 s t defining a factorization of the reduced order controller k 1 s n1 s d1 1 s we show that reducing the controller order in this way is motivated in a natural way which leads to the expectation of both good stability properties and good accuracy of approximation of closed loop behaviour this is also demonstrated in some examples

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

performance bounds for coprime factor controller reductions this paper investigates the performance of coprime factor based controller reduction methods we show that under some mild conditions closed loop performance bounds can be derived for observer based controllers the results in this paper lay a mathematical foundation for using coprime factor controller reduction methods in certain classes of controller reduction problems

l p l p estimates for overdetermined radon transforms we prove several variations on the results of f ricci and g travaglini 2001 concerning l p l p bounds for convolution with all rotations of arc length measure on a fixed convex curve in r 2 estimates are obtained for averages over higher dimensional convex nonsmooth hypersurfaces smooth k dimensional surfaces and nontranslation invariant families of surfaces we compare ricci and travaglini s approach based on average decay of the fourier transform with an approach based on l 2 boundedness of fourier integral operators and show that essentially the same geometric condition arises in proofs using the two techniques

hilbert integrals singular integrals and radon transforms i on introduit deux nouvelles classes d operateurs on etudie leur relation et on montre comment on peut les appliquer a l etude des problemes aux valeurs limites

uniform bounds for convolution and restricted x ray transforms along degenerate curves abstract we establish endpoint lebesgue space bounds for convolution and restricted x ray transforms along curves satisfying fairly minimal differentiability hypotheses with affine and euclidean arclengths we also explore the behavior of certain natural interpolants and extrapolants of the affine and euclidean versions of these operators

a generalization of bourgain s circular maximal theorem where d r is the normalized surface measure on r s1 it is easy to see that m is not bounded on l2 see example 1 1 below a well known result of bourgain 1 asserts that m is bounded on lp for 2 p we will consider the question of boundedness of m and m from lp to lq unless stated to the contrary we will be dealing only with functions defined on r2 absolute constants will be denoted by c and the notation will mean up to a constant

radon transforms and finite type conditions the purpose of this paper is to study averaging operators of radon transform type we shall formulate suitable finite type conditions and prove lp sobolev and lplq estimates the results will be essentially sharp for operators associated with families of curves in the plane let xc and t be smooth manifolds dimx3 nl dim2 nr and let m be a submanifold in xc x t with conormal bundle n m we denote by t the codimension of m we shall always assume that the projections lry m x and 7rt m 42 are surjective with rank dirx nl rank dr9 nr this in particular implies that n m c t x o x t o where 0 refers to the zero sections in t x and t 2 respectively this is the usual assumption for the canonical relation associated with lagrangian distributions arising as kernels of fourier integral operators the assumptions on dirx d7rt imply that for fixed x e x y e 2 the sets amx y e t x y e mi m x ye2x x y em i

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

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

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

a property of internally balanced and low noise structures given an n th order minimal asymptotically stable system h z we consider the class e of minimal state space realizations which minimize the trace of gamma 2 k w where k is the controllability and w the observability grammian we show that the optimal structures are defined by w gamma 2 k with optimal cost 2 gammas where s is the sum of singular values for gamma 1 the internally balanced structure belongs to e while if gamma s n the optimal noise structures of mullis and roberts 1 belong to e in particular both structures are optimal when gamma 1 and s n

nonlocal inversion formulas for the x ray transform etude de la reconstruction d une fonction f x sur r n a partir de ses integrales sur des plans k dimensionnels cas de k 1

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

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

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

fourier integral operators with cusp singularities we study the boundedness properties on lebesgue and sobolev spaces of fourier integral operators associated with canonical relations such that at least one of the projections is a simple whitney cusp in the process we obtain decay estimates for oscillatory integral operators whose symplectic relations have the same singular structure such singularities occur generically for averages over lines and curves in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i on l 2 we show that the operators lose 1 3 derivative to obtain sharp results off of l 2 we need to impose an additional transversality condition satisfied by many geometric averaging operators which leads to the notion of a strong cusp these estimates can be further improved if we impose curvature conditions on the cusp surface one application is the l 2 comp l 2 1 6 loc and l 12 7 comp l 2 loc boundedness of restrictions of the x ray transform on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i to four dimensional families of lines satisfying a natural curvature and torsion condition

another balanced controller reduction algorithm the problem of controller reduction has been addressed by a number of authors 1 4 using the concept of balanced coordinates originally proposed by moore 5 this note establishes a more direct link between the riccati balancing methods in 2 4 and moore s work and proposes yet another candidate method as a logical extension of 2 4

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

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

norm estimates for k plane transforms and geometric inequalities abstract the article is devoted to remarkable interrelation between the norm estimates for k plane transforms in weighted and unweighted l p spaces and geometric integral inequalities for cross sections of measurable sets in r n a similar interrelation is studied for more general j plane to k plane transforms on affine grassmannians and their compact modifications the article contains a series of new integral geometric inequalities with sharp constants explicit equalities conjectures and open problems

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

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

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

endpoint lp lq bounds for integration along certain polynomial curves we establish strong type endpoint lp rd lq rd bounds for the operator given by convolution with affine arclength measure on polynomial curves for d 4 the bounds established depend only on the dimension d and the degree of the polynomial

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

balanced gains and their significance for l 2 model reduction the singular values do not give enough information for model reduction in the l2sense other invariants are introduced which help describe the contribution of each state to the l2magnitude of the impulse response this leads to another suboptimal balanced truncation criterion

controller reduction by component cost analysis component cost analysis 1 is used to develop a method for controller reduction the reduction of the controller is based upon the participation of the controller states in the value of a quadratic performance metric the controller states which have the smallest contribution to the performance metric are truncated to produce the reduced controllers an error index is defined to evaluate the reduced controller compared to the optimal lqg controller and bounds on this index are derived a numerical example is included to illustrate the procedure

controller reduction using canonical interactions a conceptual approach to controller reduction is proposed that is based on closed loop considerations the method assesses the interactions between the plant and a high order controller as described by canonical correlation coefficients a free parameter in the algorithm of the reduced controller makes it possible to trade stability robustness versus performance several examples demonstrate the efficacy of the method

weak type 1 1 bounds for rough operators ii in this paper we consider four operators arising in harmonic analysis in r n 1 they share one feature each is known to be bounded on lp for all 1 p o but for each it has been an open question whether a weak type 1 1 bound also holds in two cases we show that the weak 1 1 bound does hold and for the others we establish weaker estimates which are still sharper than the lp boundedness although the four operators are quite distinct they all involve convolution kernels which are related to the singular integrals of calderon and zygmund but which lack the smoothness required for the usual analysis in each case this defect is mitigated by the presence of some form of curvature consider the bochner riesz means

l p improving bounds for averages along curves we establish local l p l q mapping properties for averages on curves the exponents are sharp except for endpoints let n 2 and let m1 and m2 be two smooth n 1 dimensional manifolds 1 each containing a preferred origin 0m1 and 0m2 we shall abuse notation and write 0 for both 0m1 and 0m2 for the purposes of integration we shall place a smooth riemannian metric on m1 and m2 although the exact choice of this metric will not be relevant all our considerations shall be local to the origin 0 we are interested in the local l p improving properties of averaging operators on curves before we give the rigorous description of these operators let us first give an informal discussion informally we assume that we have a smooth assignment x2 7 x2 taking points in m2 to curves in m1 with a corresponding dual assignment x1 x1 taking points in m1 to curves in m2 such that x1 x2 x2 1 we then form the operator r taking functions 2 on m1 to functions on m2 defined

non linear frequency response characteristics of an impact pair abstract the method of harmonic balance is used to develop approximate analytical solutions to the equations of motion of a single impact pair including a general discussion of the existence and stability of these solutions non linear frequency response characteristics are obtained for a single frequency excitation and primary resonance no subharmonic or superharmonic response and the results are validated using an analog computer simulation it has been found that the use of digital simulation numerical integration may lead to inaccurate results over certain frequency ranges it is also shown that although the shapes of the force displacement relationship for clearance non linearity and the classic cubic non linearity are similar the resulting non linear frequency response curves can be quite different various issues related to the clearance non linearity problem are discussed

the periodicity of chaotic impact oscillators in hausdorff phase spaces it is well known that non periodic behavior is one of the most puzzling characteristics of chaotic oscillators so far chaotic dynamical systems have been investigated in euclidean spaces in this paper the concept of non autonomous dynamical systems and that of hausdorff phase spaces are proposed the behavior of chaotic impact oscillators is investigated in hausdorff phase spaces it is discovered that although the non autonomous dynamical systems described by chaotic impact oscillators are non periodic in euclidean phase spaces they are periodic in hausdorff phase spaces this shows that euclidean spaces in which we stayed for hundreds of years may no longer be suitable for the investigation into chaotic phenomena in addition the periodicity of chaotic dynamical systems in hausdorff metric spaces induces a new class of strange invariant sets in euclidean spaces such strange invariant sets may be an ideal symbol of chaotic dynamical systems