discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations abstract nonlinear nonstationary problems arising in elastodynamics have naturally a nonconservative form for these systems it is not possible to define a weak solution according to distribution theory in order to define the concept of discontinuous solutions for these systems one is confronted with multiplications of distributions this is unavoidable from a physical viewpoint since discontinuous solutions are needed to represent shock waves the theory of generalized functions of the second author in which one may multiply arbitrary distributions gives a concept of generalized solutions that may be discontinuous functions existence of such global solutions of the cauchy problem for a system of two equations is proved by a compactness argument from a convergent numerical scheme in the case under consideration the generalized solutions are bounded variation functions the results thus obtained agree with the results expected by the engineers and lead to the development of new numerical schemes

some aspects of the spline smoothing approach to non parametric regression curve fitting non parametric regression using cubic splines is an attractive flexible and widely applicable approach to curve estimation although the basic idea was formulated many years ago the method is not as widely known or adopted as perhaps it should be the topics and examples discussed in this paper are intended to promote the understanding and extend the practicability of the spline smoothing methodology particular subjects covered include the basic principles of the method the relation with moving average and other smoothing methods the automatic choice of the amount of smoothing and the use of residuals for diagnostic checking and model adaptation the question of providing inference regions for curves and for relevant properties of curves is approached via a finite dimensional bayesian formulation

asymptotic optimality of cl and generalized cross validation in ridge regression with application to spline smoothing the asymptotic optimality of mallows c l and generalized cross validation is demonstrated in the setting of ridge regression an application is made to spline smoothing in nonparametric regression a counterexample is given to help understand why sometimes gcv may not be asymptotically optimal the coefficient of variation for the eigenvalues of the information matrix must be large in order to guarantee the optimality of gcv the proff is based on the connection between gcv and stein s unbiased risk estimate

localization effects and measure source terms in numerical schemes for balance laws this paper investigates the behavior of numerical schemes for non linear conservation laws with source terms we concentrate on two significant examples relaxation approximations and genuinely nonhomogeneous scalar laws the main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using riemann based numerical schemes some illustrative and relevant computational results are provided

disturbance generated bifurcation in a prototype adaptive system with e sub 1 modification law the e sub 1 modification law has been proposed by k s narendra and a m annaswamy ibid vol 32 no 2 pp 134 5 1987 for robust model reference adaptive control in the presence of bounded disturbances using this law it is shown by the present authors that in the case of regulation a prototype adaptive control system with a single unknown pole and an unknown constant disturbance at its input possesses multiple equilibria and undergoes various bifurcations simulations and local analysis give a complete characterization of the possible dynamics exhibited by the disturbance adaptive system as the constant disturbance is varied saddle node bifurcation subcritical hopf bifurcation and saddle connection bifurcation occur in that order consequently in this case the e sub 1 modification law is qualitatively equivalent to the sigma modification law

a remark on chaotic behavior in adaptive control systems this note deals with the control of a linear discrete time system designed on the assumptions that we do not have explicit knowledge of the order of the plant the objective is to cause the output to track a given constant reference y when the initial conditions are in a neighborhood of y in previous papers the presence of chaos was conjectured for an adaptive scheme where the plant to be controlled was a second order one undermodeled by a first order one we show that with an adequate choice of the parameter estimator local regulation with exponential convergence of both the output and the parameter is obtained

multivariate smoothing spline functions given data z i g t i varepsilon i 1 leqq i leqq n where g is the unknown function the t i are known d dimensional variables in a domain omega and the varepsilon i are i

nonstandard analysis and jump conditions for converging shock waves nonstandard analysis is an area of modern mathematics that studies abstract number systems containing both infinitesimal and infinite numbers this article applies nonstandard analysis to derive jump conditions for one dimensional converging shock waves in a compressible inviscid perfect gas it is assumed that the shock thickness occurs on an infinitesimal interval and the jump functions in the thermodynamic and fluid dynamic parameters occur smoothly across this interval predistributions of the heaviside function and the dirac delta measure are introduced to model the flow parameters across a shock wave the equations of motion expressed in nonconservative form are then applied to derive unambiguous relationships between the jump functions for the flow parameters

why nonconservative schemes converge to wrong solutions error analysis this paper attempts to give a qualitative and quailtitative description of the numerical error introduced by using finite difference schemes in nonconservative form for scalar conservation laws we show that these schemes converge strongly in li norm to the solution of an inhomogeneous conservation law containing a borel measure source term moreover we analyze the properties of this borel measure and derive a sharp estimate for the l1 error between the limit function given by the scheme and the correct solution in general the measure source term is of the order of the entropy dissipation measure associated with the scheme in certain cases the error can be small for short times which makes it difficult to detect numerically but generically such an error will grow in time and this would lead to a large error for large time calculations finally we show that a local correction of any high order accurate scheme in nonconservative form is sufficient to ensure its convergence to the correct solution

lmi optimization approach to stabilization of genesio tesi chaotic system via dynamic controller abstract in this paper a design method for a new controller to control the genesio tesi chaotic systems is proposed in this work a dynamic outputs feedback controller for the system is developed for the first time for stability analysis a well known lyapunov stability theorem combining with lmi linear matrix inequality optimization approach is utilized a numerical simulation is presented to show the usefulness of the proposed control scheme

numerical continuation of bifurcations of limit cycles in matlab rm cl matcont and matcont are matlab continuation packages for the interactive numerical study of a range of parameterized nonlinear dynamical systems in particular odes matcont is an interactive graphical package and rm cl matcont is a commandline version both packages allow us to compute curves of equilibria limit points hopf points limit cycles flip fold and torus bifurcation points of limit cycles we discuss computational details of the continuation of limit cycles and flip fold and torus bifurcations of limit cycles in matcont and rm cl matcont using orthogonal collocation instead of the more commonly used fully extended systems we use minimally extended systems we further describe the use of the matlab sparse matrix routines and the initialization and adaptation of the bordering vectors that are essential in minimally extended systems finally we compare the use of the minimally and the fully extended systems in the matlab environment

multiplications of distributions in elasticity and hydrodynamics it is shown how a mathematical theory of generalized functions in which the multiplications of distributions appearing in nonlinear equations of physics make sense gives new formulas and new numerical results the new methods shown here are quite general but since each particular problem requires its own study this paper is limited to elasticity and hydrodynamics in elasticity hooke s law gives systems in a nonconservative form the study of shock waves for these systems gives nonclassical multiplications of distributions of the form y y heaviside function dirac mass at the origin using this new mathematical tool new formulas are obtained more generally new numerical schemes in a first step ambiguous results are obtained then the ambiguity is removed in hydrodynamics a formulation is obtained that has a nonconservative form and is at the basis of efficient new numerical schemes strictly speaking the reader is not assumed to know anything either on distributions or on elasticity and hydrodynamics since the basic equations are recalled all computations done in this paper are rigorous from the mathematical viewpoint

simple feedback systems with chaotic behaviour abstract an example is given of a second order dynamical system containing a relay with hysteresis which can display chaotic motion it is shown that this system can be approximated by a third order model with a continuous non linearity which appears to behave in a similar manner

non linear dynamics in adaptive control chaotic and periodic stabilization abstract in this paper the non linear features of adaptive control algorithms are highlighted using a simple linear discrete time example based on classical design it is demonstrated that in the presence of undermodelling errors non linear phenomena in the feedback gain such as limit cycles and even chaos arise despite these complicated dynamics robust stabilization of the plant can still occur for the class of undermodelling errors considered the set of plants stabilizable by this adaptive controller is completely characterized

a well balanced flux vector splitting scheme designed for hyperbolic systems of conservation laws with source terms abstract we propose a way to construct robust numerical schemes for the computations of numerical solutions of one and two dimensional hyperbolic systems of balance laws in order to reduce the computational cost we selected the family of flux vector splitting schemes we reformulate the source terms as nonconservative products and treat them directly in the definition of the numerical fluxes by means of generalized jump relations this is applied to a 1d shallow water system with topography and to a 2d simplified model of two phase flows with damping effects numerical results and comparisons with a classical centered discretizations scheme are supplied

numerical periodic normalization for codim 1 bifurcations of limit cycles explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous odes are derived they include second order coefficients for the fold limit point bifurcation as well as third order coefficients for the flip period doubling and neimark sacker torus bifurcations the formulas are independent of the dimension of the phase space and involve solutions of certain boundary value problems on the interval 0 t where t is the period of the critical cycle as well as multilinear functions from the taylor expansion of the right hand sides near the cycle the formulas allow us to distinguish between sub and supercritical bifurcations in agreement with earlier asymptotic expansions of the bifurcating solutions our formulation makes it possible to use robust numerical boundary value algorithms based on orthogonal collocation rather than shooting techniques which greatly expands its applicability the actual implementation is described in detail we include three numerical examples in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients

convergence of an antidiffusion lagrange euler scheme for quasilinear equations this paper presents an explicit scheme which may be broken into several steps the first one is a transport phase for a time step with a projection on a lagrangian mesh then a new projection is performed on a fixed eulerian mesh lastly a corrected antidiffusion step of the shasta type occurs a stability condition appears to make the scheme easily computable this condition allows larger timesteps than the well known courant friedrichs lewy condition a more complex version of the scheme is proposed which works without any stability condition convergence towards the entropy solution is proved a few numerical experiments are reported showing the quality of the approximation of shocks or giving some information about the precision

multicomponent flow calculations by a consistent primitive algorithm the dynamics of inviscid multicomponent fluids may be modelled by the euler equations augmented by one or more additional species equation s attempts to compute solutions for extended euler models in conservation form show strong oscillations and other computational inaccuracies near material interfaces these are due to erroneous pressure fluctuations generated by the conservative wave model this problem does not occur in single component computations and arises only in the presence of several species a nonconservative primitive euler formulation is proposed which results in complete elimination of the oscillations the numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation numerical experiments show clean monotonic solution profiles with acceptably small conservation error for shocks of weak to moderate strengths

synchronizing the noise perturbed unified chaotic system by sliding mode control abstract in this paper synchronization between unidirectionally coupled unified chaotic systems with noise perturbation is investigated sufficient conditions of synchronization between these noise perturbed systems are established by means of the so called sliding mode control method some numerical simulations are also included to visualize the effectiveness and the feasibility of the developed approach

a simple adaptive feedback control method for chaos and hyper chaos control abstract this paper investigates the problem of chaos and hyper chaos control and proposes a simple adaptive feedback control method for chaos control under a reasonable assumption in comparison with previous methods the present control technique is simple both in the form of the controller and its application several illustrative examples with numerical simulations are studied by using the results obtained in this paper study of examples shows that our control method works very well in chaos control

microscopic profiles of shock waves and ambiguities in multiplications of distributions a rigorous mathematical method is introduced for studying discontinuous solutions for systems in nonconservative form the evidence of the agreement between the theoretical and numerical results is shown on simplified models of physics

conditions for preservation of stability of adaptive controllers for systems with unmodeled delay some recent results from the mathematical literature are used to derive conditions under which stability of an adaptive controlled system first order plant with output feedback is preserved in spite of the presence of a time delay in the plant input bounds on the delay for which stability is preserved are derived for three robust versions of the gradient update law fixed sigma modification with and without normalization and a time varying sigma modification

bayesian confidence intervals for the cross validated smoothing spline summary we consider the model y t g ti ei i 17 2 n where g t t 0 1 is a smooth function and the ei are independent n 0 a2 errors with g2 unknown the cross validated smoothing spline can be used to estimate g non parametrically from

an iterative approach to two dimensional laplacian smoothing with application to image restoration abstract smoothing methods are increasingly important in modern graphically oriented data analysis there are several fast and reliable univariate cross validated smoothing algorithms but comparable methods are not available in higher dimensions this article develops an iterative approach to two dimensional laplacian spline smoothing based on tensor product cubic b splines estimating equations are developed and solved iteratively using two line symmetric successive over relaxation with conjugate gradient acceleration an asymptotic approximation for the generalized cross validation score is described the resulting algorithm is naturally suited to two dimensional scatterplot smoothing but has potential value for image restoration as well this is illustrated on a multicolor image example along with an image restoration an uncertainty assessment based on the entropy of the marginal posterior pixel value distribution is produced this analysis particularly highlights the statistical uncertainty in resol

a split godunov scheme for solving one dimensional hyperbolic systems in a nonconservative form in this paper we developed a theoretical study for nonconservative sytems in one dimension in order to construct numerical schemes for solving the riemann problem the nonconservative form of our model system required the use of a well adapted theory in order to give us a sense of our problem we chose a framework of generalized functions for solving a scalar hyperbolic equation with a discontinuous coefficient sigma t u sigma x approx 0 where u is the velocity solution of a burgers s equation after an explicit solution of the riemann problem we derived godunov split schemes for computing an approximate solution of the cauchy problem we applied our study to a system modeling elasticity and a system modeling gas dynamics some stability properties of a scheme and its convergence to a generalized solution are proved for the first model numerical experiments confirmed this convergence result for the second model calculations of flows containing weak to moderate shocks showed that conservation errors are reduced when the mesh is refined but were not entirely eliminated

high order wave propagation algorithms for hyperbolic systems we present a finite volume method that is applicable to hyperbolic pdes including spatially varying and semilinear nonconservative systems the spatial discretization like that of the well known clawpack software is based on solving riemann problems and calculating fluctuations not fluxes the implementation employs weighted essentially nonoscillatory reconstruction in space and strong stability preserving runge kutta integration in time the method can be extended to arbitrarily high order of accuracy and allows a well balanced implementation for capturing solutions of balance laws near steady state this well balancing is achieved through the f wave riemann solver and a novel wave slope weno reconstruction procedure the wide applicability and advantageous properties of the method are demonstrated through numerical examples including problems in nonconservative form problems with spatially varying fluxes and problems involving near equilibrium solutions of balance laws

on spiking models for synaptic activity and impulsive differential equations we illustrate the problems that can arise in writing differential equations that include dirac delta functions to model equations with state dependent impulsive forcing specifically difficulties arise in the interpretation of the products of distributions with discontinuous functions we suggest several methods to resolve these ambiguities such as using limiting sequences and asymptotic analysis with applications of the results given for discrete maps these suggestions are applied to a popular model describing synaptic connections in the brain

systems with distributions and viability theorem abstract the approach to the consideration of the ordinary differential equations with distributions in the classical space d of distributions with continuous test functions has certain insufficiencies the notations are incorrect from the point of view of distribution theory the right hand side has to satisfy the restrictive conditions of equality type in the present paper we consider an initial value problem for the ordinary differential equation with distributions in the space of distributions with dynamic test functions t where the continuous operation of multiplication of distributions by discontinuous functions is defined v derr d kinzebulatov distributions with dynamic test functions and multiplication by discontinuous functions preprint arxiv math ca 0603351 2006 and show that this approach does not have the aforementioned insufficiencies we provide the sufficient conditions for viability of solutions of the ordinary differential equations with distributions a generalization of the nagumo theorem and show that the consideration of the distributional impulse controls in the problem of avoidance of encounters with the set the maximal viability time problem allows us to provide for the existence of solution which may not exist for the ordinary controls

bifurcation analysis via harmonic balance in periodic systems with feedback structure the harmonic balance approach is applied to a class of periodic dynamical systems with a specific but quite general feedback structure in order to analyse approximately tangent saddle node and flip period doubling bifurcations of periodic solutions by exploiting the system structure an algebraic system of equations whose unknowns are the coefficients of a truncated fourier series is defined and the relationship between the bifurcations of the roots of this algebraic system and the tangent and flip bifurcations of the periodic solutions is pointed out an example concerning an adaptive control system illustrates the proposed methodology

convergence of numerical schemes involving powers of the dirac delta function abstract developments of the hull elastoplastic numerical method lead to nonconservative versions which in the case of shock waves involve multiplications of distributions of the type of powers of the dirac delta function in the one dimensional case of the shock wave equation u t uu x 0 the numerical solutions will converge to the solution of a different equation if the convergence and the latter equation are considered within the nonlinear theory of generalized functions introduced recently by the second author the study of this phenomenon presented here in one of its relevant particular cases offers for the first time a rigorous understanding of important similar situations encountered in industrial applications when numerical solutions may show either agreement with or deviations from the expected solutions

on the statistical analysis of dirty pictures may 7th 1986 professor a f m smith in the chair summary a continuous two dimensional region is partitioned into a fine rectangular array of sites or pixels each pixel having a particular colour belonging to a prescribed finite set the true colouring of the region is unknown but associated with each pixel there is a possibly multivariate record which conveys imperfect information about its colour according to a known statistical model the aim is to reconstruct the true scene with the additional knowledge that pixels close together tend to have the same or similar colours in this paper it is assumed that the local characteristics of the true scene can be represented by a nondegenerate markov random field such information can be combined with the records by bayes theorem and the true scene can be estimated according to standard criteria however the computational burden is enormous and the reconstruction may reflect undesirable largescale properties of the random field thus a simple iterative method of reconstruction is proposed which does not depend on these large scale characteristics the method is illustrated by computer simulations in which the original scene is not directly related to the assumed random field some complications including parameter estimation are discussed potential applications are mentioned briefly

the harmonic balance method for bifurcation analysis of large scale nonlinear mechanical systems abstract the harmonic balance hb method is widely used in the literature for analyzing the periodic solutions of nonlinear mechanical systems the objective of this paper is to exploit the method for bifurcation analysis i e for the detection and tracking of bifurcations of nonlinear systems to this end an algorithm that combines the computation of the floquet exponents with bordering techniques is developed a new procedure for the tracking of neimark sacker bifurcations that exploits the properties of eigenvalue derivatives is also proposed the hb method is demonstrated using numerical experiments of a spacecraft structure that possesses a nonlinear vibration isolation device

harmonic balance analysis of period doubling bifurcations with implications for control of nonlinear dynamics the harmonic balance method is applied to the analysis of period doubling bifurcations in a general class of nonlinear feedback systems compact conditions for the prediction and stability analysis of period doubling bifurcations are obtained specializations of these conditions for systems in which the nonlinear subsystem is static are given the implications of the results to control design are illustrated through the development of a design procedure for solving a representative control problem the objectives of the control design can include delay of a given period doubling bifurcation as well as stability objectives the results are illustrated with a detailed example

nonclassical shocks and kinetic relations finite difference schemes we consider hyperbolic systems of conservation laws that are not genuinely nonlinear the solutions generated by diffusive dispersive regularizations may include nonclassical n c shock waves that do not satisfy the classical liu entropy criterion we investigate the numerical approximation of n c shocks via conservative difference schemes constrained only by a single entropy inequality the schemes are designed by comparing their equivalent equations with the continuous model and include discretizations of the diffusive and dispersive terms limits of these schemes are characterized via the kinetic relation introduced earlier by the authors we determine the kinetic function numerically for several examples of systems and schemes this study demonstrates that the kinetic relation is a suitable tool for the selection of unique n c solutions and for the study of their sensitive dependence on the critical parameters the ratios of diffusion dispersion and diffusion mesh size the shock strength and the order of discretization of the flux

anti synchronization of two different chaotic systems abstract in this paper the anti synchronization of two different chaotic systems is investigated on the basis of a nonlinear control scheme and lyapunov theory sufficient conditions for the stability of the error dynamics are derived where the controllers are designed by using the sum of the relevant variables in chaotic systems numerical simulations are performed for the genesio rossler system to demonstrate the effectiveness of the proposed control strategy

the elastoplastic shock problem as an example of the resolution of ambiguities in the multiplication of distributions one dimensional collisions between two isotropic solids in which the equations of physics lead to multiplications of distributions are considered based on this example a general physicomathematical method to be adapted to each particular case is proposed to resolve the ambiguity inherent in such products this can be achieved with the aid of a new mathematical theory of generalized functions which permits dealing with mathematical phenomena of a microscopic nature that govern products of distributions having singularities at the same point this tool has recently been applied in various situations in continuum mechanics in which the equations of physics lead to heuristic products of distributions one obtains new algebraic formulas in the simplest cases and new numerical schemes in more general cases the key to the resolution of ambiguities lies in more precise statements of the laws of physics than are permitted within distribution theory and have no analog in classical analysis so that in general a resolution cannot be obtained from formal calculations

viscous shock profiles and primitive formulations weak solutions of hyperbolic systems in primitive nonconservative form for which a consistent conservation form exists are considered it is shown that for primitive formulations shock relations are not uniquely defined by the states to either side of the shock but also depend on the viscous path connecting the two consistent viscous shock profiles are enforced by adding scheme dependent small viscous perturbations that account for leading order conservation errors the resulting primitive algorithm is conservative to the order of the approximation one dimensional euler calculations of flows containing weak to moderate shocks show that conservation errors in primitive calculations are substantially reduced by including the viscous perturbation terms while not eliminating conservation errors entirely it is found that for a wide range of problems both conservative and primitive flow calculations are of comparable quality

continuation techniques and interactive software for bifurcation analysis of odes and iterated maps physics abstract we present a numerical technique for the analysis of local bifurcations which is based on the continuation of structurally unstable invariant sets in a suitable phase parameter space the invariant sets involved in our study are equilibrium points and limit cycles of autonomous odes periodic solutions of time periodic nonautonomous odes fixed points and periodic orbits of iterated maps the more general concept of a continuation strategy is also discussed it allows the analysis of various singularities of generic systems and of their mutual relationships the approach is extended to codimension three singularities we introduce several bifurcation functions and show how to use them to construct well posed continuation problems the described continuation technique is supported by an interactive graphical program called locbif we discuss briefly the concepts of the locbif interface and give some examples of typical applications

on nonconservative algorithms for grid interfaces in computations of fluid flows by domain decomposition methods the necessity of conservation at grid interfaces is now widely claimed in this paper we consider nonconservative algorithms for the interfaces our investigation begins with discussions about typical nonconservative interface treatments for one dimensional calculations the analysis shows that the conservation error of a numerical solution caused by a nonconservative interface matching has an upper bound when the solution itself is bounded furthermore if the numerical solution converges as the mesh size goes to zero it converges to a weak solution of the problem under certain conditions that may be detected numerically also we acquire similar results for two dimensional calculations on grids intersecting with each other in an arbitrary way in order to illustrate the theoretical results we present numerical examples and demonstrate that under those conditions conservation error reduces and accuracy for jumps as well as locations of discontinuities improves as the mesh size decreases

adaptive backstepping synchronization between chaotic systems with unknown lipschitz constant abstract the error between two nonlinear terms is a key point of many synchronization problems however the lipschitz constant of the nonlinear term is not always easy to calculate for the stability analysis of the controlled error system thus the nonlinear systems with unknown parameters and unknown lipschitz constant is considered in this paper their scalar synchronous controller is proposed based on the thought of backstepping design without the need to evaluate the invariant set and calculate the lipschitz constant an assistant adaptive estimator is designed for the lipschitz constant what s more as a problem solving skill two different estimators are used on the same unknown parameter finally the synchronization control for both chaotic autonomous van der pol duffing advp systems and chaotic genesio systems with unknown parameters are given as examples to verify the control effect

controlling chaotic and hyperchaotic systems via a simple adaptive feedback controller this paper investigates the problem of chaos control and proposes a simple adaptive feedback control method for it under a reasonable assumption in comparison with previous methods the present control technique is simple both in the form of the controller and in its application a numerical example using the genesio chaotic system verifies the effectiveness and correctness of the proposed approach

nonlinear generalized functions and jump conditions for a standard one pressure liquid gas model abstract the mixture of a liquid and a gas is classically represented by one pressure models these models are a system of pdes in nonconservative form and shock wave solutions do not make sense within the theory of distributions they give rise to products of distributions that are not defined within distribution theory but they make sense by applying a theory of nonlinear generalized functions to these equations in contrast to the familiar case of conservative systems the jump conditions cannot be calculated a priori jump conditions for these nonconservative systems can be obtained using the theory of nonlinear generalized functions by inserting some adequate physical information into the equations the physical information that we propose to insert for the one pressure models of a mixture of a liquid and a gas is a natural mathematical expression in the theory of nonlinear generalized functions of the fact that liquids are practically incompressible while gases are very compressible and so they do not satisfy equally well their respective state laws on the shock waves this modelization gives well defined explicit jump conditions the great numerical difficulty for solving numerically nonconservative systems is due to the fact that slightly different numerical schemes can give significantly different results the jump conditions obtained above permit to select the numerical schemes and validate those that give numerical solutions that satisfy these jump conditions which can be an important piece of information in the absence of other explicit discontinuous solutions and of precise observational results we expose with care the mathematical originality of the theory of nonlinear generalized functions an original abstract analysis issued by the leopoldo nachbin team on infinite dimensional holomorphy that permits to state mathematically physical facts that cannot be formulated within distribution theory and are the key for the removal of ambiguities that classically appear when one tries to calculate on multiplications of distributions that occur in the differential equations of physics

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

chaotification via system immersion by introducing a definition of partially chaotic for a kind of generalized chaotic system this paper discusses the chaotification problem with a new approach based on system immersion and manifold invariance the basic idea is to immerse the plant system into a reduced order chaotic system the proposed approach is also applied to chaotify systems with uncertain parameters illustrative examples with simulation results are presented to validate the proposed chaotification schemes

displacement in the parameter space versus spurious solution of discretization with large time step in order to investigate a possible correspondence between differential and difference equations it is important to possess discretization of ordinary differential equations it is well known that when differential equations are discretized the solution thus obtained depends on the time step used in the majority of cases such a solution is considered spurious when it does not resemble the expected solution of the differential equation this often happens when the time step taken into consideration is too large in this work we show that even for quite large time steps some solutions which do not correspond to the expected ones are still topologically equivalent to solutions of the original continuous system if a displacement in the parameter space is considered to reduce such a displacement a judicious choice of the discretization scheme should be made to this end a recent discretization scheme based on the lie expansion of the original differential equations proposed by monaco and normand cyrot will be analysed such a scheme will be shown to be sufficient for providing an adequate discretization for quite large time steps compared to the pseudo period of the underlying dynamics

harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems abstract the paper considers the problem of determining the conditions under which a nonlinear dynamical system can give rise to a chaotic behaviour on the basis of the harmonic balance principle which is widely used in the frequency analysis of nonlinear control systems two practical methods are presented for predicting the existence and the location of chaotic motions this is formulated as a function of the system parameters when the system structure is fixed by rather general input output or state equation models several examples of application are presented to show the rather straightforward computations involved in the proposed methods the kind of results which can be obtained and due to the heuristic approach to the problem their corresponding approximation

short note a comment on the computation of non conservative products we are interested in the solution of non conservative hyperbolic systems and consider in particular the so called path conservative schemes see e g 2 3 which rely on the theoretical work in 1 the example of the standard euler equations for a perfect gas is used to illuminate some computational issues and shortcomings of this approach

paper instability analysis and improvement of robustness of adaptive control the effects of unmodeled high frequency dynamics and bounded disturbances on stability and performance of adaptive control schemes are analyzed five possible types of instability mechanisms parameter drift linear instability fast adaptation instability high frequency instability and throughput instability are analyzed using simple examples a procedure is used to construct lyapunov like functions for a modified adaptive controller applied to a dominant plant of relative degree one in the presence of parasitics and disturbances and obtain sufficient conditions under which none of the five types of instability can occur the modified scheme is robust in the sense that it guarantees the existence of a large region of attraction from which all the trajectories remain bounded and the state errors converge exponentially to a much smaller residual set the size of the region of attraction depends on the speed of parasitics in such a way that as the parasitics become infinitely fast the region of attraction becomes the whole space

the riemann problem for pressureless fluid dynamics with distribution solutions in colombeau s sense the riemann problem for the equations of pressureless fluid dynamics was considered solutions of this problem were constructed by employing the viscosity vanishing approach for some initial data solutions showed high singularity around the shock waves a new mathematical theory of generalized functions initiated by j f colombeau was applied to dealing with the multiplication of singular distributions as a byproduct the entropy condition was obtained for singular distribution solutions

smoothing noisy data with spline functions a procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m 1 fitted ton distinct not necessarily equally spaced or uniformly weighted data points is presented the procedure requires orderm 2 n operations and therefore permits efficient orderm 2 n calculation of statistics associated with a polynomial smoothing spline including the generalized cross validation the method is a significant improvement over an existing method which requires ordern 3 operations

smoothing spline anova for exponential families with application to the wisconsin epidemiological study of diabetic retinopathy the 1994 neyman memorial lecture let y i i 1 n be independent observations with the density of y i of the form h y i f i exp y i f i b f i c y i where b and c are given functions and b is twice continuously differentiable and bounded away from 0 let i t f t i where t t 1 t d t 1 t d t the t are measurable spaces of rather general form and f is an unknown function on t with some assumed smoothness properties given y i t i i 1 n it is desired to estimate f t for t in some region of interest contained in t we develop the fitting of smoothing spline anova models to this data of the form f t c f t f t t the components of the decomposition satisfy side conditions which generalize the usual side conditions for parametric anova the estimate of f is obtained as the minimizer in an appropriate function space of l y f j f j f where l y f is the negative log likelihood of y y 1 y n given f the j j are quadratic penalty functionals and the anova decomposition is terminated in some manner there are five major parts required to turn this program into a practical data analysis tool 1 methods for deciding which terms in the anova decomposition to include model selection 2 methods for choosing good values of the smoothing parameters 3 methods for making confidence statements concerning the estimate 4 numerical algorithms for the calculations and finally 5 public software in this paper we carry out this program relying on earlier work and filling in important gaps the overall scheme is applied to bernoulli data from the wisconsin epidemiologic study of diabetic retinopathy to model the risk of progression of diabetic retinopathy as a function of glycosylated hemoglobin duration of diabetes and body mass index it is believed that the results have wide practical application to the analysis of data from large epidemiologic studies

generalized solutions to a cauchy problem for a nonconservative hyperbolic system in this paper we show the existence of generalized solutions in the sense of colombeau to a cauchy problem for a nonconservative system of hyperbolic equations which has applications in elastodynamics

disturbance generated bifurcations in a simple adaptive system simulation evidence abstract we present complete characterization based on computer simulation of the possible dynamics exhibited by an adaptive control system where the first order plant with a single unknown pole has constant disturbance at the input the adaptive system is taken from the current adaptive control systems literature here we show that the system in fact undergoes bifurcations and exhibits rich dynamics as the constant disturbance is varied the system undergoes saddle node bifurcation subcritical hopf bifurcation and a saddle connection bifurcation the last bifurcation means that the system acquires a homoclinic orbit for a specific disturbance while the first two bifurcations are local the presence of the homoclinic orbit has the potential when periodically disturbed to generate nonlocal complicated behavior this latter behavior is often referred to as horseshoe chaos

a high resolution godunov method for compressible multi material flow on overlapping grids a numerical method is described for inviscid compressible multi material flow in two space dimensions the flow is governed by the multi material euler equations with a general mixture equation of state composite overlapping grids are used to handle complex flow geometry and block structured adaptive mesh refinement amr is used to locally increase grid resolution near shocks and material interfaces the discretization of the governing equations is based on a high resolution godunov method but includes an energy correction designed to suppress numerical errors that develop near a material interface for standard conservative shock capturing schemes the energy correction is constructed based on a uniform pressure velocity flow and is significant only near the captured interface a variety of two material flows are presented to verify the accuracy of the numerical approach and to illustrate its use these flows assume an equation of state for the mixture based on the jones wilkins lee jwl forms for the components this equation of state includes a mixture of ideal gases as a special case flow problems considered include unsteady one dimensional shock interface collision steady interaction of a planar interface and an oblique shock planar shock interaction with a collection of gas filled cylindrical inhomogeneities and the impulsive motion of the two component mixture in a rigid cylindrical vessel

disturbance instabilities in an adaptive system two types of disturbance instability of simple adaptive schemes are analyzed a drift equation shows how stable equilibria are forced to infinity by infrequent but persistent switches of a constant disturbance the modification proposed by p ioannou counteracts these instabilities

non linear dynamics in adaptive control periodic and chaotic stabilization ii analysis abstract a particular non linear equation arising in an adaptive control problem is presented in the first part of this paper this equation displays vividly a wealth of non linear phenomena depending on the value of one crucial parameter characterizing the extent and presence of non design circumstances we demonstrate the existence of asymptotically periodic behaviour or chaotic dynamics in the feedback controller this paper contains the mathematical analysis associated with the earlier results and as such serves two main purposes to provide the theoretical proofs and support for these results and to demonstrate more fully the extraordinary complexity and difficulty of conducting a thorough global analysis of even this simplest of robust adaptive problems by performing this latter task we aim to delineate better what are reasonable objectives to be expected of robust adaptive control theory

conservation laws with discontinuous coefficients abstract in order to model phenomena arising in matter flows in electromagnetic fields engineers join systems of conservation laws with discontinuous coefficients the simplest quasi linear equation is u t a f u x 0 where a is a given discontinuous coefficient function and f is a smooth function we begin our study by solving the riemann problem and then we construct a numerical scheme of godunov type

numerical solutions of one pressure models in multifluid flows flows of a mixture of several fluids can be described by one pressure models their formulation is formally simple and their efficiency has been recognized but from the mathematical viewpoint they present severe peculiarities they are not hyperbolic appear to be ill posed and are in nonconservative form so they show ambiguous products of distributions in case of shock waves a resolution of these ambiguities is proposed it gives jump conditions similar to the classical rankine hugoniot jump conditions of conservative systems this resolution attempts to solve the riemann problem and then the cauchy problem by godunov type schemes following nonconservative techniques initiated by a y le roux and collaborators

burgers equation with a passive scalar dissipation anomaly and colombeau calculus a connection between dissipation anomaly in fluid dynamics and colombeau s theory of products of distributions is exemplified by considering burgers equation with a passive scalar besides the well known viscosity independent dissipation of energy in the steadily propagating shock wave solution the lesser known case of passive scalar subject to the shock wave is studied an exact dependence of the dissipation rate of the passive scalar on the prandtl number pr is given by a simple analysis we show in particular 1 pr for large pr the passive scalar profile is shown to have a form of a sum of tanh2n 1 x with suitably scaled x thereby implying the necessity to distinguish h from hn when pr is large where h is the heaviside function and n is a positive integer an incorrect result of 1 pr would otherwise be obtained this is a typical example where colombeau calculus for products of weak solutions is required for a correct interpretation a cole hopf type transform is also given for the case o