melnikov method for discontinuous planar systems abstract this paper treats the occurrence of homoclinic solutions in planar systems with discontinuous right hand side more precisely we deal with a t periodic perturbed system such that the unperturbed system is an autonomous possessing homoclinic orbit by means of the so called non smooth melnikov function there is shown the existence of a homoclinic solution for a perturbed system the non smooth melnikov function is derived and the method of how to find it in concrete problems is also introduced

two parameter degenerate sliding bifurcations in filippov systems this paper is concerned with the extension to the case of codimension 2 degenerate sliding bifurcations of the theory of sliding bifurcations in filippov systems presented in m di bernardo p kowalczyk a nordmark bifurcations of dynamical systems with sliding derivation of normal form mappings physica d 170 2002 175 205 these bifurcations were detected in experimental systems such as the dry friction oscillator and turn out to be organising centres for branches of codimension 1 sliding bifurcations the analysis is carried out for generic n dimensional piecewise smooth systems the possible degenerate scenarios are classified it is shown that several branches of codimension 1 sliding bifurcations originate from the degenerate codimension 2 points such branches are appropriately classified in the degenerate crossing sliding case a friction oscillator is used as a representative example to illustrate and confirm the theoretical derivations the importance is discussed of the unfolding of the degenerate sliding bifurcations for the development of continuation techniques

an asymptotic result for laguerre sobolev orthogonal polynomials abstract let s n denote the sequence of polynomials orthogonal with respect to the sobolev inner product f g s 0 f x g x x e x d x 0 f x g x x e x d x where 1 0 and the leading coefficient of the s n is equal to the leading coefficient of the laguerre polynomial l n then if x c s 0 lim n s n x l n 1 x is a constant depending on

simultaneous bifurcation of limit cycles from a cubic piecewise center with two period annuli abstract we study the number of periodic orbits that bifurcate from a cubic polynomial vector field having two period annuli via piecewise perturbations the cubic planar system x y y x 1 2 y 2 x x 1 2 y 2 has simultaneously a center at the origin and at infinity we study up to first order averaging analysis the bifurcation of periodic orbits from the two period annuli first separately and second simultaneously this problem is a generalization of 24 to the piecewise systems class when the polynomial perturbation has degree n we prove that the inner and outer abelian integrals are rational functions and we provide an upper bound for the number of zeros when the perturbation is cubic the same degree as the unperturbed vector field the maximum number of limit cycles up to first order perturbation from the inner and outer annuli is 9 and 8 respectively when the simultaneous bifurcation problem is considered 12 limit cycles exist these limit cycles appear in three types of configurations 9 3 6 6 and 4 8 in the non piecewise scenario only 5 limit cycles were found

on limit cycles bifurcating from the infinity in discontinuous piecewise linear differential systems in this paper we consider the linear differential center x y y x perturbed inside the class of all discontinuous piecewise linear differential systems with two zones separated by the straight line y 0 using the bendixson transformation we provide sufficient conditions to ensure the existence of a crossing limit cycle coming purely from the infinity we also study the displacement function for a class of discontinuous piecewise smooth differential system

regularization and singular perturbation techniques for non smooth systems abstract this paper is concerned with some aspects of the qualitative geometric theory of non smooth systems we present a survey of the state of the art on the connection between the regularization process of non smooth vector fields and the singular perturbation problems as a matter of fact the main results in our setting fill a gap between these areas exploiting such results applications to constrained differential equations are discussed

dangerous bifurcation at border collision when does it occur it has been shown recently that border collision bifurcation in a piecewise smooth map can lead to a situation where a fixed point remains stable at both sides of the bifurcation point and yet the orbit becomes unbounded at the point of bifurcation because the basin of attraction of the stable fixed point shrinks to zero size such bifurcations have been named dangerous bifurcations in this paper we provide explanation of this phenomenon and develop the analytical conditions on the parameters under which such dangerous bifurcations will occur

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

on the number of limit cycles in general planar piecewise linear systems of node node types abstract we investigate the existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with node node dynamics using the lienard like canonical form with seven parameters some sufficient and necessary conditions for the existence of limit cycles are given by studying the fixed points of proper poincare maps in particular we prove the existence of at least two nested limit cycles and describe some parameter regions where two limit cycles exist the main results are applied to the pwl morris lecar neural model to determine the existence and stability of the limit cycles

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

averaging theory for discontinuous piecewise differential systems we develop the averaging theory of first and second order for studying the periodic solutions of discontinuous piecewise differential systems in arbitrary dimension and with an arbitrary number of systems with the minimal conditions of differentiability we also provide two applications

zeros of sobolev orthogonal polynomials following from coherent pairs let sn denote the monic orthogonal polynomial sequence with respect to the sobolev inner product f g s fgd 0 f g d 1 where d 0 d 1 is a so called coherent pair and 0 then sn has n different real zeros the position of these zeros with respect to the zeros of other orthogonal polynomials in particular laguerre and jacobi polynomials is investigated coherent pairs are found where the zeros of sn 1 separte the zeros of sn

discontinuity induced bifurcations of equilibria in piecewise smooth and impacting dynamical systems a rich variety of dynamical scenarios has been shown to occur when a fixed point of a non smooth map undergoes a border collision this paper concerns a closely related class of discontinuity induced bifurcations those involving equilibria of n dimensional piecewise smooth flows specifically transitions are studied which occur when a boundary equilibrium that is one lying within a discontinuity manifold is perturbed it is shown that such equilibria can either persist under parameter variations or can disappear giving rise to different bifurcation scenarios conditions to classify among the possible simplest scenarios are given for piecewise smooth continuous filippov and impacting systems also we investigate the possible birth of other attractors e g limit cycles at a boundary equilibrium bifurcation

zeros of sobolev orthogonal polynomials of hermite type let s n n denote a set of polynomials orthogonal with respect to the sobolev inner product s f x g x d j 0 x l f x g x d j 1 where l 0 and d j 0 d j 1 is a so called symmetrically coherent pair with d j 0 or d j 1 the hermite measure e x 2dx if d j 1 is the hermite measure then s n has n different real zeros if d j 0 is the hermite measure then s n has at least n 2 different real zeros we determine conditions for s n to have complex zeros

connection coefficients for laguerre sobolev orthogonal polynomials laguerre sobolev polynomials are orthogonal with respect to an inner product of the form p qs 0 p x q x x e x dx 0 p x q x d x w here 1 0 and p q p the linear space of polynomials with real coefficients if d x x e x dx the corresponding sequence of monic orthogonal polynomials q x has been studied by marcellan et al j com put appl math 71 1996 245 265 while if d x x dx the sequence of monic orthog onal polynomials l x was introduced by koekoek and meijer siam j math anal 24 1993 768 782 for each of these two families of laguerre sobolev polynomials here we give the explicit expression of the connection coefficients in their expansion as a series of standard la guerre polynomials the inverse connection problem of expanding laguerre polynomials in series of laguerre sobolev polynomials and the connection problem relating two families of laguerre sobolev polynomials with different parameters are also considered

sobolev orthogonal polynomials on product domains orthogonal polynomials on the product domain a 1 b 1 a 2 b 2 with respect to the inner product f g s a 1 b 1 a 2 b 2 f x y g x y w 1 x w 2 y d x d y f c 1 c 2 g c 1 c 2 are constructed where w i is a weight function on a i b i for i 1 2 0 and c 1 c 2 is a fixed point the main result shows how an orthogonal basis for such an inner product can be constructed for certain weight functions in particular for product laguerre and product gegenbauer weight functions which serve as primary examples

on the application of kam theory to discontinuous dynamical systems abstract so far the application of kolmogorov arnold moser kam theory has been restricted to smooth dynamical systems since there are many situations which can be modeled only by differential equations containing discontinuous terms such as state dependent jumps e g in control theory or nonlinear oscillators it is shown by a series of transformations how kam theory can be used to analyze the dynamical behaviour of such discontinuous systems as well the analysis is carried out for the example x x a sgn x p t with p c 6 being periodic it is known that all solutions are unbounded for small a 0 we prove that all solutions are bounded for a 0 sufficiently large and that there are infinitely many periodic and quasiperiodic solutions in this case

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

bifurcations of small limit cycles in lienard systems with cubic restoring terms abstract in this paper we study bifurcations of small amplitude limit cycles of lienard systems of the form x y f x y g x where g x is a cubic polynomial and f x is a smooth or piecewise smooth polynomial of degree n by using involutions we obtain sharp upper bounds of the number of small amplitude limit cycles produced around a singular point for some systems of this type

hermite sobolev and closely connected orthogonal polynomials abstract in this paper a bilinear functional a g m o n m c m m g m which corresponds to the hermite sobolev and closely connected orthogonal polynomials is studied the domain of m m 1 n in which a is positive definite is given and the hermite sobolev and closely connected formal orthogonal polynomials are given in the nonnormal case

nonsmooth dynamics in spiking neuron models large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue one approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models an alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies in this latter case the single neuron model of choice is often a variant of the classic integrate and fire model which is described by a nonsmooth dynamical system in this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate ranging from tonic to burst firing we show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis particularly those for treating grazing bifurcations importantly we highlight one particular single neuron model capable of generating realistic spike trains that is both computationally cheap and analytically tractable this is a planar nonlinear integrate and fire model with a piecewise linear vector field and a state dependent reset upon spiking we call this the pwl if model and analyse it at both the single neuron and network level the techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model as well as to develop the notion of liapunov exponents we also show how to construct the phase response curve for this system emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems the stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times at the network level we consider linear coupling between voltage variables as would occur in neurobiological networks with gap junction coupling and show how to analyse the properties existence and stability of both the asynchronous and synchronous states in the former case we use a phase density technique that is valid for any large system of globally coupled limit cycle oscillators whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics

stability conditions for discontinuous vector fields abstract in this work we describe the qualitative behavior of generic discontinuous vector fields around a point p r 3

existence of limit cycles in general planar piecewise linear systems of saddle saddle dynamics abstract the existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with saddle saddle dynamics are investigated using the lienard like canonical form with seven parameters the parametric regions of the existence of limit cycles are given by constructing proper poincare maps in particular the existence of at least two limit cycles is proved and some parameter regions where two nested limit cycles exist are given

δ sobolev orthogonal polynomials of meixner type asymptotics and limit relation let qn x n be the sequence of monic polynomials orthogonal with respect to the sobolev type inner product p x r x s u0 p x r x u1 p x r x where 0 f x f x 1 f x denotes the forward difference operator and u0 u1 is a coherent pair of positive definite linear functionals being u1 the meixner linear functional in this paper relative asymptotics for the qn x n sequence with respect to meixner polynomials on compact subsets of c 0 is obtained this relative asymptotics is also given for the scaled polynomials in both cases we deduce the same asymptotics as we have for the self coherent pair that is when u0 u1 is the meixner linear functional furthermore we establish a limit relation between these orthogonal polynomials and the laguerre sobolev orthogonal polynomials which is analogous to the one existing between meixner and laguerre polynomials in the askey scheme

inner products involving q differences the little q laguerre sobolev polynomials in this paper polynomials which are orthogonal with respect to the inner product p r s k 0 p qk r qk aq k aq q q q k k 0 dqp qk dqr qk aq k aq q q q k where dq is the q difference operator 0 0 q 1 and 0 aq 1 are studied for these polynomials algebraic properties and q difference equations are obtained as well as their relation with the monic little q laguerre polynomials some properties about the zeros of these polynomials are also deduced finally the relative asymptotics qn x pn x a q n on compact subsets of c 0 1 is given where qn x is the nth degree monic orthogonal polynomial with respect to the above inner product and pn x a q denotes the monic little q laguerre polynomial of degree n

the regularity and local bifurcation of steady periodic water waves steady periodic water waves on infinite depth satisfying exactly the kinematic and dynamic boundary conditions on the free surface with or without surface tension are given by solutions of a rather tidy nonlinear pseudo differential operator equation for a 2 periodic function of a real variable being an euler lagrange equation this formulation has the advantage of gradient structure but is complicated by the fact that it involves a non local operator namely the hilbert transform and is quasi linear

on sobolev orthogonal polynomials with coherent pairs the laguerre case type 1 we study the asymptotic properties of orthogonal polynomials with sobolev inner product formula the pair d d is called a coherent pair if there exists nonzero constantsdnsuch that formula wherepn x andqn x are thenth monic orthogonal polynomials with respect tod andd respectively one can divide the coherent pairs into two cases the jacobi case and the laguerre case there are two types in each case we consider thenth root asymptotics and the zero distribution for the laguerre case type 1

asymptotics of sobolev orthogonal polynomials for coherent pairs of laguerre type abstract let s n n denote a sequence of polynomials orthogonal with respect to the sobolev inner product f g s f x g x d 0 x f x g x d 1 x where 0 and d 0 d 1 is a so called coherent pair with at least one of the measures d 0 or d 1 a laguerre measure we investigate the asymptotic behaviour of sn x outside the supports of d 0 and d 1 and n

mixed mode oscillations in a stochastic piecewise linear system abstract we analyse a piecewise linear fitzhugh nagumo model the system exhibits a canard near which both small amplitude and large amplitude periodic orbits exist the addition of small noise induces mixed mode oscillations mmos in the vicinity of the canard point we determine the effect of each model parameter on the stochastically driven mmos in particular we show that any parameter variation such as a modification of the piecewise linear function in the model that leaves the ratio of noise amplitude to time scale separation unchanged typically has little effect on the width of the interval of the primary bifurcation parameter over which mmos occur in that sense the mmos are robust furthermore we show that the piecewise linear model exhibits mmos more readily than the classical fitzhugh nagumo model for which a cubic polynomial is the only nonlinearity by studying a piecewise linear model we are able to explain results using analytical expressions and compare these with numerical investigations

on extended chebyshev systems with positive accuracy a classical necessary condition for an ordered set of n 1 functions f to be an ect system in a closed interval is that all the wronskians do not vanish with this condition all the elements of span f have at most n zeros taking into account the multiplicity here the problem of bounding the number of zeros of span f is considered as well as the effectiveness of the upper bound when some wronskians vanish for this case we also study the possible configurations of zeros that can be realized by elements of span f an application to count the number of isolated periodic orbits for a family of nonsmooth systems is performed

properties of solutions to a class of differential models incorporating preisach hysteresis operator abstract we consider a class of equations which have been recently proposed as a mathematical tool for modelling dynamics of hydrological economic and biological systems exhibiting hysteresis and related memory effects the detail of the modelling approach is illustrated by an example from the hydrological context where a balance equation is coupled with a hysteretic constitutive relationship between the water content in the soil and the matric potential m of the soil matrix and where the preisach hysteresis operator is used as a model of this constitutive relationship in particular we present assumptions which eliminate spatial variation and lead to balance equations in the form of odes two examples of such hydrological models are considered followed by a less detailed discussion of applications of similar modelling approach and equations in economics and biology in the proposed formalism the closed system is described by an operator differential equation where the rate of change of the output of the preisach operator is a function of its input and time in the main part of the paper for such operator differential equations we study the initial value problem uniqueness existence extendability of solutions their dependence on initial data and the structure of the projection of the phase portrait onto the t m plane solutions are characterised by jumps of the derivative induced by either of the two reasons one is the memory of past extremum values of the solution the other is a singularity at the lines of zero flow we analyse the singularity and calculate the value of the jumps thus providing an important input to numerical solution of the equation furthermore we identify possible non uniqueness points and points of sensitivity to small perturbations of initial data as well as conditions that ensure uniqueness and stability to such perturbations regularisation of the equation and natural monotonicity conditions ensuring global stability of a periodic solution for the equation with periodic input are discussed rigorous analysis of the well posedness of the models is presented for the first time

a model reference adaptive control system for a class of discrete linear time varying systems with time delay abstract this paper presents a new method for designing a model reference adaptive control system for a class of discrete linear time varying systems with time delay based on lyapunov s stability theorem adaptive algorithms guarantee the asymptotic stability of the error between the plant output and the reference sequence without using anticipated values of the plant output the validity of the theoretical result is illustrated by numerical examples

connection problems for polynomial solutions of nonhomogeneous differential and difference equations abstract we consider nonhomogeneous hypergeometric type differential difference and q difference equations whose nonhomogeneity is a polynomial q n x the polynomial solution of these problems is expanded in the q n x basis and also in a basis p n x related in a natural way with the homogeneous hypergeometric equation we give an algorithm building a recurrence relation for the expansion coefficients in both bases that we solve explicitly in many cases involving classical orthogonal polynomials finally some concrete applications and extensions are given

small scale instabilities in dynamical systems with sliding we demonstrate with a minimal example that in filippov systems dynamical systems governed by discontinuous but piecewise smooth vector fields stable periodic motion with sliding is not robust with respect to stable singular perturbations we consider a simple dynamical system that we assume to be a quasi static approximation of a higher dimensional system containing a fast stable subsystem we tune a system parameter such that a stable periodic orbit of the simple system touches the discontinuity surface this is the so called grazing sliding bifurcation the periodic orbit remains stable and its local return map becomes piecewise linear however when we take into account the fast dynamics the local return map of the periodic orbit changes qualitatively giving rise to for example period adding cascades or small scale chaos

orthogonal polynomials on sobolev spaces old and new directions abstract during the last years orthogonal polynomials on sobolev spaces have attracted considerable attention algebraic properties distribution of their zeros and fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case in this paper we present an introductory survey about the subject the origin of the problems and their development show the interest and the promising future of this field

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

a model reference adaptive control system for discrete multivariable systems with time delays this paper presents a new method for designing a model reference adaptive control system for multivariable plants with time delays in the input and output variables an adaptive algorithm guarantees the asymptotic stability of the error between the plant output and the reference sequence using no anticipative value of the plant output and unknown plant parameter the validity of the theoretical result is illustrated by a numerical example

limit cycles of discontinuous piecewise polynomial vector fields abstract when the first average function is non zero we provide an upper bound for the maximum number of limit cycles bifurcating from the periodic solutions of the center x y x 2 y 2 2 m and y x x 2 y 2 2 m with m 1 when we perturb it inside a class of discontinuous piecewise polynomial vector fields of degree n with k pieces the positive integers m n and k are arbitrary the main tool used for proving our results is the averaging theory for discontinuous piecewise vector fields

bifurcation and chaos near sliding homoclinics we study the chaotic behaviour of a time dependent perturbation of a discontinuous differential equation whose unperturbed part has a sliding homoclinic orbit that is a solution homoclinic to a hyperbolic fixed point with a part belonging to a discontinuity surface we assume the time dependent perturbation satisfies a kind of recurrence condition which is satisfied by almost periodic perturbations following a functional analytic approach we construct a melnikov like function m in such a way that if m has a simple zero at some point then the system has solutions that behave chaotically applications of this result to quasi periodic systems are also given

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

a numerical study of an impact oscillator with the addition of dry friction abstract this paper presents an investigation into the non linear behaviour of an impact oscillator with the addition of dry friction the equations that govern the relative motion of the impact oscillator including the effects of dry friction are formulated resulting in a non conservative piecewise linear ordinary differential equation of motion the system is excited by a single component sinusoidal base motion a computer model is used to numerically simulate the motion of the system and to identify periodic solutions and the stability of these solutions trends of the periodic solutions are then interpreted using bifurcation theory to identify the non linear behaviour of this system as a function of both the excitation amplitude and the excitation frequency for two levels of the dry friction force these results illustrate a rich behaviour that includes turning point bifurcations symmetry breaking pitchfork bifurcation period doubling bifurcation cascades and bifurcations that can be characterized as boundary grazing the results also include the existence of a subharmonic resonance motion that encompasses three periods of the forcing during each period of the solution also presented are samples of time histories and phase protraits and a plot of the stable response regions for clamping force as a function of excitation amplitude

on the origin and bifurcations of stick slip oscillations a recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis particular emphasis is put on the bifurcations associated with the appearance of stick

what is beyond coherent pairs of orthogonal polynomials abstract usually coherent pairs of orthogonal polynomials have been considered in the wider context of sobolev orthogonality in this paper we focus our attention on the problem of coherence between two orthogonal polynomial sequences in terms of the corresponding linear functionals we deduce some conditions about the linear functionals in order that the corresponding orthogonal polynomial sequences constitute a coherent pair

piecewise smooth dynamical systems persistence of periodic solutions and normal forms abstract we consider an n dimensional piecewise smooth vector field with two zones separated by a hyperplane which admits an invariant hyperplane transversal to containing a period annulus a fulfilled by crossing periodic solutions for small discontinuous perturbations of these systems we develop a melnikov like function to control the persistence of periodic solutions contained in a when n 3 we provide normal forms for the piecewise linear case finally we apply the melnikov like function to study discontinuous perturbations of the given normal forms

some properties of the sobolev laguerre polynomials abstract in this paper we study the case 0 of the sobolev laguerre polynomials we determine a generating function for the polynomials and an expansion formula

bowen ruelle measures for certain piecewise hyperbolic maps we consider a class of piecewise c 2 lozi like maps and prove the existence of invariant measures with absolutely continuous conditional mea sures on unstable manifolds

smallest zeros of some types of orthogonal polynomials asymptotics we establish mehler heine type formulas for orthogonal polynomials related to rational modifications of hermite weight on the real line and for hermite sobolev orthogonal polynomials these formulas give us the asymptotic behaviour of the smallest zeros of the corresponding orthogonal polynomials furthermore we solve a conjecture posed in a previous paper about the asymptotics of the smallest zeros of the hermite sobolev polynomials as well as an open problem concerning the asymptotics of these sobolev orthogonal polynomials

a geometric analysis of front propagation in an integrable nagumo equation with a linear cut off abstract we investigate the effects of a linear cut off on front propagation in the nagumo equation at a so called maxwell point where the corresponding front solution in the absence of a cut off is stationary we show that the correction to the propagation speed induced by the cut off is positive in this case moreover we determine the leading order asymptotics of that correction in terms of the cut off parameter and we calculate explicitly the corresponding coefficient our analysis is based on geometric techniques from dynamical systems theory and in particular on the method of geometric desingularization blow up

melnikov analysis in nonsmooth differential systems with nonlinear switching manifold abstract we study the family of piecewise linear differential systems in the plane with two pieces separated by a cubic curve our main result is that 7 is a lower bound for the hilbert number of this family in order to get our main result we develop the melnikov functions for a class of nonsmooth differential systems which generalizes up to order 2 some previous results in the literature whereas the first order melnikov function for the nonsmooth case remains the same as for the smooth one i e the first order averaged function the second order melnikov function for the nonsmooth case is different from the smooth one i e the second order averaged function we show that in this case a new term depending on the jump of discontinuity and on the geometry of the switching manifold is added to the second order averaged function

coherent pairs of measures and markov bernstein inequalities abstract all the coherent pairs of measures associated to linear functionals c 0 and c 1 introduced by iserles et al in 1991 have been given by meijer in 1997 there exist seven kinds of coherent pairs all these cases are explored in order to give three term recurrence relations satisfied by polynomials the smallest zero 1 n of each of them of degree n has a link with the markov bernstein constant m n appearing in the following markov bernstein inequalities c 1 p 2 m n 2 c 0 p 2 p p n where m n 1 1 n the seven kinds of three term recurrence relations are given in the case where c 0 e x d x 0 and c 1 e x d x explicit upper and lower bounds are given for 1 n and the asymptotic behavior of the corresponding markov bernstein constant is stated except in a part of one case lim n 1 n 0 is proved in all the cases

laguerre sobolev orthogonal polynomials abstract in this paper polynomials that are orthogonal with respect to the inner product f g s 0 f x g x x e x dx 0 f x g x x e x dx where 1 and 0 are studied for these nonstandard orthogonal polynomials algebraic and differential properties as well as the relation with the classical laguerre polynomials are obtained finally some properties concerning the localization and separation of the zeros of these polynomials are deduced

nonsmooth homoclinic orbits melnikov functions and chaos in discontinuous systems abstract we first study the problem of chaotic behaviour in time perturbed discontinuous systems whose unperturbed part has a piecewise c 1 homoclinic solution transversally crossing the discontinuity manifolds we show that if a certain melnikov function has a simple zero at some point then the system has solutions that behave chaotically the melnikov function is explicitly constructed for two dimensional systems and extends the usual melnikov function for the smooth case in the second part we extend these results to sliding homoclinic bifurcations we also mention some possibilities for further research

bifurcation of periodic orbits by perturbing high dimensional piecewise smooth integrable systems abstract this paper deals with the problem of periodic orbit bifurcations for high dimensional piecewise smooth systems under the assumption that the unperturbed system has a family of periodic orbits which are transversal to the switch plane a formula for the first order melnikov vector function is developed which can be used to study the number of periodic orbits bifurcated from the periodic orbits we especially can use the function to study the number of periodic orbits both in degenerate hopf bifurcations and in degenerate homoclinic bifurcations finally we present two examples to illustrate an application of the theoretical results

on hopf bifurcation in non smooth planar systems abstract as we know for non smooth planar systems there are foci of three different types called focus focus ff focus parabolic fp and parabolic parabolic pp type respectively the poincare map with its analytical property and the problem of hopf bifurcation have been studied in coll et al 2001 3 and filippov 1988 6 for general systems and in zou et al 2006 13 for piecewise linear systems in this paper we also study the problem of hopf bifurcation for non smooth planar systems obtaining new results more precisely we prove that one or two limit cycles can be produced from an elementary focus of the least order order 1 for foci of ff or fp type and order 2 for foci of pp type theorem 2 3 different from the case of smooth systems for piecewise linear systems we prove that 2 limit cycles can appear near a focus of either ff fp or pp type theorem 3 3

on normal form calculations in impact oscillators normal form calculations are useful for analysing the dynamics close to bifurcations however the application to non smooth systems is a topic for current research here we consider a class of imp

averaging theory at any order for computing limit cycles of discontinuous piecewise differential systems with many zones abstract this work is devoted to study the existence of periodic solutions for a class of e family of discontinuous differential systems with many zones we show that the averaged functions at any order control the existence of crossing limit cycles for systems in this class we also provide some examples dealing with nonsmooth perturbations of nonlinear centers

on the positivity of some bilinear functionals in sobolev spaces abstract the positivity of a bilinear functional a a f g n m 0 m c m f m g m is studied as a function of coefficients m the concerned cases are those of laguerre gegenbauer and jacobi for c m c 0 m 1 n the domain m m 1 n where a is positive definite is given as a consequence when n 1 the corresponding markov bernstein inequalities are given

novel routes to chaos through torus breakdown in non invertible maps the paper describes a number of new scenarios for the transition to chaos through the formation and destruction of multilayered tori in non invertible maps by means of detailed numerically calculated phase portraits we first describe how three and five layered tori arise through period doubling and or pitchfork bifurcations of the saddle cycle on an ordinary resonance torus we then describe several different mechanisms for the destruction of five layered tori in a system of two linearly coupled logistic maps one of these scenarios involves the destruction of the two intermediate layers of the five layered torus through the transformation of two unstable node cycles into unstable focus cycles followed by a saddle node bifurcation that destroys the middle layer and a pair of simultaneous homoclinic bifurcations that produce two invariant closed curves with quasiperiodic dynamics along the sides of the chaotic set other scenarios involve different combinations of local and global bifurcations including bifurcations that lead to various forms of homoclinic and heteroclinic tangles we finally demonstrate that essentially the same scenarios can be observed both for a system of nonlinearly coupled logistic maps and for a couple of two dimensional non invertible maps that have previously been used to study the properties of invariant sets

new lower bound for the hilbert number in piecewise quadratic differential systems abstract we study the number of limit cycles bifurcating from a piecewise quadratic system all the differential systems considered are piecewise in two zones separated by a straight line we prove the existence of 16 crossing limit cycles in this class of systems if we denote by h p n the extension of the hilbert number to degree n piecewise polynomial differential systems then h p 2 16 as fas as we are concerned this is the best lower bound for the quadratic class moreover in the studied cases all limit cycles appear nested bifurcating from a period annulus of a isochronous quadratic center

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

grazing bifurcations and basins of attraction in an impact friction oscillator abstract this paper describes some interesting global dynamic behavior in the response of a double sided harmonically forced impact oscillator including the influence of coulomb damping the system under study is modeled as piecewise linear in both its force deflection and force velocity characteristics grazing bifurcations caused by this latter effect are a new feature the paper has two distinct but related foci first the study of basins of attraction provides information regarding the complete solution set for the system given a specific set of parameters second grazing bifurcations represent the primary source of sudden change in qualitative behavior as a system parameter is varied the numerical technique of cell to cell mapping provides a useful insight into the relation between these two thus both local and global issues are addressed indeed it is the interplay of periodic attractors and their basins of attraction that dominates bifurcational behavior

transitions from phase locked dynamics to chaos in a piecewise linear map recent work has shown that torus formation in piecewise smooth maps can take place through a special type of border collision bifurcation in which a pair of complex conjugate multipliers for a stable cycle abruptly jump out of the unit circle transitions from an ergodic to a resonant torus take place via border collision fold bifurcations we examine the transition to chaos through torus destruction in such maps considering a piecewise linear normal form map we show that this transition by virtue of the interplay of border collision bifurcations with period doubling and homoclinic bifurcations can involve mechanisms that differ qualitatively from those described by afraimovich and shilnikov

model reference adaptive control system for discrete linear time varying systems with periodically varying parameters and time delay this paper presents a new method for designing a model reference adaptive control system for a class of discrete linear time varying systems with periodically varying parameters and time delay based on lyapunov s stability theorem adaptive control algorithms guarantee the asymptotic stability of the error between the plant output and the reference sequence using no anticipative value of the plant output the validity of the theoretical result is illustrated by numerical examples

some remarks on a paper by l carlitz we study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by l carlitz we show that they are a special case of the sheffer polynomials and point out some interesting connections with certain sobolev orthogonal polynomials

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

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