detection of grazing orbits and incident bifurcations of a forced continuous piecewise linear oscillator abstract a critical phenomenon in a continuous piecewise linear oscillator subjected to periodic excitation is considered in which a periodic orbit happens to graze a switching plane between two linear regions of the oscillator at zero velocity the analysis shows that if the piecewise linearity of the oscillator changes dramatically on the switching plane the phenomenon turns the stability trend of the periodic orbit so abruptly that it may be hard to predict an incident bifurcation although it is of a local type according to the concepts of smooth dynamical systems to detect the critical phenomenon induced by the non smooth nature of the oscillator a numerical scheme of searching for all periodic grazing orbits is presented as well as a test function for grazing phenomena in tracing a branch of a periodic orbit there follows a numerical simulation based on the scheme and the test function which reveals an abundance of grazing phenomena and incident bifurcations of the oscillator

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

models of social networks based on social distance attachment we propose a class of models of social network formation based on a mathematical abstraction of the concept of social distance social distance attachment is represented by the tendency of peers to establish acquaintances via a decreasing function of the relative distance in a representative social space we derive analytical results corroborated by extensive numerical simulations showing that the model reproduces the main statistical characteristics of real social networks large clustering coefficient positive degree correlations and the emergence of a hierarchy of communities the model is confronted with the social network formed by people that shares confidential information using the pretty good privacy pgp encryption algorithm the so called web of trust of pgp

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

teixeira singularities in 3d switched feedback control systems this paper is concerned with the analysis of a singularity that can occur in three dimensional discontinuous feedback control systems the singularity is the two fold a tangency of orbits to both sides of a switching manifold particular attention is placed on the teixeira singularity which previous literature suggests as a mechanism for dynamical transitions in this class of systems we show that such a singularity cannot occur in classical single input single output systems in the lur e form it is however a potentially dangerous phenomenon in multiple input multiple output switched control systems the theoretical derivation is illustrated by means of a representative example

bifurcations of dynamical systems with sliding derivation of normal form mappings this paper is concerned with the analysis of so called sliding bifurcations in n dimensional piecewise smooth dynamical systems with discontinuous vector field these novel bifurcations occur when the system trajectory interacts with regions on the discontinuity set where sliding is possible the derivation of appropriate normal form maps is detailed it is shown that the leading order term in the map depends on the particular bifurcation scenario considered this is in turn related to the possible bifurcation scenarios exhibited by a periodic orbit undergoing one of the sliding bifurcations discussed in the paper a third order relay system serves as a numerical example

sliding bifurcations in the dynamics of mechanical systems with dry friction remarks for engineers and applied scientists the dynamics of mechanical systems with dry friction is affected by non smooth bifurcations which have been recently partially classified as sliding bifurcations in applied science a bifurcation is usually seen as the point in which the number of fixed points and or quasi periodic solutions changes the paper describes with several detailed examples that sliding bifurcations do not always correspond to such definition

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

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

continuation theorems for periodic perturbations of autonomous systems it is first shown in this paper that whenever it exists the coincidence degree of the left hand member of an autonomous differential equation x g x 0 in the space of periodic functions with fixed period omega can be computed in terms of the brouwer degree of g this result provides efficient continuation theorems specially for omega periodic perturbations of autonomous systems extensions to differential equations in flow invariant enr s are also given

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

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

low dimensional maps for piecewise smooth oscillators abstract dynamics of the piecewise smooth nonlinear oscillators is considered for which general methodology of reducing multidimensional flows to low dimensional maps is proposed this includes a definition of piecewise smooth oscillator and creation of a global iterative map providing an exact solution the global map is comprised of local maps which are constructed in the smooth sub regions of phase space to construct this low dimensional map it is proposed to monitor the points of intersections of a chosen boundary between smooth subspaces by a trajectory the dimension reduction is directly related to the dimension of the chosen boundary and the lower its dimension is the larger dimension reduction can be achieved full details are given for a drifting impact oscillator where the five dimensional flow is reduced to one dimensional 1d approximate analytical map first an exact two dimensional map has been formulated and analysed a further reduction to 1d approximate map is introduced and discussed standard nonlinear dynamic analysis reveals a complex behaviour ranging from periodic oscillations to chaos and co existence of multiple attractors accuracy of the constructed maps is examined by comparing with the exact solutions for a wide range of the system parameters

uniqueness for solutions of differential complementarity problems in this paper we consider the question of which matrices m give unique solutions for differential complementarity problems mandelbaum 1989 unpublished manuscript of the form begin array ll frac dw dt m z q t quad w 0 w 0 k ni z t perp w t in k quad rm for all t end array with applications to complementarity and discretized l1 regularization problems assuming semi smoothness it is shown that super linearly convergent newton methods can be globalized if appropriate descent directions are used for the merit function f x 2 special attention is paid to directions obtained from the primal dual active set strategy

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

bounds on the number of switchings for trajectories of piecewise analytic vector fields abstract we study the trajectories of systems x x x where x is a continuous extendably piecewise analytic vector field i e a continuous vector field x such that the domain of admits a locally finite partition i into sets such that for each a i there is a vector field xa which is analytic on a neighborhood of the closure of a and whose restriction to a coincides with that of x we prove that the trajectories are piecewise analytic with a priori bounds on the number of switchings for all trajectories that stay in a fixed compact set and whose duration does not exceed a fixed number t this result implies the existence of a regular synthesis for optimal control problems with a strictly convex lagrangian and a linear dynamics with polyhedral constraints on the controls

evolving network with different edges we propose a scale free network similar to barabasi albert networks but with two different types of edges this model is based on the idea that in many cases there are more than one kind of link in a network and when a new node enters the network both old nodes and different kinds of links compete to obtain it the degree distribution of both the total degree and the degree of each type of edge is analyzed and found to be scale free simulations are shown to confirm these results

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

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

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

border collision bifurcations an explanation for observed bifurcation phenomena recently physical and computer experiments involving systems describable by continuous maps that are nondifferentiable on some surface in phase space have revealed novel bifurcation phenomena these phenomena are part of a rich new class of bifurcations which we call border collision bifurcations a general criterion for the occurrence of border collision bifurcations is given illustrative numerical results including transitions to chaotic attractors are presented these border collision bifurcations are found in a variety of physical experiments

melnikov method for homoclinic bifurcation in nonlinear impact oscillators based on an inverted pendulum impacting on rigid walls under external periodic excitation a class of nonlinear impact oscillators is discussed for its homoclinic bifurcation the melnikov method established for smooth dynamical systems is extended to be applicable to the nonsmooth one for nonlinear impact systems closed form solutions between impacts are generally unavailable the absence of closed form solutions makes difficulties in estimation of the gap between the stable manifold and unstable manifold in this paper we give a method to compute the melnikov functions up to the n t h order so as to obtain conditions of parameters for the persistence of homoclinic cycles which are formed via the identification given by the impact rule

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

normal form maps for grazing bifurcations in n dimensional piecewise smooth dynamical systems abstract this paper presents a unified framework for performing local analysis of grazing bifurcations in n dimensional piecewise smooth systems of odes these occur when a periodic orbit has a point of tangency with a smooth n 1 dimensional boundary dividing distinct regions in phase space where the vector field is smooth it is shown under quite general circumstances that this leads to a normal form map that contains to lowest order either a square root or a 3 2 type singularity according to whether the vector field is discontinuous or not at the grazing point in particular contrary to what has been reported in the literature piecewise linear local maps do not occur generically first the concept of a grazing bifurcation is carefully defined using appropriate non degeneracy conditions next complete expressions are derived for calculating the leading order term in the normal form poincare map at a grazing bifurcation point in arbitrary systems using the concept of a discontinuity mapping finally the theory is compared with numerical examples including bilinear oscillators a relay feedback controller and general third order systems

melnikov s method for a general nonlinear vibro impact oscillator abstract melnikov s method is presented for a general nonlinear vibro impact oscillator and melnikov function for homoclinic orbits is obtained analytically a duffing vibro impact oscillator is given to illustrate the application of the procedures the obtained results are verified by the phase portrait poincare surface of section and bifurcation diagrams

bifurcation analysis of an impact oscillator with a one sided elastic constraint near grazing in this paper a linear oscillator undergoing impact with a secondary elastic support is studied experimentally and semi analytically for near grazing conditions the experimentally observed bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces smooth as well as nonsmooth bifurcations are observed and the resulting atypical bifurcations are explained often as an interplay between them in order to understand the observed bifurcation scenarios a global analysis is required due to the influence of stable and unstable orbits which are born in distant bifurcations but become important at near grazing conditions the good degree of correspondence between experiment and theory fully justifies the modelling approach

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

signatures of finite classical lie algebra representations the paper deals with the real classical lie algebras of types br cr and their arbitrary irreducible representations hermitian forms invariant relative to these representations are considered signature formulas for these forms are obtained

asymmetric type ii periodic motions for nonlinear impact oscillators abstract in this paper a general class of nonlinear impact oscillators is considered for type ii periodic motions this system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation the unperturbed system possesses a pair of homoclinic cycles and three separate families of periodic orbits inside and outside the homoclinic cycles via the identification given by the impact law by approximating the poincare map to o e directly a general method of melnikov type for detecting the existence of asymmetric type ii subharmonic orbits outside the homoclinic cycles is presented

generic bifurcation of sliding vector fields abstract the main aim of this paper is to classify the singularities which generically occur in one parameter families of sliding vector fields

continuous and discontinuous grazing bifurcations in impacting oscillators this paper seeks to formulate conditions for the persistence of a local attractor in the immediate vicinity of periodic and quasiperiodic grazing trajectories in an impacting mechanical system a local analysis based on the discontinuity mapping approach is employed to derive a normal form description of the dynamics near the grazing trajectory in agreement with previous studies of grazing periodic trajectories it is found that the catastrophic loss of a local attractor and strong instability characteristic of grazing bifurcations is directly associated with the repeated application of a square root term that appears to lowest order in the normal form expansion specifically it is found that the square root term is absent in the description of the dynamics normal to a quasiperiodic trajectory covering a co dimension one invariant torus resulting in a piecewise linear description of the normal dynamics and at most a weak instability in contrast for co dimension two or higher the square root term is generically present in the normal dynamics here however the quasiperiodicity of the grazing motion implies that there is no upper limit to the time between impacts on nearby trajectories suggesting the persistence of a local attractor for some interval about the parameter value corresponding to grazing the results of the analysis are illustrated through a series of model examples

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

global dynamics of a vibro impacting linear oscillator abstract the steady state vibro impacting responses of one dimensional harmonically excited linear oscillators are studied by using a modern dynamical systems approach allied with numerical simulation the steady state motions are attracting sets in the system phase space and capture initial conditions in their domains of attraction unlike the free harmonically excited oscillator the phase space of a vibro impacting system may be inhabited by many attracting sets for example there are sub harmonic multi impact periodic orbits and chaotic steady state responses in order to build a qualitative understanding of vibro impact response an attempt is made to build generic topological models of their phase spaces for physically significant parameter ranges use is made of the poincare section or stroboscopic mapping technique essentially following an initial impact forwards or backwards in time to subsequent or previous impacts using a computer the qualitative understanding gained from the analysis and simulations is discussed in an engineering context

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

the bandcount increment scenario i basic structures bifurcation structures in two dimensional parameter spaces formed only by chaotic attractors are still far away from being understood completely in a series of three papers we investigate the chaotic domain without periodic inclusions for a map which is considered by many authors as some kind of one dimensional canonical form for discontinuous maps in the first part we report a novel bifurcation scenario formed by crises bifurcations which includes multi band chaotic attractors with arbitrary high bandcounts and determines the basic structure of the chaotic domain

strongly nonlinear vibrations of damped oscillators with two nonsmooth limits a family of strongly nonlinear oscillators with a generalized power form elastic force and viscous damping is considered an explicit analytical solution is obtained as a combination of smooth and nonsmooth functions two different nonsmooth functions involved into the solution are associated with two different nonsmooth limits of the oscillator as the exponent becomes either zero or infinity as a result the solution is drastically simplified to give the best match with numerical tests if approaching any of the two limits

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

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

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

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

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

bifurcations of equilibrium sets in mechanical systems with dry friction the presence of dry friction in mechanical systems induces the existence of an equilibrium set consisting of infinitely many equilibrium points the local dynamics of the trajectories near an equilibrium set is investigated for systems with one frictional interface in this case the equilibrium set will be an interval of a curve in phase space it is shown in this paper that local bifurcations of equilibrium sets occur near the endpoints of this curve based on this result sufficient conditions for structural stability of equilibrium sets in planar systems are given and two new bifurcations are identified the results are illustrated by application to a controlled mechanical system with friction

generalized boundary equilibria in n dimensional filippov systems the transition between persistence and nonsmooth fold scenarios abstract we analyze a codimension two bifurcation in generic n dimensional filippov autonomous piecewise smooth discontinuous systems when a hyperbolic equilibrium collides with the discontinuity boundary at generic codim 1 boundary equilibria be two different scenarios can occur the persistence of the stationary solution that changes from a standard equilibrium on one side of the discontinuity boundary into a pseudo equilibrium and the nonsmooth fold between the standard equilibrium and a coexisting pseudo equilibrium generalized codim 2 boundary equilibria gbe separate persistence and nonsmooth fold scenarios and are here shown to occur together with a fold bifurcation between pseudo equilibria plp in a two parameter plane the plp bifurcation curve emanates from the gbe point tangentially to the be curve the asymptotic of the plp curve locally to the gbe point is also derived and tested on a 4 dimensional ecological example

bifurcations from nondegenerate families of periodic solutions in lipschitz systems abstract the paper addresses the problem of bifurcation of periodic solutions from a normally nondegenerate family of periodic solutions of ordinary differential equations under perturbations the approach to solve this problem can be described as transforming by a lyapunov schmidt reduction the initial system into one which is in the standard form of averaging and subsequently applying the averaging principle this approach encounters a fundamental problem when the perturbation is only lipschitz nonsmooth as we do not longer have smooth lyapunov schmidt projectors the situation of lipschitz perturbations has been addressed in the literature lately and the results obtained conclude the existence of the bifurcated branch of periodic solutions motivated by recent challenges in control theory we are interested in the uniqueness problem we achieve this in the case when the lipschitz constant of the perturbation obeys a suitable estimate

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

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

topology of vibro impact systems in the neighborhood of grazing abstract the grazing bifurcation is considered for the newtonian model of vibro impact systems a brief review on the conditions sufficient for the existence of a grazing family of periodic solutions is given the properties of these periodic solutions are discussed a plenty of results on the topological structure of attractors of vibro impact systems is known however since the considered system is strongly nonlinear these attractors must be very sensitive to changes of parameters of the system on the other hand they are observed in experiments and numerical simulations we offer theorem 2 an approach which allows to explain this contradiction and give a new robust mathematical model of the non hyperbolic dynamics in a neighborhood of grazing

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

bifurcation of periodic orbits in a class of planar filippov systems abstract in this paper we discuss the perturbations of a general planar filippov system with exactly one switching line when the system has a limit cycle we give a condition for its persistence when the system has an annulus of periodic orbits we give a condition under which limit cycles are bifurcated from the annulus we also further discuss the stability and bifurcations of a nonhyperbolic limit cycle when the system has an annulus of periodic orbits we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching

signatures of finite representation of real simple lie algebras the paper deals with the arbitrary irreducible representations of simple lie algebras of types g2 f4 br cr dr and the hermitian forms being invariant relative to this representation formulas and tables for calculating hermitian form signatures are obtained

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

universal limit mapping in grazing bifurcations grazing bifurcations are nonsmooth bifurcations that occur in impacting mechanical oscillators a sign of the lack of smoothness is a square root expression that arises in mappings describing the dynamics certain types of grazing bifurcations involve chaotic bands or period adding cascades with or without chaotic bands there is also a characteristic scaling behavior present here this type of bifurcation is investigated and the self similarity under scaling is used to derive a renormalized limit mapping a study of the dynamics of the limit mapping identifying all attractors for all parameter values is also presented

signatures of finite representations of real outer lie algebras the paper deals with the real lie algebras of outer type and their arbitrary irreducible representations hermitian forms being invariant relative to these representations are considered signature formulae for these forms are obtained

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

aspects of bifurcation theory for piecewise smooth continuous systems abstract systems that are not smooth can undergo bifurcations that are forbidden in smooth systems we review some of the phenomena that can occur for piecewise smooth continuous maps and flows when a fixed point or an equilibrium collides with a surface on which the system is not smooth much of our understanding of these cases relies on a reduction to piecewise linearity near the border collision we also review a number of codimension two bifurcations in which nonlinearity is important

mean field theory for scale free random networks random networks with complex topology are common in nature describing systems as diverse as the world wide web or social and business networks recently it has been demonstrated that most large networks for which topological information is available display scale free features here we study the scaling properties of the recently introduced scale free model that can account for the observed power law distribution of the connectivities we develop a mean eld method to predict the growth dynamics of the individual vertices and use this to calculate analytically the connectivity distribution and the scaling exponents the mean eld method can be used to address the properties of two variants of the scale free model that do not display power law scaling c 1999 elsevier science b v all rights reserved

impact oscillations linear theory of stability and bifurcations abstract a unified approach to analysis of impact oscillations is proposed its mathematical essence is the continuous representation of the impulsive motion in some auxiliary variables as a result an explicit formula for the fundamental matrix is obtained which allows one to construct the characteristic equation then the stability and bifurcations can be studied by the usual techniques the exceptional case of bifurcations related to grazing impacts is investigated by taking into account a non zero impact duration due to this approach linearization is also possible and the complex bifurcation becomes a sequence of ordinary ones some algebraic conditions are obtained which allow one to determine the type of the resulting motion a mechanical example is considered a disc with an offset centre of gravity bouncing on an oscillating surface