symbolic dynamics and characterization of complexity abstract symbolic dynamics deals with robust properties of dynamics without digging into numbers for one dimensional maps there have been some new results periodic window theorem generalized composition rule construction of median words without using harmonics and antiharmonics incorporation of discrete symmetry to analyze symmetry breaking and symmetry restoration etc this new understanding has been applied to ordinary differential equations in particular the systematics of stable periodic solutions in the lorenz model has been shown to be given basically by symbolic dynamics of the cubic map symbolic dynamics may be used to extract invariant characteristics from time series its relation with grammatical complexity will also be commented on

breaking of ergodicity in expanding systems of globally coupled piecewise affine circle maps to identify and to explain coupling induced phase transitions in coupled map lattices cml has been a lingering enigma for about two decades in numerical simulations this phenomenon has always been observed preceded by a lowering of the lyapunov dimension suggesting that the transition might require changes of linear stability yet recent proofs of co existence of several phases in specially designed models work in the expanding regime where all lyapunov exponents remain positive in this paper we consider a family of cml composed by piecewise expanding individual map global interaction and finite number n of sites in the weak coupling regime where the cml is uniformly expanding we show mathematically for n 3 and numerically for n ge 3 that a transition in the asymptotic dynamics occurs as the coupling strength increases the transition breaks the milnor attractor into several chaotic pieces of positive lebesgue measure with distinct empiric averages it goes along with various symmetry breaking quantified by means of magnetization type characteristics despite that it only addresses finite dimensional systems to some extend this result reconciles the previous ones as it shows that loss of ergodicity symmetry breaking can occur in basic cml independently of any decay in the lyapunov dimension

the structure of symmetric attractors we consider discrete equivariant dynamical systems and obtain results about the structure of attractors for such systems we show for example that the symmetry of an attractor cannot in general be an arbitrary subgroup of the group of symmetries in addition there are group theoretic restrictions on the symmetry of connected components of a symmetric attractor the symmetry of attractors has implications for a new type of pattern formation mechanism by which patterns appear in the time average of a chaotic dynamical system

the dynamic analysis of a beam mass system due to the occurrence of two component parametric resonance abstract the objective of this paper is an analytical and numerical study of the dynamics of a beam mass system special attention is given to the phenomena arising due to the motion of the attached mass and modal interactions produced by the existence of multi component specifically two component parametric resonance under primary resonance the two component parametric resonance occurs when the sums or the differences among internal frequencies are the same or close as the dimensionless speed parameter of the moving mass the effects of two component parametric resonance post on dynamic condition are investigated resonance generated by more than two component modes are neglected due to its remote probability of occurrence in nature the mechanics of the problem is newtonian based on the assumption that when the moving mass is set in motion the mass is assumed to be rolling on the beam the mechanics including the effects due to friction and convective accelerations of the interface between the moving mass and the beam are determined based on the bernoulli euler beam theory the coupled non linear equations of motion of an inextensible beam with an attached moving mass are derived by employing galerkin procedure the partial differential equations which describe the motion of a beam mass system are reduced to an initial value problem with finite dimensions the method of multiple time scales is applied to consider the solutions and the occurrence of internal resonance of the resulting multi degree of freedom beam mass system with time dependent coefficients

nonlinear vibrations and imperfection sensitivity of a cylindrical shell containing axial fluid flow abstract the high imperfection sensitivity of cylindrical shells under static compressive axial loads is a well known phenomenon in structural stability on the other hand less is known of the influence of imperfections on the nonlinear vibrations of these shells under harmonic axial loads the aim of this work is to study the simultaneous influence of geometric imperfections and an axial fluid flow on the nonlinear vibrations and instabilities of simply supported circular cylindrical shells under axial load the fluid is assumed to be non viscous and incompressible and the flow to be isentropic and irrotational the behavior of the thin walled shell is modeled by donnell s nonlinear shallow shell equations it is subjected to a static uniform compressive axial pre load plus a harmonic axial load a low dimensional modal expansion which satisfies the relevant boundary and continuity conditions and takes into account all relevant nonlinear modal interactions observed in the past in the nonlinear vibrations of cylindrical shells with and without flow is used together with the galerkin method to derive a set of eight coupled nonlinear ordinary differential equations of motion which are in turn solved by the runge kutta method the shell is considered to be initially at rest in a position corresponding to a pre buckling configuration then a harmonic excitation is applied and conditions for parametric instability and dynamic snap through are sought the results clarify the marked influence of geometric imperfections and fluid flow on the dynamic stability boundaries bifurcations and basins of attraction

integrity measures quantifying the erosion of smooth and fractal basins of attraction abstract the phase portraits of a dissipative dynamical system are characterized by the existence of one or more stable attractors which typically include point equilibria periodic oscillations harmonic and subharmonic quasi periodic solutions and chaotic attractors each attractor is embedded in its own domain or basin of attraction bounded by a separatrix associated with an unstable saddle solution under the variation of a control parameter as the attractors move and bifurcate the basins also undergo corresponding changes and metamorphoses these changes in size and shape are usually continuous but can be discontinuous as when an attractor vanishes along with its basin at a saddle node bifurcation associated with the homoclinic tangling of the invariant manifolds of the saddle solution basin boundaries can also change in nature from smooth to fractal in this paper the escape of a driven oscillator from a cubic potential well as an archetypal example is used to explore the engineering significance of the basin erosions that occur under increased forcing various measures of engineering integrity of the constrained attractor are introduced a global measure assesses the overall basin area a local measure assesses the distance from the attractor to the basin boundary and a velocity measure is related to the size of impulse that could be sustained without failure since engineering systems may be subjected to pulse loads of finite duration attention is given to both the absolute and transient basin boudaries the significant erosion of these at homoclinic tangencies is particularly highlighted in the present study the fractal basins having a severely reduced integrity under all three criteria

hopf bifurcation with cubic symmetry and instability of abc flow we examine the dynamics of generic hopf bifurcation in a system that is symmetric under the action of the rotational symmetries of the cube we classify the generic branches of periodic solutions at bifurcation there are generically 27 branches corresponding to maximal symmetries organized into five symmetry types there are also up to 22 periodic solution branches of two other symmetry types these results are found by examination of the normal form with s 1 normal form symmetry for the bifurcation truncated at the third order in addition to the periodic branches whose branching and stability we find there are several branches of tori homoclinic bifurcations and chaotic attractors in the dynamics of the third order normal form since many of these features are not amenable to analysis we give some numerical examples on breaking the normal form symmetry there may be breakup of the branches of tori but the predictions for the periodic solutions will be reliable for the navier stokes equations with a particular forcing an abc flow is a dynamically stable solution for small reynolds numbers r for the most symmetric case a b c 1 the first instability of this system is a hopf bifurcation at r 13 04 with rotational symmetry of the cube we use our normal form analysis to explain the observed behaviour of solutions at this primary instability numerical simulations show that there is supercritical branching to rotating waves that alternate between the three axes which undergo secondary hopf bifurcation to a 2 torus at approximately r 13 09 the eight symmetrically related tori break up and then merge to form a chaotic attractor with full symmetry we can explain all these features by use of the generic third order normal form and s 1 normal form symmetry breaking terms

yet another induction scheme for non uniformly expanding transformations abstract we introduce a new induction scheme for non uniformly expanding maps f of compact riemannian manifolds relying upon ideas of 33 and 10 we use this induction approach to prove that the existence of a gibbs markov young structure is a necessary condition for f to preserve an absolutely continuous probability with all its lyapunov exponents positive

limit sets of s unimodal maps with zero entropy one dimensional mappings at the limit of period doubling are studied in this paper without the use of the renormalization theory of feigenbaum and others the principal result is that the attracting part of the nonwandering set is a cantor set of measure zero under the additional assumption that the map has negative schwarzian derivative

an intermittency route to chaos via attractor merging in the laser kuramoto sivashinsky equation abstract we perform numerical simulations of the laser kuramoto sivashinsky equation which describes transverse effects in a mean field class a laser near threshold solition like structures that are formed by self focusing are found to govern the spatio temporal dynamics we present and discuss the bifurcation route in the low dimensional regime where only few of these structures exist there we discovered an intermittency route to chaos via attractor merging where infinitely many coexisting nonchaotic attractors merge into a single chaotic attractor these phenomena are analyzed by applying generalized concepts from low dimensional systems to our spatially extended system

axisymmetric vibrations of imperfect shallow spherical caps under pressure loading abstract the influence of axisymmetric initial geometric imperfections and a non linear compressive stress state on the natural frequencies and non linear oscillations of a pressure loaded shallow spherical shell is investigated the analysis is based on a solution of marguerre s shallow shell equations by the galerkin method the displacements and stress functions are described by linear combinations of bessel functions and modified bessel functions that satisfy all the relevant boundary and continuity conditions it is found that the natural frequencies and buckling loads depend strongly on the shape and magnitude of the initial imperfection and that the non linear frequency amplitude relationship may be either of the softening or hardening type it is also shown how measured vibration frequencies can be used to estimate safe buckling loads of spherical caps non destructively

the non linear vibration and dynamic instability of thin shallow shells in this paper the non linear vibration and dynamic instability of thin shallow spherical and conical shells subjected to periodic transverse and in plane loads are investigated the marguerre type dynamic equations used for the analysis of shallow shells when treated by the galerkin method will result in a system of total differential equations in the time functions known as duffing and mathieu equations from which the various kinds of non linear vibration and dynamic instability are determined by using numerical methods numerical results are presented for axisymmetric vibrations and dynamic instabilities of shallow spherical and conical shells with a clamped and b supported edge conditions as numerical examples non linear vibration frequencies and instability regions for shells are determined the effects of static load as well as static snap through buckling on the instability are also investigated

experiments and analysis on chaotic vibrations of a shallow cylindrical shell panel abstract detailed experimental results and analytical results are presented on chaotic vibrations of a shallow cylindrical shell panel subjected to gravity and periodic excitation the shallow shell panel with square boundary is simply supported for deflection in plane displacement at the boundary is elastically constrained by in plain springs in the experiment the cylindrical shallow shell panel with thickness 0 24 mm square form of length 140 mm and mean radius 5150 mm is used for the test specimen all edges around the shell boundary are simply supported by adhesive flexible films first to find fundamental properties of the shell panel linear natural frequencies and characteristics of restoring force of the shell panel are measured these results are compared with the relevant analytical results then geometrical parameters of the shell panel are identified exciting the shell panel with lateral periodic acceleration nonlinear frequency responses of the shell panel are obtained by sweeping the frequency of periodic acceleration in typical ranges of the exciting frequency predominant chaotic responses are generated time histories of the responses are recorded for inspection of the chaos in the analysis the donnell equation with lateral inertia force is introduced assuming mode functions the governing equation is reduced to a set of nonlinear ordinary differential equations by the galerkin procedure periodic responses are calculated by the harmonic balance method chaotic responses are integrated numerically by the runge kutta gill method the chaotic responses which are obtained by the experiment and the analysis are inspected with the fourier spectra the poincare projections the maximum lyapunov exponents and the lyapunov dimension it is found that the dominant chaotic responses of the shell panel are generated from the responses of the sub harmonic resonance of 1 2 order and of the ultra sub harmonic resonance of 2 3 order by the convergence of the maximum lyapunov exponent to the embedding dimension the number of predominant vibration modes which contribute to the chaos is found to be three or four fairly good agreements are obtained between the experimental results and the analytical results

timely communication symmetry and the karhunen loeve analysis the karhunen loeve k l analysis is widely used to generate low dimensional dynamical systems which have the same low dimensional attractors as some large scale simulations of pdes if the pde is symmetric with respect to a symmetry group g the dynamical system has to be equivariant under g to capture the full phase space it is shown that symmetrizing the k l eigenmodes instead of symmetrizing the data leads to considerable computational savings if the k l analysis is done in the snapshot method the feasibility of the approach is demonstrated with an analysis of kolmogorov flow

on the unfolding of a blowout bifurcation abstract suppose a chaotic attractor a in an invariant subspace loses stability on varying a parameter at the point of loss of stability the most positive lyapunov exponent of the natural measure on a crosses zero at what has been called a blowout bifurcation we introduce the notion of an essential basin of an attractor a this is the set of points x such that accumulation points of the sequence of measures 1 n n 1 k 0 f k x are supported on a we characterise supercritical and subcritical scenarios according to whether the lebesgue measure of the essential basin of a is positive or zero we study a drift diffusion model and a model class of piecewise linear mappings of the plane in the supercritical case we find examples where a lyapunov exponent of the branch of attractors may be positive hyperchaos or negative depending purely on the dynamics far from the invariant subspace for the mappings we find asymptotically linear scaling of lyapunov exponents average distance from the subspace and basin size on varying a parameter we conjecture that these are general characteristics of blowout bifurcations

nonlinear dynamics of harmonically excited circular cylindrical shells containing fluid flow abstract in the present study the geometrically nonlinear vibrations of circular cylindrical shells subjected to internal fluid flow and to a radial harmonic excitation in the spectral neighbourhood of one of the lowest frequency modes are investigated for different flow velocities the shell is modelled by donnell s nonlinear shell theory retaining in plane inertia and geometric imperfections the fluid is modelled as a potential flow with the addition of unsteady viscous terms obtained by using the time averaged navier stokes equations a harmonic concentrated force is applied at mid length of the shell acting in the radial direction the shell is considered to be immersed in an external confined quiescent liquid and to contain a fluid flow in order to reproduce conditions in previous water tunnel experiments for the same reason complex boundary conditions are applied at the shell ends simulating conditions intermediate between clamped and simply supported ends numerical results obtained by using pseudo arclength continuation methods and bifurcation analysis show the nonlinear response at different flow velocities for i a fixed excitation amplitude and variable excitation frequency and ii fixed excitation frequency by varying the excitation amplitude bifurcation diagrams of poincare maps obtained from direct time integration are presented as well as the maximum lyapunov exponent in order to classify the system dynamics in particular periodic quasi periodic sub harmonic and chaotic responses have been detected the full spectrum of the lyapunov exponents and the lyapunov dimension have been calculated for the chaotic response they reveal the occurrence of large dimension hyperchaos

the dynamics of a piping system with internal unsteady flow abstract the dynamics of a pipeline conveying one dimensional unsteady flow is considered the dynamics of the fluid pipe system is represented by a set of partial differential equations which consists of the axial and transverse equations of motion of the pipeline and the equations of momentum and continuity of the internal flow the vibration equations of the pipeline are derived by use of newton s law of motion while the fluid equations are derived based on the concept of a deformable moving control volume the equations are fully coupled to each other and thus can be applied to various fluid pipe interaction problems which can be generated by the diverse operations of valves and pumps attached to the pipeline the equations are applied to a simply supported inclined straight pipeline to investigate the stabilities and dynamic responses of the pipeline the stability analysis is conducted by use of bolotin s method the dynamic responses are numerically investigated for both stable and unstable conditions it is observed that unstable regions in the stability chart expand as the flow velocity and the mass density of the fluid increase and that natural frequencies of the pipeline increase as the fluid friction acting on the inner surface of pipeline increases

symmetries and dynamics for 2 d navier stokes flow abstract simulations of forced 2 d navier stokes equations are analyzed the forcing is spatially periodic and temporally steady a karhunen loeve analysis is used to identify the structures in phase space that generate the pde behavior their relationship to the invariant subspaces generated by the symmetry group is discussed it is shown that certain modes that are in the stable eigenspace of the kolmogorov flow solution play an essential role for the dynamics of the attractor for the 2 d navier stokes equations below a reynolds number of about 30 in this regime all stable solutions are identified and thier relation to the symmetry structure is elucidated a new type of gluing bifurcation generated by the symmetry is found and analyzed a mechanism for the generation of bursting behavior is suggested

on a connection between kam and nekhoroshev s theorems abstract using the iteration of nekhoroshev s theorem as a basic tool we point out the existence of a hierarchic structure of nested domains underlying the phenomenon of diffusion at each level we find that the diffusions speed is exponentially small with respect to the previous level the set of kam tori is the domain characterized by a zero diffusion speed

a formula for a symmetry detective detectives are used to determine the symmetry of attractors in finite dimensional systems we derive an explicit formula for a symmetry detective for each representation of a finite group

spectral theory zeta functions and the distribution of periodic points for collet eckmann maps we study unimodal interval mapst with negative schwarzian derivative satisfying the collet eckmann condition dtn tc k cn for some constantsk 0 and c 1 c is the critical point oft we prove exponential mixing properties of the unique invariant probability density oft describe the long term behaviour of typical in the sense of lebesgue measure trajectories by central limit and large deviations theorems for partial sum processes of the form s n sigma i 0 n 1 f t i x and study the distribution of typical periodic orbits also in the sense of a central limit theorem and a large deviations theorem

phase space reconstruction for symmetric dynamical systems abstract since the pioneering work of packard et al and takens it has become customary to reconstruct the topology of attractors in phase space from a time series of one dimensional experimental observations by using delay coordinates many practical refinements of the original methods have been developed many experimental systems possess symmetry and bifurcations can cause changes in the symmetry of observed states these changes are quite subtle when the dynamics is chaotic it is therefore important to reconstruct not just the topology of the attractor but its symmetry we indicate how this can be done by extending the packard takens approach to a single equivariant observation taking values not in the real numbers r but in a linear representation v of the symmetry group g in effect a single set of symmetrically related observations is required our central point is that not all plausible choices for such a set can generate embeddings in order for the method to produce an embedding it is necessary that v should be sufficiently complicated more precisely the phase space m must be subordinate to v in a sense introduced by wassermann this concept is technical but unavoidable in this context and it greatly clarifies the issue of embeddability using it we state a symmetric version of the takens embedding theorem and sketch the proof we also discuss the issue of setwise versus pointwise symmetry of an attractor and relate this to the transition from spatial order to spatial disorder in temporally chaotic systems

dynamic response of a rotating flexible arm carrying a moving mass abstract a clamped free flexible arm rotating in a horizontal plane and carrying a moving mass is studied in this paper the arm is modelled by the euler bernoulli beam theory in which rotatory inertia and shear deformation effects are ignored the assumed mode method in conjunction with hamilton s principle is used to derive the equation of motion of the system which takes into account the effect of centrifugal stiffening due to the rotation of the beam the eigenfunctions of a cantilever beam which satisfy the prescribed geometric boundary conditions are used as basis functions in the assumed mode method the equation of motion is expressed in non dimensional matrix form pre designed transformed cosine profiles are used as trajectory inputs for the hub angle and the moving mass the equation of motion is solved numerically using the fourth order runge kutta method graphical results are presented to show the influence of centrifugal stiffening effect moving mass values mass travelling time hub angle and mass trajectory profile on the deflection of the beam

fractal erosion of basins of attraction in coupled non linear systems abstract an examination is presented of how basins of attraction evolve in a coupled non linear oscillator that has the ability to escape from a two dimensional potential well this work may be considered an extension of earlier studies on a single degree of freedom system 1 in which it was shown that under small parameter changes there may exist a rapid erosion and stratification of the safe basin of attraction here it is shown that basin boundaries in the four dimensional phase space may become fractal and hence have a non integer dimension by analyzing cross sections of the four dimensional phase space as system parameters are varied the phenomenon of fractal basin erosion is illustrated such a phenomenon has important implications when assessing the global transient stability of the system

a chaotic hopf bifurcation in coupled maps abstract three dimensional maps with a one dimensional invariant subspace are considered in which the dynamics in the invariant subspace is chaotic such maps arise from three coupled one dimensional maps if the coupling is uni directional and identical then the system has a z3 symmetry and this forces the two normal lyapunov exponents to coincide it is then possible to define a linearised average rotation about the invariant subspace when the normal lyapunov exponents change sign a chaotic hopf bifurcation occurs by considering similar coupled systems but with different coupling strengths the z3 symmetry is lost but there is still an so 2 symmetry on the normal linearisation and so similar results for the lyapunov exponents hold if there is no symmetry on the linearisation either then the multiple lyapunov exponents split although it is still possible to define a linearised average rotation in many cases these three different scenarios are illustrated with numerical examples

detection of symmetry of attractors from observations i theory abstract barany et al 1993 proposed a method for determining the symmetries of attractors of equivariant systems by averaging certain classes of equivariant maps we use an idea in barany et al to re cast definitions of symmetry detectives assuming that we only have access to equivariant observations from the system detecting from observations allows one to perform averaging in spaces that may have much lower dimension than the phase space this paper generalises and develops their suggestion among the generalisations we consider is the use of nonpolynomial detectives and we show using the notion of prevalence of hunt et al 1992 that our detectives and the detectives of barany et al 1993 dellnitz et al 1994 and golubitsky and nicol 1995 give the correct symmetry of attractors almost certainly in a measure theoretic sense we show that detectives can persistently give incorrect symmetries at isolated points in parametrised systems and discuss how to overcome this we show how one can find the symmetry of an attractor from examination of a poincare section in part ii of this article ashwin and tomes apply these results to find symmetries of attractors in a physical system of four coupled electronic oscillators with s4 symmetry

finding lie groups that reduce the order of discrete dynamical systems discrete dynamical systems of the form are considered where is an n component vector equations x f x define a mapping f from an n dimensional projective space into itself each component of f is a rational function i e a ratio of polynomials in n dynamical variables maeda showed that when f commutes with each transformation of a lie group a reduction in the order of the dynamical system results given a discrete dynamical system the difficulty is to find its continuous symmetries we present a way of using f invariant sets to find these symmetries the approach taken is to arrange groups in order of increasing complication and to characterize the set of dynamical systems admitting each group criteria are given for recognizing and reducing the order of systems admitting subgroups of the projective general linear group in n variables pgl n or certain lie subgroups of the cremona group of birational transformations in n variables quispel et al demonstrated the use of canonical group variables for achieving this reduction we develop canonical coordinates for several groups with elementary lie algebras and demonstrate reduction of order in each case results are used to reduce the order of several examples of recursion formulae taken from the literature on renormalizable lattice models

centroidal frames in dynamical systems i point vortices the dynamics of point vortices is studied in part i of the paper it is well known that the translational centre of mass frame decomposes the motion of a mechanical system into simpler components it is less known however that special rotational frames have also been suggested for the same purpose in contrast to the centre of mass frame the angular velocities of these rotational frames are not given explicitly that limits their application to small perturbations of rigid body rotations a new class of centroidal frames cf related to different groups such as translation rotation dilation etc is introduced in this paper the cfs decompose the motion of point vortices into a group and a relative components without restriction to small perturbations of pure group motions the definition of the cfs is based on an averaging of motion or on minimization of energy of the relative motion where an appropriate energy function is expressed through generators of the group action as a result the linear and angular velocities as well as other characteristics of the cfs can be obtained explicitly part i of the paper presents application of the cfs to a hamiltonian system of point vortices examples of integrable and chaotic motions in the cfs visualize dynamical patterns that are completely hidden in the conventional fixed frame ff motions which look like chaotic in the ff reveal a variety of stable and unstable structures in the cfs quasi periodic and chaotic motions coexist for all energies and the cfs permit to clearly distinguish between them a new phenomenon of asymptotic symmetries in rotational cfs of some chaotic motions is discovered this is related to a permutation symmetry of the hamiltonian

the entropy profile a function describing statistical dependences in an attempt to find parameters of a time series which are absolutely robust with respect to nonlinear distortion we introduce a function called the entropy profile which measures in some sense the distance between the given process and white noise this concept combines a clear definition and a simple algorithm which apply to arbitrary stationary time series with an informative graphical representation similar to the fourier spectrum for sequences derived from one dimensional maps the entropy profile indicates periodic and almost periodic behavior and the presence of markov partitions

spatial period multiplying instabilities of hexagonal faraday waves abstract a recent faraday wave experiment with two frequency forcing reports two types of superlattice patterns that display periodic spatial structures having two separate scales physica d 123 1998 99 these patterns both arise as secondary states once the primary hexagonal pattern becomes unstable in one of these patterns so called superlattice two the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2 3 from the original scale of the hexagons in contrast the time averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale 3 larger than the original scale and apparently has 60 rotation symmetry we present a symmetry based approach to the analysis of this bifurcation taking as our starting point only the observed instantaneous symmetry of the superlattice two pattern presented in physica d 123 1998 99 and the subharmonic nature of the secondary instability we show a that a pattern with the same instantaneous symmetries as the superlattice two pattern can bifurcate stably from standing hexagons b that the pattern has a spatio temporal symmetry not reported in physica d 123 1998 99 and c that this spatio temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time averaged pattern but not for the apparent 60 rotation symmetry the approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns and does not rely on the primary pattern having small amplitude

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

detecting the symmetry of attractors abstract this paper addresses the issue of how to determine numerically the symmetry of an attractor for dynamical systems the symmetries of attractors in phase space are related to patterns in the time average of the solution our approach to this question proceeds in two parts first we prove a general theorem based on group theoretic and differential topological ideas which states that generically the symmetry of a thickened attractor can be computed from the symmetries of a point in an auxilliary space this theorem proceeds by integrating an equivariant mapping over the thickened attractor once this is done the numerical computation of symmetries reduces to showing that a certain nonnegative number is zero numerically demonstrating that this number is zero can be difficult thus the second part of the algorithm is to consider how this number varies with parameters and noting that sudden jumps towards zero can be associated with increases in symmetry the paper is divided into two parts in the first we prove the general theorem and in the second we illustrate how the numerical techniques work on several examples including discrete dynamical systems with tetrahedral symmetry in r 3 and systems of three coupled cells in high dimensions the integral mentioned previously is difficult to compute for such examples we assume that an ergodic theorem is valid and that symmetries can be computed using a time average we compare both of these methods on the low dimensional examples as well as detect points of symmetry creation for a reaction diffusion equation on an interval this technique can also be used in principle to compute the symmetries of an attractor in an experiment from a time series

symmetry groups of attractors for maps equivariant under the action of a finite group on n the possible symmetries of fixed points are known and correspond to the isotropy subgroups this paper investigates the possible symmetries of arbitrary possibly chaotic attractors and finds that the necessary conditions of melbourne dellnitz golubitsky 15 are sufficient at least for continuous maps

motion of a moving elastic beam carrying a moving mass analysis and experimental verification abstract in this paper vibrational motion of an elastic beam fixed on a moving cart and carrying a moving mass is investigated the equations of motion of the beam mass cart system are derived and the coupled dynamic equations are solved by the unconstrained modal analysis in modal analysis the exact normal mode solutions corresponding to the eigenfrequencies for each position of the moving mass and the ratios of the weight of the beam mass cart system are used proper transformation of time solutions between the normal modes for a position and those for the next position of the moving mass is also considered numerical simulations are carried out to obtain open loop responses of the system in tracking pre designed paths of the moving mass the simulation results show that the model predicts the dynamic behavior of the beam mass cart system well experiments are carried out to show the validity of the proposed analytical method

symmetry breaking bifurcation for coupled chaotic attractors the authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one dimensional mappings they show that the probability density of the asymmetric component satisfies a scaling law the exponent in this scaling law varies continuously with the distance from the bifurcation point and is determined by the spectrum of local lyapunov exponents of the uncoupled map finally they show that the topology of the invariant set is rather unusual though the attractor for supercritical coupling is a line it is surrounded by a strange invariant set which is dense in a two dimensional neighbourhood of the attractor

natural frequencies and ope loop responses of an elastic beam fixed on a moving cart and carrying an intermediate lumped mass abstract in this paper the motion of a bernoulli euler cantilever beam clamped on a moving cart and carrying an intermediate lumped mass is considered the equations of motion of the beam mass cart system are analyzed through unconstrained modal analysis and a unified characteristic equation for calculating the natural frequencies of the system is established the changes of natural frequencies and the corresponding mode shapes with respect to the changes in the ratios between the beam mass the lumped mass and the cart mass and to the concentrated position of the lumped mass are investigated with the frequency equation which can be generally applied to this kind of system the exact and assumed mode solutions including the dynamics of the base cart are obtained and the open loop responses of the system by arbitrarily designed forcing functions are given by numerical simulations the results match well with physical phenomena even in the extreme cases where the mass is attached to the bottom and to the top of the beam

induced maps markov extensions and invariant measures in one dimensional dynamics a way to study ergodic and measure theoretic aspects of interval maps is by means of the markov extension this tool which ties interval maps to the theory of markov chains was introduced by hofbauer and keller more generally known are induced maps i e maps that restricted to an element of an interval partition coincide with an iterate of the original map

symmetry increasing bifurcation of chaotic attractors abstract bifurcation in symmetric is typically associated with spontaneous symmetry breaking that is bifurcation is associated with new solution having less symmetry in this paper we show that symmetry increasing bifurcation in the discrete dynamics of symmetric mappings is possible and is perhaps generic the reason for these bifurcations may be understood as follows the existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry typically in computer experiments what we see is a sequence of symmetry breaking bifurcations leading to the existence of conjugate chaotic attractors as the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry we prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m gon in the last section we discuss period doubling in the presence of symmetry

chaotic behavior resulting in transient and steady state instabilities of pressure loaded shallow spherical shells abstract in this paper the axisymmetric dynamic behavior and snap through buckling of thin elastic shallow spherical shells under harmonic excitation is investigated based on marguerre kinematical assumptions the governing partial differential equations of motion for a pre loaded cap are presented in the form of a compatibility equation and a transverse motion equation the continuous model is reduced to a finite degree of freedom system using the galerkin method and a fourier bessel approach results show that pre loaded shells may exhibit co existing stable equilibrium states and that with the application of sufficiently large dynamic loads the structure escapes from the well corresponding to pre buckling configurations to another this escape load may be much lower than the corresponding quasi static buckling load indeed complex resonances can occur until the system snaps through often signalling the loss of stability as parameters are slowly varied steady state instabilities may occur these can include jumps to resonance subharmonic period doubling bifurcations cascades to chaos etc moreover a sudden pulse of excitation may lead to a transient failure of the system in this paper we examine how spherical caps under harmonic loading may be assessed in an engineering context with a view to design against steady state instabilities as well as the various modes of transient failure steady state and transient stability boundaries are presented in which special attention is devoted to the determination of the critical load conditions from this theoretical analysis dynamic buckling criteria can be properly established which may constitute a consistent and rational basis for design of these shell structures under harmonic loading

modelling instability of abc flow using a mode interaction between steady and hopf bifurcations with rotational symmetries of the cube abstract we consider hydrodynamic stability of symmetric abc flow with a b c 1 regarded as a steady state of the three dimensional navier stokes equation with appropriate forcing numerical investigations have shown that its first instability on increasing the reynolds number is a hopf bifurcation at r 13 044 and that at this bifurcation the second dominant eigenvalue is real motivated by this we study generic interaction of steady state and hopf bifurcations in systems with rotational symmetry of the cube o with an eight dimensional normal form the generic branching and bifurcation behavior of the third order truncated normal form is investigated this is used to analyse a range of the bifurcations for particular values of the coefficients obtained by center manifold reduction from the hydrodynamic system the normal form system shows a sequence of bifurcations and attractors that closely follows the sequence observed for the original hydrodynamic system up to about r 13 91 this includes a torus breakdown to a chaotic attractor and a crisis of attractors that leads to a change in symmetry we discuss numerical simulations of the hydrodynamic system for larger values of r finally we present evidence that the system has robust heteroclinic cycles between fully symmetric abc flow and six steady states with broken symmetry for a range of parameter values near r 15

bifurcation from discrete rotating waves discrete rotating waves are periodic solutions that have discrete spatiotemporal symmetries in addition to their purely spatial symmetries we present a systematic approach to the study of local bifurcation from discrete rotating waves the approach centers around the analysis of diffeomorphisms that are equivariant with respect to distinct group actions in the domain and the range our results are valid for dynamical systems with finite symmetry group and more generally for bifurcations from isolated discrete rotating waves in dynamical systems with compact symmetry group

modal interaction in chaotic vibrations of a shallow double curved shell panel abstract experimental results and analytical results are presented on chaotic vibrations of a shallow double curved shell panel subjected to gravity and periodic excitation modal interactions in the chaotic responses are discussed the shell panel with square boundary is simply supported for deflection in plane displacement at the boundary is elastically constrained in the experiment time histories of the chaotic responses at the spatial multiple positions of the shell panel are measured for the inspection of modal interaction in the analysis the shallow shell panel is assumed to have constant curvatures along to orthogonal directions and geometric initial imperfection the donnell mushtari vlasov type equation is used as governing equation with lateral inertia force assuming deflection with multiple modes of vibration the governing equation is reduced to a set of nonlinear ordinary differential equations by the bubnov galerkin procedure chaotic responses are integrated numerically the chaotic responses which are obtained by the experiment and the analysis are inspected with the fourier spectra the poincare projections the maximum lyapunov exponents and the lyapunov dimension contribution of modes of vibration to the chaotic responses is analyzed by the principal component analysis i e karhunen loeve transformation

parameter uncertainties in models of equivariant dynamical systems abstract we examine the advantages of using equivariant versus nonequivariant phenomenological models for chaotic systems that possess an inversion symmetry numerical experiments used the minimum description length mdl criteria for model selection they indicated that the selection between equivariant versus nonequivariant models should be made on an a priori basis we also examine the relationship between the optimal truncation accuracy of the parameters of the model as predicted by the mdl criteria and the true uncertainty in the parameters of the model numerical experiments indicate that the true uncertainties are poorly approximated by the optimal truncation accuracy they also indicate that the true uncertainties may be useful for determining the presence of an inversion symmetry in reconstructed attractors

dynamic stability criteria of nonlinear elastic damped undamped systems under step loading the nonlinear dynamic stability of simple nonlinearly elastic systems with or without viscous damping under step loading of infinite or finite duration is discussed the analysis refers to discrete structural systems with one or two degrees of freedom which under the action of the same loading applied statically lose their stability by snapping considering the stability of motion in the large in the sense of lagrange conditions for a bounded or an unbounded motion and thereby dynamic buckling criteria are properly established for one degree of freedom systems it is found that regardless of the length of time of loading application as well as regardless of whether or not damping is included the motion ceases to be bounded when the phase point velocity vanishes using this criterion exact or approximate lower bound dynamic buckling loads for damped or undamped systems under step loading of infinite or finite duration can be obtained without solving the strongly nonlinear differential equation of motion

detecting symmetry breaking bifurcations in the system describing the dynamics of coupled arrays of josephson junctions symmetry breaking and symmetry increasing bifurcations in a system of globally coupled josephson junctions which possesses sn symmetry have been investigated by constructing a detective for the group sn the permutation group of n elements in addition with help of this detective several bifurcations are followed numerically in two parameters an algorithm to obtain a detective for any finite subgroup of o n is given

reversing symmetries in dynamical systems dynamical systems may possess in addition to symmetries that leave the equations of motion invariant reversing symmetries that invert the equations of motion such dynamical systems are called weakly reversible some consequences of the existence of reversing symmetries for dynamical systems with discrete time mappings are discussed a reversing symmetry group is introduced and it is shown that every weakly reversible mapping l can be decomposed into two mappings k0 and k1 of the same order 2l limit l to infinity included such that k02 k12 i some applications are discussed briefly

chaotic phenomena triggering the escape from a potential well this paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold the aim is to show how the well known atoms of dissipative dynamics saddle node folds period doubling flips cascades to chaos boundary crises etc assemble to form molecules of overall response hierarchies of cusps incomplete feigenbaum trees etc particular attention is given to the basin of attraction and the loss of engineering integrity that is triggered by a homoclinic tangle the latter being accurately predicted by a melnikov analysis after escape chaotic transients are shown to conform to recent scaling laws analytical constraints on the mapping eigenvalues are used to demonstrate that sequences of flips and folds commonly predicted by harmonic balance analysis are in fact physically inadmissible

linearizability conditions of a time reversible quartic like system in this paper we study the linearizability problem of polynomial like complex differential systems we give a reduction of linearizability problem of such non polynomial systems to the problem of polynomial systems applying this reduction we find some linearizability conditions for a time reversible quartic like complex system and derive from them conditions of isochronous center for the corresponding real system

crystallographic symmetries of stochastic webs in studies of the dynamics of a kicked pendulum webs along which stochastic diffusion takes place have been observed to occur at resonance the author deals with stochastic webs with crystallographic symmetries via an analysis of the reversing symmetries of the equations of motion symmetry properties of stochastic webs are revealed furthermore the threshold for stochastic diffusion is discussed in relation to the reversing symmetries of the equations of motion

bursts in oscillatory systems with broken d 4 symmetry a new mechanism responsible for generating regular and irregular bursts of large dynamic range near onset of an oscillatory instability is identified the bursts are present in systems with nearly square symmetry and are the result of heteroclinic cycles involving infinite amplitude states created when the square symmetry is broken all possible cycles of this type are identified and the resulting bursts described global connections involving finite amplitude states are also present the intricate sequence of bifurcations that result is described in several cases 2000 elsevier science b v all rights reserved

the transition to chaotic attractors with riddled basins abstract recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby the basin is riddled with holes here we consider the dynamics near the transition to this situation as a parameter is varied using a simple analyzable model we obtain the characteristic behaviors near this transition numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered

continuous measures in one dimension families of unimodal maps satisfying 1 t 1 1 1 1 witht 1 1 and t 1 1 2 t x isc2 inx2 and and symmetric inx 3 t0 0 0 t1 0 1 with frac d d lambda t 0 0 are considered the results of guckenheimer 1982 are extended to show that there is a positive measure of for whicht has a finite absolutely continuous invariant measure

a structure theorem in one dimensional dynamics on considere la classe 7b a des applications c f 0 1 0 1 telles que f 0 f 1 0 et f a un unique point critique c 0 1 si le point critique de f 7b a est non plat alors f n a pas d intervalle errant

detection of symmetry of attractors from observations ii an experiment with s 4 symmetry abstract in part i of this paper we adapt the detectives of barany et al 1993 for symmetries of attractors of symmetric dynamical systems to apply to experiments as an illustration of this theory we consider the dynamics of a system of four forced coupled electronic oscillators with full permutation symmetry s4 on varying parameters in the circuit we obtain attractors with many different symmetry types and detect these using the methods described in part i

topological conditions for the existence of absorbing cantor sets this paper deals with strange attractors of s unimodal rnaps f it generalizes earlier results in the sense that very general topological conditions are given that either i guarantee the existence of an absorbing cantor set provided t he critical point of f is sufficiently degenerate or ii prohibit the existence of an absorbing cantor set altogether as a by product we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for f

statistical properties of the noisy on off intermittency the dependence of the escape time on the deviation from the critical point is investigated for the noisy on off intermittency the power law scaling behaviour is destroyed by the noise in a wide region above the critical point in the general case the width of the region decreases logarithmically with the decrease in the noise amplitude below the critical point the noise induced escape time is sensitive to the statistical properties of the chaotic driving signal universal exponential dependence holds only for the gaussian approximation

symmetry restoration in a class of forced oscillators abstract this paper is concerned with a class of symmetric forced oscillators modeled by nonlinear ordinary differential equations the forced duffing oscillator and the forced pendulum belong to this class of oscillators another example is found in an electric power system and is associated with a phenomenon known as ferroresonance solutions for this class of systems can be symmetric or non symmetric sometimes as a physical parameter is varied solutions lose or gain the symmetry at a bifurcation point it will be shown that bifurcations which might otherwise be called symmetry breaking symmetry creation via collision and symmetry creation via explosion are all the result of a collision between conjugate attractors i e attractors that relate to each other by the symmetry and a symmetric limit set the same mechanism seems plausible for at least some bifurcations involving non attracting sets this point is illustrated with some examples

reversing k symmetries in dynamical systems we generalize the concept of reversing symmetries of a dynamical system i e we study dynamical systems that possess symmetry properties only if considered on a proper time scale in particular considering dynamical systems with discrete time the kth iterate of a map may possess more reversing symmetries than the map itself in this way the concepts of reversing symmetries and reversing symmetry groups are generalized to reversing k symmetries and reversing k symmetry groups furthermore a method is studied for finding orbits that are k symmetric with respect to reversing k symmetries firstly an existing method for finding orbits that are symmetric with respect to one reversing symmetry is extended to the case of more than one reversing symmetry and secondly a generalization of this method to the case of reversing k symmetries is introduced some physically relevant examples of dynamical systems possessing reversing k symmetries are discussed briefly

wild attractors of polymodal negative schwarzian maps we study wild attractors of polymodal negative schwarzian interval maps and prove that they are it super persistently recurrent a polymodal version of persistent recurrence we also prove that if a map has an attractor which is a cycle of intervals then at almost every point of this cycle the map has properties similar to the markov property introduced by martens thus the lack of super persistent recurrence at a critical point c can be considered as a mild topological expanding property and this expansion prevents c from being a wild attractor in the previous paper we have shown that it also prevents the map from being c2 stable

symmetry breaking bifurcation with on off intermittency in chaotic dynamical systems when a dynamical system possesses certain symmetry there can be an invariant subspace in the phase space in the invariant subspace there can be a chaotic attractor as a parameter changes through a critical value the chaotic attractor can lose stability with respect to perturbations transverse to the invariant subspace we show that the loss of the transverse stability can lead to a symmetry breaking bifurcation characterized by lack of the system symmetry in the asymptotic attractor an accompanying physical phenomenon is an extreme type of temporally intermittent bursting behavior the mechanism for this type of symmetry breaking bifurcation is elucidated s1063 651x 96 51505 9

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

on symmetric attractors in reversible dynamical systems abstract let o n be a finite group acting orthogonally on r n we say that is a reversing symmetry group of a homeomorphism diffeomorphism or flow f t r n r n t z or t r if has an index two subgroup whose elements commute with ft and for all elements and all t ft x f t x we give necessary group and representation theoretic conditions for subgroups of reversing symmetry groups to occur as symmetry groups of attractors lyapunov stable limit sets these conditions arise due to topological obstructions in dimensions 1 and 2 we present a complete description of possible symmetry groups of asymptotically stable attractors for homeomorphisms and diffeomorphisms these attractors cannot possess reversing symmetries we also have a fairly complete description in the context of subgroups which contain reversing symmetries for all dimensions n we present complete results on the possible symmetry groups of connected asymptotically stable attractors in addition we summarize our results in the context of reversible equivariant flows which are complete in dimensions 1 and 2 as well as for subgroups of o n in any dimension n a survey of the corresponding results in the equivariant context is also given

on the approximation of complicated dynamical behavior we present efficient techniques for the numerical approximation of complicated dynamical behavior in particular we develop numerical methods which allow us to approximate sinai ruelle bowen srb measures as well as almost cyclic behavior of a dynamical system the methods are based on an appropriate discretization of the frobenius perron operator and two essentially different mathematical concepts are used our idea is to combine classical convergence results for finite dimensional approximations of compact operators with results from ergodic theory concerning the approximation of srb measures by invariant measures of stochastically perturbed systems the efficiency of the methods is illustrated by several numerical examples

observing the symmetry of attractors abstract we show how the symmetry of attractors of equivariant dynamical systems can be observed by equivariant projections of the phase space equivariant projections have long been used but they can give misleading results if used improperly and have been considered untrustworthy we find conditions under which an equivariant projection generically shows the correct symmetry of the attractor

fractal control boundaries of driven oscillators and their relevance to safe engineering design when a metastable damped oscillator is driven by strong periodic forcing the catchment basin of constrained motions in the space of the starting conditions x 0 0 develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds the four dimensional basin in the phase control space spanned by x 0 0 f where f is the magnitude and the frequency of the excitation will likewise acquire a fractal boundary and we here explore the engineering significance of the control cross section corresponding for example to x 0 0 0 the fractal boundary in this section is a failure locus for a mechanical or electrical system subjected while resting in its ambient equilibrium state to a sudden pulse of excitation we assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well absolute and transient basins are examined giving control space maps analogous to familiar pictures of the mandelbrot set generalizing from this prototype study it is argued that in engineering design against boat capsize or earthquake damage for example a study of safe basins should augment and perhaps entirely replace conventional analysis of the steady state attracting solutions

pattern selection with o 3 symmetry abstract for each irreducible representation of so 3 and o 3 we determine up to conjugacy all isotropy subgroups and identify in particular the maximal isotropy subgroups each isotropy subgroup corresponds to a possible planform for the spherical benard problem using an equivariant branching lemma of cicogna 1 we prove for each of these representations the existence of solutions corresponding to a number of different planforms thus extending substantially the work of busse 2 3 and sattinger 4 we also give a useful criterion for showing when solutions obtained by the equivariant branching lemma must be unstable