lyapunov exponents for a duffing oscillator with the help of a parallel computer we perform a systematic computation of lyapunov exponents for a duffing oscillator driven externally by a force proportional to cos t in contrast to the familiar situation in discrete time systems where one finds windows of regularity embedded in intervals of chaos we find the continuous time duffing oscillator to contain a quite regular epetition of relatively self similar islands of chaos i e regions characterized by positive exponents embedded in large seas of regularity negative exponents we also investigate the effect of driving the oscillator with a jacobian elliptic function cn t m for m 0 one has cn t 0 cos t the usual trigonometric pumping for m 1 one has cn t 1 sech t a hyperbolic pumping when 0 m 1 the jacobian function is an intermediary double periodic function with periodicity depending on m and with fourier spectrum consisting of a regular train of narrow lines with varying envelope using m to tune the drive appropriately one may displace the islands of chaos in parameter space thus jacobian pumping provides a possible way of cleaning chaos in regions of the parameter space for periodically driven systems

on dirac s conjecture for systems having only first class constraints dirac s conjecture is proved for a general class of systems having only first class constraints in other words it is shown for that kind of problem that all the primary or secondary first class constraints generate equivalence transformations between physical states

lagrangian and hamiltonian many time equations the lagrangian and hamiltonian many time equations are derived for a finite dimensional system with an arbitray number of primary and secondary first class constraints assuming that all the secondary constraints are generators of gauge transformations the general form of the lagrangian gauge algebra is given

chaotic phase similarities and recurrences in a damped driven duffing oscillator we report strong evidence of remarkably close periodic repetitions of the structuring of the parameter space of a damped driven duffing oscillator as the amplitude of the drive increases families of period adding cascades and some intricate networks of periodic oscillations embedded in chaotic phases are also found to recur closely as the driving force grows such surprising regularities suggest that some hitherto unknown renormalization mechanism may be operating in higher codimension controlling the alternation of chaos and order in parameter space of certain flows

lagrangian and hamiltonian constraint structure coefficients we introduce the notion of lagrangian and hamiltonian constraint structure coefficients these coefficients appear in the process of investigating the consistency condition on constraints in both formulations then we show that for a first class hamiltonian constrained system one can find all the lagrangian coefficients in terms of hamiltonian ones

non linear dynamics of gear pair systems with periodic stiffness and backlash the present work investigates dynamics of a gear pair system involving backlash and time dependent mesh stiffness in addition the system is under the action of external excitation caused by torsional moments and gear geometry errors first the equation of motion is established in a strongly non linear form then the emphasis is laid on a specific forcing frequency range corresponding to conditions of simultaneous fundamental parametric resonance and principal external resonance for these conditions several types of periodic steady state response are identified and determined by employing suitable methodologies including techniques applicable to piecewise linear systems and to oscillators with time periodic coefficients moreover these methodologies are complemented by appropriate procedures revealing the stability properties of the located periodic solutions in the second part of the work numerical results are presented these results verify the validity and effectiveness of the new analytical methodology and provide information on the gear pair dynamics first series of typical response diagrams are obtained illustrating the effect of the mesh stiffness variation the damping and the forcing parameters on the gear pair periodic response these response diagrams are accompanied by results obtained with direct integration of the equation of motion in this way it is demonstrated that for some parameter combinations the dynamical system examined can exhibit more complicated and irregular response including crises and intermittent chaos

forced periodic vibration of unsymmetric piecewise linear systems the forced steady response of a single degree of freedom system involving a large non linearity represented by unsymmetric piecewise linear stiffness is determined by a harmonic balance newton raphson method with the application of the fast fourier transformation fft algorithm all possible subharmonic harmonic and superharmonic responses are sought the responses of two systems involving subharmonic and superharmonic dominant motions are calculated by the newly developed method and compared with previously published findings the results obtained by using this method reveal the details of the response of the systems more efficiently than previous methods

the dynamics of a vibromachine with parametric excitation the dynamics of a vibration machine with piecewise linear elastic ties under parametric harmonic excitation is investigated different designs of elastic elements with periodically time varying elasticity are described specific non linear features of parametric oscillations in the system under study are revealed the invariance of parametric vibration regime to possible disturbance of phase co ordinates conditions of limitedness of amplitude of parametric vibrations spectral features of non linear parametric regimes etc by the utilization of these non linear effects a procedure for the design of the main parameters of a parametric vibromachine is proposed

poincare maps of duffing type oscillators and their reduction to circle maps i analytic results bifurcation diagrams and plots of lyapunov exponents in the r omega plane for duffing type oscillators x 2rx v x omega t 0 exhibit a regular pattern of repeating self similar tongues with complex internal structure the authors demonstrate here that this behaviour is easily understood qualitatively and quantitatively from the poincare map of the system in action angle variables this map approaches the one dimensional form phi n 1 a c e rt cos phi n t pi omega provided e rt but not necessarily ce rt r and omega are small the authors derive asymptotic for small r omega formulae for a and c for a special class of potentials v they argue that these special cases contain all the information needed to treat the general case of potentials which obey v or 0 at all times the essential tools of the derivation are the use of action angle variables the adiabatic approximation and the introduction of a non oscillating reference solution of duffing s equation with respect to which the action angle variables have to be determined these allow the explicit construction of the poincare map in powers of e rt to first order the authors obtain the phi map which survives asymptotically to second order they obtain the two dimensional i phi map in the i direction it contracts by a factor e rt upon each iteration

local symmetries in systems with constraints for arbitrary systems with first class constraints the local gauge transformations are constructed in phase and configuration spaces i e a method for obtaining symmetry transformations in the second noether s theorem is given

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

on the invalidity of dirac s conjecture for a system with a singular higher order lagrangian based on the canonical nother s theorem and poincare cartan integral invariant for a system with a singular higher order lagrangian we present a counterexample with any higher order derivatives in which no linearizations of constraints are made to the system showing that dirac s conjecture is invalid

the super separability of the three body inverse square calogero system the geometrical theory of the variable separation for the hamilton jacobi equation is applied to the classical three body inverse square calogero system it is proved that this system is separable in infinitely many inequivalent ways related to five different kinds of separable webs in the euclidean three space and the corresponding systems of independent first integrals in involution are computed

n symplectic algebra of observables in covariant lagrangian field theory n symplectic geometry on the adapted frame bundle l e e of an n m k dimensional fiber bundle e m is used to set up an algebra of observables for covariant lagrangian field theories using the principle bundle l e j1 we lift a lagrangian l j1 r to a lagrangian l l l e r and then use l to define a modified n symplectic potential l on l e the cartan hamilton poincare chp rn valued 1 form if the lifted lagrangian is nonzero then l e d l is an n symplectic manifold to characterize the observables we define a lifted legendre transformation l from l e into le the image ql l l e is a submanifold of le and ql d ql is shown to be an n symplectic manifold we prove the theorem that l l ql and pull back the reduced canonical n symplectic geometry on ql to l e to define the algebras of observables on the n symplectic manifold l e d l to find the reduced n symplectic algebra on ql we set up the equations of n symplectic reduction and apply the general theory to the mo

dissecting shrimps results for some one dimensional physical models this paper describes how certain shrimp like clusters of stability organize themselves in the parameter space of dynamical systems clusters are composed of an infinite affine similar repetition of a basic elementary cell containing two primary noble points a head and a tail defining an axis of approximate symmetry knowledge of the axis and the skewness of the k periodic main cell of a k 2 cluster is enough to define the orientation of the whole cluster in space peculiarly simple directions along which shrimp like clusters align are formed by the locus of doubly degenerate saddle zero multipliers corresponding to the main shrimp head in addition we report a family of models having the boundaries of all isoperiodic domains of stability totally degenerate and describe different aspects of their mathematical arrangement and some of their consequences for example that shrimps are diffeomorphic copies of shrimps

period one motions of a mechanical oscillator with periodically time varying piecewise nonlinear stiffness the dynamic behavior of a piecewise nonlinear mechanical oscillator with parametric and external excitations is investigated the viscously damped single degree of freedom oscillator is subjected to a periodically time varying piecewise nonlinear restoring function defined by a clearance surrounded by continuously nonlinear quadratic and cubic regions typical applications represented by this oscillator are highlighted a multi term harmonic balance formulation is used in conjunction with discrete fourier transforms and a parametric continuation scheme to determine steady state period 1 motions of the system due to both parametric and external excitations the accuracy of the analytical solutions is demonstrated by comparing them to direct numerical integration solutions floquet theory is applied to determine the stability of the steady state harmonic balance solutions at the end detailed parametric studies are presented to quantify the combined influence of clearance quadratic and cubic nonlinearities within reasonable ranges of all other system parameters a comparison between time varying and time invariant systems is also provided to demonstrate the influence of the parametric and external excitations on a piecewise nonlinear system

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

gauge invariance of systems with first class constraints the infinitesimal canonical transformations which map solutions of the total hamiltonian equations of motion into each other are investigated for that the generating function of such transformations should satisfy certain conditions in general is a function which depends on the coordinates the momenta and the lagrange multipliers however it is shown that the requirement of independence of the function on the lagrange multipliers is sufficient for the existence of gauge invariant transformations in the lagrange formalism it is shown that the condition that the poisson brackets between and all the primary first class constraints are a linear combination of the latter ones provides the independence of the function the existence of such a independent function is proven for some systems in particular this is proven for the relevant case of systems having primary and secondary first class constraints the authors suggest the possibility that for some specific systems a independen

a method for studying stability domains in physical models abstract we present a method for investigating the simultaneous movement of all zeros of equations of motions defined by discrete mappings the method is used to show that knowledge of the interplay of all zeros is of fundamental importance for establishing periodicities and relative stability properties of the various possible physical solutions the method is also used i to show that the frontiere set of fatou is defined primarily by zeros of functions leading to an entire invariant limiting function which underlies every dynamical system ii to identify cyclotomic polynomials as components of the limiting function obtained for a parameter value supporting a particular superstable orbit of the quadratic map iii to describe highly symmetric periodic cycles embedded in these components and iv to provide an unified picture about which mathematical objects form basin boundaries of dynamical systems in general the closure of all zeros not belonging to stable orbits

covariant field theory on frame bundles of fibered manifolds we show that covariant field theory for sections of e m lifts in a natural way to the bundle of vertically adapted linear frames l e our analysis is based on the fact that l e is a principal fiber bundle over the bundle of 1 jets j1 on l e the canonical soldering 1 forms play the role of the contact structure of j1 a lifted lagrangian l l e r is used to construct modified soldering 1 forms which we refer to as the cartan hamilton poincare 1 forms these 1 forms on l e pass to the quotient to define the standard cartan hamilton poincare m form on j1 we derive generalized hamilton jacobi and hamilton equations on l e and show that the hamilton jacobi and canonical equations of caratheodory rund and de donder weyl are obtained as special cases

a non linear analysis of the interactions between parametric and external excitations abstract solutions of a non linear equation having a form often encountered in the study of structural dynamics and in which both parametric and external excitation is present have been obtained analytically and numerically all the asymptotic analytical results can be verified by numerical integration for small values of some other solutions obtained numerically cannot be approximated by conventional asymptotic methods both types of solutions can exist simultaneously in these situations the initial conditions determine which solution describes the actual response of the system the solutions that can be predicted only by numerical means exhibit period multiplications i e the response has a period that is an integral multiple of the period of the excitation and chaos in fact in a certain range of one set of initial conditions can lead to period doubling in the response while another set can lead to a response very close to the asymptotic prediction very flexible structures can exhibit non linear characteristics in their response at very small amplitudes of the excitation the present results serve to illustrate some of the extraordinary behavior that such structures can exhibit and suggest that it may be extremely difficult to develop control systems for them

k symplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k symplectic geometry we discuss the relation between the lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied in particular systems with symmetry and poisson sigma models

on the n symplectic structure of faithful irreducible representations each faithful irreducible representation of an n dimensional vector space v1 on an n dimensional vector space v2 is shown to define a unique irreducible n symplectic structure on the product manifold v1 v2 the basic details of the associated poisson algebra are developed for the special case n n2 and 2n dimensional symplectic submanifolds are shown to exist

dynamics of a piecewise non linear system subject to dual harmonic excitation using parametric continuation abstract the dynamic behavior of piecewise linear and or non linear vibratory systems subject to a single harmonic excitation has been recently analyzed by using a one or multi term harmonic balance or galerkin s method piecewise linear techniques analog simulation and direct numerical integration digital simulation instead this paper proposes to utilize the technique of parametriccontinuationto study the steady state response and global dynamics of a two degree of freedom piecewise non linear system with backlash or multi valued springs and impact damping the physical system is under the influence of a mean load and is subject to a dual harmonic external excitation where the second frequency is either twice or three times the fundamental excitation frequency the effects of various parameters such as impact damping mean and alternating loads the piecewise stiffness ratios and the relative phase between the excitation harmonics on the system response are thoroughly examined system or excitation parameter regimes that exhibit sub harmonic super harmonic quasi periodic and chaotic solutions are obtained efficiently and systematically by using the proposed scheme this investigation also clarifies the differences between viscous and non linear impact damping models which have been proposed earlier in the literature for studying clearance non linearities limited experimental data validate our modelling strategy finally selected predictions from this technique match very well with a multi term harmonic balance method and with direct numerical integration wherever applicable

noether s theorem and gauge transformations application to the bosonic string and cp n 12 model new results on the theory of constrained systems are applied to characterize the generators of noether s symmetry transformations as a byproduct an algorithm to construct gauge transformations in hamiltonian formalism is derived this is illustrated with two relevant examples

gunther s formalism k symplectic formalism in classical field theory skinner rusk approach and the evolution operator the first aim of this paper is to extend the skinner rusk formalism on classical mechanics for first order field theories the second is to generalize the definition and properties of the evolution k operator on classical mechanics for first order field theories using in both cases gunther s formalism k symplectic formalism

new resampling method to assess the accuracy of the maximal lyapunov exponent estimation abstract in this paper we propose a novel method to assess standard errors and confidence intervals for the maximal lyapunov exponent estimated on time continuous chaotic systems the method is based on resampling the original series by means of spline interpolation in the time domain in such a way new time series of increased size are obtained and the sample distribution of the estimators is constructed the method is explained and tested on the basis of computer simulations both for clean and noisy series we give evidence that the distribution of the maximal lyapunov exponent calculated by this method fairly agrees with the one obtained by true series with different initial conditions an empirical criterion for the choice of the parameters of the resampling is also suggested

periodically forced duffing s equation abstract sufficient conditions are established for an equation of the type x x x 3 p t to have periodic solutions where p t is periodic the results are applied to analyze forced vibrations of a mass supported by a non linear spring to verify sufficiency of the conditions a computer program has been prepared and used to provide results for some examples the program searches for periodic solutions of the generalized problem of non linear non autonomous second order differential equations and draws x t and phase trajectory diagrams for the periodic solutions obtained these numerical results are in agreement with theoretical expectations

gromov s theorem in n symplectic geometry on lrn if m is an n dimensional manifold then the associated bundle of linear frames lm of m supports the canonically defined rn valued soldering 1 form the pair lm d is an n symplectic manifold where d is the n symplectic 2 form we adapt the proofs of de gosson and mcduff and salamon of gromov s non squeezing theorem on r2n to give a proof of gromov s theorem for affine n symplectomorphisms on lrn

subharmonic resonances of a mechanical oscillator with periodically time varying piecewise nonlinear stiffness abstract the subharmonic period 1 motions of a piecewise nonlinear pn mechanical oscillator having parametric and external excitations are investigated the system is formed by a viscously damped single degree of freedom oscillator subjected to a periodically time varying pn stiffness defined by a clearance surrounded by continuous forms of nonlinearity a multiterm harmonic balance formulation in conjunction with discrete fourier transforms is used to determine steady state period motions of the system near the parametric instability regions the accuracy of analytical solutions is verified through a comparison with direct numerical integration results a parametric study is also presented to demonstrate the combined influence of a clearance and of cubic nonlinearities on period motions within typical ranges of system and excitation parameters

the response of multidegree of freedom systems with quadratic non linearities to a harmonic parametric resonance abstract an analysis is presented of the response of multidegree of freedom systems with quadratic non linearities to a harmonic parametric excitation in the presence of an internal resonance of the combination type 3 2 1 where the n are the linear natural frequencies of the systems in the case of a fundamental resonance of the third mode i e 3 where is the frequency of the excitation one can identify two critical values 1 and 2 where 2 1 of the amplitude f of the excitation the value f 2 corresponds to the transition from stable to unstable solutions when f 2 the motion grows exponentially with time according to the linear theory but the non linearity limits the motion to a finite amplitude steady state the amplitude of the third mode which is directly excited is independent of f whereas the amplitudes of the first and second modes which are indirectly excited through the internal resonance are functions of f when 1 f 2 the motion decays or achieves a finite amplitude steady state depending on the initial conditions according to the non linear theory whereas it decays to zero according to the linear theory this is an example of subcritical instability in the case of a fundamental resonance of either the first or second mode the trivial response is the only possible steady state when f 2 the motion decays to zero according to both linear and non linear theories when f 2 the motion grows exponentially with time according to the linear theory but it is aperiodic according to the non linear theory experiments are being planned to check these theoretical results

full and partial gauge fixing gauge fixing may be done in different ways we show that using the chain structure to describe a constrained system enables us to use either a full gauge in which all gauged degrees of freedom are determined or a partial gauge in which some first class constraints remain as subsidiary conditions to be imposed on the solutions of the equations of motion we also show that the number of constants of motion depends on the level in a constraint chain in which the gauge fixing condition is imposed the relativistic point particle electromagnetism and the polyakov string are discussed as examples and full or partial gauges are distinguished

poincare maps of duffing type oscillators and their reduction to circle maps ii methods and numerical results bifurcation diagrams and plots of lyapunov exponents in the r plane for duffing type oscillators exhibit a regular pattern of repeating self similar tongues with complex internal structure we demonstrate here how this behaviour is easily understood qualitatively and quantitatively from a poincare map of the system in action angle variables in the limit of large driving force or equivalently small driving frequency this map approaches the one dimensional form as derived in paper i this second paper describes our approach to calculating the various constants and functions introduced in paper i it gives numerical applications of the theory and tests its range of validity by comparison with results from the numerical integration of duffing type equations finally we show how to extend the range in the parameter space where the map is applicable

classification of gauge groups in terms of algebraic structure of first class constraints properties of gauge transformations for singular lagrangians are investigated to classify types of gauge groups a general method of the classification is proposed based on properties of structure functions of the poisson brackets or the commutators of first class constraints a remarkable result is that the algebraic structure of the gauge group is essentially determined by the first class constraints of the final step of constraint series which are required successively from the stationarity conditions of the constraints owing to this consequence the classification of gauge groups is made simple and transparent the structure and property of the gauge group can be characterized in terms of the algebraic structure functions among the final step constraints and the number of the steps of the constraints series the formulation proposed will give a clue to find new types of gauge groups

local brst cohomology in the antifield formalism i general theorems we establish general theorems on the cohomologyh s d of the brst differential modulo the spacetime exterior derivative acting in the algebra of localp forms depending on the fields and the antifields sources for the brst variations it is shown thath k s d is isomorphic toh k d in negative ghost degree k k 0 where is the koszul tate differential associated with the stationary surface the cohomology grouph1 d in form degreen is proved to be isomorphic to the space of constants of the motion thereby providing a cohomological reformulation of noether s theorem more generally the grouph k d in form degreen is isomorphic to the space ofn k forms that are closed when the equations of motion hold the groupsh k d k 2 are shown to vanish for standard irreducible gauge theories the grouph2 d is then calculated explicitly for electromagnetism yang mills models and einstein gravity the invariance of the groupsh k s d under the introduction of non minimal variables and of auxiliary fields is also demonstrated in a companion paper the general formalism is applied to the calculation ofh k s d in yang mills theory which is carried out in detail for an arbitrary compact gauge group

general quadratic gauge theory constraint structure symmetries and physical functions how can we relate the constraint structure and constraint dynamics of the general gauge theory in the hamiltonian formulation to specific features of the theory in the lagrangian formulation especially relate the constraint structure to the gauge transformation structure of the lagrangian action how can we construct the general expression for the gauge charge if the constraint structure in the hamiltonian formulation is known whether we can identify the physical functions defined as commuting with first class constraints in the hamiltonian formulation and the physical functions defined as gauge invariant functions in the lagrangian formulation the aim of the present paper is to consider the general quadratic gauge theory and to answer the above questions for such a theory in terms of strict assertions to fulfil such a programme we demonstrate the existence of the so called superspecial phase space variables in terms of which the quadratic hamiltonian action takes a simple canonical form on the basis of such a representation we analyse a functional arbitrariness in the solutions of the equations of motion of the quadratic gauge theory and derive the general structure of symmetries by analysing a symmetry equation we then use these results to identify the two definitions of physical functions and thus prove the dirac conjecture

dynamics of unsymmetric piecewise linear non linear systems using finite elements in time the dynamic response and stability of a single degree of freedom system with unsymmetric piecewise linear non linear stiffness are analyzed using the finite element method in the time domain based on a hamilton s weak principle this method provides a simple and efficient approach for predicting all possible fundamental and sub periodic responses the stability of the steady state response is determined by using floquet s theory without any special effort for calculating transition matrices this method is applied to a number of examples demonstrating its effectiveness even for a strongly non linear problem involving both clearance and continuous stiffness non linearities close agreement is found between available published findings and the predictions of the finite element in time approach which appears to be an efficient and reliable alternative technique for non linear dynamic response and stability analysis of periodic systems

the polysymplectic hamiltonian formalism in field theory and calculus of variations i the local case on presente un formalisme hamiltonien geometrique invariant pour des problemes variationnels a integrale multiple et des theories de champ le formalisme est base sur la notion de forme polysymplectique qui est une generalisation a valeur vectorielle des formes symplectiques on etudie des equations hamiltoniennes des transformations canoniques des systemes de lagrange des symetries des applications en pulsion de theorie du champ et une classification des systemes de theorie du champ g homogenes sur une generalisation des orbites coadjointes

symmetry in a constrained hamiltonian system with singular higher order lagrangian for a canonical formalism with a higher order derivative the corresponding generalized first noether theorem gfnt for a constrained hamiltonian system and the generalized noether identities gni for a system with a non invariant action integral are derived which may be useful to analyse the dirac constraint for such a system using the gfnt another example is given in which dirac s conjecture fails using the gni the strong and weak conservation laws are deduced and it is pointed out that for certain variant systems there is also a dirac constraint suppose that there are only first class constraints fcc in a system then an algorithm for the construction of a gauge generator is developed once the hamiltonian and the fcc of the system with a higher order lagrangian are given

harmonic balance and continuation techniques in the dynamic analysis of duffing s equation duffing s equation with a negative linear stiffness is investigated a harmonic balance technique coupled with a continuation scheme is utilized to trace the system response to changes in the magnitude of the applied harmonic forcing with the aid of floquet theory the stability of the resulting solutions is ascertained the use of the analysis technique described herein allows for the identification of previously undetected strange attractors that coexist with the stable period one solutions these results explain the discrepancies between the observed and theoretically predicted levels of applied loading required for chaotic motion

on the noether identities for a class of systems with singular lagrangians a straightforward method for obtaining the generators of lagrangian gauge transformations within the lagrangian formalism is presented this procedure can be carried out completely without the need of developing the dirac hamilton formalism singular lagrangians which may generate lagrangian constraints are considered it is assumed that the consistency conditions on the lagrangian constraints do not generate new independent equations for the accelerations from the structure of these consistency conditions the noether identities are explicitly constructed and the lagrangian generators of gauge transformations are obtained it is shown that in the generic case of systems having chains of lagrangian constraints up to a level k k 2 the variations q under which the action is invariant depend on q dq dt and higher time derivatives up to dkq dtk the relation between this lagrangian approach and the standard dirac hamiltonian approach is discussed

on the attracting set for duffing s equation ii a geometrical model for moderate force and damping abstract after a brief review of some earlier work on duffing s equation in the small force and damping regions we use the results of numerical integrations to construct a geometrically defined poincare map which captures the qualitative features of the attracting set of larger force and damping levels this map has a small constant jacobian determinant and can be regarded as a perturbation of a non invertible one dimensional map we give a partial analysis of the map and pose some important open questions regarding perturbations of one dimensional maps and the creation of strange attractors during bifurcation to horseshoes

schouten nijenhuis brackets the poisson and graded poisson schouten nijenhuis algebras of symmetric and antisymmetric contravariant tensor fields respectively on an n dimensional manifold m are shown to be n symplectic this is accomplished by showing that both brackets may be defined in a unified way using the n symplectic structure on the bundle of linear frames lm of m new results in n symplectic geometry are presented and then used to give globally defined representations of the hamiltonian operators defined by the schouten nijenhuis brackets

steady state forced response of a mechanical oscillator with combined parametric excitation and clearance type non linearity a mechanical system exhibiting combined parametric excitation and clearance type non linearity is examined analytically and experimentally in an effort to explain complex behavior that is commonly observed in the steady state forced response of rotating machines the specific case of a preloaded mechanical oscillator having a periodically time varying stiffness function and subject to a symmetric backlash condition is considered a generalized solution methodology is proposed based on the harmonic balance method the resulting non linear algebraic equations are solved by using a direct newton raphson technique in which a closed form jacobian matrix is computed using frequency domain methods analytical solutions are validated by comparison with numerical integration results and experimental measurements obtained from a gear dynamics test rig

local symmetries and constraints the general relationship between local symmetries occurring in a lagrangian formulation of a field theory and the corresponding constraints present in a phase space formulation are studied first a prescription applicable to an arbitrary lagrangian field theory for the construction of phase space from the manifold of field configurations on space time is given next a general definition of the notion of local symmetries on the manifold of field configurations is given that encompasses as special cases the usual gauge transformations of yang mills theory and general relativity local symmetries on phase space are then defined via projection from field configuration space it is proved that associated to each local symmetry which suitably projects to phase space is a corresponding equivalence class of constraint functions on phase space moreover the constraints thereby obtained are always first class and the poisson bracket algebra of the constraint functions is isomorphic to the lie bracket algebra of the local symmetries on the constraint submanifold of phase space the differences that occur in the structure of constraints in yang mills theory and general relativity are fully accounted for by the manner in which the local symmetries project to phase space in yang mills theory all the field independent local symmetries project to all of phase space whereas in general relativity the nonspatial diffeomorphisms do not project to all of phase space and the ones that suitably project to the constraint submanifold are field dependent as by products of the present work definitions are given of the symplectic potential current density and the symplectic current density in the context of an arbitrary lagrangian field theory and the noether current density associated with an arbitrary local symmetry a number of properties of these currents are established and some relationships between them are obtained

equivalence between the lagrangian and hamiltonian formalism for constrained systems the equivalence between the lagrangian and hamiltonian formalism is studied for constraint systems a procedure to construct the lagrangian constraints from the hamiltonian constraints is given those hamiltonian constraints that are first class with respect to the hamiltonian constraints produce lagrangian constraints that are fl projectable

resonance and symmetry breaking for the pendulum abstract periodic solutions of the differential equation for periodic forcing of a lightly damped pendulum are obtained on the alternative hypotheses that i the contributions of second and higher harmonics to the average lagrangian are negligible or ii the solution is close to that for free oscillations the resonance curves amplitude or root mean energy vs driving frequency and stability boundaries for symmetric swinging oscillations and their asymmetric descendents following symmetry breaking are determined for 1 and o where is the ratio of actual to critical damping and is the ratio of the maximum external moment to the maximum gravitational moment resonance as defined by synchronism between the external moment and the damping moment is found to be impossible and the conventional resonance curve separates into two branches if 3 28 o 3 which condition is necessary for normal symmetry breaking a numerical fourier series determination of the resonance curve and bifurcation points for 1 8 and 1 2 is presented in an appendix by p j bryant

explicit form of the gauge transformation generator in terms of constraints we study the conditions that a gauge transformation generator should satisfy then considering a special class of constrained systems we have derived the explicit form of the gauge generator in terms of first class constraints of the system we have shown that the generator derived can be demonstrated by two different known methods and satisfies the conditions required in both methods

analysis of periodically forced nonlinear hill s oscillator with application to a geared system although nonlinear systems subject to combined parametric and external excitations have been examined in some depth very few of these investigations have addressed the influence of a mean time invariant load on the system response in particular the importance of the mean load effect has been highlighted in the study of whirling asymmetric shafts where significant changes in system response and stability occur under the influence of gravity the present paper intends to characterize the specific effect of mean load on the dynamic behavior of a hill s oscillator with a clearance type nonlinearity while also being subjected to a periodic base displacement excitation the parametric continuation technique and method of harmonic balance would be used for this purpose issues discussed would include the coupling between the mean load and the dynamic response amplitude interaction between the parametric excitation effect and the clearance nonlinearity and comparison between time invariant and time varying systems further a geared system has been analyzed as a practical application of the nonlinear hill s equation examined earlier studies on gear dynamics have been reexamined to provide improved correlation with prior experimental data new interpretations for some of the nonlinear phenomena in the experimental data has also been provided based on our study

periodic response and stability of oscillators with symmetric trilinear restoring force abstract an exact solution is presented for the periodic symmetric double crossing response of a single degree of freedom strongly non linear oscillator subjected to harmonic excitation the non linearity of the system appears in the restoring force which is a symmetric trilinear function of the system displacement the stability analysis of the periodic solutions is also developed and presented both period one and subharmonic motions are analyzed and investigated for some combinations of the system parameters no stable periodic motion is found to exist and the system exhibits chaotic behavior

recursive construction of the generator for lagrangian gauge symmetries we obtain for a subclass of structure functions characterizing a first class hamiltonian system recursive relations from which the general form of the local symmetry transformations can be constructed in terms of the independent gauge parameters we apply this to a non trivial hamiltonian system involving two primary constraints as well as two secondary constraints of the nambu goto type we also illustrate for a pure chern simons theory how this formalism can be extended to a system with first and second class constraints

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

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