the wssus pulse design problem in multicarrier transmission optimal link adaption to the scattering function of wide sense stationary uncorrelated scattering wssus mobile communication channels is still an unsolved problem despite its importance for next generation system design in multicarrier transmission such link adaption is performed by pulse shaping i e by properly adjusting the transmit and receive filters for example pulse shaped offset quadrature amplitude modulation oqam systems have recently been shown to have superior performance over standard cyclic prefix orthogonal frequency division multiplexing ofdm while operating at higher spectral efficiency in this paper we establish a general mathematical framework for joint transmitter and receiver pulse shape optimization for so called weyl heisenberg or gabor signaling with respect to the scattering function of the wssus channel in our framework the pulse shape optimization problem is translated to an optimization problem over trace class operators which in turn is related to fidelity optimization in quantum information processing by convexity relaxation the problem is shown to be equivalent to a convex constraint quasi convex maximization problem thereby revealing the nonconvex nature of the overall wssus pulse design problem we present several iterative algorithms for optimization providing applicable results even for large scale problem constellations we show that with transmitter side knowledge of the channel statistics a gain of 3 6 db in signal to interference and noise ratio sinr can be expected

a theorem on divergence in the general sense for continued fractions if the odd and even parts of a continued fraction converge to different values the continued fraction may or may not converge in the general sense we prove a theorem which settles the question of general convergence for a wide class of such continued fractions we apply this theorem to two general classes of q continued fraction to show that if g q is one of these continued fractions and q 1 then either g q converges or does not converge in the general sense we also show that if the odd and even parts of the continued fraction kn 1 an 1 converge to different values then limn an

truncation error bounds for the composition of limit periodic linear fractional transformations abstract baltus c truncation error bounds for the composition of limit periodic linear fractional transformations journal of computational and applied mathematics 46 1993 395 404 suppose i tn w is a sequence of linear fractional transformations tn w t w aw b w d ad b or a b 0 d where fixed points un u vn v with d v d u

normal forms for periodic systems the theory of normal forms for autonomous differential equations based on the lie bracket has proved to be an important technique in determining the behaviour of solutions of nonlinear differential equations near a stationary solution it has been widely used in the bifurcation theory for autonomous ordinary differential equations see e g 1 3 6 9 11 i in its simplest form the theory applies to an equation of the form

nonlinear oscillations and chaotic dynamics of an antisymmetric cross ply laminated composite rectangular thin plate under parametric excitation in this paper nonlinear oscillations and chaotic dynamics of a simply supported antisymmetric cross ply laminated composite rectangular thin plate under parametric excitation is investigated for the first time the governing equations of motion for the antisymmetric cross ply laminated composite plate are derived by using von karman type plate equation the geometric nonlinearity and nonlinear damping are included in the governing equations of motion a two dof parametrically excited nonlinear system including the quadratic and cubic nonlinear terms is obtained by using the galerkin method based on the fourier expansion and the temporal rescaling an asymptotic perturbation method is utilized to obtain four dimensional nonlinear averaged equations on the amplitude and the phase of nonlinear oscillations of the antisymmetric cross ply laminated composite plate based on the averaged equations the steady state nonlinear responses and their stabilities are determined by using numerical approach the relations between the steady state nonlinear responses and the amplitude and frequency of parametric excitation are obtained under the certain conditions the antisymmetric cross ply laminated composite plate may have two steady state nonzero solutions in which the jumping phenomenon occurs numerical simulation is used to discover the periodic and chaotic motions in the antisymmetric cross ply laminated composite rectangular thin plate it is observed from the numerical results that the multipulse orbits exist in the antisymmetric cross ply laminated composite rectangular thin plate

applications of the heine and bauer muir transformations to rogers ramanujan type continued fractions abstract in this paper we show that various continued fractions for the quotient of general ramanujan functions g a q b q g a b may be derived from each other via bauer muir transformations the separate convergence of numerators and denominators play a key part in showing that the continued fractions and their bauer muir transformations converge to the same limit we also show that these continued fractions may be derived from either heine s continued fraction for a ratio of 1 2 functions or other similar continued fraction expansions of ratios of 1 2 functions further by employing essentially the same methods a new continued fraction for g a q b q g a b is derived finally we derive a number of new versions of some beautiful continued fraction expansions of ramanujan for certain combinations of infinite products with the following being an example a b q a b q a b q a b q a b 1 a b 1 a 2 1 b 2 q 1 a b q 2 a b q 2 b a q 2 q 1 a b q 4 1 a 2 q 2 1 b 2 q 2 q 3 1 a b q 6 a b q 4 b a q 4 q 3 1 a b q 8

singularities in vibro impact dynamics abstract the non differentiable nature of vibro impact dynamics can lead to breakdown of the global stable manifold theorem applicable to smooth dynamical systems in one dimensional periodically excited system the breakdown leads to the shredding of the stable manifolds and this process is analyzed in detail bodily translation of filaments of stable manifolds can lead to homoclinic intersections and this topic is discussed in the context of a new bifurcation due to re entrance shredding leads to a strong similarity between the geometries of the singular subspaces and the geometries of homoclinic tangles and strange attracting sets

non similar normal oscillations in a strongly non linear discrete system abstract the free oscillations of a strongly non linear discrete oscillator are examined by computing its non similar non linear normal modes these are motions represented by curves in the configuration space of the system and they are not encountered in classical linear vibration theory or in existing non linear perturbation techniques the mikhlin manevich asymptotic methodology is used for solving the singular functional equation describing the non similar modes and approximate analytical expressions are derived for an oscillator with weak coupling stiffness and mistuning both localized and non localized modes are detected occurring in small neighborhoods of degenerate and global similar modes of the tuned system when strong coupling is considered only non localized modes were found to exist an interesting result of this work is the detection of mode localization in the tuned periodic system a result with no counterpart in existing theories on linear mode localization as a check of the analytical results numerical integrations of the equations of motion were carried out and the existence of the theoretically predicted non similar modes was verified

a recursive approach to compute normal forms normal forms are instrumental in the analysis of dynamical systems described by ordinary differential equations particularly when singularities close to a bifurcation are to be characterized however the computation of a normal form up to an arbitrary order is numerically hard this paper focuses on the computer programming of some recursive formulas developed earlier to compute higher order normal forms a computer program to reduce the system to its normal form on a center manifold is developed using the maple symbolic language however it should be stressed that the program relies essentially on recursive numerical computations while symbolic calculations are used only for minor tasks some strategies are proposed to save computation time examples are presented to illustrate the application of the program to obtain high order normalization or to handle systems with large dimension

on the failure of static stability analyses of nonconservative systems in regions of divergence instability abstract the stability of perfect bifurcational discrete dissipative systems under follower loads in regions of existence non existence of adjacent equilibria is thoroughly re examined in the light of recent progress in nonlinear dynamics a general theory for such nongradient systems described by autonomous ordinary differential equations is developed conditions for the existence of adjacent equilibria the stability of precritical critical and postcritical states as well as for different types of local bifurcations are established focusing attention on the interaction of geometric nonlinearities and vanishing damping new findings contradicting widely accepted results of the classical linear analysis are discovered in a small region of adjacent equilibria near a compound branching point which is explicitly determined an interaction of two consecutive postbuckling modes occurs related to the following phenomena in case of vanishing damping loss of stability may occur via a hopf dynamic bifurcation prior to static divergence buckling moreover the critical states of divergence instability may be associated with a double zero jacobian eigenvalue satisfying also the conditions of a hopf local bifurcation besides local dynamic bifurcations global bifurcations are also found an example is used to illustrate the qualitative findings

printed antennas analysis by a gabor frame based method of moments gabor frame based discretization is proposed for the first time as a fully rigorous and flexible tool in the context of antenna analysis a rigorous discretization procedure based on frame theory is presented and applied to integral equations solution through a method of moments mom in this approach the unknown field or current is expanded in a set of spatially and spectrally translated elementary functions the use of a gaussian window function as basis element allows for the representation of radiated fields as a superposition of shifted and rotated gaussian beams by exploiting the well understood propagation and transformation features of gaussian beams the fields can be evaluated by summations of analytic terms at any observation point this method seems well suited to model antennas embedded in complex systems including arbitrary interfaces numerical results are presented for slot antennas at the interface between two dielectric half spaces and compared to a standard mom to validate the approach and illustrate its attractive characteristics

on codimension three bifurcations in the motion of articulated tubes conveying a fluid abstract we consider the planar motion of two articulated tubes conveying a fluid near a double degeneracy in the linearized dynamical equations in particular the interaction between hopf and pitchfork symmetric saddle node bifurcations and the effects of asymmetries on that interaction are studied in detail the fourth order system of nonlinear equations is reduced to a third order system using center manifold theory the method of averaging provides the normal form which can be considered as a planar vector field with three important parameters one each for the hopf bifurcation the pitchfork bifurcation and the asymmetry this second order system is shown to possess twenty three open sets in the parameter space each with qualitatively different phase portraits some of these portraits are presented along with bifurcation analysis and the physical implications of the results are discussed

global and chaotic dynamics for a parametrically excited thin plate the global bifurcations and chaotic dynamics of a parametrically excited simply supported rectangular thin plate are analyzed the formulas of the thin plate are derived by von karman type equation and galerkin s approach the method of multiple scales is used to obtain the averaged equations based on the averaged equations theory of normal form is used to give the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues by maple program on the basis of the normal form global bifurcation analysis of the parametrically excited rectangular thin plate is given by a global perturbation method developed by kovacic and wiggins the chaotic motion of thin plate is found by numerical simulation

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

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

truncation error bounds for modified continued fractions with applications to special functions much recent work has been done to investigate convergence of modified continued fractions mcf s following the proof by thron and waadeland 35 in 1980 that a limit periodic mcfk a n 1 x 1 with mathop lim limits n to infty a n a in mathbb c infty tfrac 1 4 andnth approximant g n s n x 1 frac a 1 1 frac a 2 1 cdots frac a n 1 1 frac a n 1 x 1 converges more rapidly to its limitf than the ordinary reference continued fractionk a n 1 withnth approximantf n s n 0 herex 1 denotes the smaller in modulus of the two fixed points oft w a 1 w the present paper gives truncation error bounds for bothf n andg n that exploit the limit periodic property lima n a certain a posteriori bounds given forg n are shown to be best possible relative to the given limited information available this is the first instance in which truncation error bounds for this problem have been shown to be best possible also included in this paper are results on speed of convergence a practical method for constructing the bounds and applications to a number of special functions the given numerical examples indicate that the error bounds are indeed sharp for the function arctanz we give graphical contour maps of the number of significant digits in the approximationsf n z g n z andp n z thenth partial sum of the maclaurin series forz in a key region of the complex plane these maps help us compare the various approximations with each other and add to our understanding of their convergence behavior

discrete zak transforms polyphase transforms and applications we consider three different versions of the zak 1967 transform zt for discrete time signals namely the discrete time zt the polyphase transform and a cyclic discrete zt in particular we show that the extension of the discrete time zt to the complex z plane results in the polyphase transform an important and well known concept in multirate signal processing and filter bank theory we discuss fundamental properties relations and transform pairs of the three discrete zt versions and we summarize applications of these transforms in particular the discrete time zt and the cyclic discrete zt are important for discrete time gabor 1946 expansion weyl heisenberg frame theory since they diagonalize the weyl heisenberg frame operator for critical sampling and integer oversampling the polyphase representation plays a fundamental role in the theory of filter banks especially dft filter banks simulation results are presented to demonstrate the application of the discrete zt to the efficient calculation of dual gabor windows tight gabor windows and frame bounds

normal modes for non linear vibratory systems a methodology is presented which extends to non linear systems the concept of normal modes of motion which is well developed for linear systems the method is constructive for weakly non linear systems and provides the physical nature of the normal modes along with the non linear differential equations which govern their dynamics it also provides the non linear co ordinate transformation which relates the original system co ordinates to the modal co ordinates using this transformation we demonstrate how an approximate non linear version of superposition can be employed to reconstruct the overall motion from the individual non linear modal dynamics the results presented herein for non linear systems reduce to modal analysis for the linearized system when non linearities are neglected even though the approach is entirely different from the traditional one the tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique in this paper the basic ideas are outlined a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions

a high resolution technique for ultrasound harmonic imaging using sparse representations in gabor frames over the last few decades there were dramatic improvements in ultrasound imaging quality with the utilization of harmonic frequencies induced by both tissue and echo contrast agents the advantages of harmonic imaging cause rapid penetration of this modality to diverse clinical uses among which myocardial perfusion determination seems to be the most important application in order to effectively employ the information comprised in the higher harmonics of the received signals this information should be properly extracted a commonly used method of harmonics separation is linear filtering one of its main shortcomings is the inverse relationship between the detectability of the contrast agent and the axial resolution in this paper a novel nonlinear technique is proposed for separating the harmonic components contained in the received radio frequency images it is demonstrated that the harmonic separation can be efficiently performed by means of convex optimization it performs the separation without affecting the image resolution the procedure is based on the concepts of sparse signal representation in overcomplete signal bases a special type of the sparse signal representation that is especially suitable for the problem at hand is explicitly described the ability of the novel technique to acquire un masked second or higher harmonic images is demonstrated in series of computer and phantom experiments

analysis of critical and post critical behaviour of non linear dynamical systems by the normal form method part ii divergence and flutter abstract the normal form method nfm presented in part i 1 is applied to the analysis of systems near two basic bifurcation conditions divergence and flutter the reduced systems and the normalizing transformation obtained by means of a computer routine implementing the normalization formulae of part i are utilized to determine a the stability of the bifurcating singular point b the local equilibrium paths associated with divergence bifurcation and c the amplitude and frequency or more generally the waveform of post flutter oscillations

orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine gordon equation abstract in this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two degree of freedom hamiltonian systems our methods are fundamentally different than other global perturbation methods e g standard melnikov theory in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation our methods combine the higher dimensional melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds we apply our methods to a modified model of the forced and damped sine gordon equation developed by bishop et al we give explicit conditions in terms of the system parameters for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle focus type chaotic dynamics follow from a theorem of silnikov this provides a mechanism for chaotic dynamics geometrically similar to that observed by bishop et al namely a random jumping between two spatially dependent states with an intermediate passage through a spatially independent state however in order for this type of silnikov dynamics to exist we require a different and unphysical type of damping compared to that used by bishop et al

normal modes of vibration for non linear continuous systems a definition and a constructive methodology for normal nodes of motion are developed for a class of vibratory systems the dynamics of which are governed by non linear partial differential equations the definition for normal modes is given in terms of the dynamics on two dimensional invariant manifolds in the system phase space these manifolds are a continuation of the planes which represent the well known normal modes of the linearized system a local asymptotic approximation of the geometric structure of these manifolds can be obtained by using an approach which follows that used for generating center manifolds the procedure also provides the non linear ordinary differential equations which govern individual modal dynamics and a physical description of the system configuration when it is undergoing a modal motion in this paper the general theory is described for the application of vibrations of continuous media but it can easily be extended to other situations in order to demonstrate the power of the approach and its unique procedural aspects three examples involving beam vibrations are worked out in detail the examples are conservative simply supported beams the first demonstrates the methodology for a linear beam model the second is a beam on a non linear elastic foundation and the third example is a beam with non linear torsional springs attached at each end in these examples the simplicity of the mode shapes of the linearized model yields relatively simple calculations that do not obscure the important features of the procedure

a perturbation analysis of interactive static and dynamic bifurcations the instability behavior of a nonlinear autonomous system in the vicinity of a coincident critical point which leads to interactions between static and dynamic bifurcations is studied the critical point considered is characterized by a simple zero and a pair of pure imaginary eigenvalues of the jacobian and the system contains two independent parameters the static and dynamic bifurcations and quasiperiodic motions resulting from the interaction of the bifurcation modes and the associated invariant tori are analyzed by a novel unification technique that is based on an intrinsic perturbation procedure divergence boundary dynamic bifurcation boundary secondary bifurcations and invariant tori are determined explicitly two illustrative examples concerning control systems are presented

center manifolds normal forms and bifurcations of vector fields with application to coupling between periodic and steady motions abstract we study the instabilities known to aeronautical engineers as flutter and divergence mathematically these states correspond to bifurcations to limit cycles and multiple equilibrium points in a differential equation making use of the center manifold and normal form theorems we concentrate on the situation in which flutter and divergence become coupled and show that there are essentially two ways in which this is likely to occur in the first case the system can be reduced to an essential model which takes the form of a single degree of freedom nonlinear oscillator this system which may be analyzed by conventional phase plane techniques captures all the qualitative features of the full system we discuss the reduction and show how the nonlinear terms may be simplified and put into normal form invariant manifold theory and the normal form theorem play a major role in this work and this paper serves as an introduction to their application in mechanics repeating the approach in the second case we show that the essential model is now three dimensional and that far more complex behavior is possible including nonperiodic and chaotic motions throughout we take a two degree of freedom system as an example but the general methods are applicable to multi and even infinite degree of freedom problems

discretized gabor frames of totally positive functions in this paper a large class of universal windows for gabor frames weyl heisenberg frames is constructed these windows have the fundamental property that every overcritical rectangular lattice generates a gabor frame likewise every undercritical rectangular lattice generates a riesz sequence

sensitivities and linear stability analysis around a double zero eigenvalue a general multiparameter system admitting a double zero eigenvalue at a critical equilibrium point is considered a sensitivity analysis of the critical eigenvalues is performed to explore the neighborhood of the critical point in the parameter space because the coalescence of the eigenvalues implies that the jacobian matrix is defective or nilpotent well suited techniques of perturbation analysis must be employed to evaluate the eigenvalues and the eigenvector sensitivities different asymptotic methods are used based on perturbations both of the eigenvalue problem and the characteristic equation the analysis reveals the existence of a generic nonsingular case and of a nongeneric singular case however even in the generic case a codimension 1 subspace exists in the parameter space on which a singularity occurs by the use of the relevant asymptotic expansions linear stability diagrams arebuiltup and different bifurcation mechanisms divergence hopf doubledivergence doubledivergence hopf degeneratehopf arehighlighted the problem of e nding a uniqueexpression uniformly valid in the wholespaceis then addressed it isfound that a second degreealgebraic equation governsthebehaviorofthecriticaleigenvalues it also permits clarie cation of the geometrical meaning of the unfolding parameters which has been discussed in literature for the takens bogdanova bifurcation finally a mechanical system loaded by nonconservative forces and exhibiting a double zero bifurcation is studied as an example

a convergence question inspired by stieltjes and by value sets in continued fraction theory abstract let v be a subset of the complex plane c let fn n 1 be a sequence of self mappings of v i e fn v v the question is then under what conditions will the sequence fn w f1 f2 fn w n 1 2 3 of composite maps converge to a constant function in v in this paper we give a survey of some of the answers and open problems connected with this question such answers have applications in dynamical systems schur analysis continued fractions and other similar structures like infinite exponentials infinite radicals

a study of the limit cycles associated with a generalized codimension 3 lienard oscillator in this paper melnikov and petrov methods are used to study the limit cycles associated with the degenerate bifurcations of a representative generalized codimension 3 lienard oscillator it is shown that the coexistence of multiple limit cycles is possible and that jumping phenomena exist between limit cycles moreover it is found that the stable limit cycles oscillations and the unstable limit cycles oscillations can occur alternatively

general convergence of continued fractions on introduit la notion de convergence generale des fractions continues qu on compare a celles de convergence ordinaire et de convergence forte on etablit quelques unes de ses proprietes

continued fractions with three limit points abstract on page 45 in his lost notebook ramanujan asserts that a certain q continued fraction has three limit points more precisely if a n b n denotes its n th partial quotient and n tends to in each of three residue classes modulo 3 then each of the three limits of a n b n exists and is explicitly given by ramanujan ramanujan s assertion is proved in this paper moreover general classes of continued fractions with three limit points are established

decoupling nonlinear systems noncommutative generating power series and lie algebras the joint use of noncommutative generating power series and lie algebras gives an algebraic approach for decoupling problem of nonlinear systems we produce a coherent decoupling method using the additional notion of the immersion of a system into another a concept which makes clear the fact that two systems have a same input output behaviour thus a judicious choice of feedback which gives decoupling is able to immerse a system into a system defined on mathbb r n with eta depending on the characteristic numbers of the system

domains of validity for some of ramanujan s continued fraction formulas abstract by establishing uniform convergence of some of ramanujan s continued fractions we are able to extend the domain of validity for some of his formulas we also suggest more direct proofs for some of his results based on this technique

implementation of discretized gabor frames and their duals the usefulness of gabor frames depends on the easy computability of a suitable dual window this question is addressed under several aspects several versions of schulz s iterative algorithm for the approximation of the canonical dual window are analyzed for their numerical stability for gabor frames with totally positive windows or with exponential b splines a direct algorithm yields a family of exact dual windows with compact support it is shown that these dual windows converge exponentially fast to the canonical dual window

effects of initial geometric imperfections on the interaction between forced and parametric vibrations abstract the influence of initial geometric imperfections on the response of simply supported rectangular plates subjected to periodic in plane forces is theoretically analyzed in this work the large deflection plate theory used in the analysis is derived in terms of the stress function f and lateral displacement w the temporal response is analyzed by the first order generalized asymptotic method and principal parametric and forced resonances are considered and investigated interaction between the two types of resonances is found to depend on mode shapes and imperfection amplitudes

imperfection sensitivity of interacting hopf and steady state bifurcations and their classification abstract the bifurcation problem of interacting time periodic and stationary solutions of nonlinear evolution equations with double degeneracy is discussed in terms of singularity and imperfect bifurcation theory a complete classification up to symmetry covariant contact equivalence and codimension three of generic perturbations of interacting hopf and steady state bifurcations is presented the sensitivity of the bifurcation diagrams to imperfections is analyzed normal forms describing sequences of secondary and tertiary bifurcations leading to motions on tori are determined a variety of phenomena such as gaps in hopf branches periodic motions not stably connected to steady states and the formation of islands is discovered which one can expect to find in perturbed evolution equations on pure geometric grounds implications for physical systems are discussed

modal interaction in circular plates abstract the dynamic analogue of the von karman equations in polar form is used to study the symmetric response of a circular plate to a harmonic external excitation the method of multiple scales is used to determine a set of ordinary differential equations governing the modulation of the phases and amplitudes of oscillation for the case of an internal or autoparametric resonance the results show that the internal resonance is responsible for coupling of the modes involved and that the excited mode is not necessarily the dominant one furthermore the results show that for a choice of parameters the multi mode response loses its stability through a hopf bifurcation resulting in periodically or chaotically modulated motions of the plate

using the extended melnikov method to study the multi pulse global bifurcations and chaos of a cantilever beam the aim of this paper is to investigate the multi pulse global bifurcations and chaotic dynamics for the nonlinear non planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended melnikov method in the resonant case first the extended melnikov method for studying the shilnikov type multi pulse homoclinic orbits and chaos in high dimensional nonlinear systems is briefly introduced in the theoretical frame then this method is utilized to investigate the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non planar oscillations of the cantilever beam how to employ this method to analyze the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics of high dimensional nonlinear systems in engineering applications is demonstrated through this example finally the results of numerical simulation are given and also show that the shilnikov type multi pulse chaotic motions can occur for the nonlinear non planar oscillations of the cantilever beam which verifies the analytical prediction

on the bauer muir transformation for continued fractions and its applications abstract we show that the bauer muir transformation is useful also for continued fractions with non positive elements some examples of applications of this transformation are studied

on the divergence of the rogers ramanujan continued fraction on the unit circle this paper studies ordinary and general convergence of the rogers ramanujan continued fraction let the continued fraction expansion of any irrational number t 0 1 be denoted by 0 e 1 t e 2 t and let the i th convergent of this continued fraction expansion be denoted by c i t d i t let s t 0 1 e i 1 t d i t infinitely often where 5 1 2 let y s exp 2 it t e s it is shown that if y e y s then the rogers ramanujan continued fraction r y diverges at y s is an uncountable set of measure zero it is also shown that there is an uncountable set of points g c y s such that if y g then r y does not converge generally it is further shown that r y does not converge generally for y 1 however we show that r y does converge generally if y is a primitive 5m th root of unity for some m e n combining this result with a theorem of i schur then gives that the continued fraction converges generally at all roots of unity

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

analysis of critical and post critical behaviour of non linear dynamical systems by the normal form method part i normalization formulae abstract a method based on normal form theory is presented to study the dynamical behaviour of a system in the neighbourhood of a nearly critical equilibrium state associated with a bifurcation condition explicit formulae for the normalization procedure are derived these formulae can be numerically programmed avoiding usual complicated algebraic calculations and making the method effectively applicable for n dimensional systems rather general bifurcations can be included e g non linear flutter hopf bifurcation divergence and internal resonance

bifurcation analysis of double pendulum with a follower force abstract the stability and bifurcations of a double pendulum with a follower force are considered the focus is placed on a doubly degenerate system which possesses a zero eigenvalue and a pair of pure imaginary eigenvalues that is coupled flutter and divergence bifurcation the local qualitative behavior of the system is examined in the neighborhood of the degenerate system the four dimensional equations of motion are reduced to the two dimensional ones by using some qualitative reduction theory for dynamical systems and it is shown that two distinct bifurcations can occur in the system according to the ratio of two damping coefficients of the double pendulum numerical simulations have been performed using the original four dimensional equations to confirm the analytical results some global forms of behavior of the solutions have also been investigated numerically

multi pulse chaotic dynamics in non planar motion of parametrically excited viscoelastic moving belt abstract this paper investigates the multi pulse global bifurcations and chaotic dynamics for the nonlinear non planar oscillations of the parametrically excited viscoelastic moving belt using an extended melnikov method in the resonant case using the kelvin type viscoelastic constitutive law and hamilton s principle the equations of motion are derived for the viscoelastic moving belt with the external damping and parametric excitation applying the method of multiple scales and galerkin s approach to the partial differential governing equation the four dimensional averaged equation is obtained for the case of 1 1 internal resonance and primary parametric resonance from the averaged equations obtained the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues based on the explicit expressions of normal form the extended melnikov method is used for the first time to investigate the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics the paper demonstrates how to employ the extended melnikov method to analyze the shilnikov type multi pulse homoclinic bifurcations and chaotic dynamics of high dimensional nonlinear systems in engineering applications numerical simulations show that for the nonlinear non planar oscillations of the viscoelastic moving belt the shilnikov type multi pulse chaotic motions can occur overall both theoretical and numerical studies suggest that the chaos for the smale horseshoe sense in motion exists

homoclinic tangencies in the dynamics of articulated pipes conveying fluid abstract numerical evidence is presented showing homoclinic tangencies to periodic orbits occuring in a four dimensional system of ordinary differential equations that describes the dynamics of a pair of articulated pipes conveying fluid bifurcation sequences of mixed mode oscillations are discovered in two separate regions of parameter space in one region the equations have z 2 symmetry and a symmetric pair of heteroclinic tangencies is implied in both cases satisfactory agreement is found between the observed bifurcation sequences and the local theory of homoclinic tangencies global branches of mixed mode orbits are computed it is discovered that some orbits are generated by either sil nikov or figure of eight homoclinic orbits to stationary points whereas others form isolas that feature characteristic towers of saddle node and period doubling bifurcations

global stabilty analysis of a classical non conservative system under a follower load abstract the bifurcations critical conditions and global stability of the classical two degree of freedom ziegler s dissipative model under a partial follower load are thoroughly discussed with the aid of a complete non linear dynamic analysis attention is mainly focused on the range of values of the non conservativeness loading parameter for which adjacent equilibria do not exist various characteristic findings are established by studying the nature of the jacobian matrix eigenvalues associated with the above non linear autonomous dissipative system it is found that the system exhibits stable hopf bifurcations global stability for the entire region of non adjacent equilibria while the classical linear stability analysis leads to flutter instability local instability moreover in an explicit general form the significant effect of damping on the critical dynamic bifurcation load is established on the basis of the latter the critical loads for the case of vanishing damping are found to be completely different from those of the classical local stability analysis all findings of this analysis are checked against numerical solution of the equations of motion

bifurcations and chaotic motions in the autonomous system of a restrained pipe conveying fluid the stability and dynamics of a cantilevered pipe conveying fluid with motion limiting constraints and a linear spring support have been investigated emphasis is placed on analyzing local qualitative behavior of the system in the neighborhood of a doubly degenerate point using some qualitative reduction methods of dynamical system theory the four dimensional differential equation of motion is reduced to a two dimensional one and then the possible motions of the pipe are predicted through analyzing bifurcations of the solution to the reduced equation of motion the unfolding result is found to be in good agreement with the result obtained using the numerical method it is also found that there exist the quasi periodic motions and route to chaos through breakup of the quasi periodic torus surface in some parameter region of the system which differs from that of periodic doubling bifurcation route found earlier in this system numerical simulations have been performed using the four dimensional equation of motion to confirm the analytical results

flow induced bifurcations to three dimensional oscillatory motions in continuous tubes three dimensional motions of a cantilever tube carrying an incompressible fluid and having rotational symmetry about the vertical axis are examined for bifurcating oscillatory solutions the system behavior depends on two parameters rho the flow velocity and beta the mass ratio of the fluid and the tube as the flow rate is increased past a critical value the zero solution becomes unstable by a double pair of complex conjugate eigenvalues of the linearized operator crossing the imaginary axis using ideas from the center manifold theory and the method of integral averaging the governing partial differential equations are reduced to a set of four nonlinear ordinary differential equations it is then shown that the system bifurcates into two qualitatively distinct oscillatory motions these oscillatory solutions themselves for some values of the mass ratio beta bifurcate into more complex amplitude modulated oscillations

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

non periodic motion caused by grazing incidence in an impact oscillator the motion of a single degree of freedom periodically forced oscillator subjected to a rigid amplitude constraint is considered using analytical methods the singularities caused by grazing impact are studied it is shown that as a stable periodic orbit comes to grazing impact under the control of a single parameter a special type of bifurcation occurs the motion after the bifurcation may be non periodic and a criterion for this based on orientation and eigenvalues is given

local birth of homoclinic chaos abstract local generation of chaotic behaviours associated with a sil nikov homoclinic orbit is studied near the codimension two bifurcation at the double stationary oscillatory instability of eigenvalues 0 we develop a perturbative calculation of the sil nikov homoclinic orbit and of its bifurcation locus starting from a conservative vector field flat nonanalytic behaviours due to non axisymmetric perturbations of the normal form are discussed and we show that the codimension one locus of sil nikov homoclinicity is unavoidable near bicriticality under certain conditions the results are applied to rossler s model quasiperiodic regimes born at the double instability are also studied and illustrated in the rossler model

continued fractions with circular twin value sets we prove that if the continued fraction k a n 1 has circular twin value sets v 0 v 1 then k a n 1 converges except in some very special cases the results generalize previous work by jones and thron

bifurcations in a two degree of freedom elastic system with follower forces abstract the stability and bifurcations of a two degree of freedom elastic system subjected to follower forces are studied attention is mainly concentrated on local behavior small amplitude motions about a centre equilibrium for the parameter values in the neighborhood of some point on the stability boundaries by using centre manifold theory and some codimension two bifurcations it is shown that a number of distinct bifurcations may occur in such a system according to the ratio of two damping coefficients numerical integrations of the full equations of motion have been performed to confirm the analytical aspects of the local solutions as well as to understand some aspects of the global behavior of the solutions

bifurcations and chaos in autonomous self excited oscillators with impact damping dynamics and bifurcations of a class of single degree of freedom self excited oscillators with an impact damper have been studied the basic motivation is to determine a persistent birfurcation structure of co dimension one which is independent of the exact functional form of the self excitation mechanism in the process an analytical method for studying the qualitative dynamical features is presented a new numerical scheme for analyzing the eigenproperties of a solution is developed a detailed numerical study revealing the superstructure of the bifurcation phenomena has been performed

modal analysis of mechanical systems with impact non linearities limitations to a modal superposition this paper presents an attempt to generalize the modal superposition formula to mechanical systems with impact type non linearities following the procedure introduced for smooth non linearities the study is restricted to simple one and two degree of freedom systems with a unilateral constraint on one of the degrees of freedom for which the response can be analytically determined generalized frequencies modes and masses are built in the procedure the results obtained for various sets of parameters point out some limitations to the validity of a general modal superposition formula

repeated modifications of limitk periodic continued fractions the advantages of using modified approximants for continued fractions can be enhanced by repeating the modification process ifk a n b n is limitk periodic a natural choice for the modifying factors is ak periodic sequence of right or wrong tails of the correspondingk periodic continued fraction if it exists if the modified approximants thus obtained are ordinary approximants of a new limitk periodic continued fraction we repeat the process if possible some examples where this process is applied to obtain a convergence acceleration are also given