parametrically excited vibrations of an oscillator with strong cubic negative nonlinearity in this paper the parametrically excited vibrations of an oscillator with strong cubic negative nonlinearity are analyzed the two dimensional lindstedt poincare perturbation technique applied for finding an approximate solution of linear parametrically excited systems is extended for analyzing a strong nonlinear oscillator based on the solution of a nonlinear differential equation with constant coefficients an approximative solution is introduced the transition curves and transient surfaces along which periodic solutions exist are obtained their strong dependence on the initial conditions is evident to prove the analytical solution the numerical experiment is done for certain values initial conditions and parameter values the time history diagrams for the oscillator are plotted

subharmonic and homoclinic bifurcations in a parametrically forced pendulum abstract depending on the parameters of a parametrically forced pendulum system the boundaries of subharmonic and homoclinic bifurcations are calculated on the basis of the melnikov method and of averaging methods it is shown that as a parameter is varied repeated resonances of successively higher periods occur culminating in homoclinic orbits according to the theorem of smale homoclinic bifurcation is the source of the unstable chaotic motions observed for some selected parameter sets the theoretical predictions are tested by numerical calculations very good agreement is found between analytical and numerical results

asymptotic solutions for mathieu instability under random parametric excitation and nonlinear damping a theoretical analysis is presented of the response of a lightly and nonlinearly damped mass spring system in which the spring constant contains a small randomly fluctuating component damping is represented by a combination of linear and nonlinear power law damping system response to some initial disturbance at time zero is described by a sinusoidal wave whose amplitude and phase vary slowly and randomly with time leading order formulations for the equations of amplitude and phase are obtained through the application of methods of stochastic averaging of stratonovich the equations of amplitude and phase are given in two versions fokker planck equations for transient probability and langevin equations for response in the time domain solutions in closed form of these equations are derived by methods of mathematical and theoretical physics involving higher transcendental functions they are used to study the behavior of system response for ever increasing time applying asymptotic methods of analysis such as the method of steepest descent or saddle point method it is found that system behavior depends on the power density of the parametric excitation at twice the natural frequency and on the magnitude and form of the damping depending on these parameters different types of system behavior are found to be possible response which decays exponentially to zero response which leads to a stationary state of random behavior and response which can either grow unboundedly or which approaches zero in a finite time

relative negligibility of linear automorphisms let u and v be finite dimensional vector spaces over a field k gl u gl v and i be the identity transformation on v denote by and i the induced linear automorphisms on u v and i can also be regarded as k automorphisms on the function field k u v it is elementary to check whether and i are conjugate within gl u v by examining their rational canonical forms in this paper we shall give necessary and sufficient conditions for and i to be conjugate within autk k u v for this characterization we introduce the concept of the generalized order through this invariant we also settle the question of when two different polynomials are minimal polynomials of the same linear automorphism of a rational function field

on the solution of strong nonlinear oscillators by applying a rational elliptic balance method a rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using jacobi elliptic functions to illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators we first investigate the exact solution of the non homogenous undamped duffing equation then we introduce first and second order rational elliptic form solutions to obtain approximate solutions of two nonlinear oscillators at the end of the paper we compare the numerical integration values of the angular frequencies with approximate solution results based on the proposed rational elliptic balance method

extension and improvement to the krylov bogoliubov methods using elliptic functions abstract it is shown that the krylov bogoliubov methods that give the approximate oscillatory solution of the equation in terms of jacobi elliptic functions are applicable not only when c 1 0 and c 3 0 but also when c 1 0 and c 3 0 in particular the most precise of these methods the christopher brocklehurst method is discussed in detail its accuracy has been improved by utilizing the transformation properties of elliptic functions with respect to their parameters

topology and resonances in a quasiperiodically forced oscillator in this paper we analyze resonance behavior in a quasiperiodically forced nonlinear mathieu equation we develop a perturbation technique based on the method of multiple scales to find both a criterion for resonance and approximations to solutions in the neighborhood of a resonance we compare the perturbation results to numerical solutions to validate both the resonance criterion and the approximate solutions we also investigate the implications of resonance for the topology of attractors in the four dimensional phase space we show that a resonance occurs due to topological torus bifurcations ttbs and that resonant trajectories lie on topologically interesting knotted tori we have recently described elsewhere topological bifurcations of attracting 2 tori of quasiperiodically driven nonlinear oscillators in review the perturbation approximations capture both ttbs and the topology of invariant manifolds near resonance

solution of a mixed parity nonlinear oscillator harmonic balance abstract the method of harmonic balance is used to calculate first order approximations to the periodic solutions of a mixed parity nonlinear oscillator first the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction then the two auxiliary equations where the restoring force functions are odd are solved by using a first harmonic term without a constant therefore we obtain the two approximate solutions to the mixed parity nonlinear oscillator one is expressed in terms of the exact amplitude in the negative direction the other in terms of the approximate amplitude these solutions are more accurate than the second approximate solution obtained by the lindstedt poincare method for large amplitudes

semi discretization method for delayed systems summary the paper presents an ecient numerical method for the stability analysis of linear delayed systems the method is based on a special kind of discretization technique with respect to the past eect only the resulting approximate system is delayed and also time periodic but still it can be transformed analytically into a high dimensional linear discrete system the method is applied to determine the stability charts of the mathieu equation with continuous time delay copyright 2002 john wiley sons ltd

convergence rate analysis of a multiplicative schwarz method for variational inequalities this paper derives a linear convergence for the schwarz overlapping domain decomposition method when applied to constrained minimization problems the convergence analysis is based on a minimization approach to the corresponding functional over a convex set a general framework of convergence is established for some multiplicative schwarz algorithm the abstract theory is particularly applied to some obstacle problems which yields a linear convergence for the corresponding schwarz overlapping domain decomposition method of one and two levels numerical experiments are presented to confirm the convergence estimate derived in this paper

a perturbative algorithm for quasi periodic linear systems close to constant coefficients a perturbative procedure is proposed to formally construct analytic solutions for a linear differential equation with quasi periodic but close to constant coefficients the scheme constructs the necessary linear transformations involved in the reduction process up to an arbitrary order in the perturbation parameter it is recursive can be implemented in any symbolic algebra package and leads to accurate analytic approximations sharing with the exact solution important qualitative properties this algorithm can be used in particular to carry out systematic stability analyses in the parameter space of a given system by considering variational equations

a note on algebraic automorphisms abstract let v be a vector space over the field k of a possibly infinite dimension and let t be an automorphism of v of a possibly infinite order assume that t is algebraic with minimal polynomial p t k t let k v be the field of quotients of the symmetric k algebra of v and let s be the k automorphism on k v induced by t we show that s is completely determined by p t and dimk v

comments on the method of harmonic balance the advantages and limitations of the harmonic balance or describing function approximation scheme for solving nonlinear ordinary differential equations of oscillatory motion are discussed advantages include appicability to equations of any order and with large degrees of nonlinearity ease of determining limit cycle behavior and its stability and overall speed and efficiency the limitation rules are essentially those described by mickens 1983 it is pointed out that perturbation procedures provide better results when the degree of nonlinearity is small

a generalized galerkin s method for non linear oscillators by allowing for frequency variation in the virtual displacement co ordinates a generalized galerkin s procedure is developed this leads to a compact form of galerkin s conditions which in turn yield as many algebraic equations as the number of unknown parameters and frequency ies the independence of the equations is not guaranteed but depends on the nature of the oscillators when they are not independent the possible contradiction serves as a measure of the approximation of the solution the two well known non linear oscillators van der pol s and duffing s are used as the basic examples for the application of the method

an empirical approach for delayed oscillator stability and parametric identification this paper investigates a semi empirical approach for determining the stability of systems that can be modelled by ordinary differential equations with a time delay this type of model is relevant to biological oscillators machining processes feedback control systems and models for wave propagation and reflection where the motion of the waves themselves is considered to be outside the system model a primary aim is to investigate the extension of empirical floquet theory to experimental or numerical data obtained from time delayed oscillators more specifically the reconstructed time series from a numerical example and an experimental milling system are examined to obtain a finite number of characteristic multipliers from the reduced order dynamics a secondary goal of this work is to demonstrate a benefit of empirical characteristic multiplier estimation by performing system identification on a delayed oscillator the principal results from this study are the accurate estimation of delayed oscillator characteristic multipliers and the utilization the empirical results for parametric identification of model parameters combining these results with previous research on an experimental milling system provides a particularly relevant result the first approach for identifying all model parameters for stability prediction directly from the cutting process vibration history

on the comparison of semi analytical methods for the stability analysis of delay differential equations abstract this paper provides the first comparison of the semi discretization spectral element and legendre collocation methods each method is a technique for solving delay differential equations ddes as well as determining regions of stability in the dde parameter space we present the necessary concepts assumptions and equations required to implement each method to compare the relative performance between the methods the convergence rate and computational time for each method is compared in three numerical studies consisting of a ship stability example the delayed damped mathieu equation and a helicopter rotor control problem for each study we present one or more stability diagrams in the parameter space and one or more convergence plots the spectral element method is demonstrated to have the quickest convergence rate while the legendre collocation method requires the least computational time the semi discretization method on the other hand has both the slowest convergence rate and requires the most computational time

applications of a generalized galerkin s method to non linear oscillations of two degree of freedom systems abstract a generalized galerkin s method is formulated for multi degree of freedom holonomic systems two autonomous non linear two d o f oscillators are used as the examples for the applications of the method i free oscillations of a non linear mass spring system with two d o f ii two weakly non linearly coupled identical van der pol oscillators the accuracy of the approximate solutions is discussed the effect of different time intervals of integration on the results is also investigated

on the higher order semi discretizations for periodic delayed systems semi discretization techniques of periodic delayed systems are presented using zeroth first and higher order approximations of the delayed term it is shown that if the time periodic coefficients in the equation are approximated by piecewise constant functions then there is no need to use higher than first order approximations of the delayed term the results are demonstrated on construction of the stability chart of the delayed mathieu equation

nonlinear vibration of buckled beams some exact solutions in this paper exact solutions are obtained for the dynamics of buckled beams with different types of end conditions the differential equation for the vibrations of buckled beams with restrictions on axial stretch is nonlinear using the modes of the corresponding linear problem which readily satisfy the boundary conditions the natural frequencies are obtained for the nonlinear problem by using jacobi elliptic functions analytical solutions for the dynamics of buckled beams with various rotational restraints are presented

amplitude decay of damped non linear oscillators studied with jacobian elliptic functions approximate bounded solutions of the equation x bx c 1 x c 3 x 3 0 with b 0 c 1 0 and c 3 0 are developed in terms of the jacobian elliptic functions cn cd and dn the solutions are found by following the method developed by christopher in 1973 for the case with c 1 0 and c 3 0 formulas for the amplitude decay are given in two different approximations the solutions are compared with runge kutta numerical integration results and shown to be accurate for a wide range of b c 1 c 3 and initial conditions

generalized fourier series and limit cycles of generalized van der pol oscillators abstract a very simple generalization using jacobian elliptic functions of the usual fourier series appropriate for non linear systems is used to study the first order approximate solution of the generalized van der pol oscillators x ax 2bx 3 z 3 z 2 x 2 z 1 x 4 dot x 0 a generalized harmonic balance method is used to determine the limit cycles the cases a 0 b 0 and a 0 are considered in some detail for given values of the parameters zi the values of a and b for which limit cycles exist are found as functions of m numerical values for the radius frequency and energy of the limit cycles are given the presence of zero one two three or four limit cycles depends on the value of the parameters of the equation stability bifurcations of fixed points and the limit cycle stability are studied qualitatively

combining the methods of harmonic balance and kryloff bogoliuboff in a recent work mickens has described a set of conditions under which the harmonic balance method hbm is implementable the aim here is to broaden the scope of hbm analysis by drawing attention to the possible combination of such methods with those of kryloff bogoliuboff surprisingly these issues are not widely appreciated and as such provide an incentive for this note

nonoscillation of mathieu equations with two frequencies abstract as is well known mathieu s equation is a representative of mathematical models describing parametric excitation phenomena this paper deals with the oscillation problem for mathieu s equation with two frequencies the ratio of these two frequencies is not necessarily a rational number when the ratio is an irrational number the coefficient of mathieu s equation is quasi periodic but not periodic for this reason the basic knowledge for linear periodic systems such as floquet theory is not useful whether all solutions of mathieu s equation oscillate or not is determined by parameters and frequencies our results provide parametric conditions to guarantee that all solutions are nonoscillatory the advantage of the obtained parametric conditions is that it can be easily checked parametric nonoscillation region is drawn to understand these results easily finally several simulations are carried out to clarify the remaining problems

error bounds for approximate eigenvalues of periodic coefficient linear delay differential equations we describe a new chebyshev spectral collocation method for systems of variable coefficient linear delay differential equations with a single fixed delay computable uniform a posteriori bounds are given for this method when the coefficients are periodic the system has a unique compact nonnormal monodromy operator whose spectrum determines the stability of the system the spectral method approximates this operator by a dense matrix of modest size in cases where the coefficients are smooth we observe spectral convergence of the eigenvalues of that matrix to those of the operator our main result is a computable a posteriori bound on the eigenvalue approximation error in the case that the coefficients are analytic

hill s equation with small fluctuations cycle to cycle variations and stochastic processes hill s equations arise in a wide variety of physical problems and are specified by a natural frequency a periodic forcing function and a forcing strength parameter this classic problem is generalized here in two ways a to random hill s equations which allow the forcing strength qk the oscillation frequency k and the period k of the forcing function to vary from cycle to cycle and b to stochastic hill s equations which contain at least one additional term that is a stochastic process this paper considers both random and stochastic hill s equations with small parameter variations so that pk qk qk lk k k and are all o e where e 1 we show that random hill s equations and stochastic hill s equations have the same growth rates when the parameter variations pk and lk obey certain constraints given in terms of the moments of for random hill s equations the growth rates for the solutions are given by the growth rates of a matrix transformation under matrix multiplicat

analysis of a beam column system under varying axial forces of elliptic type the exact solution of lame s equation abstract we investigate the dynamical response of a beam column system with hinged ends subjected to an axial pulsating force of elliptic type it is shown that the resulting equation is of the form d 2 y 1 d 2 y 1 d 1 d 2 cn 2 k 2 0 which is the well known lame equation higher transcendental functions bateman manuscript project edited by mcgraw hill new york vol 3 in this paper we obtain the general exact solution of this equation that reveals stable behavior of the beam column system if the assigned initial conditions are of the form y1 0 y10 and y 1 0 0 it is also found that at a certain value of the modulus of the elliptic force the lateral vibrational frequency is independent of the material properties of the beam column system

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

periodic solutions of strongly quadratic non linear oscillators by the elliptic lindstedt poincare method abstract the elliptic lindstedt poincare method is used employed to study the periodic solutions of quadratic strongly non linear oscillators of the form x c 1 x c 2 x 2 e f x x in which the jacobian elliptic functions are employed instead of the usual circular functions in the classical lindstedt poincare method the generalized van de pol equation with f x x 0 1 x 2 x 2 is studied in detail comparisons are made with the solutions obtained by using the lindstedt poincare method and runge kutta method to show the efficiency of the present method

exact solution of the quadratic mixed parity helmholtz duffing oscillator abstract in this paper the exact solution of the quadratic mixed parity helmholtz duffing oscillator is derived by using jacobi elliptic functions it is also shown that the exact period of oscillation is given as a function of the complete elliptic integral of the first kind at the end of the paper we examine the stability of the system and determine the regions for which periodic and unbounded motions take place

chaotic motion in the resonant separatrix bands of a mathieu duffing oscillator with a twin well potential the m 1 resonant bands in the left and right potential wells are skew symmetric and the 2m 1 resonant bands of the large orbit motion are symmetric the analytical conditions for the onset and destruction of a resonant band are developed through the incremental energy approach the numerical predictions of such onset and destruction are also completed by the energy increment spectrum method the sub resonance interaction occurs for strong excitations which needs to be further investigated these results are applicable to the small and large orbit motions of post buckled structure under a parametric excitation

transition curves for the quasi periodic mathieu equation in this work we investigate an extension of mathieu s equation the quasi periodic qp mathieu equation given by ddot psi delta eps cos t cos omega t psi 0 for small eps and irrational omega of interest is the generation of stability diagrams that identify the points or regions in the delta omega parameter plane for fixed eps for which all solutions of the qp mathieu equation are bounded numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots oflyapunov exponents in addition we derive approximate analytic expressions for transition curves using two distinct techniques 1 a regular perturbation method under which transition curves delta delta omega eps are each expanded in powers of eps and 2 the method of harmonic balance utilizing hill s determinants both analytic methods deliver results in good agreement with those generated numerically in the sense that pre

averaging method using generalized harmonic functions for strongly non linear oscillators abstract an averaging method in which generalized harmonic functions are used is applied to study the approximate solutions of the strongly non linear oscillators g x e x and g x ef x t where g x is an arbitrary non linear function the method gives the approximate solutions in terms of generalized harmonic functions these functions are also periodic and are exact solutions of strongly non linear differential equations some phenomena considered include limit cycles of strongly non linear autonomous oscillators and steady state response of strongly non linear oscillators subject to weak harmonic excitation the procedure is simple and easy to apply

convergence rate of a schwarz multilevel method for the constrained minimization of nonquadratic functionals in l badea convergence rate of a multiplicative schwarz method for strongly nonlinear inequalities v barbu i lasiecka d tiba and c varsan eds kluwer academic publishers boston 2003 the convergence of a subspace correction method applied to the constrained minimization of a functional in a general reflexive banach space has been proved provided that the convex set verifies a certain assumption this assumption is weaker than that in which the convex set is decomposed according to the space decomposition as a sum of subsets in the sobolev spaces the proposed method becomes a multiplicative schwarz method for the solution of the variational inequalities coming from the minimization of nonquadratic functionals we prove in this paper that this assumption holds for the one two and multilevel multiplicative schwarz methods in the finite element space and we explicitly write the constants in the error estimations depending on the overlapping and mesh parameters our error estimates are similar with those obtained for the minimization of quadratic functionals in l badea x c tai and j wang siam j numer anal 2003 pp 1052 1073 or with those obtained for the one obstacle problem in x c tai numer math 93 2003 pp 755 786

some results on the dynamics of the linear parametric oscillator with general time varying frequency the dynamics of single degree of freedom linear oscillators under parametric excitation of general type periodic or non periodic is considered an analytical approach based on complexification and amplitude phase decomposition of the response is developed leading to an exact set of first order ordinary differential equations governing the amplitude and phase of the motion perturbation schemes for solving this set are developed and the solutions of the derived modulation equation are outlined in addition some results and comments on the stability of the derived solutions are provided

jacobi elliptic functions a review of nonlinear oscillatory application problems abstract this review paper is concerned with the applications of jacobi elliptic functions to nonlinear oscillators whose restoring force has a monomial or binomial form that involves cubic and or quadratic nonlinearity first geometric interpretations of three basic jacobi elliptic functions are given and their characteristics are discussed it is shown then how their different forms can be utilized to express exact solutions for the response of certain free conservative oscillators these forms are subsequently used as a starting point for a presentation of different quantitative techniques for obtaining an approximate response for free perturbed nonlinear oscillators an illustrative example is provided further two types of externally forced nonlinear oscillators are reviewed i those that are excited by elliptic type excitations with different exact and approximate solutions ii those that are damped and excited by harmonic excitations but their approximate response is expressed in terms of jacobi elliptic functions characteristics of the steady state response are discussed and certain qualitative differences with respect to the classical duffing oscillator excited harmonically are pointed out parametric oscillations of the oscillators excited by an elliptic type forcing are considered as well and the differences with respect to the stability chart of the classical mathieu equation are emphasized the adjustment of the melnikov method to derive the general condition for the onset of homoclinic bifurcations in a system parametrically excited by an elliptic type forcing is provided and compared with those corresponding to harmonic excitations advantages and disadvantages of the use of jacobi elliptic functions in nonlinear oscillatory application problems are discussed and some suggestions for future work are given

improvement of a krylov bogoliubov method that uses jacobi elliptic functions abstract an improved version of a krylov bogoliubov method that gives the approximate solution of the non linear cubic oscillator x c 1 x c 3 x 3 ef x x dot 0 in terms of jacobi elliptic functions is described compact general expressions are given for the time derivatives of the amplitude and phase similar to those obtained by the usual krylov bogoliubov method which gives the approximate solution in terms of circular functions these expressions are especially simple for quasilinear c3 0 and quasi pure cubic c1 0 oscillators two types of cubic oscillators have been used as examples the linear damped oscillator f x x dot x dot and the van der pol oscillator f x x dot x 2 x dot the approximate solutions of these quasilinear and quasi pure cubic oscillators are simple and accurate the influence of the non linearity on the rate of variation of the amplitude of these two types of cubic oscillators was also studied

an additive schwarz method for variational inequalities this paper proposes an additive schwarz method for variational inequalities and their approximations by finite element methods the schwarz domain decomposition method is proved to converge with a geometric rate depending on the decomposition of the domain the result is based on an framework of convergence analysis established for general variational inequalities in hilbert spaces

application of a domain decomposition method to elastoplastic problems abstract in this paper parallel computing is applied to the solution of elastoplastic problems with strain hardening after a brief discussion of the classical variational and finite element formulations of the plasticity problem an algorithm is proposed it is based on a variant of the johnson weak formulation and on the schwarz domain decomposition method the domain is triangulated and the subdomains are defined as the union of the elements having a node in common this leads to many small subproblems which can be solved in a simple and efficient manner the algorithm has been implemented on a parallel computer the numerical results obtained for two examples a sheared plate and a perforated plate in tension compare quite favorably with simulations using a classical finite element code the computing times obtained show that the proposed method utilizes the parallel structure of the computer very efficiently

improvement to the averaging method using the jacobian elliptic function the averaging method was improved by using the jacobian elliptic sine sn cosine cn and delta dn functions as generating solutions in order to obtain a highly accurate periodic solution for a strongly nonlinear dynamical system the proposed method can be applied to a relatively general nonlinear system based on the single degree of freedom duffing equation two methods of stability analysis were applied and evaluated for the approximate solutions obtained by the proposed method the approximate solutions obtained by the proposed method for practical examples were compared to the solutions obtained by the shooting method the results confirmed that the proposed method provides a more accurate solution and more accurate stability analysis results than those obtained by the conventional averaging method that uses trigonometric functions as the generating solution

negligibility of projective linear automorphisms abstract letkbe any field s aij 1 i j n 1 gln 1 k k k x1 xn 1 xn xn 1 y1 yn 1 yn yn 1 a field extension of transcendence degree 2noverk the diagonal action ofsonkis defined as equation equation theorem there existz1 zn ksuch thatk k x1 xn 1 xn xn 1 z1 zn ands zi zifor 1 i n in particular ifk x1 xn 1 xn xn 1 s is rational purely transcendental overk so isk s overk

fractional delayed damped mathieu equation this paper investigates the dynamical behaviour of the fractional delayed damped mathieu equation this system includes three different phenomena fractional order time delay parametric resonance the method of harmonic balance is employed to achieve approximate expressions for the transition curves in the parameter plane the n 0 and n 1 transition curves both lower and higher order approximations are obtained the dependencies of these curves on the system parameters and fractional orders are determined previous results for the transition curves reported for the damped mathieu equation delayed second order oscillator and fractional mathieu equation are confirmed as special cases of the results for the current system

approximations for motion of the oscillators with a non negative real power restoring force abstract oscillators with a non negative real power restoring force are considered in this paper this type of restoring force is related to systems with a quasi zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature examples of these types of restoring force are grounded in real physical and engineering systems periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and hamilton s variational principle then the approach is extended to non conservative oscillators by adjusting the elliptic krylov bogoliubov method the methods proposed for the conservative and non conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force several examples the majority of which are so far unsolved are given to illustrate the methods proposed and to demonstrate their generality which permits unforeseen solutions for motion containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force the results obtained are compared with numerical results and have excellent accuracy

some actions of symmetrical groups abstract let k be any field v the n dimensional vector space over k on which the symmetric group sn acts by permuting a basis then the fixed fields k skv sn and k k v sn are rational over k where skv and k v are the kth symmetric power and exterior power of v respectively the rationality of some other fixed fields are established also

stability of elastic columns with periodic retarded follower forces the objective of this work is to present a stability analysis for elastic columns under the influence of periodically varying follower forces whose orientation is retarded i e depends on the position of the system at a previous time one and two degree of freedom dof discretized systems under the simultaneous influence of both parametric excitation and time delay whose effects on such systems have previously been only considered separately are studied by employing an orthogonal polynomial approximation the infinite dimensional floquet transition matrix associated with the time periodic differential delay system is approximated the stability criteria that all the eigenvalues floquet multipliers of this matrix must lie within the unit circle is then applied the stability charts for different combinations of the remaining system parameters are shown and the previously reported results for the special cases where either the parametric excitation or the time delay vanishes are verified two cases when the parametric forcing period is equal to or twice the delay period are taken into consideration in this work for special cases of the single dof system the numerical stability plots are verified by considering the analytical expressions for the corresponding stability boundaries for an analogous delayed mathieu equation

a generalization of the schwarz alternating method to an arbitrary number of subdomains we give a generalization of the schwarz alternating method to an arbitrary number of subdomains as applications we prove that the method converges for the main second order boundary value problems as well as for the system of linear elasticity for the dirichlet problems it is proved that the method converges geometrically and some numerical examples are given

updated semi discretization method for periodic delay differential equations with discrete delay an updated version of the semi discretization method is presented for periodic systems with a single discrete time delay the delayed term is approximated as a weighted sum of two neighbouring discrete delayed state values and the transition matrix over a single period is determined stability charts are constructed for the damped and delayed mathieu equation for different time period time delay ratios the convergence of the method is investigated by examples stability charts are constructed for 1 and 2 degree of freedom milling models the codes of the algorithm are also attached in the appendix copyright 2004 john wiley sons ltd

a study of a nonlinear vibration isolator with a quasi zero stiffness characteristic a vibration isolator consisting of a vertical linear spring and two nonlinear pre stressed oblique springs is considered in this paper the system has both geometrical and physical nonlinearity firstly a static analysis is carried out the softening parameter leading to quasi zero dynamic stiffness at the equilibrium position is obtained as a function of the initial geometry pre stress and the stiffness of the springs the optimal combination of the system parameters is found that maximises the displacement from the equilibrium position when the prescribed stiffness is equal to that of the vertical spring alone it also satisfies the condition that the dynamic stiffness only changes slightly in the neighbourhood of the static equilibrium position for these values a dynamical analysis of the isolator under asymmetric excitation is performed to quantify the undesirable effects of the nonlinearities it includes considering the possibilities of the appearance of period doubling bifurcation and its development into chaotic motion for this purpose approximate analytical methods and numerical simulations accompanied with qualitative methods including phase plane plots poincare maps and lyapunov exponents are used finally the frequency at which the first period doubling bifurcation appears is found and the effect of damping on this frequency determined

improved lindstedt poincare method for the solution of nonlinear problems abstract we apply the linear delta expansion lde to the lindstedt poincare distorted time method to find improved approximate solutions to nonlinear problems we find that our method works very well for a wide range of parameters in the case of the anharmonic oscillator duffing equation of the non linear pendulum and of more general anharmonic potentials the approximate solutions found with this method converge more rapidly to the exact ones than in the simple lindstedt poincare method

construction of approximate analytical solutions to a new class of non linear oscillator equation the principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire by assuming that harmonic balance will hold solutions are devised for a steady state limit cycle and or limit point motion a method of slowly varying amplitudes then allows derivation of approximate solutions by determining the form of the exact solutions and substituting into them the lowest order terms of their respective fourier expansions the latter technique is actually a generalization of the method proposed by kryloff and bogoliuboff 1943

a generalization of the method of harmonic balance a procedure is provided for generalizing the method of harmonic balance to obtain higher order approximations to the periodic solutions of differential equations for two systems one a conservative system and the other nonconservative the procedure is currently being applied to investigate the possible solution behaviors of singular nonlinear oscillators where the usual perturbation methods do not work

stability of linear time periodic delay differential equations via chebyshev polynomials this paper presents a new technique for studying the stability properties of dynamic systems modeled by delay differential equations ddes with time periodic parameters by employing a shifted chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period the dynamic system can be reduced to a set of linear difference equations for the chebyshev expansion coefficients of the state vector in the previous and current intervals this defines a linear map which is the infinite dimensional floquet transition matrix u two different formulas for the computation of the approximate u whose size is determined by the number of polynomials employed are given the first one uses the direct integral form of the original system in state space form while the second uses a convolution integral variation of parameters formulation additionally a variation on the former method for direct application to second order systems is also shown an error analysis is presented which allows the number of polynomials employed in the approximation to be selected in advance for a desired tolerance an extension of the method to the case where the delay and parametric periods are commensurate is also shown stability charts are produced for several examples of time periodic ddes including the delayed mathieu equation and a model for regenerative chatter in impedance modulated turning the results indicate that this method is an effective way to study the stability of time periodic ddes

vibrations in a parametrically excited system abstract this paper deals with vibrations of parametrically excited non linear systems with one degree of freedom the non linearity is cubic and is of the same order as the linear terms the parametric vibrations are excited by a periodical force of jacobi elliptic type the mathematical model of the system is a special type of non linear hill s equation the analytical approximate solution of the equation is obtained applying the elliptic krylov bogolubov method method of variable phase and amplitude developed for strong non linear differential equation of duffing type it enables the regions of unbounded solution to be defined approximately the parameters of a dynamic absorber which transforms the motion to regular are calculated in this paper

generalized fourier series for the study of limit cycles the approximate solution to first order of non linear differential equations is studied using the method of harmonic balance with generalized fourier series and jacobian elliptic functions as an interesting use of the series very good analytic approximations to the limit cycles of lienard s ordinary differential equation ode x g x f x x are presented specifically it is shown that contrary to an opinion given in a well known textbook on non linear oscillations g x not only modifies the period but influences the topology in the generalized van der pol equation with f x e 1 x2 and g x ax 2bx3 for 0 1 the presence of zero one or three limit cycles is found to depend on the value of ab

a modified elliptic lindstedt poincare method for certain strongly non linear oscillators abstract a modified elliptic lindstedt poincare method is presented for the steady state analysis of strongly non linear oscillators of the form x c 1 x c 3 x 3 ef x x in which a new parameter e is employed such that the value of is always small regardless of the magnitude of the original parameter e therefore the above strongly non linear oscillators with large parameter e is transformed into a small parameter system with respect to

an efficient computational scheme for the analysis of periodic systems abstract in this paper a new efficient numerical scheme for the stability analysis of linear systems with periodic parameters is suggested the approach is based on the idea that the state vector and the periodic matrix of the system can be expanded in terms of chebyshev polynomials over the principal period such an expansion reduces the original problem to a set of linear algebraic equations from which the solution in the interval of one period can be obtained furthermore the technique is combined with floquet theory to yield the transition matrix at the end of one period and provide the stability conditions via an eigen analysis procedure two formulations are presented the first formulation is suitable for a set of equations written in state space form while the second can be applied directly to a set of second order equations the application is demonstrated through practical illustrative examples including the parametric excitation problem of a fixed fixed column the numerical results thus obtained are compared with those obtained from dverk a runge kutta code available in the imsl library an error bound analysis is also included it is concluded that the suggested schemes not only provide accurate results but are also computationally very efficient in particular the second formulation is found to be several times faster than the standard runge kutta type codes

on the schwarz alternating method with more than two subdomains for nonlinear monotone problems in a hilbert space an iterative process is defined which generalizes the schwarz alternating method for nonlinear monotone problems to more than two subdomains and the convergence of this process is proved in the case of the problems with the solution in a convex of h 0 1 omega it is proved that the method converges geometrically numerical results are given concerning the problem of elastic plastic torsion of prismatic bars and the problem of seepage of fluids through porous media

a note on a relaxation schwarz alternating method abstract a relaxation schwarz alternating method for elliptic problems and their finite element solutions is proposed and studied the method with suitable relaxation parameters is shown to be much better than the conventional schwarz alternating method

computing the stability regions of hill s equation a numerical scheme is presented for computing the boundaries between the stable and unstable regions of hill s equation the approach based on formulating an integral representation of hill s equation thus the problem of finding the boundary between the stable and unstable regions in parameter space is then a spectral problem of an integral operator several properties of the operator are derived and an iteration scheme for numerical calculation of the stability boundary is presented the numerical scheme has a number of advantages over the classical methods for locating the stability boundary several examples are presented

application of the elliptic balance method to a nonlinear singular oscillator abstract new analytical approximate solutions based on jacobi elliptic functions are formulated to compute the angular frequency value of a nonlinear singular oscillator it is shown that when our derived analytical angular frequency solutions are compared to the exact angular frequency value the third order elliptic balance solution provides the best estimate to the exact value since the percentage error value is about 0 0059 which is 216 times lower than the best estimate percentage error value reported in the literature

irregular behaviour arising from quasiperiodic forcing of simple quantum systems insight from perturbation theory the perturbation of simple quantum systems with almost periodic time dependent forces can produce irregular dynamics which is manifested in i broadband fourier spectra and ii rapid decay of autocorrelation functions some classical perturbation methods for differential equations provide insight into this behaviour by showing the existence of a dense spectrum on the real line such irregular behaviour is shown also to occur in some model linear systems of differential equations e g a quasiperiodic mathieu type equation

stability analysis of the non linear mathieu equation abstract the non linear mathieu equation is analyzed within the framework of the method of normal forms analytical conditions for explosive instability are obtained and expressions for the period as well as the amplitude of the stable response are derived

oscillator with fraction order restoring force abstract in this paper a new analytical method for solving the differential equations which describe the motion of the oscillator with fraction order elastic force is introduced using the first integral of motion the exact period of vibration in the form of the euler beta function is obtained based on that value the approximate solution of the strong nonlinear fraction order differential equation is formed the suggested procedure is applied for various fraction values but also for the pure quadratic cubic and quintic oscillators the advantage of the suggested method is that is valid for all fraction values 1 the accuracy of the approximate solution is very high as the period of vibration is exactly analytically determined and is independent on the time in the paper the fraction order differential equation with small linear and nonlinear terms is also considered the krylov bogolubov method is extended due to the fact that frequency of vibration depends on the amplitude of vibration the approximate solution is with time variable amplitude phase and frequency the linear damped fraction order oscillator is widely discussed the approximate solution of the differential equation which describes the vibrations of such oscillator is compared with exact numerical one and shows a good agreement

rationality of some projective linear actions let k be any field which may not be algebraically closed v a finite dimensional vector space over k gl v where the order of can be finite or infinite theorem if dimk v 3 then both k v and k p v are rational purely transcendental over k similar results hold for a cyclic affine action

quasi pure cubic oscillators studied using a krylov bogoliubov method abstract a method of krylov bogoliubov type which gives the approximate solution in terms of jacobi elliptic functions is used to study the quasi pure cubic oscillators x c 3 x 3 ef x x 0 explicit approximate solutions with the perturbative terms x x 2 x x quadratic damping and sgn coulomb damping are given and are compared with the numerical or exact solution a simple and accurate expression is derived for the time at which the oscillations with coulomb damping cease

efficient generation of the ring of invariants we shall use the binet minc formula in the theory of permanents to prove david richman s theorem letgbe a finite group acting ona r a1 ar whereris any commutative ring with 1 g r then the ring of invariantsagis generated overrby g a 11a 22 a rr where 1 r g applications of permanents to other problems related to invariants are given also

analysis of a parallel schwarz algorithm for elliptic problems a parallel algorithm which is derived directly from the schwarz alternating algorithm with an overlapping domain decomposition for solving boundary value problems of elliptic partial differential equations is analyzed a new general analysis framework different from lions ones is established to describe the recursive relations of error functions between approximate solutions and the exact solution at the different iterative steps some abstract theories on convergence and convergent rate of the synchronously parallel algorithm are established under different conditions

transition curves in a parametrically excited pendulum with a force of elliptic type this article investigates the equilibria and stability of a pendulum when the support has a prescribed motion defined by an elliptic function stability charts are generated in the parameter plane for different values of the elliptic function modulus numerical integration and floquet theory are used to generate stability charts that are later obtained through harmonic balance analysis it is shown that the size and location of the instability tongues is directly linked to the elliptic function modulus comparisons are also made between the stability charts of mathieu s equation and those of the pendulum when the prescribed motion is defined by an elliptic function

stability chart for the delayed mathieu equation in the space of system parameters the closedform stability chart is determined for the delayed mathieu equation defined as t cost x t bx t2 this stability chart makes the connection between t