failure of the method of slowly varying amplitude and phase for non linear singular oscillators it is shown that the method of slowly varying amplitude and phase yields erroneous results in the study of the mathematical properties of nonlinear singular oscillator systems the analytical solution is described in which the phase function is constant and for which a special limiting behavior exists when the wavelength is zero the previous method based on the condition of boundedness cannot be satisfied for nonlinear singular characteristics and the erroneous designation of the expansion parameter is identified

discrete dynamical systems associated with root systems of indefinite type a geometric charactrization of the equation found by hietarinta and viallet which satisfies the singularity confinement criterion but which exhibits chaotic behavior is presented it is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be a realization of a cremona isometry on the picard group of the surface it is also shown that the group of cremona isometries is isomorphic to an extended weyl group of indefinite type a method to construct the mappings associated with some root systems of indefinite type is also presented

frequency limited h controller order reduction for linear parameter varying systems abstractin this paper a controller order reduction method for linear parameter varying systems is presented the proposed method is based on the frequency weighted balanced truncation technique which has the advantage to reduce the order in a specific frequency range the approach is discussed and is proved to preserve the closed loop stability with a guaranteed upper error bound effectiveness and performance of the obtained reduced order controller are investigated by applying it to an automotive semi active suspension control the obtained simulation results show that objectives such as the road handling and the passenger comfort realised with the reduced order controller are kept in the same performance level as with the full order controller moreover a comparison with an other order reduction method is shown and confirms the advantage of the developed method

a comparison of model reduction techniques from structural dynamics numerical mathematics and systems and control in this paper popular model reduction techniques from the fields of structural dynamics numerical mathematics and systems and control are reviewed and compared the motivation for such a comparison stems from the fact that the model reduction techniques in these fields have been developed fairly independently in addition the insight obtained by the comparison allows for making a motivated choice for a particular model reduction technique on the basis of the desired objectives and properties of the model reduction problem in particular a detailed review is given on mode displacement techniques moment matching methods and balanced truncation whereas important extensions are outlined briefly in addition a qualitative comparison of these methods is presented hereby focusing both on theoretical and computational aspects finally the differences are illustrated on a quantitative level by means of application of the model reduction techniques to a common example

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

approximate analytic solutions for singular non linear oscillators mickens 1981 1984 has considered analytic techniques for obtaining approximate solutions to one dimensional nonlinear oscillatory systems x double dot x lambda f x x dot lambda where lambda is a small positive parameter and f is a nonlinear polynomial function of its arguments however in certain cases there is interest in the analysis of physical systems for which the nonlinear function f x x dot lambda is singular for finite values of x or x dot the present investigation is concerned with the use of existing approximate analytic schemes to obtain solutions to singular nonlinear oscillatory differential equations

linearisable ultradiscrete systems with sign variables and the confinement of singularities we present the singularity analysis of the ultradiscrete analogue of linearisable mappings of quispel roberts thompson qrt type the ultradiscretisation method used here is one which keeps track of signs and thus can be applied without the positivity restrictions of the classical ultradiscretisation approach we show that in all cases the mappings possess confined singularities the same is true for two non autonomous equations which are equally linearisable in the discrete case we construct explicitly the solutions of the ultradiscrete mappings analysed here

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

third order integrable difference equations generated by a pair of second order equations we show that the third order difference equations proposed by hirota kimura and yahagi are generated by a pair of second order difference equations in some cases the pair of the second order equations are equivalent to the quispel robert thomson qrt system but in the other cases they are irrelevant to the qrt system we also discuss an ultradiscretization of the equations

a simple frequency weighted model reduction using structurally balanced truncation existence of solutions this paper examines weighted model reduction using structurally balanced truncation for a given weighted single or double sided transfer function a state space realization with the linear fractional transformation form is obtained then we prove that two block diagonal lmi linear matrix inequality solutions always exist and it is possible to get a reduced order model with guaranteed stability and a priori error bound finally two examples are used to show the validity of proposed weighted reduction method and the method is compared with other existing methods

a new frequency weighted balanced truncation method and an error bound in this paper a new frequency weighted model reduction method with an a priori error bound is proposed the method is a generalization of enns technique 1984 and yields stable models even when both input and output weightings are included the proposed method is compared with other existing methods using numerical examples

on the complexity of some birational transformations using three different approaches we analyse the complexity of various birational maps constructed from simple operations inversions on square matrices of arbitrary size the first approach comprises the study of the images of lines and relies mainly on univariate polynomial algebra the second approach is a singularity analysis and the third method is more numerical using integer arithmetics these three methods have their own domain of application but they give corroborating results and lead us to a conjecture on the algebraic entropy of a class of maps constructed from matrix inversions

controller reduction concepts and approaches the problem of passing from a linear time invariant high order controller designed for a linear time invariant plant of presumably high order to a low order approximation of the controller is discussed the approximation problem is often best posed as a frequency weighted l sub infinity approximation problem many different controller representations are possible giving different performances of the various reduction algorithms

on the algebraic structure of rational discrete dynamical systems we show how singularities shape the evolution of rational discrete dynamical sys tems the stabilisation of the form of the iterates suggests a description providing among other things generalised hirota form exact evaluation of the algebraic entropy as well as remarkable polynomial factorisation properties we illustrate the phenomenon explicitly with examples covering a wide range of models

algebraic entropy and the space of initial values for discrete dynamical systems a method to calculate the algebraic entropy of a mapping which can be lifted to an isomorphism of a suitable rational surface the space of initial values is presented it is shown that the degree of the nth iterate of such a mapping is given by its action on the picard group of the space of initial values it is also shown by construction that the degree of the nth iterate of every painleve equation in sakai s list is o n2 and therefore its algebraic entropy is zero

error bound for transfer function order reduction using frequency weighted balanced truncation abstract an error bound for transfer function order reduction is derived when frequency weighted balanced truncation is the order reduction method the bound is valid for both one sided input or output and two sided weighted balancing approximations with stable weights which can otherwise be arbitrary the error bound formula is valid for both discrete time and continuous time problems examples are studied to demonstrate effectiveness of the error bound

improved bounds for the eigenvalues of solutions of lyapunov equations bounds on the solution and on the eigenvalues of the solution of the discrete and of the continuous matrix lyapunov equations are presented these bounds are nontrivial in the case of system matrices having multiple eigenvalues especially in the latter case these bounds are better than bounds found in the literature further these bounds are sharp i e there exist systems such that the lower and upper bounds coincide yielding the solution or its eigenvalues respectively

singularity confinement test for ultradiscrete equations with parity variables we present a new ultradiscretization method which does not require that the solutions of the discrete equation have a fixed sign we construct an ultradiscrete analogue of the singularity confinement test using this method and thereby propose an integrability test for ultradiscrete equations

a finite extensibility nonlinear oscillator abstract the dynamics of a finite extensibility nonlinear oscillator feno is studied analytically by means of two different approaches a generalized decomposition method gdm and a linearized harmonic balance procedure lhb from both approaches analytical approximations to the frequency of oscillation and periodic solutions are obtained which are valid for a large range of amplitudes of oscillation within the generalized decomposition method two different versions are presented which provide different kinds of approximate analytical solutions in the first version it is shown that the truncation of the perturbation solution up to the third order provides a remarkable degree of accuracy for almost the whole range of amplitudes the second version which expands the nonlinear term in taylor s series around the equilibrium point exhibits a little lower degree of accuracy but it supplies an infinite series as the approximate solution on the other hand a linearized harmonic balance method is also employed and the comparison between the approximate period and the exact one numerically calculated is slightly better than that obtained by both versions of the gdm in general the agreement between the results obtained by the three methods and the exact solution numerically integrated for amplitudes a between 0 a 0 9 is very good both for the period and the amplitude of oscillation for the rest of the amplitude range 0 9 a l 2 error demonstrates that all three approximations do not represent a good description for the feno and higher order perturbation solutions are needed instead as a complement very accurate asymptotic representations of the period are provided for the whole range of amplitudes of oscillation

properties of finite difference models of non linear conservative oscillators finite difference fd approaches to the numerical solution of the differential equations describing the motion of a nonlinear conservative oscillator are investigated analytically a generalized formulation of the duffing and modified duffing equations is derived and analyzed using several fd techniques and it is concluded that although it is always possible to contstruct fd models of conservative oscillators which are themselves conservative caution is required to avoid numerical solutions which do not accurately reflect the properties of the original equation

elementary exact calculations of degree growth and entropy for discrete equations second order discrete equations are studied over the field of rational functions c z where z is a variable not appearing in the equation the exact degree of each iterate as a function of z can b

explicit solution and eigenvalue bounds in the lyapunov matrix equation an explicit solution to the algebraic lyapunov matrix equation is obtained in terms of the controllability matrix of the pair of coefficient matrices this enables us to determine the number of positive eigenvalues of the positive semidefinite solution through the controllability matrix based on this explicit formula upper and lower bounds for each eigenvalue of the solution are derived which always give nontrivial estimates

a survey of model reduction by balanced truncation and some new results balanced truncation is one of the most common model reduction schemes in this note we present a survey of balancing related model reduction methods and their corresponding error norms and also introduce some new results five balancing methods are studied 1 lyapunov balancing 2 stochastic balancing 3 bounded real balancing 4 positive real balancing and 5 frequency weighted balancing for positive real balancing we introduce a multiplicative type error bound moreover for a certain subclass of positive real systems a modified positive real balancing scheme with an absolute error bound is proposed we also develop a new frequency weighted balanced reduction method with a simple bound on the error system based on the frequency domain representations of the system gramians two numerical examples are illustrated to verify the efficiency of the proposed methods

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

rational surfaces associated with affine root systems and geometry of the painleve equations we present a geometric approach to the theory of painleve equations based on rational surfaces our starting point is a compact smooth rational surface x which has a unique anti canonical divisor d of canonical type we classify all such surfaces x to each x there corresponds a root subsystem of e 1 8 inside the picard lattice of x we realize the action of the corresponding affine weyl group as the cremona action on a family of these surfaces we show that the translation part of the affine weyl group gives rise to discrete painleve equations and that the above action constitutes their group of symmetries by backlund transformations the six painleve differential equations appear as degenerate cases of this construction in the latter context x is okamoto s space of initial conditions and d is the pole divisor of the symplectic form defining the hamiltonian structure

continuous families of rational surface automorphisms with positive entropy for any k we construct k parameter families of rational surface automorphisms with positive entropy these are automorphisms of surfaces mathcal x which are constructed from iterated blowups over the projective plane in certain cases we are able to determine the exact automorphism group of mathcal x as well as when two of these surfaces are inequivalent

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

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

algebraic entropy of an extended hietarinta viallet equation we introduce a series of discrete mappings which is considered to be an extension of the hietarinta viallet mapping with one parameter we obtain the algebraic entropy for this mapping by obtaining the recurrence relation for the degrees of the iterated mapping for some parameter values the mapping has a confined singularity in which case the mapping is equivalent to a recurrence relation between six irreducible polynomials for other parameter values the mapping does not pass the singularity confinement test the properties of irreducibility and co primeness of the terms play crucial roles in the discussion

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

do integrable cellular automata have the confinement property we analyse a criterion introduced by joshi and lafortune for the integrability of cellular automata obtained from discrete systems through the ultradiscretization procedure we show that while this criterion can be used in order to single out integrable ultradiscrete systems there do exist cases where the system is nonintegrable and still the criterion is satisfied conversely we show that for ultradiscrete systems that are derived from linearizable mappings the criterion is not satisfied we investigate this phenomenon further in the case of a mapping which includes a linearizable subcase and show how the violation of the criterion comes to be finally we comment on the growth properties of ultradiscrete systems

a geometric approach to singularity confinement and algebraic entropy a geometric approach to the equation found by hietarinta and viallet which satisfies the singularity confinement criterion but exhibits chaotic behaviour is presented it is shown that this equation can be lifted to an automorphism of a certain rational surface and can therefore be considered to be the action of an extended weyl group of indefinite type a method to calculate its algebraic entropy by using the theory of intersection numbers is presented

the linearization method based on the equivalence of dissipated energies for nonlinearly damped structural systems even though the nonlinear damping characteristics inherently pertain to most structural systems many useful dynamic models are still linear as a consequence a method that is able to equivalently linearize eql the nonlinear damping so that it can be directly applied in these existing linear models is essential in the present paper a novel eql method for nonlinearly damped and single degree of freedom systems is developed the method is theoretically derived by modulating the steady state responses of the original nonlinear system during the linearization the new eql method requires the dissipated energy of the target linear system to equal that of the original nonlinear one in effect this criterion is equivalent to forcing both the phase angles and amplitudes of the two systems to be equal or at least to within a small allowable error furthermore the present paper proves that it is possible for the dissipative energy to be expressed in terms of the fourier coefficients of the modulated signal thus the equivalent viscous damping ratio can be computed from these fourier coefficients this new eql method is numerically tested using examples of bi linear damping models and subjected to experimental measurements both the simulation results and the experimental data verify the validity of the method they also prove that the current method gives the equivalent viscous damping with good accuracy in addition a quality index that signifies how well the eql system reaches is appropriately added

a systematic method for constructing discrete painleve equations in the degeneration cascade of the e8 group we present a systematic and quite elementary method for constructing discrete painleve equations in the degeneration cascade for e8 1 starting from the invariant for the autonomous limit of the e8 1 equation one wishes to study the method relies on choosing simple homographies that will cast this invariant into certain judiciously chosen canonical forms these new invariants lead to mappings the deautonomisations of which allow us to build up the entire degeneration cascade of the original mapping we explain the method on three examples two symmetric mappings and an asymmetric one and we discuss the link between our results and the known geometric structure of these mappings