analysis on the laguerre formula for approximating delay systems this paper provides a detailed analysis on the commonly employed laguerre formula for approximating delay systems convergence proofs are provided and error bounds are constructed with respect to the l spl infin and l sub 2 norms

a parameter robust finite difference method for singularly perturbed delay parabolic partial differential equations a dirichlet boundary value problem for a delay parabolic differential equation is studied on a rectangular domain in the x t plane the second order space derivative is multiplied by a small singular perturbation parameter which gives rise to parabolic boundary layers on the two lateral sides of the rectangle a numerical method comprising a standard finite difference operator centred in space implicit in time on a rectangular piecewise uniform fitted mesh of n xxn t elements condensing in the boundary layers is proved to be robust with respect to the small parameter or parameter uniform in the sense that its numerical solutions converge in the maximum norm to the exact solution uniformly well for all values of the parameter in the half open interval 0 1 more specifically it is shown that the errors are bounded in the maximum norm by c n x 2ln 2n x n t 1 where c is a constant independent not only of n x and n t but also of the small parameter numerical results are presented which validate numerically this theoretical result and show that a numerical method consisting of the standard finite difference operator on a uniform mesh of n xxn t elements is not parameter robust

kolmogorov sinai entropy and lyapunov spectra of a hard sphere gas abstract the mixing behavior of a hard sphere gas has its origin in the exponential growth of small perturbations in phase space this instability is characterized by the so called lyapunov exponents in this work we compute full spectra of lyapunov exponents for the hard sphere gas for a wide range of densities and particle numbers by using a recently developed algorithm in the dilute gas regime the maximum lyapunov exponent is found to obey the krylov relation ln a formula exactly derived for the low density lorentz gas by dorfman and van beijeren we study the system size dependence and the effect of the fluid solid phase transition on the spectra in the second part of this work we describe and test a direct simulation monte carlo method dsmc for the computation of lyapunov spectra and present results for dilute hard sphere gases excellent agreement is obtained with the results of the deterministic simulations this suggests that the lyapunov instability of a hard sphere gas may be analyzed within the framework of kinetic theory

new solitons for the hirota equation and generalized higher order nonlinear schrodinger equation with variable coefficients in this paper multisoliton solutions of the hirota equation with variable coefficients are obtained by the darboux transformation based on the ablowitz kaup newell segur technology as an example we discuss the evolutional behaviour of a two soliton solution in a soliton control fibre system the results reveal that one may control the interaction between the pulses by choosing the third order dispersion parameters d4 and h appropriately meanwhile more generalized forms of bright soliton and dark soliton solutions of generalized higher order nonlinear schrodinger equations ghonlse with variable coefficients are obtained by the extended tanh function method moreover new bright and dark combined solitary wave kink solitary wave and m shaped solitary wave to ghonlse with variable coefficients are firstly reported in this paper especially the term proportional to 1 resulting from the group velocity decides the group velocity and the phase shift of these new solitary waves

renormalization of correlations in a quasiperiodically forced two level system quadratic irrationals generalizing from the case of golden mean frequency to a wider class of quadratic irrationals we extend our renormalization analysis of the self similarity of correlation functions in a quasiperiodically forced two level system we give a description of all piecewise constant periodic orbits of an additive functional recurrence generalizing that present in the golden mean case we establish a criterion for periodic orbits to be globally bounded and also calculate the asymptotic height of the main peaks in the correlation function

universal criterion for the breakup of invariant tori in dissipative systems the transition from quasiperiodicity to chaos is studied in a two dimensional dissipative map with the inverse golden mean rotation number on the basis of a decimation scheme it is argued that the minimal slope of the critical iterated circle map is proportional to the effective jacobian determinant approaching the zero jacobian determinant limit the factor of proportion becomes a universal constant numerical investigation on the dissipative standard map suggests that this universal number could become observable in experiments

solitonic excitations and interactions in an α helical protein modeled by three coupled nonlinear schrodinger equations with variable coefficients governing the molecular excitations associated with bioenergy transport in an helical protein through the soliton modes three coupled higher order nonlinear schrodinger equations with variable coefficients are investigated via symbolic computation using bell polynomials a bilinear form and backlund transformation are derived for this model furthermore explicit n soliton solutions are constructed in terms of the double wronskian determinant solitonic excitations are found to remain stable against the disorders of the parameters under certain constraints finally propagation characteristics and interactions of the solitonic excitations are discussed soliton amplitudes are related to the energy modulation coefficients and both the soliton properties and energy distribution during the interactions are affected by the inhomogeneity coefficients of the protein with time dependent disorders

kolmogorov sinai entropy for dilute gases in equilibrium we consider the density expansion of the kolmogorov sinai ks entropy per particle for a dilute gas in equilibrium and use methods from the kinetic theory of gases to compute the leading term for an equilibrium system the ks entropy hks is the sum of all of the positive lyapunov exponents characterizing the chaotic behavior of the gas we compute hks n where n is the number of particles in the gas this quantity has a density expansion of the form hks n5an 2ln n 1b1o n where n is the single particle collision frequency and nis the reduced number density of the gas the theoretical values for the coefficients a and b are compared with the results of computer simulations with excellent agreement for a and less than satisfactory agreement for b possible reasons for this difference in b are discussed s1063 651x 97 06911 0

localization problem in one dimension mapping and escape a one dimensional schr odinger equation in a discontinuous quasiperiodic potential is reduced to a recursion relation for transfer matrices and then to one for traces of these matrices when the potential is periodic the bandwidth goes to zero as an algebraic function of the period with a critical index which depends upon the potential strength this critical index is also evaluated as the solution to an escape rate problem for the recursion relations

time domain model reduction of time delay systems via orthogonal polynomial expansions abstract this paper investigates model reduction of time delay systems in the time domain based on the expansion of systems over orthogonal polynomials time delay systems are first expanded over generalized laguerre polynomials the nice properties of generalized laguerre polynomials lead to a direct system expansion and a sylvester equation with special structures which enables an efficient calculation of laguerre coefficients of systems projection methods are then adopted to generate reduced models and we show that a desired number of laguerre coefficients are preserved by reduced models further we extend the proposed method to general orthogonal polynomials where the relationship between taylor expansion and orthogonal polynomial expansion is examined to achieve the expansion of time delay systems systems with multiple delays and delays appearing in the derivative of the states are also discussed two numerical examples are simulated to showcase the efficiency of our approach

renormalization analysis of correlation properties in a quasiperiodically forced two level system we give a rigorous renormalization analysis of the self similarity of correlation functions in a quasiperiodically forced two level system more precisely the system considered is a quantum two level system in a time dependent field consisting of periodic kicks with amplitude given by a discontinuous modulation function driven in a quasiperiodic manner at golden mean frequency mathematically our analysis consists of a description of all piecewise constant periodic orbits of an additive functional recurrence we further establish a criterion for such orbits to be globally bounded functions in a particular example previously only treated numerically we further calculate explicitly the asymptotic height of the main peaks in the correlation function

multi soliton solutions and a backlund transformation for a generalized variable coefficient higher order nonlinear schrodinger equation with symbolic computation abstract in this paper by virtue of symbolic computation the investigation is made on a generalized variable coefficient higher order nonlinear schrodinger equation with varying higher order effects and gain or loss which can describe the femtosecond optical pulse propagation in a monomode dielectric waveguide a modified dependent variable transformation is introduced into the bilinear method to transform such an equation into a variable coefficient bilinear form based on the formal parameter expansion technique the multi soliton solutions of this equation are obtained through the bilinear form under sets of parametric constraints a backlund transformation in bilinear form is also obtained for the first time in this paper finally discussions on the analytic soliton solutions are given and various propagation situations are illustrated

renormalization for the harper equation for quadratic irrationals in this paper we use renormalization methods to study self similarity in the fluctuations i of the harper equation in the strong coupling limit for quadratic irrationals of the form a2 4 a 2 for a n using the decimation method we obtain a second order functional recurrence which we prove rigorously has an entire fixed point this fixed point governs the scaling of the fluctuations i in the strong coupling limit

types of solutions of the variable coefficient nonlinear schrodinger equation with symbolic computation by using hirota s bilinear method and symbolic computation solutions for a variable coefficient nonlinear schrodinger equation are obtained theoretically it is found that the type of the solutions changes with the different choices of the group velocity dispersion coefficient beta 2 z according to those solutions the relevant properties and features of physical and optical interest are illustrated in addition an effective technique for controlling the shape of the pulses is presented the results of this paper will be valuable to the study of the future development of ultrahigh rate and long distance optical communication systems

lyapunov instability local curvature and the fluid solid phase transition in two dimensional particle systems abstract we study the relation between the lyapunov spectrum and the multidimensional geometry of the potential energy surface in terms of the distribution of stable and unstable modes for different models for this purpose we determined lyapunov exponents for the so called correlated cell model and its smooth generalization as a function of the density for various energies in the smooth case averaged structural quantities such as the fraction of unstable modes the gaussian curvature and the riemannian curvature were computed and compared to the mechanical instability of the system in the sense of lyapunov a similar analysis was also carried out for 36 disk systems representing various fluid and solid states in all studied cases the most positive lyapunov exponent exhibits a maximum at the phase transition in agreement with results for the fluid solid transition in many particle systems

fluctuations and the many body lyapunov exponent an ab initio theoretical method is derived for calculating the maximal lyapunov exponent of an n body system obeying hamilton s equations the theory is developed in detail for a dilute gas it shows the lyapunov exponent to be a function of the time integral of the correlation function for fluctuations in the second derivative of the interparticle potential approximately a power frac 1 3 law we apply the theory to a one component plasma and derive the dependence of the lyapunov exponent on a plasma parameter

harper equation the dissipative standard map and strange nonchaotic attractors relationship between an eigenvalue problem and iterated maps the almost periodic eigenvalue problem described by the harper equation is connected to other classes of quasiperiodic behavior the dissipative dynamics on critical invariant tori and quasiperiodically driven maps firstly the strong coupling limit of the supercritical harper equation and the strong dissipation limit of the critical standard map play equivalent role in describing the universal characteristics of these systems secondly a simple transformation is used to relate the harper equation to a quasiperiodically forced one dimensional map in this case the localized eigenstates of the supercritical harper equation correspond to strange but nonchaotic attractors of the driven map furthermore the existence of localization in the eigenvalue problem is associated with the appearance of homoclinic points in the corresponding map

renormalization of correlations and spectra of a strange non chaotic attractor we study a simple nonlinear mapping with a strange nonchaotic attractor characterized by a singular continuous power spectrum we show that the symbolic dynamics is exactly described by a language generated from a suitable inflation rule we derive renormalization transformations for both the power spectrum and the autocorrelation function thus obtaining a quantitative description of the scaling properties the multifractal nature of the spectrum is also discussed

a garden of orchids a generalized harper equation at quadratic irrational frequencies we consider a generalized harper equation at quadratic irrational flux showing in the strong coupling limit the fluctuations of the exponentially decaying eigenfunctions are governed by the dynamics of a renormalization operator on a renormalization strange set this work generalizes previous analyses which have considered only the golden mean case projections of the renormalization strange sets are illustrated analogous to the orchid present in the golden mean case

a note on the numerical approximate solutions for generalized sylvester matrix equations with applications abstract in the present paper we propose a krylov subspace method for solving large and sparse generalized sylvester matrix equations the proposed method is an iterative projection method onto matrix krylov subspaces as a particular case we show how to adapt the ilu and the ssor preconditioners for solving large sylvester matrix equations numerical examples and applications to some pde s will be given

renormalization of correlations in a quasiperiodically forced two level system for a general class of modulation function we provide a renormalization analysis of correlations in a quasi periodically forced two level system in a time dependent field with periodic kicks whose amplitude is given by a general class of discontinuous modulation function for certain intensities of modulation we give a complete understanding of the autocorrelation function furthermore once the locations of the discontinuities of the modulation function are known aperiodic orbits lead to correlations on renormalization strange sets which are determined by two specified features of the modulation function of which there are only a finite number of variations

dimensions of structurally stable and critical strange non chaotic attractors we provide evidence that the box counting dimension of a structurally stable strange non chaotic attractor sna of pinched skew product type is equal to 2 by showing that it has non negligible area the argument presented is made more accurate in the study of a piecewise linear sna furthermore we provide evidence that the fractal dimension of a critical sna is not equal to 2 but in fact lies between 1 and 2 we numerically calculate the box counting dimension for several critical snas providing further evidence to support this conjecture

interpolation method for adapting reduced order models and application to aeroelasticity reduced order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real time analysis although most reduced order models can operate in near real time their construction can be computationally expensive as it requires accumulating a large number of system responses to input excitations furthermore reduced order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation together these two issues underline the need for a fast and robust method for adapting precomputed reduced order models to new sets of physical or modeling parameters to this effect this paper presents an interpolation method based on the grassmann manifold and its tangent space at a point that is applicable to structural aerodynamic aeroelastic and many other reduced order models based on projection schemes this method is illustrated here with the adaptation of computational fluid dynamics based aeroelastic reduced order models of complete fighter configurations to new values of the freestream mach number good correlations with results obtained from direct reduced order model reconstruction high fidelity nonlinear and linear simulations are reported thereby highlighting the potential of the proposed reduced order model adaptation method for near real time aeroelastic predictions using precomputed reduced order model databases

multi solitonic solutions for the variable coefficient variant boussinesq model of the nonlinear water waves abstract for the nonlinear and dispersive long gravity waves traveling in two horizontal directions with varying depth of the water we consider a variable coefficient variant boussinesq vcvb model with symbolic computation we construct the connection between the vcvb model and a variable coefficient ablowitz kaup newell segur vcakns system under certain constraints using the n fold darboux transformation of the vcakns system we present two sets of multi solitonic solutions for the vcvb model which are expressed in terms of the vandermonde like and double wronskian determinants respectively dynamics of those solutions are analyzed and graphically discussed such as the parallel solitonic waves shape changing collision head on collision fusion fission behavior and elastic fusion coupled interaction

chaotic correlations in barrier billiards with arbitrary barriers we study autocorrelation functions in symmetric barrier billiards for golden mean trajectories with arbitrary barriers renormalization analysis reveals the presence of a chaotic invariant set and thus that for a typical barrier there are chaotic correlations the chaotic renormalization set is the analogue of the so called orchid that arises in a generalized harper equation

solitonic solutions and backlund transformation for the inhomogeneous n coupled nonlinear schrodinger equations abstract governing the simultaneous propagation of n fields in optical fibers and plasmas the inhomogeneous n coupled nonlinear schrodinger equations are with symbolic computation analytically investigated in this paper as one of the important integrable properties the explicit one and two solitonic solutions for the inhomogeneous n coupled nonlinear schrodinger equations are derived using the bilinear method with illustrated graphics additionally the backlund transformation in bilinear form is constructed

on a direct bilinearization method kaup s higher order water wave equation as a modified nonlocal boussinesq equation a systematic procedure for the bilinearization of classes of soliton equations is developed with the help of a generalization of bell s exponential polynomials 1934 application of this procedure to kaup s higher order wave equation 1975 discloses several links with other soliton systems in particular it is found that the kaup equation is the modified version of a sech square soliton system which constitutes an alternative to the good boussinesq equation

correlation between the kolmogorov sinai entropy and the self diffusion coefficient in simple liquids molecular dynamics simulations were performed for soft and hard sphere systems for number densities ranging from 0 5 to 1 0 and the kolmogorov sinai entropy ks entropy and self diffusion coefficients were calculated it is found that the ks entropy when expressed in terms of average collision frequency is uniquely related to the self diffusion coefficient by a simple scaling law the dependence of the ks entropy on average collision frequency and number density was also explored numerical results show that the scaling laws proposed by dzugutov and by beijeren dorfman posch and dellago can be applied to both soft and hard sphere systems by changing to more generalized forms

an iterative svd krylov based method for model reduction of large scale dynamical systems abstract in this paper we propose a model reduction algorithm for approximation of large scale linear time invariant dynamical systems the method is a two sided projection combining features of the singular value decomposition svd based and the krylov based model reduction techniques while the svd side of the projection depends on the observability gramian the krylov side is obtained via iterative rational krylov steps the reduced model is asymptotically stable matches certain moments and solves a restricted h 2 minimization problem we present modifications to the proposed approach for employing low rank gramians in the reduction step and also for reducing discrete time systems several numerical examples from various disciplines verify the effectiveness of the proposed approach it performs significantly better than the q cover a yousouff r e skelton covariance equivalent realizations with applications to model reduction of large scale systems in c t leondes ed control and dynamic systems vol 22 academic press 1985 pp 273 348 a yousouff d a wagie r e skelton linear system approximation via covariance equivalent realizations journal of math anal and appl 196 1985 91 115 and the least squares s gugercin a c antoulas model reduction of large scale systems by least squares linear algebra appl 415 2 3 2006 290 321 methods that have a similar projection structure to the proposed method also in terms of both the h 2 and h error measures its performance is comparable to or sometimes better than that of balanced truncation moreover the method proves to be robust with respect to the perturbations due to usage of approximate gramians

a practical method for calculating largest lyapunov exponents from small data sets detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest lyapunov exponent lyapunov exponents quantify the exponential divergence of initially close state space trajectories and estimate the amount of chaos in a system we present a new method for calculating the largest lyapunov exponent from an experimental time series the method follows directly from the definition of the largest lyapunov exponent and is accurate because it takes advantage of all the available data we show that the algorithm is fast easy to implement and robust to changes in the following quantities embedding dimension size of data set reconstruction delay and noise level furthermore one may use the algorithm to calculate simultaneously the correlation dimension thus one sequence of computations will yield an estimate of both the level of chaos and the system complexity

on the combinatorics of the hirota d operators a generic formula is presented which relates the hirota d operators to simple combinatorics particular classes of partition polynomials bell polynomials and generalizations are found to play an important role in the characterization of bilinearizable equations as a consequence it is shown that bilinear backlund transformations for single field bilinearizable equations linearize systematically into corresponding laxpairs

completely integrable nonlinear schrodinger type equations on moving space curves using the lamb formalism we show that some completely integrable homogeneous and inhomogeneous nonlinear schr odinger nls type equations such as derivative nls extended nls higher order nls inhomogeneous nls circularly and radially symmetric nls and generalized inhomogeneous radially symmetric nls equations can be related to certain types of moving helical space curves

correlations and spectra of strange non chaotic attractors we consider correlations and spectra of strange nonchaotic attractors in quasiperiodically driven nonlinear systems it is demonstrated that a self similar autocorrelation function and a singular continuous spectrum may be observed in such systems the properties of correlations and spectra depend very subtly on the rotation number of the quasiperiodic force and on the parameters of the system for some parameter values the usual discrete spectrum is restored

n fold darboux transformation and solitonic interactions of a variable coefficient generalized boussinesq system in shallow water abstract under consideration in this paper is a variable coefficient generalized boussinesq system for the long weakly nonlinear and weakly dispersive surface waves in shallow water with the aid of symbolic computation n fold darboux transformation n dt is constructed for that system analytic solutions of the system are obtained via the n dt elastic interactions of three bell shaped and periodic bell shaped solitons are obtained fusion interactions and periodic fusion fission interactions of the solitary waves are graphically analyzed which are inelastic

time domain model order reduction of general orthogonal polynomials for linear input output systems for a class of large linear input output systems we present a new model order reduction algorithm based on general orthogonal polynomials in the time domain the main idea of the algorithm is first to expand the unknown state variables in the space spanned by orthogonal polynomials then the coefficient terms of polynomial expansion are calculated by a recurrence formula the basic procedure is to use the coefficient terms to generate a projection matrix many classic methods with orthogonal polynomials are special cases of the general approach the proposed approach has a good computational efficiency and preserves the stability and passivity under certain condition numerical experiments are reported to verify the theoretical analysis

model order reduction methods for coupled systems in the time domain using laguerre polynomials in this paper based on laguerre polynomials we present new methods for model reduction of coupled systems in the time domain by appropriately selected projection matrices a reduced order system is produced to retain the topology structure of the original system meanwhile it preserves a desired number of laguerre coefficients of the system s output thereby providing good approximation accuracy we also study the two sided projection method in the time domain as well as the stability of reduced order systems two numerical examples are used to illustrate the efficiency of the proposed methods

singular continuous spectra in a pseudointegrable billiard the pseudointegrable barrier billiard invented by hannay and mccraw j phys a 23 887 1990 a rectangular billiard with line segment barrier placed on a symmetry axis is generalized it is demonstrated that the fourier spectrum of a typical function on the phase space exhibits a singular continuous component

pre specified performance based model reduction for time varying delay systems in fuzzy framework this paper attempts to provide a new solution to the model approximation problem for dynamic systems with time varying delays under the fuzzy framework for a given high order system our focus is on the construction of a reduced order model which approximates the original one in a prescribed error performance level and guarantees the asymptotic stability of the corresponding error system based on the reciprocally convex technique a less conservative stability condition is established for the dynamic error system with a given error performance index furthermore the reduced order model is eventually obtained by applying the projection approach which converts the model approximation into a sequential minimization problem subject to linear matrix inequality constraints by employing the cone complementary linearization algorithm finally two numerical examples are provided to illustrate the effectiveness of the proposed method

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

the development of variable step symplectic integrators with application to the two body problem the authors develop and test variable step symplectic runge kutta nystrom algorithms for the integration of hamiltonian systems of ordinary differential equations numerical experiments suggest that for symplectic formulae moving from constant to variable stepsizes results in a marked decrease in efficiency on the other hand symplectic formulae with constant stepsizes may outperform available standard nonsymplectic variable step codes for the model situation consisting in the long time integration of the two body problem our experimental findings are backed by theoretical analysis

renormalisation of correlations in a barrier billiard quadratic irrational trajectories abstract we present an analysis of autocorrelation functions in symmetric barrier billiards using a renormalisation approach for quadratic irrational trajectories depending on the nature of the barrier this leads to either self similar or chaotic behaviour in the self similar case we give an analysis of the half barrier and present a detailed calculation of the locations asymptotic heights and signs of the main peaks in the autocorrelation function then we consider arbitrary barriers illustrating that typically these give rise to chaotic correlations of the autocorrelation function which we further represent by showing the invariant sets associated with these correlations our main ingredient here is a functional recurrence which has been previously derived and used in work on the harper equation strange non chaotic attractors and a quasi periodically forced two level system

correlation between kolmogorov sinai entropy and self diffusion coefficient in simple fluids the relationship between the kolmogorov sinai entropy h mathrm ks and the self diffusion coefficient d is studied for two classical simple fluid systems with purely repulsive potentials one system with a wayne chandler anderson potential and the other with a hard sphere potential numerical simulation data for h mathrm ks and d normalized by the average collision frequency ensuremath nu and the diameter of the particle ensuremath sigma as natural units of time and distance reveal that in the region spanning from normal liquid up to near solidification 0 50l ensuremath rho l 0 93 the kolmogorov sinai entropy has a power law dependeney on the self diffusion coefficient of the form h mathrm ks ensuremath nu ensuremath propto d ensuremath sigma 2 ensuremath nu ensuremath eta in which ensuremath eta is independent of density and temperature

golden mean renormalization for the harper equation the strong coupling fixed point we construct a renormalization fixed point corresponding to the strong coupling limit of the golden mean harper equation we give an analytic expression for this fixed point establish its existence and uniqueness and verify properties previously seen only in numerical calculations the spectrum of the linearization of the renormalization operator at this fixed point is also explicitly determined this strong coupling fixed point also helps describe the onset of a strange nonchaotic attractor in quasiperiodically forced systems

data driven interpolation of dynamical systems with delay abstract we present a data driven realization for systems with delay which generalizes the loewner framework the realization is obtained with low computational cost directly from measured data of the transfer function the internal delay is estimated by solving a least square optimization over some sample data our approach is validated by several examples which indicate the need for preserving the delay structure in the reduced model

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

soliton solutions and a backlund transformation for a generalized nonlinear schrodinger equation with variable coefficients from optical fiber communications under investigation in this paper is a generalized nonlinear schrodinger model with variable dispersion nonlinearity and gain loss which could describe the propagation of optical pulse in inhomogeneous fiber systems by employing the hirota method one and two soliton solutions are obtained with the aid of symbolic computation furthermore a general formula which denotes multi soliton solutions is given some main properties of the solutions are discussed simultaneously as one important property of nonlinear evolution equation the backlund transformation in bilinear form is also constructed which is helpful on future research and as far as we know is firstly proposed in this paper

ultrashort self similar solutions of the cubic quintic nonlinear schrodinger equation with distributed coefficients in the inhomogeneous fiber by means of the similarity transformation we obtain exact self similar solutions similaritons of the generalized cubic quintic cq nonlinear schrodinger equation with spatially inhomogeneous group velocity dispersion cq nonlinearity and amplification or attenuation exact balance conditions between the dispersion nonlinearity and the gain loss have been obtained as an example we investigate their propagation dynamics in the dispersion decreasing fiber ddf considering the fluctuation of the fiber parameter in real application the exact balance conditions do not satisfy and so we perform direct numerical analysis with initial 10 white noise for the bright similariton in both the ddf and the periodic distributed amplification system numerical calculations indicate stable propagation of the bright similariton over tens of dispersion lengths these ultrashort self similar optical waves are potentially useful for all optical data processing schemes and the design of beam compressors and amplifiers