ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

spatial structure of time periodic solutions of the ginzburg landau equation abstract we study solutions of the form a x t a x ei t of the complex ginzburg landau equation i a t a xx a a 2 a near those of the integrable limit the nonlinear schrodinger equation i a t a xx a 2 a real we show that spatially periodic and quasiperiodic solutions as well as heteroclinic orbits to them exist in addition to the spatially decaying homoclinic solitions found by hocking and stewartson 1972 we speculate briefly on spatially chaotic solutions

parameter identification in noisy extended systems a hydrodynamic case abstract this paper is concerned with the robustness of parameter identification methods with respect to the noise levels typically found in experiments more precisely we focus on the case of an extended nonlinear system a system of coupled local maps akin to a discretized complex ginzburg landau equation modeling a wake experiment after a brief description of this hydrodynamic experiment as well as of the associated cost function and synthetic data generation we introduce two inversion methods a one time step approach and a more sophisticated n time step optimization procedure solved by a backpropagation method the one time step approach reduces to a small linear system for the unknown parameters while the n time step approach involves a backpropagation equation for a set of lagrage multipliers the sensitivity of each method with respect to noise is then discussed while the n time step method is very robust even with large amounts of noise the one time step approach is shown to be affected by small noise levels

a moderate deviation principle for 2 d stochastic navier stokes equations in this paper we prove a central limit theorem and establish a moderate deviation principle for twodimensional stochastic navier stokes equations with multiplicative noise the weak convergence method plays an important role

moderate deviations of dependent random variables related to clt this paper consists of three parts in the first part we find a common condition the c 2 regularity both for clt and for moderate deviations we show that this condition is verified in two important situations the lee yang theorem case and the fkg system case in the second part we apply the previous results to the additive functionals of a markov process by means of feynman kac formula and kato s analytic perturbation theory we show that the lee yang theorem holds under the assumption that 1 is an isolated simple and the only eigenvalue with modulus 1 of the operator p 1 acting on an appropriate banach space be c b e l 2 the last part is devoted to some applications to statistical mechanical systems where the c 2 regularity becomes a property of the pressure functionals and the two situations presented above become exactly the lee yang theorem case and the fkg system case we shall discuss in detail the ferromagnetic model and give some general remarks on some other models

probability and pathwise order of convergence of a semidiscrete scheme for the stochastic manakov equation it is well accepted by physicists that the manakov pmd equation is a good model to describe the evolution of nonlinear electric fields in optical fibers with randomly varying birefringence in the regime of the diffusion approximation theory an effective asymptotic dynamics has recently been obtained to describe this evolution this equation is called the stochastic manakov equation in this article we propose a semidiscrete version of a crank nicolson scheme for this limit equation and we analyze the approximation error allowing sufficient regularity of the initial data we prove that the numerical scheme has probability order 1 2 and almost sure order 1 2 epsilon for any epsilon in 0 1 2

collective synchronization as a method of learning and generalization from sparse data we propose a method for extracting general features from multivariate data using a network of phase oscillators subject to an analogue of the kuramoto model for collective synchronization in this method the natural frequencies of the oscillators are extended to vector quantities to which multivariate data are assigned the common frequency vectors of the groups of partially synchronized oscillators are interpreted to be the template vectors representing the general features of the data set we show that the proposed method becomes equivalent to the self organizing map algorithm devised by kohonen when the governing equations are linearized about their solutions of partial synchronization as a case study to test the utility of our method we applied it to care needs certification data in the japanese public long term care insurance program and found major general patterns in the health status of the elderly needing nursing care

stochastic navier stokes equations with caputo derivative driven by fractional noises abstract in this paper we consider the extended stochastic navier stokes equations with caputo derivative driven by fractional brownian motion we firstly derive the pathwise spatial and temporal regularity of the generalized ornstein uhlenbeck process then we discuss the existence uniqueness and holder regularity of mild solutions to the given problem under certain sufficient conditions which depend on the fractional order and hurst parameter h the results obtained in this study improve some results in existing literature

constructing nonautonomous differential equations from experimental time series an approach to constructing model differential equations of harmonically driven systems is proposed it is a modification of the standard global reconstruction technique an algebraic polynomial which coefficients depend on time is used for approximation efficiency and details of the approach are demonstrated by various numerical and natural examples

modulations of perturbed kdv wavetrains the modulations of n phase korteweg devries kdv wavetrains in the presence of external perturbations is investigated an invariant representation of these modulation equations in terms of differentials on a riemann surface is derived from averaged perturbed conservation laws in particular the explicit dependence of the representation on the external perturbation is obtained this invariant representation is used to place the equation in a riemann diagonal form whose dependence on the external perturbation is explicitly computed

a noise reduction method for signals from nonlinear systems abstract a numerical algorithm is presented for the purpose of reducing noise from a discretely sampled input signal where the underlying signal of interest has a broadband spectrum it is designed to be useful even if the clean signal is contaminated with 100 or more noise signal to noise ratio less than or equal to zero the method is based on time delay embedding using coordinates generated by local low pass filtering which we call a low pass embedding the singular value decomposition is then used locally in embedding space to distinguish between the dynamics and the noise the algorithm is evaluated for chaotic signals generated by the lorenz and rossler systems to which gaussian white noise has been added

chaos synchronization of unknown inputs takagi sugeno fuzzy this note deals with the observer design for unknown inputs takagi sugeno fuzzy systems with application in chaos synchronization and secure communications using lyapunov method sufficient synthesis conditions are given in linear matrix inequalities lmi terms the given design is then applied in a chaos synchronization and secure communication procedure where the plaintext message is encrypted using chaotic signals at the drive system side the plaintext is retrieved via the designed unknown input observer a pole placement in an lmi region is also considered to improve the observer performances an illustrative example is given to show the effectiveness of the given design

asymptotic analysis of weakly nonlinear bessel gaus beams abstract in this paper we investigate the propagation of conical waves in nonlinear media in particular we are interested in the effects resulting from applying a gaussian apodization to an ideal nondiffracting wave first we present a multiple scales approach to derive amplitude equations for weakly nonlinear conical waves from a governing equation of cubic nonlinear schrodinger type from these equations we obtain asymptotic solutions for the linear and the weakly nonlinear problem for which we state several uniform estimates that describe the deviation from the ideal nondiffracting solution moreover we show numerical simulations based on an implementation of our amplitude equations to support and illustrate our analytical results

building dynamical models from data and prior knowledge the case of the first period doubling bifurcation this paper reviews some aspects of nonlinear model building from data with gray box and without black box prior knowledge the model class is very important because it determines two aspects of the final model namely i the type of nonlinearity that can be accurately approximated and ii the type of prior knowledge that can be taken into account such features are usually in conflict when it comes to choosing the model class the problem of model structure selection is also reviewed it is argued that such a problem is philosophically different depending on the model class and it is suggested that the choice of model class should be performed based on the type of a priori available a procedure is proposed to build polynomial models from data on a poincare section and prior knowledge about the first period doubling bifurcation for which the normal form is also polynomial the final models approximate dynamical data in a least squares sense and by design present the first period doubling bifurcation at a specified value of parameters the procedure is illustrated by means of simulated examples

symmetry classes of variable coefficient nonlinear schrodinger equations a variable coefficient nonlinear schrodinger vcnls equation involving three arbitrary complex functions of space time in 1 1 dimensions is analysed from the point of view of its symmetries all equations of the type studied having non trivial lie point symmetry groups g are identified the symmetry group is shown to be at most five dimensional and only when the equation is equivalent to the nls equation itself or to a rather special complex ginzburg landau equation lie point transformations are used to obtain solutions of specific vcnls equations that should be of interest in nonlinear optics or other branches of physics

a universal prior for integers and estimation by minimum description length of the number of bits required to write down the observed data has been reformulated to extend the classical maximum likelihood principle the principle permits estimation of the number of the parameters in statistical models in addition to their values and even of the way the parameters appear in the models i e of the model structures the principle rests on a new way to interpret and construct a universal prior distribution for the integers which makes sense even when the parameter is an individual object truncated realvalued parameters are converted to integers by dividing them by their precision and their prior is determined from the universal prior for the integers by optimizing the precision 1 introduction in this paper we study estimation based upon the principle of minimizing the total number of binary digits required to rewrite the observed data when each observation is given with some precision instead of attempting at an absolutely shortest description which would be futile we look for the optimum relative to a class of parametrically given distributions this minimum description length mdl principle which we introduced in a less comprehensive form in 25 turns out to degenerate to the more familiar maximum likelihood ml principle in case the number of parameters in the models is fixed so that the description length of the parameters themselves can be ignored in another extreme case where the parameters determine the data it similarly degenerates to jaynes s principle of maximum entropy 14 but the main power of the new criterion is that it permits estimates of the entire model its parameters their number and even the way the parameters appear in the model i e the model structure hence there will be no need to supplement the estimated parameters with a separate hypothesis test to decide whether a model is adequately parameterized or perhaps over parameterized

information and entropy in strange attractors a technique for analyzing time series data from experiments is presented that provides estimates of four basic characteristics of a system 1 the measure theoretic entropy 2 the accuracy of the measurements 3 the number of measurements necessary to specify a system state and 4 the best delay time t to use in order to construct phase space portraits by the method of delays these characteristics are obtained by separating the entropy of measurements into a part due to noise and parts due to deterministic effects for the technique to work the noise associated with each measurement must be independent of the noise associated with all other measurements an algorithm for implementing the analysis is presented with three examples

large deviations for a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the non gaussian stochastic reaction diffusion equation srde partial t nu varepsilon mathscr l nu varepsilon f x nu varepsilon varepsilon sigma x nu varepsilon ddot w tx as a random perturbation of the deterministic rde partial t nu 0 mathscr l nu 0 f x nu 0 here the space variable takes values on the unit circle s 1 and mathscr l is a strongly elliptic second order operator with constant coefficients the functions f and sigma are sufficiently regular so that there is a unique solution to the above srde for any continuous initial condition we also assume that there are positive constants m and m such that m leq sigma x y leq m for all x in s 1 and all y in mathbb r the perturbation ddot w tx is the formal derivative of a brownian sheet it is known that if the initial condition is continuous then the solution will also be continuous and moreover if the initial condition is assumed to be holder continuous of exponent kappa for some 0 kappa frac 1 2 then the solution will be holder continuous of exponent kappa 2 as a function of t x in this paper we establish the large deviations principle for nu varepsilon in the holder norm of exponent kappa 2 when the initial condition is holder continuous of exponent kappa for any 0 kappa frac 1 2 and when the initial condition is assumed only to be continuous we establish the large deviations principle for nu varepsilon in the supremum norm moreover we prove that these large deviations principles are uniform with respect to the initial condition

convection patterns in large aspect ratio systems abstract a theory which should have widespread application is developed to treat the statics and slow dynamics of patterns of convective rolls encountered in large aspect ratio rayleigh benard boxes for case of presentation the theory is developed using model equations wavenumber selection the shape of patterns stability and the time dependence resulting from long wavelength instabilities are discussed the effects of adding the local mean drift important in low prandtl number situations are investigated our theory includes the notion of the busse stability balloon reduces near critical values of the stress parameter to the newell whitehead segel equations and contains the pomeau manneville phase equation it also gives a detailed description of the way in which the field amplitude which is slaved to the phase gradient away from threshold becomes an independent order parameter near the critical point

difference equations versus differential equations a possible equivalence for the rossler system abstract when a set of nonlinear differential equations is investigated most of time there is no analytical solution and only numerical integration techniques can provide accurate numerical solutions in a general way the process of numerical integration is the replacement of a set of differential equations with a continuous dependence on the time by a model for which the time variable is discrete in numerical investigations a fourth order runge kutta integration scheme is usually sufficient nevertheless sometimes a set of difference equations may be required and in this case standard schemes like the forward euler backward euler or central difference schemes are used the major problem encountered with these schemes is that they offer numerical solutions equivalent to those of the set of differential equations only for sufficiently small integration time steps in some cases it may be of interest to obtain difference equations with the same type of solutions as for the differential equations but with significantly large time steps nonstandard schemes as introduced by mickens nonstandard finite difference models of differential equations world scientific 1994 allow to obtain more robust difference equations in this paper using such nonstandard scheme we propose some difference equations as discrete analogues of the rossler system for which it is shown that the dynamics is less dependent on the time step size than when a nonstandard scheme is used in particular it has been observed that the solutions to the discrete models are topologically equivalent to the solutions to the rossler system as long as the time step is less than the threshold value associated with the nyquist criterion

large deviations for stochastic reaction diffusion systems with multiplicative noise and non lipshitz reaction term following classical work by freidlin trans amer math soc 1988 305 665 657 and subsequent works by sowers ann probab 1992 20 504 537 and peszat probab theory related fields 1994 98 113 136 we prove large deviation estimates for the small noise limit of systems of stochastic reaction diffusion equations with globally lipschitz but unbounded diffusion coefficients however assuming the reaction terms to be only locally lipschitz with polynomial growth this generalizes results of the above mentioned authors our results apply in particular to systems of stochastic ginzburg landau equations with multiplicative noise

degenerate hyperbolic conservation laws with dissipation reduction to and validity of a class of burgers type equations a conservation law is said to be degenerate or critical if the jacobian of the flux vector evaluated on a constant state has a zero eigenvalue in this paper it is proved that a degenerate conservation law with dissipation will generate dynamics on a long time scale that resembles burger s dynamics the case of k fold degeneracy is also treated and it is shown that it leads to a reduction to a quadratically coupled k fold system of burgers type equations validity of the reduction and existence for the reduced system is proved in the class of uniformly local spaces thereby capturing both finite and infinite energy solutions the theory is applied to some examples from stratified shallow water hydrodynamics that model the birth of hydraulic jumps

phase dynamics in the complex ginzburg landau equation abstract for 1 stable time periodic solutions a x t aqeiqx i qt are the locally preferred planform for the complex ginzburg landau equation t a a 1 i x 2 a 1 i a a 2 in order to describe the spatial global behavior an evolution equation for the local wave number q can be derived formally the local wave number q satisfies approximately a conservation law q h q it is the purpose of this paper to explain the extent to which the conservation law is valid by proving estimates for this formal approximation

an envelope soliton problem the direct scattering problem of zakharov and shabat for the nonlinear schrodinger equation iu t u xx kappa u 2 u 0 is solved for the initial condition u text sech x the asymptotic solution t uparrow infty is a bound state of n centered solitons all moving at the same speed where n is the integral part of kappa 2 1 2

large deviation principles for 2 d stochastic navier stokes equations driven by levy processes abstract in this paper we establish a large deviation principle for the two dimensional stochastic navier stokes equations driven by levy processes which involves the study of the levy noise and the investigation of the effect of the highly nonlinear unbounded drifts

moderate deviations of minimum contrast estimators under contamination since statistical models are simplifications of reality it is important in estimation theory to study the behavior of estimators also under distributions slightly different from the proposed model in testing theory when dealing with test statistics where nuisance parameters are estimated knowledge of the behavior of the estimators of the nuisance parameters is needed under alternatives to evaluate the power in this paper the moderate deviation behavior of minimum contrast estimators is investigated not only under the supposed model but also under distributions close to the model a particular example is the multivariate maximum likelihood estimator determined within the proposed model the set up is quite general including for instance also discrete distributions the rate of convergence under alternatives is determined both when comparing the minimum contrast estimator with a natural parameter in the parameter space and when comparing it with the proposed true value in the parameter space it turns out that under the model the asymptotic optimality of the maximum likelihood estimator in the local sense continues to hold in the moderate deviation area

stochastic 2d primitive equations central limit theorem and moderate deviation principle abstract in this paper we establish a central limit theorem and a moderate deviation for two dimensional stochastic primitive equations driven by multiplicative noise this is the first result about the limit theorem and the moderate deviations for stochastic primitive equations the proof of the results relies on the weak convergence method and some delicate and careful a p r i o r i estimates

mean field effects for counterpropagating traveling wave solutions of reaction diffusion systems in many problems e g in combustion or solidification one observes traveling waves that propagate with constant velocity and shape in the x direction say are independent of y and z and describe transitions between two equilibrium states e g the burned and the unburned reactants as parameters of the system are varied these traveling waves can become unstable and give rise to waves having additional structure such as traveling waves in the y and z directions which can themselves be subject to instabilities as parameters are further varied to investigate this scenario we consider a system of reaction diffusion equations with a traveling wave solution as a basic state we determine solutions bifurcating from the basic state that describe counterpropagating traveling waves in directions orthogonal to the direction of propagation of the basic state and determine their stability specifically we derive long wave modulation equations for the amplitudes of the counterpropagating traveling waves that

high order exponential integrators for nonlinear schrodinger equations with application to rotating bose einstein condensates this article deals with the numerical integration in time of nonlinear schrodinger equations the main application is the numerical simulation of rotating bose einstein condensates the authors perform a change of unknown so that the rotation term disappears and they obtain as a result a nonautonomous nonlinear schrodinger equation they consider exponential integrators such as exponential runge kutta methods and lawson methods they provide an analysis of the order of convergence and some preservation properties of these methods in a simplified setting and they supplement their results with numerical experiments with realistic physical parameters moreover they compare these methods with the classical split step methods applied to the same problem

analytical and numerical aspects of certain nonlinear evolution equations ii numerical nonlinear schrodinger equation abstract various numerical methods are employed in order to approximate the nonlinear schrodinger equation namely i the classical explicit method ii hopscotch method iii implicit explicit method iv crank nicolson implicit scheme v the ablowitz ladik scheme vi the split step fourier method f tappert and vii pseudospectral fourier method fornberg and whitham comparisons between the ablowitz ladik scheme which was developed using notions of the inverse scattering transform and the other utilized schemes are obtained

modeling and nonlinear parameter estimation with kronecker product representation for coupled oscillators and spatiotemporal systems we focus here on the question of global modeling that is building an ordinary differential equation ode of minimal dimensions which models a given multivariate time dependent data set we give proof of convergence of the given method we will furthermore introduce a new method which adapts the least squares best approximation by a kronecker product representation c f van loan n pitsianis approximation with kronecker products www math tkk fi mhuhtane kronecker ps to analyze underlying structure when it exists as a system of coupled oscillators several examples are presented including diffusively coupled chaotic systems some interesting prospective problems will be briefly discussed in the conclusions

pseudo spectral solution of nonlinear schro dinger equations abstract we compare four pseudo spectral split step methods for solving a class of nonlinear schrodinger nls equations the importance of observing the l 2 invariance of the continuous problem is demonstrated through numerical experiments the best performance is obtained by transforming the given equation to an nls equation where two of the coefficients satisfy a simple algebraic relationship the problem can be solved efficiently in terms of the new variables and the l 2 norm of the computed solution is time invariant

spectral splitting method for nonlinear schrodinger equations with singular potential we consider the time dependent one dimensional nonlinear schrodinger equation with pointwise singular potential by means of spectral splitting methods we prove that the evolution operator is approximated by the lie evolution operator where the kernel of the lie evolution operator is explicitly written this result yields a numerical procedure which is much less computationally expensive than multi grid methods previously used furthermore we apply the lie approximation in order to make some numerical experiments concerning the splitting of a soliton interaction among solitons and blow up phenomenon

modeling and synchronizing chaotic systems from time series data the problem of obtaining ordinary differential equations ode s from chaotic time series data is addressed the vector fields for the ode s are polynomials constructed from a basis set that is orthonormal on the data the method for constructing the model is similar to the integration of ode s using adams predictor corrector integration the method is compared to the usual euler model and is shown to be accurate for much larger sampling intervals in addition the method used to construct the model is capable of determining the optimal polynomial vector field for the given data finally we demonstrate that it is possible to synchronize in the sense of fujisaka and yamada prog theor phys 69 32 1983 as well as pecora and carroll phys rev lett 64 821 1990 phys rev a 44 2374 1991 the model to a time series synchronization is used as a nontrivial test to determine how close the model vector field is to the true vector field implications and possible applications of synchronization are discussed

numerical methods based on additive splittings for hyperbolic partial differential equations we derive and analyze several methods for systems of hyperbolic equations with wide ranges of signal speeds these techniques are also useful for problems whose coefficients have large mean values about which they oscillate with small amplitude our methods are based on additive splittings of the operators into components that can be approximated independently on the different time scales some of which are sometimes treated exactly the efficiency of the splitting methods is seen to depend on the error incurred in splitting the exact solution operator this is analyzed and a technique is discussed for reducing this error through a simple change of variables a procedure for generating the appropriate boundary data for the intermediate solutions is also presented

split step methods for the solution of the nonlinear schro dinger equation a split step method is used to discretize the time variable for the numerical solution of the nonlinear schrodinger equation the space variable is discretized by means of a finite difference and a fourier method these methods are analyzed with respect to various physical properties represented in the nls in particular it is shown how a conservation law dispersion and instability are reflected in the numerical schemes analytical and numerical instabilities of wave train solutions are identified and conditions which avoid the latter are derived these results are demonstrated by numerical examples

on the non equivalence of observables in phase space reconstructions from recorded time series in practical problems of phase space reconstruction it is usually the case that the reconstruction is much easier using a particular recorded scalar variable this seems to contradict the general belief that all variables of a dynamical system are equivalent in phase space reconstruction problems this paper will argue that in many cases the choice of a particular scalar time series from which to reconstruct the original dynamics could be critical it is argued that different dynamical variables do not provide the same level of information observability of the underlying dynamics and as a consequence the quality of a global reconstruction critically depends on the recorded variable examples in which the choice of observables is critical are discussed and the level of information contained in a given variable is quantified in the case where the original system is known a clear example of such a situation arises in the r ossler system for which the performance of a global vector field reconstruction technique is investigated using time series of variables x y or z taken one at a time

sparse model from optimal nonuniform embedding of time series an approach to obtaining a parsimonious polynomial model from time series is proposed an optimal minimal nonuniform time series embedding schema is used to obtain a time delay kernel this scheme recursively optimizes an objective functional that eliminates a maximum number of false nearest neighbors between successive state space reconstruction cycles a polynomial basis is then constructed from this time delay kernel a sparse model from this polynomial basis is obtained by solving a regularized least squares problem the constraint satisfaction problem is made computationally tractable by keeping the ratio between the number of constraints to the number of variables small by using fewer samples spanning all regions of the reconstructed state space this helps the structure selection process from an exponentially large combinatorial search space a forward stagewise algorithm is then used for fast discovery of the optimization path results are presented for the mackey glass system

model order reduction for nonlinear schrodinger equation we apply the proper orthogonal decomposition pod to the nonlinear schrodinger nls equation to derive a reduced order model the nls equation is discretized in space by finite differences and is solved in time by structure preserving symplectic mid point rule a priori error estimates are derived for the pod reduced dynamical system numerical results for one and two dimensional nls equations coupled nls equation with soliton solutions show that the low dimensional approximations obtained by pod reproduce very well the characteristic dynamics of the system such as preservation of energy and the solutions

numerical studies on the split step finite difference method for nonlinear schrodinger equations in this study we use the split step finite difference method to solve various nonlinear schrodinger equations including coupled ones we study the numerical accuracy of the method detailed numerical results including higher dimensions show that the method provides accurate and stable solutions for nonlinear schrodinger equations

order estimates in time of splitting methods for the nonlinear schrodinger equation in this paper we consider the nonlinear schrodinger equation u t i delta u f u 0 in two dimensions we show by an operator theoretic proof that the well known lie and strang formulae which are splitting methods are approximations of the exact solution of order 1 and 2 in time

justification of the coupled mode asymptotics for localized wavepackets in the periodic nonlinear schrodinger equation abstract we consider wavepackets composed of two modulated carrier bloch waves with opposite group velocities in the one dimensional periodic nonlinear schrodinger gross pitaevskii equation these can be approximated by first order coupled mode equations cmes for the two slowly varying envelopes under a suitably selected periodic perturbation of the periodic structure the cmes possess a spectral gap of the corresponding spatial operator and allow families of exponentially localized solitary waves parametrized by velocity this leads to a family of approximate solitary waves in the periodic nonlinear schrodinger equation besides a formal derivation of the cmes a rigorous justification of the approximation and an error estimate in the supremum norm are provided several numerical tests corroborate the analysis

analytical and numerical aspects of certain nonlinear evolution equations i analytical abstract nonlinear partial difference equations are obtained which have as limiting forms the nonlinear schrodinger korteweg devries and modified korteweg devries equations these difference equations have a number of special properties they are constructed by methods related to the inverse scattering transform they can be used as a basis for numerical schemes to the associated nonlinear evolution equations experiments have shown that they compare very favorably with other known numerical methods papers ii iii in paper ii the ablowitz ladik scheme for the nonlinear schrodinger equation is compared to other known numerical schemes and generally proved to be faster than all utilized finite difference schemes but somewhat slower than the finite fourier pseudospectral methods in paper iii a proposed scheme for the korteweg de vries equation proved to be faster than both the finite difference and finite fourier methods already considered

slow time periodic solutions of the ginzburg landau equation in this paper we study the behaviour of solutions of the form if z t q z e i wt e 1 of the rescaled ginzburg landau equation k 1 1 ib l kl2 k 1 ia kzz for a ca b eb w plays the role of free parameter this leads to a perturbation analysis on a complex duffing equation similar to the analysis of holmes physica d 23 1986 84 we show that the spatial quasiperiodic solutions of the unperturbed e 0 case disappear due to the perturbation and prove the existence of degenerated periodic solutions which oscillate through the origin we also establish the existence of several types of heteroclinic orbits connecting counterrotating periodic patterns

the structure of spiral domain patterns and shocks in the 2d complex ginzburg landau equation abstract cellular patterns appear spontaneously in a number of nonequilibrium systems governed by the dynamics of a complex field in the case of the complex ginzburg landau equation disordered cells of effectively frozen spirals appear separated by thin walls shocks on a scale much larger than the basic wavelength of the spirals we show that these structures can be understood in very simple terms in particular we show that the walls are to a good approximation segments of hyperbolae and this allows us to construct the wall pattern given the vortex centers and a phase constant for each vortex the fact that the phase is only defined up to an integer multiple of 2 introduces a quantization condition on the sizes of the smallest spiral domains the transverse structure of the walls is analyzed by treating them as heteroclinic connections of a system of ordinary differential equations the structure depends on the angle the wall makes with the local phase contours and the behavior can be either monotonic or oscillatory depending on the parameters

spiral wave solutions for reaction diffusion equations in a fast reaction slow diffusion limit abstract a two species reaction diffusion equation is studied in a distinguished limit of fast reaction and slow diffusion asymptotic matching yields a spiral solution whose rotation frequency is a unique function of the physical parameters and the constant pitch of the spiral using an idea of fife the problem is first reduced to a free boundary problem the solution is obtained by matching the core where diffusion balances the curvature of the free boundary to the far field where the effects of diffusion are small the equations of motion are averaged over the rotation frequency to yield steady equations in the two regions helically symmetric solutions to the averaged equation reduce to solving a fourth order ode in the core region matched to a second order ode in the far field speculations on the dynamics of the system and coupling between the core and far field are presented

mass conservative fourier spectral methods for solving the fractional nonlinear schrodinger equation we propose three fourier spectral methods i e the split step fourier spectral ssfs the crank nicolson fourier spectral cnfs and the relaxation fourier spectral refs methods for solving the fractional nonlinear schrodinger nls equation all of them are mass conservative and time reversible and they have the spectral order accuracy in space and the second order accuracy in time in addition the cnfs and refs methods are energy conservative the performance of these methods in simulating the plane wave and soliton dynamics is discussed the ssfs method preserves the dispersion relation and thus it is more accurate for studying the long time behaviors of the plane wave solutions furthermore our numerical simulations suggest that the ssfs method is better in solving the defocusing nls but the cnfs and refs methods are more effective for the focusing nls

evaluation of dynamical models dissipative synchronization and other techniques some recent developments for the validation of nonlinear models built from data are reviewed besides giving an overall view of the field a procedure is proposed and investigated based on the concept of dissipative synchronization between the data and the model which is very useful in validating models that should reproduce dominant dynamical features like bifurcations of the original system in order to assess the discriminating power of the procedure four well known benchmarks have been used namely duffing ueda duffingholmes and van der pol oscillators plus the henon map the procedure developed for discrete time systems is focused on the dynamical properties of the model rather than on statistical issues for all the systems investigated it is shown that the discriminating power of the procedure is similar to that of bifurcation diagrams which in turn is much greater than say that of correlation dimension but at a much lower computational cost

parameter uncertainties in models of equivariant dynamical systems abstract we examine the advantages of using equivariant versus nonequivariant phenomenological models for chaotic systems that possess an inversion symmetry numerical experiments used the minimum description length mdl criteria for model selection they indicated that the selection between equivariant versus nonequivariant models should be made on an a priori basis we also examine the relationship between the optimal truncation accuracy of the parameters of the model as predicted by the mdl criteria and the true uncertainty in the parameters of the model numerical experiments indicate that the true uncertainties are poorly approximated by the optimal truncation accuracy they also indicate that the true uncertainties may be useful for determining the presence of an inversion symmetry in reconstructed attractors

pseudospectral method for the good boussinesq equation we prove the nonlinear stability and convergence of a fully discrete pseudospectral scheme for the good boussinesq equation utt uxxxx uxx u2 xx numerical comparisons with finite difference schemes are also reported

on exact solutions of modified complex ginzburg landau equation abstract the one dimensional modified complex ginzburg landau equation has been studied by the use of the conte and musette method this method permits us to derive all the known exact solutions in a unified natural scheme these solutions are expressed in terms of solitary wave periodic unbounded wave and shock type wave we also find previously unknown exact propagating hole the degeneracies of modified complex ginzburg landau equation have also been examined as well as several of their solutions

learning dynamics from nonstationary time series analysis of electroencephalograms we propose an empirical modeling technique for a nonstationary time series analysis proposed methods include a high dimensional n 3 dynamical model construction in the form of delay differential equations a nonparametric method of respective time delay calculation the detection of quasistationary regions of the process by recurrence analysis in the space of model coefficients and final fitting of the model to quasistationary segments of observed time series we also demonstrate the effectiveness of our approach for nonstationary signal classification in the space of model coefficients applying the empirical modeling technique to electroencephalogram eeg records analysis we find evidence of high dimensional nonlinear dynamics in quasistationary eeg segments recurrence analysis of model parameters reveals long term correlations in nonstationary eeg records using the dynamical model as a nonlinear filter we find that different emotional states of subjects can be clearly distinguished in the space of model coefficients

numerical experience with the nonlinear schrodinger equation abstract increasing the magnitude of the parameter multiplying the nonlinear term of the nonlinear schrodinger equation without changing the initial condition u x 0 sech x leads to bound states of an increasing number of solitons this results in very steep gradients in space and time and so provides a more severe test of numerical methods than before in particular we find that methods which satisfy various conservation laws theoretically may now fail to do so in practice various analytical and numerical results relevant to this situation are discussed and illustrated by numerical examples

observability of multivariate differential embeddings the present paper extends some results recently developed for the analysis of observability in nonlinear dynamical systems the aim of the paper is to address the problem of embedding an attractor using more than one observable a multivariate nonlinear observability matrix is proposed which includes the monovariable nonlinear and linear observability matrices as particular cases using the developed framework and a number of worked examples it is shown that the choice of embedding coordinates is critical moreover in some cases to reconstruct the dynamics using more than one observable could be worse than to reconstruct using a scalar measurement finally using the developed framework it is shown that increasing the embedding dimension observability problems diminish and can even be eliminated this seems to be a physically meaningful interpretation of the takens embedding theorem

an explicit finite difference scheme with exact conservation properties abstract a finite difference scheme for the numerical study of the korteweg de vries equation is constructed it is explicit and yet conserves exactly the energy of the computed solutions the underlying idea can also be applied to more general equations or systems numerical experiments are included

generalized synchronization of chaos in electronic circuit experiments abstract two examples of one way coupled electronic circuits displaying generalized synchronization of chaos are considered in one of them the dynamics of both the response and the driving systems represent a double scroll chaos oscillator in another example the double scroll oscillator is driven by an electronic analog of the mackey glass system to detect the generalized synchronization an auxiliary response system that is a replica of the original one is used in these systems we have discovered two types of generalized synchronization namely a strong and a weak synchronization which correspond to the existence of a smooth and a nonsmooth map from the trajectories of the driving attractor to the trajectories of the response system respectively

problems in estimating dynamics from data abstract this paper discusses some difficulties in estimating dynamics from time delay embeddings of experimental data that can be characterized as low dimensional a new procedure is described to reduce noise by exploiting the properties of saddle periodic orbits on the reconstructed attractor

boundedness and synchronization of y coupled lorenz systems with or without controllers abstract the purpose of this paper is to investigate the boundedness and synchronization of y coupled lorenz systems when the coupling term is only added to the second variable we call them y coupled lorenz systems in this paper we first prove the boundedness of y coupled lorenz systems which ensures the existence and uniqueness of the solution when t based on the boundedness we prove that for y coupled lorenz systems quad condition is satisfied it should be pointed out that quad condition plays an essential role in the discussion of synchronization based on the boundedness and quad condition we prove that if the coupling is strong enough the y coupled lorenz systems can achieve the complete synchronization globally and exponentially

moderate deviations and associated laplace approximations for sums of independent random vectors let xj be an i i d sequence of banach space valued r v s and let sn z 1 xj for certain positive sequences bn oo we determine the exact asymptotic behavior of e exp b2 n d sn bn where d is a smooth function we also prove a large deviation principle for 2y sn bn

unknown inputs observer design for fuzzy systems with application to chaotic system reconstruction this paper deals with the observer design for nonlinear systems in takagi sugeno fuzzy representation based on the lyapunov method and linear matrix inequalities lmi formulation sufficient conditions have been derived for observers design unknown inputs can result either from model uncertainty faults or due to the presence of unknown external excitation these different results have been widely applied in the field of fault diagnosis and fault tolerance based on unknown inputs observer design secure communication and chaotic system reconstruction problems have been also studied examples are given to illustrate a chaotic cryptosystem procedure where the plaintext message is encrypted using chaotic signals at the drive system side and the plaintext is retrieved via the designed unknown input observer

energy conserving methods for the nonlinear schrodinger equation in this paper we further develop recent results in the numerical solution of hamiltonian partial differential equations pdes brugnano et al 2015 by means of energy conserving methods in the class of line integral methods in particular the runge kutta methods named hamiltonian boundary value methods hbvms we shall use hbvms for solving the nonlinear schrodinger equation nlse of interest in many applications we show that the use of energy conserving methods able to conserve a discrete counterpart of the hamiltonian functional confers more robustness on the numerical solution of such a problem

fully nonlinear global modes in spatially developing media abstract global modes on a doubly infinite one dimensional domain wkbj approximations in the linear and nonlinear regions through appropriate transition layers the real global frequency is determined by requiring that spatial branches issuing from x and x be continuously connected at a saddle point of the local nonlinear dispersion relation n l k r 2 x between the frequency the wave number k and amplitude r at a given station x the results constitute a fully nonlinear generalization of the linear frequency selection criteria previously obtained by chomaz et al 1991 monkewitz et al 1993 and le dizes et al 1996

slowly varying fully nonlinear wavetrains in the ginzburg landau equation abstract following the ideas of howard and kopell 9 a perturbation theory is developed for slowly varying fully nonlinear wavetrains i e solutions which appear locally as travelling waves but with frequencies and wavelengths which may vary widely on long length and time scales this perturbation theory is applied to the ginzburg landau equation the motion and stability of slowly varying wavetrains is shown to be governed by a simple wave equation which can develop shocks corresponding to rapid changes in wavenumber numerical results supporting this theory are presented a shock structure is proposed and numerically verified these results together with a winding invariant valid in the limit of slow variation suggest that over a large range of parameters many initial conditions relax to uniform wavetrains the evolution of a marginally diffusively stable wavetrain is also examined it is argued that the evolution is governed by a perturbed korteweg de vries equation

n th order operator splitting schemes and nonreversible systems this paper is concerned with partitioned nth order accurate split operator schemes built using m distinct solution operators for semidiscrete equations of the form frac d dt u j f j 1 u k cdots f j m u k which arise among others from constant coefficient parabolic partial differential equations we prove that for every n geq 3 and m geqslant 2 each solution operator must be applied for at least one backward fractional time step during each complete time step this result has important consequences for applications to the complex ginzburg landau equation with periodic boundary conditions and other partial differential equations with both reversible and nonreversible components furthermore for the special case of n 3 and m 2 we analytically determine all possible schemes

nonequilibrium statistical behavior of nonlinear schrodinger equations this paper explores the equilibrium and near equilibrium statistical behavior of nonlinear schrodinger equations through numerical simulations the investigation centers on a class of finite dimensional hamiltonian systems obtained by discretizing nls equations defined on a bounded interval with saturated focusing nonlinearities these nonlinear systems govern a wave dynamics that is nonintegrable and nonsingular and for which typical solutions are composed of large scale coherent structures interacting within a background of small scale fluctuations the equilibrium statistics of such a system is most naturally defined by a gibbs distribution that is canonical in energy and microcanonical in particle number by sampling these equilibrium distributions using a metropolis type algorithm the typical excitations of the system are determined over the full range of system parameters nonequilibrium relaxations from initial perturbations of these equilibrium distributions are then computed by means of linear response theory the computations reveal a surprising dependence of the qualitative properties of relaxation toward equilibrium on the temperature and the nonlinearity on the one hand a simple exponential decay to equilibrium is observed for sufficiently high temperature or sufficiently strong nonlinearity for low temperature and moderate nonlinearity on the other hand the system rapidly adjusts to a metaequilibrium state that equilibrates extremely slowly this two stage thermalization may be related to quasiperiodicity in the underlying dynamics indeed the metaequilibria are found to occur in a range of system parameters for which approximately quasiperiodic solutions are prevalent in equilibrium

delta method in large deviations and moderate deviations for estimators the delta method is a popular and elementary tool for deriving limiting distributions of transformed statistics while applications of asymptotic distributions do not allow one to obtain desirable accuracy of approximation for tail probabilities the large and moderate deviation theory can achieve this goal motivated by the delta method in weak convergence a general delta method in large deviations is proposed the new method can be widely applied to driving the moderate deviations of estimators and is illustrated by examples including the wilcoxon statistic the kaplan meier estimator the empirical quantile processes and the empirical copula function we also improve the existing moderate deviations results for m estimators and l statistics by the new method some applications of moderate deviations to statistical hypothesis testing are provided

on fully discrete galerkin methods of second order temporal accuracy for the nonlinear schrodinger equation we approximate the solutions of an initial and boundary value problem for nonlinear schrodinger equations with emphasis on the cubic nonlinearity by two fully discrete finite element schemes based on the standard galerkin method in space and two implicit crank nicolson type second order accurate temporal discretizations for both schemes we study the existence and uniqueness of their solutions and provel 2 error bounds of optimal order of accuracy for one of the schemes we also analyze one step of newton s method for solving the nonlinear systems that arise at every time step we then implement this scheme using an iterative modification of newton s method that at each time stept n requires solving a number of sparse complex linear systems with a matrix that does not change withn the effect of this inner iteration is studied theoretically and numerically

observation of a strange attractor abstract phase space portraits have been constructed and analyzed for noisy nonperiodic data obtained in an experiment on a nonequilibrium homogeneous chemical reaction the phase space trajectories define a limit set that is an attractor following a perturbation the trajectory quickly returns to the attracting set this attracting set is shown to be strange nearby trajectories separate exponentially on the average moreover the poincare sections exhibit the stretching and folding that is characteristic of strange attractors

a moderate deviation principle for 2 d stochastic navier stokes equations driven by multiplicative levy noises abstract in this paper we establish a moderate deviation principle for two dimensional stochastic navier stokes equations driven by multiplicative levy noises the weak convergence method introduced by budhiraja dupuis and ganguly in 3 plays a key role

the numerical treatment of the nonlinear schrodinger equation abstract the implicit midpoint rule for the integration in time of a fourth order accurate semidiscretization in space of the nonlinear schrodinger equation is studied the stability analysis for the present method is carried out by the conservation of the energy in time three increasingly difficult problems are tested by the method mentioned comparison with other methods is presented the results obtained here seem to be in better agreement with the exact solutions

periodic solutions of the ginzburg landau equation abstract spatially periodic solutions to the ginzburg landau equation are considered in particular we obtain criteria for primary and secondary bifurcation limit cycle solutions nonlinear dispersion relations relating spatial and temporal frequencies only relatively simple tools appear in the treatment and as a result a wide range of parameter cases are considered finally we briefly treat the case of spatial bifurcations

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

any order propagation of the nonlinear schrodinger equation we derive an exact propagation scheme for nonlinear schrodinger equations this scheme is entirely analogous to the propagation of linear schrodinger equations we accomplish this by defining a special operator whose algebraic properties ensure the correct propagation as applications we provide a simple proof of a recent conjecture regarding higher order integrators for the gross pitaevskii equation extend it to multicomponent equations and to a new class of integrators

choice of dynamical variables for global reconstruction of model equations from time series the success of modeling from an experimental time series is determined to a significant extent by the choice of dynamical variables we propose a method for preliminary investigation of a time series whose purpose is to find out whether a global dynamical model with smooth functions can be constructed for the chosen variables the method consists in the estimation of single valuedness and continuity of relations between dynamical variables and variables to enter left hand sides of model equations the method is explained with numerical examples its efficiency is demonstrated by modeling a real nonlinear electric circuit