parameter estimation of delay dynamical system from a scalar time series under external noise we introduce an adaptive learning rules for estimating all unknown parameters of delay dynamical system using a scalar time series sufficient condition for synchronization is derived using krasovskii lyapunov theory this scheme is highly applicable in secure communication since multiple messages can be transmitted through multiple parameter modulations one of the advantage of this method is that parameter estimation is also possible even when only one time series of the transmitter is available we present numerical examples for mackey glass system with periodic delay time which are used to illustrate the validity of this scheme the corresponding numerical results and the effect of external noise are also studied

existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations in this paper the banach fixed point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference email protected n dx n b n x email protected email protected n x h 1 n x h 2 n x h k n f n x f 1 n x f 2 n x f k n c n n n 0 where t email protected n n email protected n 0 a b c n n 0 r with a n 0 for email protected n n 0 h f n n 0xr k r and h l f l n n 0 z withlimn h l n limn f l n email protected 1 2 k a few mann type iterative approximation schemes with errors are suggested and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed seven nontrivial examples are given to illustrate the advantages of our results

gauss and the invention of least squares the most famous priority dispute in the history of statistics is that between gauss and legendre over the discovery of the method of least squares new evidence both documentary and statistical is discussed and an attempt is made to evaluate gauss s claim it is argued though not conclusively that gauss probably possessed the method well before legendre but that he was unsuccessful in communicating it to his contemporaries data on the french meridian arc are presented that could conceivably permit a definitive verification of the claim

oscillation criteria for second order quasi linear neutral difference equations abstract in this paper we obtain some oscillation criteria for the second order quasi linear neutral delay difference equation where 0 0 and 0 are constants a n p n q n are nonnegative sequences and f c r r

oscillation theorems for higher order nonlinear delay difference equations abstract some new oscillation results for difference equations of the form 1 1 and forced difference equations of the form 1 2 are established the results obtained are discrete analogues of some known results for delay differential equations

estimating parameters of a nonlinear dynamical system a method based on a modified newton raphson scheme is presented to estimate parameters of a nonlinear dynamical system from the time series data of the variables the method removes some of the problems associated with the standard synchronization based methods an important achievement of this method is that it is possible to determine the exact form of dynamical equations for systems with quadratic nonlinearity

oscillation for second order nonlinear neutral delay difference equations in this paper we obtain sufficient conditions for the oscillation of all solutions of the second order nonlinear delay difference email protected a n d x n p nx g n a f n x s n 0 where dx n x n 1 x n is the forward difference operator p n is a sequence of nonnegative real numbers and a 1 is a quotient of positive odd integers our results extend and improve the known results in the literature an example that dwell up the advantage of our results is included

adaptive approximation method for joint parameter estimation and identical synchronization of chaotic systems we introduce a numerical approximation method for estimating an unknown parameter of a primary chaotic system which is partially observed through a scalar time series specifically we show that the recursive minimization of a suitably designed cost function that involves the dynamic state of a fully observed secondary system and the observed time series can lead to the identical synchronization of the two systems and the accurate estimation of the unknown parameter the salient feature of the proposed technique is that the only external input to the secondary system is the unknown parameter which needs to be adjusted we present numerical examples for the lorenz system which show how our algorithm can be considerably faster than some previously proposed methods

markov chain monte carlo method in bayesian reconstruction of dynamical systems from noisy chaotic time series the presence of a noise component in the time series generated by a dynamical system ds results in finite accuracy of reconstruction of the evolution operator eo any model that can be constructed will generally speaking differ from the original system one of the ways to take this fact into consideration is to describe parameters of the model eo as random quantities 1 two limiting cases may be distinguished 2 the perfect model scenario when the form of the operator describing the ds evolution is reliably known and the imperfect model scenario when eo is unknown the perfect model scenario is realized for instance when the problem of hidden information transmission is solved using chaotic time series ts generated by the known ds 3 in this case we are interested in the characteristics of the system perfect model parameters the most probable value mean dispersion distribution in investigations of many natural atmospheric oceanic tectonic biological systems the eo is unknown which corresponds to the imperfect model scenario then information about model parameters is of no value as the physical meaning of the parameters is unknown and we are interested in the properties of the model defined by these parameters and reflecting the properties of the reconstructed ds nevertheless in this case too evaluation of statistical characteristics of model parameters is the key element without which the problem of reconstructing a ds generating the original ts cannot be solved the mathematical body used to reconstruct the ds is determined by a specific application for example for finding the most probable set of model perfect or imperfect parameters it suffices to determine the maximum of posterior probability density of model parameters 4 6 however it should be remembered that in the presence of noise the inverse problem of reconstruction becomes ill posed i e it admits an infinite set of solutions selection of the most correct solutions demands regularization that is a physically justified constraint on admissible values of parameters based solely on a priori information 7 this approach is usually called bayesian within the bayesian approach there are methods which make it possible to estimate dispersions and mathematical expectations of the model parameters 8 a significant advantage of such methods is their relatively low computational resource requirements however due to the model nonlinearity the form of parameter distribution may differ strongly from the normal one in this case estimation of expectation and dispersion is insufficient to construct a correct model and one must use the bayesian approach in full in other words we must construct models which include distributions of parameters and take a priori information about the system into account correctly in this formulation the problem can be solved using the markov chain monte carlo mcmc algorithms 9 however in the case of reconstruction of dss by noisy chaotic time series computational resources required for these algorithms strongly depend on distribution dimensionality this limits applicability of the algorithms to a narrow class of problems e g the case of short ts 10 the method proposed in this paper makes it possible to expand the applicability domain of the mcmc algorithms to the case of arbitrarily long ts

estimation of parameters in nonlinear systems using balanced synchronization using synchronization between observations and a model with undetermined parameters is a natural way to complete the specification of the model the quality of the synchronization a cost function to be minimized typically is evaluated by a least squares difference between the data time series and the model time series if the coupling between the data and the model is too strong this cost function is small for any data and any model and the variation of the cost function with respect to the parameters of interest is too small to permit selection of an optimal value of the parameters we introduce two methods for balancing the competing desires of a small cost function for the quality of the synchronization and the numerical ability to determine parameters accurately one method of balanced synchronization adds to the synchronization cost function a requirement that the conditional lyapunov exponent of the model system conditioned on being driven by the data remain negative but small in magnitude the other method allows the coupling between the data and the model to vary in time according to the error in synchronization this method succeeds because the data and the model exhibit generalized synchronization in the region where the parameters of the model are well determined examples are explored which have deterministic chaos with and without noise in the data signal

oscillatory and asymptotic properties of higher order nonlinear neutral difference equations abstract in this paper we study the oscillatory and asymptotic properties of solutions of higher order nonlinear neutral difference equations using krasnoselskii s fixed point theorem schauder s fixed point theorem and lebesgue s dominated convergence theorem we obtain some necessary and sufficient conditions for all bounded solutions to be oscillatory or tend to zero our results extend and improve some known results

forced oscillation of higher order nonlinear difference equations this paper considers the oscillation problem for forced nonlinear difference equations of the email protected mx n q nf x n t e n we study three cases q n 0 q n 0 and q n is oscillatory no restriction assumed in known literatures is imposed on the forcing term e n

on existence of positive solutions and bounded oscillations for neutral difference equations abstract a criterion for the existence of a positive solution for the first order difference equation xn cxn m pnxn k 0 pn 0 is established results are also obtained for the oscillation and nonoscillation of solutions of a second order difference equation 2 xn cxn m pnxn k where the pn are either of constant sign or are oscillatory some of the results are sharp

parameter estimation through ignorance dynamical modeling lies at the heart of our understanding of physical systems its role in science is deeper than mere operational forecasting in that it allows us to evaluate the adequacy of the mathematical structure of our models despite the importance of model parameters there is no general method of parameter estimation outside linear systems a relatively simple method of parameter estimation for nonlinear systems is introduced based on variations in the accuracy of probability forecasts it is illustrated on the logistic map the henon map and the 12 dimensional lorenz96 flow and its ability to outperform linear least squares in these systems is explored at various noise levels and sampling rates as expected it is more effective when the forecast error distributions are non gaussian the method selects parameter values by minimizing a proper local skill score for continuous probability forecasts as a function of the parameter values this approach is easier to implement in practice than alternative nonlinear methods based on the geometry of attractors or the ability of the model to shadow the observations direct measures of inadequacy in the model the implied ignorance and the information deficit are introduced

oscillation of certain second order nonlinear difference equations abstract in this paper we consider the second order nonlinear difference equation formula where y n y n 1 y n is the forward difference operator f c r r f is nondecreasing and uf u 0 as u 0 q n is a real sequence some new sufficient conditions for the oscillation of all solutions of 1 are obtained

upper bounds to the spin correlation functions of the random bond ising model and n vector model with continuously distributed random interactions obtains upper bounds to spin correlation functions in the thermodynamic limit of the zero external field limit for the ising model of general spins and for the n vector model assuming that the exchange integrals are quenched random variables and that their probability distributions are continuous the upper bounds involve as a factor the corresponding spin correlation function of a uniform ferromagnetic ising model of spin or 1 in which the exchange integral is determined by the distributions of the random exchange integrals of the original system using these upper bounds the authors find a sufficient condition on the probability distributions of the exchange integrals for the disappearance of the ferromagnetic and antiferromagnetic states in the system

parameter estimation of dynamical systems via a chaotic ant swarm through the construction of suitable objective function the parameter estimation of the dynamical system could be converted to the problem of parameter optimization based on the chaotic ant swarm optimization approach we investigate the problem of parameter optimization for the dynamical systems in the presence of noise we systematically analyze the basic relationships among the complexity of objective function the length of time series and the performance of the searching algorithm furthermore we consider the effect of measurable additive noise on the objective function numerical simulations are also provided to show the effectiveness and feasibility of the proposed methods

oscillation criteria for difference equations with damping terms by means of generalized riccati transformation techniques we establish some new oscillation criteria for second order nonlinear difference equations with damping

oscillation criteria for second order nonlinear neutral delay dynamic equations in this paper we will establish some oscillation criteria for the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t here 0 is a quotient of odd positive integers with r t and p t real valued positive functions defined on t to the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales so this paper initiates the study

failure of parameter identification based on adaptive synchronization techniques in this paper several examples as well as their numerical simulations are provided to show some possible failures of parameter identification based on the so called adaptive synchronization techniques these failures might arise not only when the synchronized orbit produced by the driving system is designed to be either some kind of equilibrium or to be some kind of periodic orbit but also when this orbit is deliberately designed to be chaotic the reason for emergence of these failures is theoretically analyzed in the paper and the boundedness of all trajectories generated by the coupled systems is rigorously proved moreover synchronization techniques are proposed to realize complete synchronization and unknown parameter identification in a class of systems where nonlinear terms are not globally lipschitz in addition unknown parameter identification is studied in coupled systems with time delays

existence of the ferromagnetic phase in competing random bond ising models on the square lattice for several types of probability distribution of exchange interaction the authors generalise the arguments given in a preceding letter unpublished 1982 by the present authors to give a sufficient condition for the existence of the spontaneous magnetisation in the random bond ising model on the square lattice they study in detail the system with a gaussian distribution of exchange interaction the system with the exchange interactions j 0 0 and j and the system with the exchange interactions ja 0 and jb 0 it is shown that in each of these systems the ferromagnetic phase certainly exists in a region in the space of the parameters describing the probability distribution of the exchange interaction at sufficiently low temperatures the general condition for the existence of the ferromagnetic phase is given for the system without specifying the form of the probability distribution of the exchange interaction

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

numerical oscillation and non oscillation for differential equation with piecewise continuous arguments of mixed type the paper deals with the oscillation and non oscillation of the runge kutta methods for a differential equation with piecewise continuous arguments of mixed type the conditions of the oscillation and non oscillation for the runge kutta method are obtained it is proved that oscillation of the analytic solution is not preserved by the runge kutta method under any conditions the conditions under which the non oscillation of analytic solutions is preserved by the runge kutta method are obtained some numerical experiments are given

recovering unknown model parameters contained in a class of time variant chaotic dynamical systems abstract this work deals mainly with the problem of recovering all unknown parameters for a class of time variant chaotic dynamical systems from given time sequence based on synchronization between a chaotic sender system and an additional receiver system a procedure which combines a linear feedback technique with updated feedback gain and an adapted control strategy associated with the law of estimated parameters is developed to dynamically determine the values of unknown parameters contained in the sender system to promote widespread applications the structure of the receiver system can be independent of that of the sender system the effectiveness of this procedure is guaranteed by the periodic version of the classical lasalle invariance principle of differential equations illustrations are presented for a harmonically excited duffing oscillator and a four dimensional chaotic oscillator the numerical results reveal the present procedure not only can precisely recover unknown model parameters but also can rapidly response to sudden changes in unknown parameters in addition it has great robustness against the disturbance of noise

single shooting homotopy method for parameter identification in dynamical systems an algorithm for identifying parameters in dynamical systems is developed in this work using homotopy transformations and the single shooting method the equations governing the dynamics of the mathematical model are augmented with observer like homotopy terms that smooth the objective function as a result premature convergence to a local minimum is avoided and the obtained parameter estimates are globally optimal numerical examples are presented to demonstrate the application of the proposed approach to chaotic systems

oscillation of the emden fowler difference systems abstract in this paper we are concerned with some new criteria for the oscillation of the discrete emden fowler system

properties of a simple bilinear stochastic model estimation and predictability we analyze the properties of arguably the simplest bilinear stochastic multiplicative process proposed as a model of financial returns and of other complex systems combining both nonlinearity and multiplicative noise by construction it has no linear predictability zero two point correlation but a certain nonlinear predictability non zero three point correlation it can thus be considered as a paradigm for testing the existence of a possible nonlinear predictability in a given time series we present a rather exhaustive study of the process including its ability to produce fat tailed distributions from gaussian innovations the unstable characteristics of the inversion of the key nonlinear parameters and of the two initial conditions necessary for the implementation of a prediction scheme and an analysis of the associated super exponential sensitivity of the inversion of the innovations in the presence of a large impulse our study emphasizes the conditions under which a degree of predictability can be achieved and describe a number of different attempts which overall illuminates the properties of the process in conclusion notwithstanding its remarkable simplicity the bilinear stochastic process exhibits remarkably rich and complex behavior which makes it a serious candidate for the modeling of financial time series among others

use of synchronization and adaptive control in parameter estimation from a time series a technique is introduced for estimating unknown parameters when a time series of only one variable from a multivariate nonlinear dynamical system is given the technique employs a combination of two different control methods linear feedback for synchronizing system variables and adaptive control and is based on dynamic minimization of synchronization error the technique is shown to work even when the unknown parameters appear in the evolution equations of the variables other than the one for which the time series is given the technique not only establishes that explicit detailed information about all system variables and parameters is contained in a scalar time series but also gives a way to extract it out under suitable conditions illustrations are presented for lorenz and rossler systems and a nonlinear dynamical system in plasma physics also it is found that the technique is reasonably stable against noise in the given time series and the estimated value of a parameter fluctuates around the correct value with the error of estimation growing linearly with the noise strength for small noise

a second order nonlinear difference equation oscillation and asymptotic behavior discrete analogues are investigated for well known results on oscillation growth and asymptotic behavior of solutions of y q t y 0 for q t 0 and for q t 0 the analogue of atkinson s oscillation criterion is shown to be true for 2yn 1 qn yn 0 but the analogue for atkinson s nonoscillation criterion is shown to be false

on a recursive method for the estimation of unknown parameters of partially observed chaotic systems we investigate a recently proposed method for on line parameter estimation and synchronization in chaotic systems this novel technique has been shown effective to estimate a single unknown parameter of a primary chaotic system with known functional form that is only partially observed through a scalar time series it works by periodically updating the parameter of interest in a secondary system with the same functional form as the primary one but no explicit coupling between their dynamic variables in order to minimize a suitably defined cost function in this paper we review the basics of the method and investigate its robustness and new extensions in particular we study the performance of the novel technique in the presence of noise either observational i e an additive contamination of the observed time series or dynamical i e a random perturbation of the system dynamics and when there is a mismatch between the primary and secondary systems numerical results including comparisons with other techniques are presented finally we investigate the extension of the original method to perform the estimation of two unknown parameters and illustrate its effectiveness by means of computer simulations

better nonlinear models from noisy data attractors with maximum likelihood a new approach to nonlinear modelling is presented which by incorporating the global behaviour of the model lifts shortcomings of both least squares and total least squares parameter estimates although ubiquitous in practice a least squares approach is fundamentally flawed in that it assumes independent normally distributed ind forecast errors nonlinear models will not yield ind errors even if the noise is ind a new cost function is obtained via the maximum likelihood principle superior results are illustrated both for small data sets and infinitely long data streams

a hierarchical bayes ensemble kalman filter abstract a new ensemble filter that allows for the uncertainty in the prior distribution is proposed and tested the filter relies on the conditional gaussian distribution of the state given the model error and predictability error covariance matrices the latter are treated as random matrices and updated in a hierarchical bayes scheme along with the state the hyper prior distribution of the covariance matrices is assumed to be inverse wishart the new hierarchical bayes ensemble filter hbef assimilates ensemble members as generalized observations and allows ordinary observations to influence the covariances the actual probability distribution of the ensemble members is allowed to be different from the true one an approximation that leads to a practicable analysis algorithm is proposed the new filter is studied in numerical experiments with a doubly stochastic one variable model of truth the model permits the assessment of the variance of the truth and the true filtering error variance at each time instance the hbef is shown to outperform the enkf and the henkf by myrseth and omre 2010 in a wide range of filtering regimes in terms of performance of its primary and secondary filters

oscillation by impulses for a second order delay differential equation we consider a certain second order nonlinear delay differential equation and prove that the all solutions oscillate when proper impulse controls are imposed an example is given

statistical methods of parameter estimation for deterministically chaotic time series we discuss the possibility of applying some standard statistical methods the least square method the maximum likelihood method and the method of statistical moments for estimation of parameters to deterministically chaotic low dimensional dynamic system the logistic map containing an observational noise a segmentation fitting maximum likelihood ml method is suggested to estimate the structural parameter of the logistic map along with the initial value x 1 considered as an additional unknown parameter the segmentation fitting method called piece wise ml is similar in spirit but simpler and has smaller bias than the multiple shooting previously proposed comparisons with different previously proposed techniques on simulated numerical examples give favorable results at least for the investigated combinations of sample size n and noise level besides unlike some suggested techniques our method does not require the a priori knowledge of the noise variance we also clarify the nature of the inherent difficulties in the statistical analysis of deterministically chaotic time series and the status of previously proposed bayesian approaches we note the trade off between the need of using a large number of data points in the ml analysis to decrease the bias to guarantee consistency of the estimation and the unstable nature of dynamical trajectories with exponentially fast loss of memory of the initial condition the method of statistical moments for the estimation of the parameter of the logistic map is discussed this method seems to be the unique method whose consistency for deterministically chaotic time series is proved so far theoretically not only numerically

synchronization based estimation of all parameters of chaotic systems from time series by a simple combination of adaptive scheme and linear feedback with the updated feedback strength for a large class of chaotic systems it is proved rigorously by using the invariance principle of differential equations that all unknown model parameters can be estimated dynamically this approach supplies a systematic and analytical procedure for estimating parameters from time series and it is simple to implement in practice in addition this method is quite robust against the effect of noise and able to respond rapidly to changes in operating parameters of the experimental system lorenz and r ossler hyperchaos systems are used to illustrate the validity of this technique

new oscillation criteria for second order nonlinear neutral delay difference equations in this paper we will establish some new oscillation criteria for the second order nonlinear neutral delay difference equation d a n dx n p nx n t c f n x n s 0 n 0 1 2 where c 0 is a quotient of odd positive integers

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

oscillation for a second order neutral differential equation with impulses we consider a certain type of second order neutral delay differential systems and we establish two results concerning the oscillation of solutions after the system undergoes controlled abrupt perturbations called impulses as a matter of fact some particular non impulsive cases of the system are oscillatory already thus we are interested in finding adequate impulse controls under which our system remains oscillatory

comparison theorems for second order nonlinear difference equations abstract new oscillation results are obtained for the second order nonlinear difference equation r n f x n 1 g n x n 0 and its functional form r n f x n 1 g n x n 0 the role played by the argument n on the oscillation of the functional equation is explored in particular we characterize a class of sequences n which have a harmless effect on the oscillation of this type of equations some of our results generalize improve or unify known fundamental oscillation results for several particular cases of the above equations

seeker optimization algorithm for parameter estimation of time delay chaotic systems time delay chaotic systems have some very interesting properties and their parameter estimation has received increasing interest in the recent years it is well known that parameter estimation of a chaotic system is a nonlinear multivariable and multimodal optimization problem for which global optimization techniques are required in order to avoid local minima in this work a seeker optimization algorithm soa based method is proposed to address this issue in the soa search direction is based on the empirical gradients by evaluating the response to the position changes and step length is based on uncertainty reasoning by using a simple fuzzy rule the performance of the algorithm is evaluated on two typical test systems moreover two state of the art algorithms i e particle swarm optimization and differential evolution are also considered for comparison the simulation results show that the proposed algorithm is better than or at least as good as the other two algorithms and can effectively solve the parameter estimation problem of time delay chaotic systems

upper bounds to the zero field susceptibilities and magnetisations for the random bound ising model and the random bond n vector model the self avoiding walk approximation is used to give upper bounds to the zero field spin pair correlation function zero field susceptibilities and the spontaneous magnetisation for the systems of the random bond ising model of general spin s and the random bond n vector model where the exchange integrals are assumed to be random variables using these bounds the authors find a sufficient condition for the disappearance of the long range orders the condition is stronger for some cases than the one obtained previously by the authors the present one applies to some cases for which the preceding one does not

oscillation of second order nonlinear neutral differential equations consider the second order nonlinear neutral delay differential equation r t x t p t x email protected a f t x email protected 0 e where t and s are nonnegative constants and a is a quotient of positive odd integers some new oscillatory criteria for eq e are established several examples which dwell upon the importance of our results are also illustrated

oscillation theorems for some nonlinear difference equations abstract some new oscillation theorems for the difference equations of the form r n u n q n f u n k 0 are established

oscillation of second order perturbed nonlinear difference equations using the riccati transformation techniques we will establish some new oscillation criteria for the second order perturbed nonlinear difference equation d a n 1 dx n 1 c f n x n g n x n dx n n 1where c 0 is a quotient of odd positive integers some comparison between our theorems and those previously known results are indicated examples are interested in the text to illustrate the relevance of our results

oscillation theorems for second order neutral difference equations abstract oscillation criteria for the second order difference equations of the form 2xn qnf xn r 0 and the neutral difference equation 2 xn pxn k qnf xn r 0 are established here xn xn 1 xn is the forward difference operator 1 p is a real number and k are nonnegative integers qn is a sequence of nonnegtive real numbers and f r r is continuous

parameter and state estimation of experimental chaotic systems using synchronization we examine the use of synchronization as a mechanism for extracting parameter and state information from experimental systems we focus on important aspects of this problem that have received little attention previously and we explore them using experiments and simulations with the chaotic colpitts oscillator as an example system we explore the impact of model imperfection on the ability to extract valid information from an experimental system we compare two optimization methods an initial value method and a constrained method each of these involves coupling the model equations to the experimental data in order to regularize the chaotic motions on the synchronization manifold we explore both time dependent and time independent coupling and discuss the use of periodic impulse coupling we also examine both optimized and fixed or manually adjusted coupling for the case of an optimized time dependent coupling function u t we find a robust structure which includes sharp peaks and intervals where it is zero this structure shows a strong correlation with the location in phase space and appears to depend on noise imperfections of the model and the lyapunov direction vectors for time independent coupling we find the counterintuitive result that often the optimal rms error in fitting the model to the data initially increases with coupling strength comparison of this result with that obtained using simulated data may provide one measure of model imperfection the constrained method with time dependent coupling appears to have benefits in synchronizing long data sets with minimal impact while the initial value method with time independent coupling tends to be substantially faster more flexible and easier to use we also describe a method of coupling which is useful for sparse experimental data sets our use of the colpitts oscillator allows us to explore in detail the case of a system with one positive lyapunov exponent the methods we explored are easily extended to driven systems such as neurons with time dependent injected current they are expected to be of value in nonchaotic systems as well software is available on request

realization of parameters identification in only locally lipschitzian dynamical systems with multiple types of time delays most of the existing results for identifying parameters in dynamical systems through adaptive coupling techniques always require a uniform lipschitz condition on system vector fields however it is not this requirement but the only locally lipschitz condition that is more suitable for many nonlinear systems in real applications also time delays sometimes are essential for modeling real systems so that in addition to the discrete form time delays in the distributed form could be considered this paper thus considers the problem of parameters identification in only locally lipschitzian dynamical systems with multiple types of time delays through adaptive coupling techniques in particular the paper proposes some linear independence conditions for realization of parameters identification in both nonautonomous and autonomous systems and draws a comparison between these conditions and the persistent of excitation condition which is used frequently in the literature moreover the paper proposes an index t

optimize design of adaptive synchronization controllers and parameter observers in different hyperchaotic systems based on the lyapunov stability and adaptive synchronization theory optimization design of adaptive controllers and parameter observers with controllable gain coefficient are investigated in detail the linear errors of corresponding variables and parameters are used to construct different appropriate positive lyapunov functions v and the parameter observers and adaptive controllers are approached analytically by simplifying the differential inequality dv dt 0 particularly an optional gain coefficient is selected in the parameter observers and positive lyapunov function which decides the transient period to identify the unknown parameters and reach synchronization the scheme is used to study the synchronization of two non identical hyperchaotic rossler systems the theoretical and numerical results confirm that the four unknown parameters in the drive system are estimated exactly and the two hyperchaotic systems reach complete synchronization when the controllers and parameter observers work on the driven system to confirm the model independence of this scheme an alternative hyperchaotic system is investigated whereby the results confirm that the five unknown parameters are identified rapidly and exactly and that the two hyperchaotic systems reach complete synchronization as well

likelihood and bayesian prediction of chaotic systems abstract there has recently been considerable interest in both applied disciplines and in mathematics as well as in the popular science literature in the areas of nonlinear dynamical systems and chaotic processes by a nonlinear deterministic dynamical system we mean a time series in which starting at some initial condition the values of the series are some fixed nonlinear function of the previous states one of the more intriguing aspects of these models is their propensity for displaying very complex apparently random behavior even when simple models are analyzed a consequence of such chaotic behavior is that it is difficult to predict the exact behavior of a chaotic system the difficulty in prediction stems from the fact that even the tiniest of errors including computer roundoff in either the specification of the function or the initial condition can lead to huge errors in prediction after a brief review of dynamical systems and the role of probability in dealing with uncertainty a com

upper bounds to the correlation functions of the random bond ising model and n vector model with competing interactions studies a correlation function which is given by the canonical average of a product of one or more spin variables for the random bond ising model in which the exchange integrals are j j o and j with probabilities p and 1 p respectively the authors show that an upper bound to the configurational average of the correlation functions calculated in the thermodynamic limit in the zero external field limit is the product of the same correlation function for the corresponding ferromagnetic ising model at the temperature under consideration and the same quantity at the temperature t1 which is determined by the condition t1 2j mod kb1n p 1 p mod where kb is the boltzmann constant the results are given for the random bond ising model of an arbitrary spin s and also for the diluted random bond ising models of an arbitrary spin s with the pair interaction and with general interaction and for the diluted and undiluted random bond n vector model

failure of maximum likelihood methods for chaotic dynamical systems the maximum likelihood method is a basic statistical technique for estimating parameters and variables and is the starting point for many more sophisticated methods like bayesian methods this paper shows that maximum likelihood fails to identify the true trajectory of a chaotic dynamical system because there are trajectories that appear to be far more infinitely more likely than truth this failure occurs for unbounded noise and for bounded noise when it is sufficiently large and will almost certainly have consequences for parameter estimation in such systems the reason for the failure is rather simple in chaotic dynamical systems there can be trajectories that are consistently closer to the observations than the true trajectory being observed and hence their likelihood dominates truth the residuals of these truth dominating trajectories are not consistent with the noise distribution they would typically have too small standard deviation and many outliers and hence the situation may be remedied by using methods that examine the distribution of residuals and are not entirely maximum likelihood based

estimation of noise levels for models of chaotic dynamical systems we investigate how far it is possible to identify and separate dynamical noise from measurement noise in observed nonlinear time series using bayesian methods we derive estimates for the two noise levels and find that given a good model of the dynamics these can give accurate results even if the dynamical noise level is orders of magnitude smaller than the measurement noise level whereas a simple calculation of root mean square error badly understates the dynamical noise we argue that this allows better estimates of the underlying dynamical time series and so better predictions of its future and of its fundamental dynamical properties

oscillatory criteria for second order quasilinear neutral delay difference equations consider the second order quasilinear neutral difference equation da n d x n p nx g n a email protected x n p nx g n f n x s n 0 the sufficient conditions are established for oscillation of the solution of this eq these generalize and improve some known results about both neutral and delay difference equations

oscillation and non oscillation for second order linear difference equations abstract we prove new oscillation and non oscillation theorems for the second order linear difference equation 2xn 1 pnxn 0 where p n n 1 is a real sequence with pn 0 these results are extensions of earlier results of zhang and zhou comput math appl 39 2000 1 7

oscillation of solutions for second order nonlinear difference equations with nonlinear neutral term abstract some riccati type difference inequalities are given for the second order nonlinear difference equations with nonlinear neutral term and using these inequalities we obtain some oscillation criteria for the above equation

some oscillation results for second order nonlinear dynamic equations of neutral type abstract in this paper some oscillation results for the second order nonlinear neutral delay dynamic equation r t y t p t y t q t y t sgn y t 0 are obtained on a time scale t under the assumption t 0 1 r t 1 t where 0 is a quotient of odd positive integers

indistinguishable states i perfect model scenario an accurate forecast of a nonlinear system will require an accurate estimation of the initial state it is shown that even under the ideal conditions of a perfect model and infinite past observations of a deterministic nonlinear system uncertainty in the observations makes exact state estimation is impossible consistent with the noisy observations there is a set of states indistinguishable from the true state this implies that an accurate forecast must be based on a probability density on the indistinguishable states this paper shows that this density can be calculated by first calculating a maximum likelihood estimate of the state and then an ensemble estimate of the density of states that are indistinguishable from the maximum likelihood state a new method for calculating the maximum likelihood estimate of the true state is presented which allows practical ensemble forecasting even when the recurrence time of the system is long in a subsequent paper the theory and practice described in this paper are extended to an imperfect model scenario 2001 published by elsevier science b v

an application of discrete inequality to second order nonlinear oscillation abstract by using some simple discrete inequalities oscillation criteria are provided for the second order difference equations 2 y n a n 1 y n 1 0 n n where the operator is defined by yn yn 1 yn an is a real sequence the function is such that u u 0 for u 0 and u v g u v u v for u v 0 for some nonnegative function g

oscillation of second order nonlinear neutral delay dynamic equations on time scales in this paper some sufficient conditions for oscillation of the second order nonlinear neutral delay dynamic equation r t y t p t y t f t y t 0 on a time scale t are established here 1 is an odd positive integer with r t and p t are rd continuous functions defined on t our results as a special case when t r and t n involve and improve some well known oscillation results for second order neutral delay differential and difference equations when t hn and t qn t t qk k n q 1 i e for generalized neutral delay difference and q neutral delay difference equations our results are essentially new and also can be applied on different types of time scales e g t n2 t2 t n and t tn tn n n0 where tn is the set of harmonic numbers some examples illustrating our main results are given

oscillatory and asymptotic behavior of second order neutral difference equations with maxima abstract in this paper the asymptotic behavior of the non oscillatory solutions of neutral difference equations with maxima is considered sufficient conditions for oscillation of all solutions are also obtained