computing two dimensional poincar maps for hyperchaotic dynamics poincar map pm is one of the felicitous discrete approximation of the continuous dynamics to compute pm the discrete relation s between the successive point of interactions of the trajectories on the suitable poincar section ps are found out these discrete relations act as an amanuensis of the nature of the continuous dynamics in this article we propose a computational scheme to find a hyperchaotic pm hpm from an equivalent three dimensional 3d subsystem of a 4d or higher hyperchaotic model for the experimental purpose a standard four dimensional 4d hyperchaotic lorenz stenflo system hlss and a five dimensional 5d hyperchaotic laser model hlm is considered equivalent 3d subsystem is obtained by comparing the movements of the trajectories of the original hyperchaotic systems with all of their 3d subsystems the quantitative measurement of this comparison is made promising by recurrence quantification analysis rqa various two dimensional 2d poincar mas are computed for several suitable poincar sections for both the systems but only some of them are hyperchaotic in nature the hyperchaotic behavior is verified by positive values of both one dimensional 1d lyapunov exponent le i and 2d lyapunov exponent le ii at the end similarity of the dynamics between the hyperchaotic systems and their 2d hyperchaotic poincar maps hpm has been established through mean recurrence time mrt statistics for both of 4d hlss and 5d hlm and the best approximated discrete dynamics for both the hyperchaotic systems are found out

spectral methods for weakly singular volterra integral equations with smooth solutions we propose and analyze a spectral jacobi collocation approximation for the linear volterra integral equations vies of the second kind with weakly singular kernels in this work we consider the case when the underlying solutions of the vies are sufficiently smooth in this case we provide a rigorous error analysis for the proposed method which shows that the numerical errors decay exponentially in the infinity norm and weighted sobolev space norms numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence

legendre multi galerkin methods for fredholm integral equations with weakly singular kernel and the corresponding eigenvalue problem abstract in this paper we consider legendre multi galerkin methods to solve fredholm integral equations of the second kind with weakly singular kernel and the corresponding eigenvalue problem we obtain the convergence rates for the approximated solution and iterated solution in weakly singular fredholm integral equations of the second kind in both l 2 and infinity norm we also establish error bounds of approximated eigenelements with exact eigenelements in the eigenvalue problem of a compact integral operator with weakly singular kernel in both infinity and l 2 norm numerical examples are presented to prove the theoretical estimates

calculation of lyapunov exponents avoiding spurious elements abstract a new approach to evaluate lyapunov exponents from time series is discussed with the help of this approach the number of calculated lyapunov exponents can be reduced to the physically relevant ones calculations in high dimensional embedding spaces are performed on smaller data bases with improved speed more direct information is obtained from the dynamics of the underlying dynamical system

entropy of weighted recurrence plots the shannon entropy of a time series is a standard measure to assess the complexity of a dynamical process and can be used to quantify transitions between different dynamical regimes an alternative way of quantifying complexity is based on state recurrences such as those available in recurrence quantification analysis although varying definitions for recurrence based entropies have been suggested so far for some cases they reveal inconsistent results here we suggest a method based on weighted recurrence plots and show that the associated shannon entropy is positively correlated with the largest lyapunov exponent we demonstrate the potential on a prototypical example as well as on experimental data of a chemical experiment

complexity and synchronization in chaotic fiber optic systems abstract in this work we investigate experimentally the complexity of chaotic attractors generated by a semiconductor laser subjected to optical feedback and associate their dimensionality with the synchronization efficiency of the corresponding chaotic transmitter receiver configuration the complexity is characterized by calculating the correlation dimension d 2 of experimental chaotic time series for different values of the optical feedback we present the effect of d 2 on the synchronization efficiency and determine the optimal operating condition that leads to the most complex chaotic carrier and simultaneously to the most successful synchronization lastly we associate and explain our experimental results with theoretical predictions in the research literature

weighted multiscale renyi permutation entropy of nonlinear time series abstract in this paper based on renyi permutation entropy rpe which has been recently suggested as a relative measure of complexity in nonlinear systems we propose multiscale renyi permutation entropy mrpe and weighted multiscale renyi permutation entropy wmrpe to quantify the complexity of nonlinear time series over multiple time scales first we apply mpre and wmpre to the synthetic data and make a comparison of modified methods and rpe meanwhile the influence of the change of parameters is discussed besides we interpret the necessity of considering not only multiscale but also weight by taking the amplitude into account then mrpe and wmrpe methods are employed to the closing prices of financial stock markets from different areas by observing the curves of wmrpe and analyzing the common statistics stock markets are divided into 4 groups 1 dji s p500 and hsi 2 nasdaq and ftse100 3 dax40 and cac40 and 4 shangzheng and shencheng results show that the standard deviations of weighted methods are smaller showing wmrpe is able to ensure the results more robust besides wmpre can provide abundant dynamical properties of complex systems and demonstrate the intrinsic mechanism

computation of optimal feedforward and feedback control by a modified reduced gradient method a computational scheme for solving problems of optimal open loop control policy and optimal state feedback control was introduced the problems were first converted into a nonlinear programming problem the conjugate reduced gradient method was then applied to find the minimum solution the sensitivity matrices of the state trajectories with respect to the coefficients of control variables could be recursively calculated so that the reduced gradients could be easily evaluated in this computational algorithm the h orthogonal property for linear optimal control problems was demonstrated such that the global minimum solution of such problems was thus achieved in finite iterations using this algorithm the computational strategy was easily and efficiently extended to determine the optimal state feedback gain matrices

on the dynamic properties of axially moving systems abstract the objective of the present paper is a deep analysis of some recent numerical and experimental results regarding the complex dynamics of axially moving systems such important mechanical systems exhibit interesting dynamic behaviors homoclinic orbits sub harmonic responses amplitude modulations and chaos these dynamics have been obtained numerically and in some cases have been experimentally observed using recent techniques of the non linear time series analysis the response of axially moving systems has been studied for a large variety of test cases the correlation dimension of the time series which is deeply related to the minimal dimension of a system able to reproduce the dynamics is estimated lyapunov exponents are evaluated in order to quantify the response regularity the present work gives a contribution towards understanding the complex dynamics observed both in conservative and dissipative systems the dynamical phenomena are analyzed within the unified framework of the non linear time series analysis in the case of experimental data the new non linear filtering techniques based on the embedding techniques have been applied to reduce high noise when classical techniques give bad results

a chebyshev expansion method for solving nonlinear optimal control problems this paper presents a numerical technique for solving the controlled duffing oscillator the control and state variables are approximated by chebyshev series a technique is provided for approximating the system dynamics boundary conditions and performance index this technique is based on using an explicit formula for the chebyshev polynomials in terms of arbitrary order of their derivatives the system dynamics and the performance index are converted into some algebraic equations then the optimal control problem is reduced to constrained optimization problem results and comparisons are given at the end of this paper

lossless convexification of non convex optimal control problems for state constrained linear systems this paper analyzes a class of finite horizon optimal control problems with mixed non convex and convex control constraints and linear state constraints a convex relaxation of the problem is proposed and it is proved that a solution of the relaxed problem is also a solution of the original problem this process is called lossless convexification and its generalization for problems with state constraints is the primary contribution of the paper doing so enables the use of interior point methods of convex optimization to obtain global optimal solutions of the original non convex problem the approach is also demonstrated on example problems

superconvergence of legendre spectral projection methods for fredholmhammerstein integral equations in this paper we consider the multi galerkin and multi collocation methods for solving the fredholmhammerstein integral equation with a smooth kernel using legendre polynomial bases we show that legendre multi galerkin and legendre multi collocation methods have order of convergence o n3r 34 and o n2r 12 respectively in uniform norm where n is the highest degree of legendre polynomial employed in the approximation and r is the smoothness of the kernel also one step of iteration method is used to improve the order of convergence and we prove that iterated legendre multi galerkin and iterated legendre multi collocation methods have order of convergence o n4r and o n2r respectively in uniform norm numerical examples are given to illustrate the theoretical results

fundamental limitations for estimating dimensions and lyapunov exponents in dynamical systems we show that values of the correlation dimension estimated over a decade from the grassberger procaccia algorithm cannot exceed the value 2 log10n if n is the number of points in the time series when this bound is saturated it is thus not legitimate to conclude that low dimensional dynamics is present the estimation of lyapunov exponents is also discussed

multiscale recurrence quantification analysis of order recurrence plots abstract in this paper we propose a new method of multiscale recurrence quantification analysis msrqa to analyze the structure of order recurrence plots the msrqa is based on order patterns over a range of time scales compared with conventional recurrence quantification analysis rqa the msrqa can show richer and more recognizable information on the local characteristics of diverse systems which successfully describes their recurrence properties both synthetic series and stock market indexes exhibit their properties of recurrence at large time scales that quite differ from those at a single time scale some systems present more accurate recurrence patterns under large time scales it demonstrates that the new approach is effective for distinguishing three similar stock market systems and showing some inherent differences

recurrence plot analysis of irregularly sampled data irregularly sampled time series usually require data preprocessing before a desired time series analysis can be applied we propose an approach for distance measuring of pairs of data points which is directly applicable to irregularly sampled time series in order to apply recurrence plot analysis to irregularly sampled time series we use this approach and detect regime transitions in prototypical models and for an application from palaeoclimatatology this approach might be useful for any method that is based on distance measuring e g correlation sum or lyapunov exponent estimation

brief numerical treatment of multiobjective optimal control problems we present a numerical procedure for solving optimal control problems with both linear terminal constraints and multiple criteria using a chebyshev spectral procedure the problem reduces to a constrained optimization problem which can be solved using hybrid penalty partial quadratic interpolation hppqi technique the proposed procedure compares quite favorably with other methods on a sample of well known examples

topological analysis and synthesis of chaotic time series abstract we have developed a topological procedure for analyzing chaotic time series which identifies the stretching and squeezing mechanisms responsible for chaotic behavior in low dimensional dynamical systems these mechanisms quantitatively described by a template or knot holder can then be used to model the processes which generate the original chaotic data set

a chebyshev technique for solving nonlinear optimal control problems a numerical technique for solving nonlinear optimal control problems is introduced the state and control variables are expanded in the chebyshev series and an algorithm is provided for approximating the system dynamics boundary conditions and performance index application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the chebyshev coefficients the optimum condition is obtained by applying the method of constrained extremum for linear quadratic optimal control problems the state and control variables are determined by solving a set of linear equations in the chebyshev coefficients applicability is illustrated with the minimum time and maximum radius orbit transfer problems

testing for chaos in deterministic systems with noise abstract recently we introduced a new test for distinguishing regular from chaotic dynamics in deterministic dynamical systems and argued that the test had certain advantages over the traditional test for chaos using the maximal lyapunov exponent in this paper we investigate the capability of the test to cope with moderate amounts of noisy data comparisons are made between an improved version of our test and both the tangent space method and direct method for computing the maximal lyapunov exponent the evidence of numerical experiments ranging from the logistic map to an eight dimensional lorenz system of differential equations the lorenz 96 system suggests that our method is superior to tangent space methods and that it compares very favourably with direct methods

an efficient chebyshev algorithm for the solution of optimal control problems in this paper we derive an efficient chebyshev algorithm for solving optimal control problems the chebyshev expansions are employed to approximate both the control and the state functions the discretizing process and the related techniques are unique compared to existing methods the optimal control problems are transformed into the resulting mathematical programming problems theoretical analysis is given to support the method further numerical examples and comparisons are presented to illustrate the efficiency of the method

legendre spectral projection methods for urysohn integral equations in this paper we consider the legendre spectral galerkin and legendre spectral collocation methods to approximate the solution of urysohn integral equation we prove that the approximated solutions of the legendre galerkin and legendre collocation methods converge to the exact solution with the same orders o n r in l 2 norm and o n 1 2 r in infinity norm and the iterated legendre galerkin solution converges with the order o n 2 r in both l 2 norm and infinity norm whereas the iterated legendre collocation solution converges with the order o n r in both l 2 norm and infinity norm n being the highest degree of the legendre polynomial employed in the approximation and r being the smoothness of the kernel we are able to obtain similar superconvergence rates for the iterated galerkin solution for urysohn integral equations with smooth kernel as in the case of piecewise polynomial basis functions

measurement of the lyapunov spectrum from a chaotic time series the exponential divergence or convergence of nearby trajectories lyapunov exponents is conceptually the most basic indicator of deterministic chaos we propose a new method to determine the spectrum of several lyapunov exponents including positive zero and even negative ones from the observed time series of a single variable we have applied the method to various known model systems and also to the rayleigh b enard experiment and have elucidated the dependence of the lyapunov exponents on the rayleigh number

fast accurate and small scale direct trajectory optimization using a gegenbauer transcription method this paper reports a novel direct gegenbauer ultraspherical transcription method gtm for solving continuous time optimal control oc problems ctocps with linear nonlinear dynamics and path constraints in elgindy et al 2012 1 we presented a gtm for solving nonlinear ctocps directly for the state and the control variables and the method was tailored to find the best path for an unmanned aerial vehicle mobilizing in a stationary risk environment this article extends the gtm to deal further with problems including higher order time derivatives of the states by solving the ctocp directly for the control u t and the highest order time derivative x n t n z the state vector and its derivatives up to the n 1 th order derivative can then be stably recovered by successive integration moreover we present our solution method for solving linear quadratic regulator lqr problems as we aim to cover a wider collection of ctocps with the concrete aim of comparing the efficiency of the current work with other classical discretization methods in the literature the proposed numerical scheme fully parameterizes the state and the control variables using gegenbauer expansion series for problems with various order time derivatives of the state variables arising in the cost function dynamical system or path terminal constraints the gtm seeks to fully parameterize the control variables and the highest order time derivatives of the state variables the time horizon is mapped onto the closed interval 0 1 the dynamical system characterized by differential equations is transformed into its integral formulation through direct integration the resulting problem on the finite interval is then transcribed into a nonlinear programming nlp problem through collocation at the gegenbauer gauss gg points the integral operations are approximated by optimal gegenbauer quadratures in a certain optimality sense the reduced nlp problem is solved in the gegenbauer spectral space and the state and the control variables are approximated on the entire finite horizon the proposed method achieves discrete solutions exhibiting exponential convergence using relatively small scale number of collocation points the advantages of the proposed direct gtm over other traditional discretization methods are shown through four well studied oc test examples the present work is a major breakthrough in the area of computational oc theory as it delivers significantly more accurate solutions using considerably smaller numbers of collocation points states and controls expansion terms moreover the gtm produces very small scale nlp problems which can be solved very quickly using the modern nlp software

a wavelet based method for surrogate data generation hypothesis testing based on surrogate data has emerged as a popular way to test the null hypothesis that a signal is a realisation of a linear gaussian stochastic process if these surrogates are constrained to the values and power spectrum of the original data there is no need to formulate a pivotal test statistic in this paper a method is presented for generating constrained surrogates using a wavelet transform introducing a threshold above which wavelet detail coefficients are pinned to their original values such surrogates avoid problems of nonstationarity for pseudo periodic data and appear to be more robust than conventional approaches for situations where period modulation is corrupting a gaussian stochastic process when used for generating ensemble realisations of a process the approach used here avoids some of the difficulties of methods based on simple randomisation of wavelet coefficients

using higher order correlations to define an embedding window abstract correlation functions of up to fourth order are investigated numerically in search of a reliable window for embedding or reconstructing chaotic flows by the method of time delays a number of these higher order correlations display coincident extrema the time at which this coincidence occurs is shown to be a reasonable embedding window for calculating the correlation dimension of a number of well known chaotic attractors

dispersive graded entropy on computing dynamical complexity abstract we propose a phase space based statistical disorder to investigate the dynamical complexity of chaotic models the statistical disorder is defined by introducing a grade function inversely maps the mean dispersion of the trajectories in the phase space we denote the associate entropy by the dispersive graded entropy dge numerical investigation shows that dge can quantify the dynamical complexity of discrete as well as continuous chaotic systems a comparative study is also made with the other phase space based entropy measures finally the proposed measure has been applied on three types of heart rate variability hrv signals the results support the clinical observations related to the dynamics of healthy and congestive hearts

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

an approach for characterizing coupling in dynamical systems abstract the study of coupling in dynamical systems dates back to christian hyugens who in 1665 discovered that pendulum clocks with the same length pendulum synchronize when they are near to each other in that case the observed synchronous motion was out of phase in this paper we propose a new approach for measuring the degree of coupling and synchronization of a dynamical system consisting of interacting subsystems the measure is based on quantifying the active degrees of freedom e g correlation dimension of the coupled system and the constituent subsystems the time delay embedding scheme is extended to coupled systems and used for attractor reconstruction of the coupled dynamical system we use the coupled lorenz rossler and henon model systems with a coupling strength variable for evaluation of the proposed approach results show that we can measure the active degrees of freedom of the coupled dynamical systems and can quantify and distinguish the degree of synchronization or coupling in each of the dynamical systems studied furthermore using this approach the direction of coupling can be determined

estimating the shannon entropy recurrence plots versus symbolic dynamics recurrence plots were first introduced to quantify the recurrence properties of chaotic dynamics a few years later the recurrence quantification analysis was introduced to transform graphical representations into statistical analysis among the different measures introduced a shannon entropy was found to be correlated with the inverse of the largest lyapunov exponent the discrepancy between this and the usual interpretation of a shannon entropy is solved here by using a new definition still based on the recurrence plots and it is verified that this new definition is correlated with the largest lyapunov exponent as expected from the pesin conjecture a comparison with a shannon entropy computed from symbolic dynamics is also provided

numerical method for a class of optimal control problems subject to nonsmooth functional constraints in this paper we consider a class of optimal control problems which is governed by nonsmooth functional inequality constraints involving convolution first we transform it into an equivalent optimal control problem with smooth functional inequality constraints at the expense of doubling the dimension of the control variables then using the chebyshev polynomial approximation of the control variables we obtain an semi infinite quadratic programming problem at last we use the dual parametrization technique to solve the problem

superconvergence of legendre projection methods for the eigenvalue problem of a compact integral operator in this paper we consider the galerkin and collocation methods for the eigenvalue problem of a compact integral operator with a smooth kernel using the legendre polynomials of degree n we prove that the error bounds for eigenvalues are of the order o n 2 r and the gap between the spectral subspaces are of the orders o n r in l 2 norm and o n 1 2 r in the infinity norm where r denotes the smoothness of the kernel by iterating the eigenvectors we show that the iterated eigenvectors converge with the orders of convergence o n 2 r in both l 2 norm and infinity norm we illustrate our results with numerical examples

mean phase coherence as a measure for phase synchronization and its application to the eeg of epilepsy patients abstract we apply the concept of phase synchronization of chaotic and or noisy systems and the statistical distribution of the relative instantaneous phases to electroencephalograms eegs recorded from patients with temporal lobe epilepsy using the mean phase coherence as a statistical measure for phase synchronization we observe characteristic spatial and temporal shifts in synchronization that appear to be strongly related to pathological activity in particular we observe distinct differences in the degree of synchronization between recordings from seizure free intervals and those before an impending seizure indicating an altered state of brain dynamics prior to seizure activity

a benchmark test of accuracy and precision in estimating dynamical systems characteristics from a time series characteristics of dynamical systems are often estimated to describe physiological processes for instance lyapunov exponents have been determined to assess the stability of the cardio vascular system respiration and more recently human gait and posture however the systematic evaluation of the accuracy and precision of these estimates is problematic because the proper values of the characteristics are typically unknown we fill this void with a set of standardized time series with well defined dynamical characteristics that serve as a benchmark estimates ought to match these characteristics at least to good approximation we outline a procedure to employ this generic benchmark test and illustrate its capacity by examining methods for estimating the maximum lyapunov exponent in particular we discuss algorithms by wolf and co workers and by rosenstein and co workers and evaluate their performances as a function of signal length and signal to noise ratio in all scenarios the precision of rosenstein s algorithm was found to be equal to or greater than wolf s algorithm the latter however appeared more accurate if reasonably large signal lengths are available and noise levels are sufficiently low due to its modularity the presented benchmark test can be used to evaluate and tune any estimation method to perform optimally for arbitrary experimental data

recurrence plot based measures of complexity and their application to heart rate variability data the knowledge of transitions between regular laminar or chaotic behaviors is essential to understand the underlying mechanisms behind complex systems while several linear approaches are often insufficient to describe such processes there are several nonlinear methods that however require rather long time observations to overcome these difficulties we propose measures of complexity based on vertical structures in recurrence plots and apply them to the logistic map as well as to heart rate variability data for the logistic map these measures enable us not only to detect transitions between chaotic and periodic states but also to identify laminar states i e chaos chaos transitions the traditional recurrence quantification analysis fails to detect the latter transitions applying our measures to the heart rate variability data we are able to detect and quantify the laminar phases before a life threatening cardiac arrhythmia occurs thereby facilitating a prediction of such an event our findings could be of importance for the therapy of malignant cardiac arrhythmias

a scaling exponent based detector of chaos in oscillatory circuits abstract in this paper a scaling exponent based approach is proposed to determine the state of chaotic circuits and the scaling exponent is calculated using detrended fluctuation analysis dfa the corresponding detector is designed using the fact that the scaling exponent changes for various states of chaotic circuits simulation examples in this paper are performed for chua s circuit and other chaotic systems and compared with the state of the art in the field the proposed detector outperforms existing techniques in ability to distinguish the chaotic and periodic states in the circuits for relatively high noise

nonlinear transformation on the transfer entropy of financial time series abstract transfer entropy te now is widely used in the data mining and economic field however te itself demands that time series intend to be stationary and meet markov condition naturally we are interested in investigating the effect of the nonlinear transformation of the two series on the te therefore the paper is designed to study the te of five nonlinear volatile transformations based on the data which are generated by the linear modeling and the logistic maps modeling as well as the dataset that come from financial markets with only one of the te of nonlinear transformations fluctuating around the te of original series the te of others all have increased with different degrees

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

a pmgop method of parameter identification for a class of continuous nonlinear systems a series of piecewise multiple general orthogonal polynomials pmgops is introduced and applied to the parameter identication problem for a class of continuous nonlinear systems after introducing pmgops and discussing their main properties an effective procedure for the parameter identification of a large class of continuous systems called parameter separable systems is proposed some relevant algorithms are presented the procedure given in the paper has the following advantages compared with other methods the identification algorithm ia obtained is computationally fast and accurate the ia can be implemented in a recursive fashion the ia is effective for a small number of data points the ia does not require a priori knowledge of the estimated parameters the ia is tolerant to choices of expanding orders when pmgops are applied some numerical examples together with an experimental example taken from a biochemical fermentation process are given for illustrative purposes

brief paper a unified framework for the numerical solution of optimal control problems using pseudospectral methods a unified framework is presented for the numerical solution of optimal control problems using collocation at legendre gauss lg legendre gauss radau lgr and legendre gauss lobatto lgl points it is shown that the lg and lgr differentiation matrices are rectangular and full rank whereas the lgl differentiation matrix is square and singular consequently the lg and lgr schemes can be expressed equivalently in either differential or integral form while the lgl differential and integral forms are not equivalent transformations are developed that relate the lagrange multipliers of the discrete nonlinear programming problem to the costates of the continuous optimal control problem the lg and lgr discrete costate systems are full rank while the lgl discrete costate system is rank deficient the lgl costate approximation is found to have an error that oscillates about the true solution and this error is shown by example to be due to the null space in the lgl discrete costate system an example is considered to assess the accuracy and features of each collocation scheme

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

brief paper lossless convexification of a class of optimal control problems with non convex control constraints we consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost convex state constraints and non convex control constraints we propose a convex relaxation of the non convex control constraints and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem which is referred to as the lossless convexification of the optimal control problem the lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non convex optimal control problem the solution approach is demonstrated on a number of planetary soft landing optimal control problems

detecting determinism in short time series using a quantified averaged false nearest neighbors approach we propose a criterion to detect determinism in short time series this criterion is based on the estimation of the parameter e 2 defined by the averaged false neighbors method for analyzing time series cao physica d 110 43 1997 using surrogate data testing with several chaotic and stochastic simulated time series we show that the variation coefficient of e 2 over a few values of the embedding dimension d defines a suitable statistic to detect determinism in short data sequences this result holds for a time series generated by a high dimensional chaotic system such as the mackey glass one different decreasing lengths of the time series are included in the numerical experiments for both synthetic and real world data we also investigate the robustness of the criterion in the case of deterministic time series corrupted by additive noise

state control spectral chebyshev parameterization for linearly constrained quadratic optimal control problems in this paper we propose a computationally attractive numerical method for determining the optimal control of constrained linear dynamic systems with a quadratic performance the method is based upon constructing the mth degree interpolating polynomials using chebyshev nodes to approximate the control and the state vectors the system dynamics are collocated at chebyshev nodes the performance index is discretized by a cell averaging method the state and control inequality constraints are converted into algebraic inequalities through collocation at the nodes the linear quadratic optimal control problem is thereby transformed into a quadratic programming one simulation studies demonstrate computational advantages relative to a standard riccati method a classical chebyshev based method fourier based method and other methods in the literature

intermittent phase synchronization of coupled spatiotemporal chaotic systems phase synchronization is studied with a discrete system formed by two coupled map lattices in which phases are measured in two dimensional vectors simulation results show that by imposing external coupling between the two lattices phase synchronization can be found in all two dimensional phase planes between them when the system is approaching the phase synchronizing state unstable phase synchronization is observed this is referred to as intermittent phase synchronization that appears when the trajectories on two interacting phase planes have opposite directions of rotation but with only a small phase difference the intermittent phase synchronization could also be observed in coupled autonomous systems with diffusive attractors although their phase concepts are inconsistent our results show that the intermittent phase synchronization of both discrete and autonomous systems relates to the diffusion or the complexity of the attractors

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

the problem of spurious lyapunov exponents in time series analysis and its solution by covariant lyapunov vectors we briefly recall the methods for the determination of lyapunov exponents from time series data and their limitations one particular problem is given by the fact that reconstructed phase spaces have usually extra dimensions compared to the true phase space of a dynamical system leading to extra so called spurious lyapunov exponents several methods to identify the true ones have been proposed which do not give satisfactory results we show that the geometric information contained in covariant lyapunov vectors can be used to identify the true exponents we illustrate its use and its limitations by applying it to experimental nmr laser data this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

can complexity decrease in congestive heart failure abstract the complexity of a signal can be measured by the recurrence period density entropy rpde from the reconstructed phase space we have chosen a window based rpde method for the classification of signals as rpde is an average entropic measure of the whole phase space we have observed the changes in the complexity in cardiac signals of normal healthy person nhp and congestive heart failure patients chfp the results show that the cardiac dynamics of a healthy subject is more complex and random compare to the same for a heart failure patient whose dynamics is more deterministic we have constructed a general threshold to distinguish the border line between a healthy and a congestive heart failure dynamics the results may be useful for wide range for physiological and biomedical analysis

a robust method to estimate the largest lyapunov exponent of noisy signals a revision to the rosenstein s algorithm abstract this study proposed a revision to the rosenstein s method of numerical calculation of the largest lyapunov exponent lye to make it more robust to noise to this aim the effect of increasing number of initial neighboring points on the lye value was investigated and compared to values obtained by filtering the time series both simulated lorenz and passive dynamic walker and experimental human walking time series were used to calculate the lye the number of initial neighbors used to calculate lye for all time series was 1 the original rosenstein s method 2 3 4 5 10 15 20 25 and 30 data points the results demonstrated that the lye graph reached a plateau at the 15 point neighboring condition implying that the lye values calculated using at least 15 neighboring points were consistent the proposed method could be used to calculate more consistent lye values in experimental time series acquired from biological systems where noise is omnipresent

quantifying time varying coordination of multimodal speech signals using correlation map analysis this paper demonstrates an algorithm for computing the instantaneous correlation coefficient between two signals the algorithm is the computational engine for analyzing the time varying coordination between signals which is called correlation map analysis cma correlation is computed around any pair of points in the two input signals thus coordination can be assessed across a continuous range of temporal offsets and be detected even when changing over time due to temporal fluctuations the correlation algorithm has two major features i it is structurally similar to a tunable filter requiring only one parameter to set its cutoff frequency and sensitivity ii it can be applied either uni directionally computing correlation based only on previous samples or bi directionally computing correlation based on both previous and future samples computing instantaneous correlation for a range of time offsets between two signals produces a 2d correlation map in which correlation is characterized as

legendre multi projection methods for solving eigenvalue problems for a compact integral operator in this paper we consider the m galerkin and m collocation methods for solving the eigenvalue problem for a compact integral operator with smooth kernels using legendre polynomial bases we obtain error bounds for the eigenvalues and the gap between the spectral subspaces in both legendre m galerkin and legendre m collocation methods we also obtain superconvergence results for the eigenvalues and iterated eigenvectors in both l 2 and infinity norm we illustrate our results with numerical examples

can stability really predict an impending slip related fall among older adults the primary purpose of this study was to systematically evaluate and compare the predictive power of falls for a battery of stability indices obtained during normal walking among community dwelling older adults one hundred and eighty seven community dwelling older adults participated in the study after walking regularly for 20 strides on a walkway participants were subjected to an unannounced slip during gait under the protection of a safety harness full body kinematics and kinetics were monitored during walking using a motion capture system synchronized with force plates stability variables including feasible stability region measurement margin of stability the maximum floquet multiplier the lyapunov exponents short and long term and the variability of gait parameters including the step length step width and step time were calculated for each subject sensitivity of predicting slip outcome fall vs recovery was examined for each stability variable using logistic regression results showed that the feasible stability region measurement predicted fall incidence among these subjects with the highest sensitivity 68 4 except for the step width with an sensitivity of 60 2 no other stability variables could differentiate fallers from those who did not fall for the sample included in this study the findings from the present study could provide guidance to identify individuals at increased risk of falling using the feasible stability region measurement or variability of the step width

visualization of coupling in time series by order recurrence plots we introduce a method to visualize dependencies between two time series by applying the concept of cross recurrence plots to the local ordinal structure we derive a measure of the coupling strength which is robust against observational noise nonlinear distortion of the amplitude and low frequency trends connections to the instantaneous phase and the determination of phase coupling of two coupled rossler systems in the standard and funnel regimes are shown an application to electroencephalogram data demonstrates that the method is robust with respect to artifacts

numerical methods based on extended one step methods for solving optimal control problems in this article we develop extended one step methods for solving the dynamic system of optimal control problems governed by ordinary differential equations the control variables are approximated by polynomial functions the proposed problem is reduced to either a constrained or unconstrained minimization problem according to the nature of the dynamic system and the given conditions numerical results and comparisons with other methods are presented

data smoothing and interpolation using eighth order algebraic splines a new type of algebraic spline is used to derive a filter for smoothing or interpolating discrete data points the spline is dependent on control parameters that specify the relative importance of data fitting and the derivatives of the spline a general spline of arbitrary order is first formulated using matrix equations we then focus on eighth order splines because of the continuity of their first three derivatives desirable for motor and robotics applications the spline s matrix equations are rewritten to give a recursive filter that can be implemented in real time for lengthy data sequences the filter is lowpass with a bandwidth that is dependent on the spline s control parameters numerical results including a simple image processing application show the tradeoffs that can be achieved using the algebraic splines

learning continuous trajectories in recurrent neural networks with time dependent weights the paper is concerned with a general learning with arbitrary criterion and state dependent constraints of continuous trajectories by means of recurrent neural networks with time varying weights the learning process is transformed into an optimal control framework where the weights to be found are treated as controls a learning algorithm based on a variational formulation of pontryagin s maximum principle is proposed this algorithm is shown to converge under reasonable conditions to an optimal solution the neural networks with time dependent weights make it possible to efficiently find an admissible solution i e initial weights satisfying state constraints which then serves as an initial guess to carry out a proper minimization of a given criterion the proposed methodology may be directly applicable to both classification of temporal sequences and to optimal tracking of nonlinear dynamic systems numerical examples are also given which demonstrate the efficiency of the approach presented

deterministic dynamics of neural activity during absence seizures in rats the study of brain electrical activities in terms of deterministic nonlinear dynamics has recently received much attention forbidden ordinal patterns fop is a recently proposed method to investigate the determinism of a dynamical system through the analysis of intrinsic ordinal properties of a nonstationary time series the advantages of this method in comparison to others include simplicity and low complexity in computation without further model assumptions in this paper the fop of the eeg series of genetic absence epilepsy rats from strasbourg was examined to demonstrate evidence of deterministic dynamics during epileptic states experiments showed that the number of fop of the eeg series grew significantly from an interictal to an ictal state via a preictal state these findings indicated that the deterministic dynamics of neural networks increased significantly in the transition from the interictal to the ictal states and also suggested that the fop measures of the eeg series could be considered as a predictor of absence seizures

evaluation of the threshold of stability for the human spine the threshold of stability tos is introduced as a new tool for evaluating spinal stability current methods have evaluated the magnitude of spinal kinematic variability stability diffusion or lyapunov exponents this present method differs by modifying task difficulty to obtain a critical value delineating regions of stability and instability conceptually as task difficulty increases kinematic variability of the system increases and the basin of stability decreases when kinematic variability exceeds the basin of stability stability cannot be maintained over time this concept is first illustrated using a mathematical model from which an effective potential function was calculated to show the relationships among kinematic variability the basin of stability and task difficulty in addition an experiment was performed to evaluate the sensitivity of the tos to changes in postural control using visual feedback as a control variable the tos was found to be more sensitive than the lyapunov exponent to removal of visual feedback suggesting it may have use as a diagnostic indicator e g for low back pain furthermore this new method has an additional advantage in which minimal instrumentation is needed its simplicity sensitivity and low cost suggest that the tos has potential as a diagnostic or prognostic tool in a clinical setting

legendre series approach to identification and analysis of linear systems in this paper the legendre operational matrix of integration is introduced and it is subsequently used for the identification and analysis of linear systems the algorithms proposed are easily implementable in a digital computer and they are analogous to those already derived for other types of orthogonal functions

a chebyshev approximation for solving optimal control problems abstract this paper presents a numerical solution for solving optimal control problems and the controlled duffing oscillator a new chebyshev spectral procedure is introduced control variables and state variables are approximated by chebyshev series then the system dynamics is transformed into systems of algebraic equations the optimal control problem is reduced to a constrained optimization problem results and comparisons are given at the end of the paper

the pseudospectral legendre method for discretizing optimal control problems this paper presents a computational technique for optimal control problems including state and control inequality constraints the technique is based on spectral collocation methods used in the solution of differential equations the system dynamics are collocated at legendre gauss lobatto points the derivative x spl dot t of the state x t is approximated by the analytic derivative of the corresponding interpolating polynomial state and control inequality constraints are collocated at legendre gauss lobatto nodes the integral involved in the definition of the performance index is discretized based on the gauss lobatto quadrature rule the optimal control problem is thereby converted into a mathematical programming program thus existing well developed optimization algorithms may be used to solve the transformed problem the method is easy to implement capable of handling various types of constraints and yields very accurate results illustrative examples are included to demonstrate the capability of the proposed method and a comparison is made with existing methods in the literature

polynomially based multi projection methods for fredholm integral equations of the second kind in this paper we propose a multi projection and iterated multi projection methods for fredholm integral equations of the second kind with a smooth kernel using polynomial bases we obtain super convergence rates for the approximate solutions more precisely we prove that in m galerkin and m collocation methods not only iterative solution u n approximates the exact solution u in the supremum norm with the order of convergence n 4 k but also the derivatives of u n approximate the corresponding derivatives of u in the supremum norm with the same order of convergence n being the degree of polynomial approximation and k being the smoothness of the kernel

a numerical solution of problems in calculus of variation using direct method and nonclassical parameterization in this paper a direct method for solving variational problems using nonclassical parameterization is presented a nonclassical parameterization based on nonclassical orthogonal polynomials is first introduced to reduce a variational problem to a nonlinear mathematical programming problem then using the lagrange multiplier technique the problem is converted to that of solving a system of algebraic equations illustrative examples are given to demonstrate the validity and applicability of the technique

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

noise induced fluctuations of period lengths of stable periodic orbits we discuss a class of one dimensional maps which possesses a globally attracting stable periodic orbit despite a strongly negative lyapunov exponent a small amount of noise can introduce fluctuations of the period length it is shown that this is a reasonable model for the observed dynamics of a bubble formation experiment in a heated capillary embedded in boiling water

optimal gegenbauer quadrature over arbitrary integration nodes this paper treats definite integrations numerically using gegenbauer quadratures the novel numerical scheme introduces the idea of exploiting the strengths of the chebyshev legendre and gegenbauer polynomials through a unified approach and using a unique numerical quadrature in particular the developed numerical scheme employs the gegenbauer polynomials to achieve rapid rates of convergence of the quadrature for the small range of the spectral expansion terms for a large scale number of expansion terms the numerical quadrature has the advantage of converging to the optimal chebyshev and legendre quadratures in the l norm and l 2 norm respectively the key idea is to construct the gegenbauer quadrature through discretizations at some optimal sets of points of the gegenbauer gauss gg type in a certain optimality sense we show that the gegenbauer polynomial expansions can produce higher order approximations to the definite integrals 1 x if x dx of a smooth function f x c 1 1 for the small range by minimizing the quadrature error at each integration point x i through a pointwise approach the developed gegenbauer quadrature can be applied for approximating integrals with any arbitrary sets of integration nodes exact integrations are obtained for polynomials of any arbitrary degree n if the number of columns in the developed gegenbauer integration matrix gim is greater than or equal to n the error formula for the gegenbauer quadrature is derived moreover a study on the error bounds and the convergence rate shows that the optimal gegenbauer quadrature exhibits very rapid convergence rates faster than any finite power of the number of gegenbauer expansion terms two efficient computational algorithms are presented for optimally constructing the gegenbauer quadrature we illustrate the high order approximations of the optimal gegenbauer quadrature through extensive numerical experiments including comparisons with conventional chebyshev legendre and gegenbauer polynomial expansion methods the present method is broadly applicable and represents a strong addition to the arsenal of numerical quadrature methods

the effect of unstable loading versus unstable support conditions on spine rotational stiffness and spine stability during repetitive lifting lumbar spine stability has been extensively researched due to its necessity to facilitate load bearing human movements and prevent structural injury the nature of certain human movement tasks are such that they are not equivalent in levels of task stability i e the stability of the external environment the goal of the current study was to compare the effects of dynamic lift instability administered through both the load and base of support on the dynamic stability maximal lyapunov exponents and stiffness emg driven model of the lumbar spine during repeated sagittal lifts fifteen healthy males performed 23 repetitive lifts with varying conditions of instability at the loading and support interfaces an increase in spine rotational stiffness occurred during unstable support scenarios resulting in an observed increase in mean and maximum euclidean norm spine rotational stiffness p 0 0011 significant stiffening effects were observed in unstable support conditions about all lumbar spine axes with the exception of lateral bend relative to a stable control lifting trial the addition of both an unstable load as well as an unstable support did not result in a significant change in the local dynamic stability of the lumbar spine p 0 5592 the results suggest that local dynamic stability of the lumbar spine represents a conserved measure actively controlled at least in part by trunk muscle stiffening effects it is evident therefore that local dynamic stability of the lumbar spine can be modulated effectively within a young healthy population however this may not be the case in a patient population

species permanence and dynamical behavior analysis of an impulsively controlled ecological system with distributed time delay in this paper on the basis of the theories and methods of ecology and ordinary differential equations an ecological system with impulsive harvest and distributed time delay is established by using the theories of impulsive equations small amplitude perturbation skills and comparison techniques we get a condition which guarantees the global asymptotical stability of the prey x eradication and predator z eradication periodic solution further the influences of the impulsive perturbation on the inherent oscillation are studied numerically and shows rich dynamics such as period doubling bifurcation chaotic bands periodic windows chaotic crises etc moreover the computation of the largest lyapunov exponent shows the chaotic dynamic behavior of the model meanwhile we investigate the qualitative nature of the strange attractor by using fourier spectra all of these results may be useful in the study of the dynamic complexity of ecosystems