measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

attempt to distinguish electric signals of a dichotomous nature three types of electric signals were analyzed ion current fluctuations in membrane channels icfmc seismic electric signals activities ses and artificial noises an the wavelet transform when applied to the conventional time domain does not allow a classification of these signals but does so in the natural time domain a classification also becomes possible if we study q versus q where chi stands for the natural time for q values approximately between 1 and 2 the signals are classified and icfmc lies between the other two types for q 1 the entropy s identical with ln of icfmc almost equals that of a uniform distribution while the an and ses have larger and smaller s values respectively the recent p varotsos n sarlis and e skordas phys rev e 67 021109 2003 finding that in short time scales both ses and an which are shown to be non markovian result in comparable detrended fluctuation analysis exponents alpha in 1 0 1 5 is revisited even a markovian dichotomous time series in short time scales leads to similar alpha exponents

minimal representations of semiseparable kernels and systems with separable boundary conditions abstract the simplest representations of triangular parts of finite rank kernels are analysed criteria for uniqueness up to similarity are given the results are applied to the problem of minimal realization of systems with separable boundary conditions

local global double algebras for slow h sup infinity adaptation i inversion and stability for pt ii see ibid vol 36 no 2 p 143 51 1991 in this two part work a common algebraic framework is introduced for the frozen time analysis of stability and h sup infinity optimization in slowly time varying systems based on the notion of a normed algebra on which local and global products are defined relations between local stability local near optimality local coprime factorization and global versions of these properties are sought the framework is valid for time domain disturbances in l sup infinity h sup infinity behavior is related to l sup infinity input output behavior via the device of an approximate isometry between frequency and time domain norms the authors presently elaborate on the double algebra concept for volterra operators which approximately commute with the shift and summarize the main algebraic properties and norm inequalities local conditions for global invertibility are also obtained classical frozen time stability conditions are incorporated in relations between local and global spectra

synchronization likelihood an unbiased measure of generalized synchronization in multivariate data sets the study of complex systems consisting of many interacting subsystems requires the use of analytical tools which can detect statistical dependencies between time series recorded from these subsystems typical examples are the electroencephalogram eeg and magnetoencephalogram meg which may involve the simultaneous recording of 150 or more time series coherency which is often used to study such data is only sensitive to linear and symmetric interdependencies and cannot deal with non stationarity recently several algorithms based upon the concept of generalized synchronization have been introduced to overcome some of the limitations of coherency estimates e g physica d 134 1999 419 brain res 792 1998 24 however these methods are biased by the degrees of freedom of the interacting subsystems physica d 134 1999 419 physica d 148 2001 147 we propose a novel measure for generalized synchronization in multivariate data sets which avoids this bias and can deal with non stationary dynamics

similarity and reduction for time varying linear systems with well posed boundary conditions the classical theory of similarity and reduction of causal and anti causal systems is generalized and extended to time varying linear systems with well posed boundary conditions two main problems are solved the first is to determine to what extent a system with boundary conditions can be simplified by similarity and reduction the second problem is to find the invariants of such a simplification special attention is given to the time invariant case

polynomial matrix fraction representations for linear time varying systems abstract this paper deals with the existence and associated realization theory of skew polynomial fraction representations for linear time varying discrete time systems it is shown that the time varying analog of the polynomial model always has a free state space necessary and sufficient conditions for the existence of bezout factorizations are obtained in terms system theoretic properties of state space realizatios

learning driver response relationships from synchronization patterns we test recent claims that causal driver response relationships can be deduced from interdependencies between simultaneously measured time series we apply two recently proposed interdependence measures that should give results similar to cross predictabilities used by previous authors the systems that we study are asymmetrically coupled simple models lorenz roessler and henon models the couplings being such that they lead to generalized synchronization if the data were perfect noise free infinitely long we should be able to detect at least in some cases which of the coupled systems is the driver and which the response this might no longer be true if the time series has finite length instead estimated interdependencies depend strongly on which of the systems has a higher effective dimension at the typical neighborhood sizes used to estimate them and causal relationships are more difficult to detect we also show that slightly different variants of the interdependence measure can have quite different sensitivities

transfer matrices realization and control of continuous time linear time varying systems via polynomial fractional representations abstract an algebraic theory of transfer matrices fractional representations and control for linear continuous time time varying systems based on the realization theory of input output maps is given it is shown for the first time that the realization of such systems specified by an abstract input output map as a module homomorphism over noncommutative polynomial rings can be established using an abstract kalman input output map defined over a ring of skew polynomials with time varying coefficients it is shown that in fact transfer matrices can be defined as formal power

on state models of linear time variant discrete time systems a decoupled full dimension form is introduced for state models in this form the dimension of the state vector is time invariant which is equal to the dimension of the state space at each time instant therefore the dimension of the state vector is minimized from the time variant point of view and all the state variables are independent moreover the state vector in a state model of the decoupled full dimension form is partitioned into two decoupled subvectors each of which alone forms a subsystem one of the subsystems is an fir system whose state vector is dependent only on inputs at finitely past instants in addition this subsystem is completely reachable the other subsystem is an iir system whose state space is equal to the force free solution set of the state vector of the subsystem it is proved that every state model in the conventional form can be mapped by a linear one to one map into an equivalent state model in the decoupled full dimension form in comparison with a state model in the

construction and classification of minimal representations of semi separable kernels abstract an algorithm to construct a minimal lower separable representation out of a lower separable representation of a kernel of an integral operator is given the minimal lower separable representations of a kernel are classified the number of non similar minimal lower separable representations of a kernel is discussed

stability stochastic stationarity and generalized lyapunov equations for two point boundary value descriptor systems the authors introduce the concept of internal stability for two point boundary value descriptor systems tpbvdss since tpbvdss are defined only over a finite interval the concept of stability is not easy to formulate the definition which is used consists of requiring that as the length of the interval of definition increases the effect of boundary conditions on states located close to the center of the interval should go to zero stochastic tpbvdss are studied and the property of stochastic stationarity is characterized in terms of a generalized lyapunov equation satisfied by the variance of the boundary vector a second generalized lyapunov equation satisfied by the state variance of a stochastically stationary tpbvds is also introduced and the existence and uniqueness of positive definite solutions to this equation are then used to characterize the property of internal stability

transfer functions and operator theory abstract the notion of a transfer function from systems theory serves as an important tool in dealing with certain problems in operator and matrix theory this is illustrated with a variety of material taken from existing literature a striking feature of the results is their high degree of explicitness

local global double algebras for slow h sup infinity adaptation ii optimization of stable plants for pt i see ibid vol 36 no 2 p 130 42 1991 the authors presently establish an explicit formula linking global and local sensitivity for systems with stable plants in which local sensitivity is a lipschitz continuous function of data frequency domain estimates of time domain sensitivity norms which become accurate as rates of time variation approach zero are obtained notions of adaptive versus nonadaptive robust control are introduced it is shown that adaptive control can achieve better sensitivity than optimal nonadaptive control it is demonstrated that in general h sup infinity optimal interpolants do not depend lipschitz continuously on data however delta suboptimal interpolants of the aak central maximal entropy type are shown to satisfy a tractable lipschitz condition

the effect of time delay on approximate sample entropy calculations abstract approximate and sample entropy are two widely used techniques to measure system complexity or regularity based on chosen parameters such as pattern length m and tolerance r in this paper we investigate how different values of the time delay parameter can be used in conjunction with standard values of m and r in the computation of approximate and sample entropy the results show that for time series generated by nonlinear dynamics that have long range correlation a time delay equal to the first zero crossing or minimum of the autocorrelation function can provide additional information into the characteristics of the time series that may be useful in comparative analysis with a unity delay we demonstrate that approximate and sample entropy are possibly measuring only the linear autocorrelation properties of the signal and these are highly invariant under surrogate data generation methods hence when this occurs the complexity measures of the surrogate and original data are not statistically different

maximum approximate entropy and r threshold a new approach for regularity changes detection abstract approximate entropy apen has been widely used as an estimator of regularity in many scientific fields it has proved to be a useful tool because of its ability to distinguish different system s dynamics when there is only available short length noisy data incorrect parameter selection embedding dimension m threshold r and data length n and the presence of noise in the signal can undermine the apen discrimination capacity in this work we show that r m a x a p e n m r m a x n a p e n m a x can also be used as a feature to discern between dynamics moreover the combined use of a p e n m a x and r m a x allows a better discrimination capacity to be accomplished even in the presence of noise we conducted our studies using real physiological time series and simulated signals corresponding to both low and high dimensional systems when a p e n m a x is incapable of discerning between different dynamics because of the noise presence our results suggest that r m a x provides additional information that can be useful for classification purposes based on cross validation tests we conclude that for short length noisy signals the joint use of a p e n m a x and r m a x can significantly decrease the misclassification rate of a linear classifier in comparison with their isolated use

on minimality and stable minimality of time varying linear systems with well posed boundary conditions for time varying linear systems with well posed boundary conditions the concept of stable minimality is introduced a complete characterization is given of stably minimal systems with analytic coefficients

minimum variance prediction for linear time varying systems in this paper we study the problem of minimum variance prediction for linear time varying systems we consider the standard time varying autoregression moving average arma model and develop a predictor which guarantees minimum variance prediction for a large class of linear time varying systems the predictor is developed based on a pseudocommutation technique for dealing with noncommutativity of linear time varying operators in a transfer operator framework we also show connections between this input output predictor and the kalman predictor via an example

on the linear stability of crystals in the schrodinger poisson model we consider the schrodinger poisson newton equations for crystals with one ion per cell we linearize this dynamics at the periodic minimizers of energy per cell and introduce a novel class of the ion charge densities that ensures the stability of the linearized dynamics our main result is the energy positivity for the bloch generators of the linearized dynamics under a wiener type condition on the ion charge density we also adopt an additional jellium condition which cancels the negative contribution caused by the electrostatic instability and provides the jellium periodic minimizers and the optimality of the lattice the energy per cell of the periodic minimizer attains the global minimum among all possible lattices we show that the energy positivity can fail if the jellium condition is violated while the wiener condition holds the proof of the energy positivity relies on a novel factorization of the corresponding hamilton functional the bloch generators are nonselfadjoint and even nonsymmetric hamilton operators we diagonalize these generators using our theory of spectral resolution of the hamilton operators with positive definite energy komech and kopylova in j stat phys 154 1 2 503 521 2014 j spectral theory 5 2 331 361 2015 the stability of the linearized crystal dynamics is established using this spectral resolution

on eigenfunction expansion of solutions to the hamilton equations we establish a spectral representation for solutions to linear hamilton equations with positive definite energy in a hilbert space our approach is a special version of m krein s spectral theory of j selfadjoint operators in the hilbert spaces with indefinite metric our main result is an application to the eigenfunction expansion for the linearized relativistic ginzburg landau equation

fractional representations for systems over a p i d a constructive technique abstract in this note we present a constructive technique for obtaining polynomial matrix fraction representations for reachable linear systems over a principal ideal domain an example is given to illustrate the algorithm

realization of acausal weighting patterns with boundary value descriptor systems this paper examines the realization of acausal weighting patterns with two point boundary value descriptor systems tpbvdss attention is restricted to the subclass oftpbvdss that are stationary so that their input output weighting pattern is shift invariant and extendible i e their weighting pattern can be extended outwards indefinitely then given an infinite acausal shift invariant weighting pattern the realization problem consists of constructing a minimal tpbvds over a fixed interval whose extended weighting pattern matches the given pattern the realization method that is proposed relies on a new transform the s t transform which is better adapted to the analysis of descriptor dynamics than the standard z transform since it handles zero and infinite frequencies in a symmetric way this new transform is used to determine the dimension of a minimal realization and then to construct a minimal realization by obtaining state space representations for two homogeneous rational matrices in s and obtained from the causal and anticausal components of the weighting pattern

characterization of integral operators with semi separable kernels with symmetries we characterize integral operators with semi separable kernels in a certain class that have different symmetries we treat the selfadjoint case the positive case the j unitary case the positive real case the dissipative case and the contractive case

the mourre theory for analytically fibered operators abstract we develop in this paper the mourre theory for an abstract class of fibered self adjoint operators which we call analytically fibered operators we construct a conjugate operator for which we prove that a mourre estimate holds examples of analytically fibered operators are given and finally perturbations of such operators are considered

reachability observability minimality and extendibility for two point boundary value descriptor systems a deterministic system theory is developed for two point boundary value descriptor systems tpbvdss in particular detailed characterizations of the properties of reachability observability and minimality are obtained in addition extendibility i e the concept of considering tpbvds as being defined on a sequence of intervals of increasing length is defined and studied this system theory generalizes our earlier results for the class of stationary systems for which the input output map weighting pattern is shift invariant

revisiting sample entropy analysis abstract we modify the definition of sample entropy saen by incorporating a time delay between the components of the block from which the densities are estimated and show that the modified method characterizes the complexity of the system better than the original version we apply the modified saen to the standard deterministic systems and stochastic processes uncorrelated and long range correlated lrc processes and show that the underlying complexity of the system is better quantified by the modified method we extend this analysis to the rr intervals of the normal and congestive heart failure chf subjects available via www physionet org and show that there is a good degree of separation between the two groups

a transfer function approach to linear time varying discrete time systems in the first part of the paper a transfer function approach is developed for the class of linear time varying discrete time systems the theory is specified in terms of skew noncommutative rings of polynomials and formal power series both with coefficients in a ring of time functions the transfer function matrix is defined to be a matrix whose entries belong to a skew ring of formal power series it is shown that various system properties such as asymptotic stability can be characterized in terms of the skew ring framework in the last part of the paper the transfer function framework is applied to the study of feedback control new results are obtained on assignability of system dynamics by using dynamic output feedback and dynamic state feedback the results are applied to the control of an armature controlled do motor with a variable loading

minimal state space realization in linear system theory an overview we give a survey of the results in connection with the minimal state space realization problem for linear time invariant systems we start with a brief historical overview and a short introduction to linear system theory next we present some of the basic algorithms for the reduction of nonminimal state space realizations and for the minimal state space realization of infinite or finite sequences of markov parameters of linear time invariant systems finally we discuss some extensions of this problem to other classes of systems and point out some related problems

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

on the dispersion decay for crystals in the linearized schrodinger poisson model abstract the schrodinger poisson newton equations for crystals with a cubic lattice and one ion per cell are considered the ion charge density is assumed i to satisfy the wiener and jellium conditions introduced in our previous paper 25 and ii to be exponentially decaying at infinity the corresponding examples are given we study the linearized dynamics at the ground state the dispersion relations are introduced via spectral resolution for the non selfadjoint hamilton generator using the positivity of the energy established in 25 our main result is the dispersion decay in the weighted sobolev norms for solutions with initial states from the space of continuous spectrum of the hamilton generator we also prove the absence of singular spectrum and limiting absorption principle the multiplicity of every eigenvalue is shown to be infinite the proofs rely on novel exact bounds and compactness for the inversion of the bloch generators and on uniform asymptotics for the dispersion relations we derive the bounds by the energy positivity from 25 we also use the theory of analytic sets

surrogate time series before we apply nonlinear techniques e g those inspired by chaos theory to dynamical phenomena occurring in nature it is necessary to first ask if the use of such advanced techniques is justified by the data while many processes in nature seem very unlikely a priori to be linear the possible nonlinear nature might not be evident in specific aspects of their dynamics the method of surrogate data has become a very popular tool to address such a question however while it was meant to provide a statistically rigorous foolproof framework some limitations and caveats have shown up in its practical use in this paper recent efforts to understand the caveats avoid the pitfalls and to overcome some of the limitations are reviewed and augmented by new material in particular we will discuss specific as well as more general approaches to constrained randomisation providing a full range of examples new algorithms will be introduced for unevenly sampled and multivariate data and for surrogate spike trains the main limitation which lies in the interpretability of the test results will be illustrated through instructive case studies we will also discuss some implementational aspects of the realisation of these methods in the tisean software package

eeg based topography analysis in string recognition task abstract vision perception and recognition is a complex process during which different parts of brain are involved depending on the specific modality of the vision target e g face character or word in this study brain activities in string recognition task compared with idle control state are analyzed through topographies based on multiple measurements i e sample entropy symbolic sample entropy and normalized rhythm power extracted from simultaneously collected scalp eeg our analyses show that for most subjects both symbolic sample entropy and normalized gamma power in string recognition task are significantly higher than those in idle state especially at locations of p4 o2 t6 and c4 it implies that these regions are highly involved in string recognition task since symbolic sample entropy measures complexity from the perspective of new information generation and normalized rhythm power reveals the power distributions in frequency domain complementary information about the underlying dynamics can be provided through the two types of indices

symbolic joint entropy reveals the coupling of various brain regions abstract the convergence and divergence of oscillatory behavior of different brain regions are very important for the procedure of information processing measurements of coupling or correlation are very useful to study the difference of brain activities in this study eeg signals were collected from ten subjects under two conditions i e eyes closed state and idle with eyes open we propose a nonlinear algorithm symbolic joint entropy to compare the coupling strength among the frontal temporal parietal and occipital lobes and between two different states instead of decomposing the eeg into different frequency bands theta alpha beta gamma etc the novel algorithm is to investigate the coupling from the entire spectrum of brain wave activities above 4hz the coupling coefficients in two states with different time delay steps are compared and the group statistics are presented as well we find that the coupling coefficient of eyes open state with delay consistently lower than that of eyes close state across the group except for one subject whereas the results without delay are not consistent the differences between two brain states with non zero delay can reveal the intrinsic inter region coupling better we also use the well known henon map data to validate the algorithm proposed in this paper the result shows that the method is robust and has a great potential for other physiologic time series

effective dynamics for bloch electrons peierls substitution and beyond we consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential x and vector potential a x with x d and 1 we prove that associated to an isolated family of bloch bands there exists an almost invariant subspace of l 2 d and an effective hamiltonian governing the evolution inside this subspace to all orders in to leading order the effective hamiltonian is given through the peierls substitution we explicitly compute the first order correction from a semiclassical analysis of this effective quantum hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics

trajectory tracking and disturbance rejection for linear time varying systems input output representation this paper considers the problem of asymptotic trajectory tracking and or disturbance rejection for linear time varying systems via input output representation inspired by the structure of repetitive control a novel feedback mechanism is proposed based on which the necessary and sufficient condition for the problem is derived this condition is a generalization to the time varying case of the result in time invariant setting based on the complete knowledge of the plant model a design is provided and it is applicable to repetitive control for time varying plants an important feature of the proposed approach is that a systematic way of constructing time varying internal model unit is provided

mutual information and global strange attractors in taylor couette flow abstract we apply a new method for the estimation of the mutual information m to experimental time series from different locations in weakly turbulent taylor couette flow we observe a dependence of m on the partition scale e and find m e dm log 1 e for small e and sufficient embedding dimension m our results confirm that the overall dynamics of the taylor couette flow is dominated by a global strange attractor of fractal information dimension d 1 d m at moderate reynolds numbers the dimension estimates obtained by our method are found to be more reliable than results from dimension algorithms based on single measurements alone a fact which makes our method most attractive for dimension analysis in extended systems

on matrix fraction representations for linear systems over commutative rings this paper deals with the problem of existence of polynomial matrix fraction representations for transfer matrices of linear systems over rings as well as the related realization theory these representations are then used in establishing new results for various classes of systems including split systems the relevance of these results to the regulation of various nonclassical classes of linear systems for example delay differential systems two dimensional systems etc is also discussed

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

stabilizability and stable proper factorizations for linear time varying systems the objective of this paper is to study in detail stabilizability and stable proper factorizations for linear time varying discrete time systems our main results are i if a linear time varying system can be stabilized by dynamic state feedback then it can also be stabilized by memoryless state feedback ii a complete characterization of the existence of stable proper factorizations for linear time varying input output operators this characterization is nontrivial there exist input output operators that do not admit stable proper factorizations